**Meet the editor**

Dr. Mueller received his Diploma degree in mathematics from the University of Applied Sciences, Mittweida, Germany, in 1997, a Diploma degree in electrical engineering from the University of Northumbria, Newcastle, U.K., in 1998, and a Diploma and Ph.D. degrees in mechanical engineering from the Technical University Chemnitz, Chemnitz, Germany, in 2001 and 2004,

respectively. From 1989 to 1993, he did professional work as an electronic technician. From 1998 to 2008, he was a research assistant with the Institute of Mechatronics, Chemnitz. Since 2008, he has been Lecturer with the Chair of Mechanics and Robotics, University Duisburg-Essen, Germany. His research interests include dynamics and control of non-linear systems, mechanisms, robotics, parallel mechanisms, singularities, differential geometric methods, advanced simulation methods, biomechanics, microelectromechanical systems, and chaotic systems.

Contents

**Preface IX** 

Chapter 1 **Parametric Robust Stability 3**  César Elizondo-González

Seddik M. Djouadi

**Part 1 Theoretical Aspects of Robust Control 1** 

Chapter 3 **A Sum of Squares Optimization Approach** 

Chapter 2 **Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 27** 

> **to Robust Control of Bilinear Systems 39**  Eitaku Nobuyama, Takahiko Aoyagi and Yasushi Kami

Chapter 5 **An Iterative Approach to the Fixed-Order Robust** *H* **Control** 

**Problem Using a Sequence of Infeasible Controllers 87** 

**with Parametric Uncertainty Using the QFT Method 139** 

Takayuki Kuwashima, Jun Imai and Masami Konishi

Chapter 4 **Spatially Sampled Robust Repetitive Control 55**  Cheng-Lun Chen and George T.-C. Chiu

Yasushi Kami and Eitaku Nobuyama

**Part 2 Robust Control of Robotic Systems 115** 

**Control of Mobile Manipulators 117**  Tatsuo Narikiyo and Michihiro Kawanishi

Chapter 6 **Optimizing the Tracking Performance in Robust Control Systems 103**  Hossein Oloomi and Bahram Shafai

Chapter 7 **Robust Adaptive Position/Force** 

Chapter 8 **Positioning Control of One Link Arm** 

## Contents

### **Preface** XI

	- **Part 2 Robust Control of Robotic Systems 115**

X Contents


## Preface

The vital condition for the practical applicability of any control system is that its basic characteristics are invariant to variations of its constituent parameters and to external disturbances, at least in some well-defined range. This is considered as Robustness, and the design of control systems with prescribed performance and stability limits is the subject of Robust Control.

The aim of this two-volume book `Robust Control' is to provide a selective overview of recent developments in the theory and application of robust control. The book is a compilation of 39 contributions by recognized experts in the broad field of robust control. Since robust control is a diverse field of research it is difficult to provide an exhaustive and at the same time balanced coverage of this topic. The chapters were selected so to equally account for recent developments in the theory as well as different application areas.

Volume I comprises 19 chapters covering selected problems in the theory of robust control and its application to robotic and electromechanical systems.

The first part of this volume consists of six chapters addressing specific theoretical issues. Chapter 1 deals with the robustness of control systems to **parametric uncertainties** of linear time invariant (LTI) systems.

Many practical problems can be classified as **time delay systems**. The time delay may in general be time dependent, which must be accounted for by the controller in order to ensure stability. This robustness problem is addressed in chapter 2.

Often robust control problems are formulated as *L*2 minimization problem. This approach fails for bilinear systems, however. Such systems are treated in section 3, where **sum of squares** formulation is used.

The principle of **repetitive control**, a variant of internal model control, is applied to spatially sampled systems in chapter 4, where attention is paid to non-linear saturation effects.

In chapter 5 the problem of robust *H* controller design for time invariant systems with **polytopic uncertainties** is addresses and an iterative design method is proposed.

Chapter 6 addresses the selection of **optimal weights in mixed sensitivity** *H***<sup>1</sup> design**. In particular the selection problem for tracking sinusoidal reference signals is studied and a procedure for the weight selection is proposed. The weight parameters are related to the tracking error specification via a functional approximation.

Preface XI

**Andreas Mueller** 

Germany

Chair of Mechanics and Robotics University Duisburg-Essen

motors. The concept of higher order sliding mode control is briefly recalled before the

Chapter 14 presents the design of a sliding mode controller for **switched reluctance** 

Chapter 15 addresses the **robust control of sensorless AC drives** using *H* control design. Adaptivity is introduced to account for the system uncertainties and for

The **identification of electrical parameters** of a three-phase permanent magnet synchronous machine is approached in section 16 by means of a robust estimation

Chapter 17 considers the effect of uncertainties in pulse with modulation (PVM) DC-DC converters. A **robust controller** is proposed **subject to linear matrix inequality** 

The **compensation of harmonic currents** in electrical mains by means of active filtering is addressed in chapter 18. A nonlinear robust controller is designed and

**Passivity** is a crucial property of non-linear control systems. This is discussed in

Given the wide spectrum covered by this monograph the editor and the authors are confident that the two volumes of 'Robust Control' will be a valuable and stimulating

chapter 19 for the control **of permanent magnet synchronous drives**.

reference for researchers from different disciplines.

case study is discussed in detail.

identification.

**(LMI) constraints**.

strategy.

tested.

**motors**. The robustness of the controller is shown.

The second part of volume I is dedicated to robust control of robotic and mobile manipulators that are inherently nonlinear.

In chapter 7 an **adaptive hybrid position-force control** scheme is proposed for mobile platforms exhibiting kinematic as well as dynamic uncertainties. While the method is developed for the particular case of a mobile manipulator it represents a general method potentially applicable to general non-holonomically constrained mechanisms.

The classical **quantitative feedback theory (QFT)** is applied to the control of a robotic arm in chapter 8 where the arm is assumed to possess uncertainties in its dynamical parameters. This case study demonstrates the application of the QFT.

Visual servoing, being an established method for the control of robotic systems, is used in section 9 for motion control of a robotic arm with three degrees of freedom. In this method the dynamics of the controlled system is represented in the image space. It is crucial that fundamental properties of the dynamics model in joint space are inherited by that in image space. This allows designing a **robust visual servoing** control scheme assuming certain bounds on the variation of the robot parameters as shown in the chapter.

Since any robust control method assumes certain **bounds of the model parameters** it is crucial to provide bounds on the uncertainties for the particular control problem. This is pursued in chapter 10 for the robotic manipulators. A bound estimation is derived in terms of the function that gives rise to a particular solution of the defining condition. Hence the main result of this chapter is a family of bound estimations. Three different particular solutions and the corresponding Lyapunov-stable control laws are presented.

Chapter 11 addresses the formation control of cooperating missiles. **Optimal control** is applied to establish and control formations.

Besides the robustness of the actual model-based controller of a robotic manipulator the robustness of the underlying model is crucial. This problem is addressed in chapter 12 for the control of parallel manipulators with actuation redundancy. It is pointed out that the problem of input-singularities, that are eliminated in the plant by the actuation redundancy, remains for the dynamics model. A globally valid dynamics formulation is proposed that does not suffer from this problem.

The third part of this volume is reserved for applications of robust control to electromechanical problems. Chapter 13 presents a robust variant of a **higher order sliding mode controller** used for the control of permanent magnet synchronous motors. The concept of higher order sliding mode control is briefly recalled before the case study is discussed in detail.

X Preface

Chapter 6 addresses the selection of **optimal weights in mixed sensitivity** *H***<sup>1</sup> design**. In particular the selection problem for tracking sinusoidal reference signals is studied and a procedure for the weight selection is proposed. The weight parameters are

The second part of volume I is dedicated to robust control of robotic and mobile

In chapter 7 an **adaptive hybrid position-force control** scheme is proposed for mobile platforms exhibiting kinematic as well as dynamic uncertainties. While the method is developed for the particular case of a mobile manipulator it represents a general method potentially applicable to general non-holonomically constrained mechanisms.

The classical **quantitative feedback theory (QFT)** is applied to the control of a robotic arm in chapter 8 where the arm is assumed to possess uncertainties in its dynamical

Visual servoing, being an established method for the control of robotic systems, is used in section 9 for motion control of a robotic arm with three degrees of freedom. In this method the dynamics of the controlled system is represented in the image space. It is crucial that fundamental properties of the dynamics model in joint space are inherited by that in image space. This allows designing a **robust visual servoing** control scheme assuming certain bounds on the variation of the robot parameters as

Since any robust control method assumes certain **bounds of the model parameters** it is crucial to provide bounds on the uncertainties for the particular control problem. This is pursued in chapter 10 for the robotic manipulators. A bound estimation is derived in terms of the function that gives rise to a particular solution of the defining condition. Hence the main result of this chapter is a family of bound estimations. Three different particular solutions and the corresponding Lyapunov-stable control

Chapter 11 addresses the formation control of cooperating missiles. **Optimal control** is

Besides the robustness of the actual model-based controller of a robotic manipulator the robustness of the underlying model is crucial. This problem is addressed in chapter 12 for the control of parallel manipulators with actuation redundancy. It is pointed out that the problem of input-singularities, that are eliminated in the plant by the actuation redundancy, remains for the dynamics model. A globally valid dynamics

The third part of this volume is reserved for applications of robust control to electromechanical problems. Chapter 13 presents a robust variant of a **higher order sliding mode controller** used for the control of permanent magnet synchronous

related to the tracking error specification via a functional approximation.

parameters. This case study demonstrates the application of the QFT.

manipulators that are inherently nonlinear.

shown in the chapter.

laws are presented.

applied to establish and control formations.

formulation is proposed that does not suffer from this problem.

Chapter 14 presents the design of a sliding mode controller for **switched reluctance motors**. The robustness of the controller is shown.

Chapter 15 addresses the **robust control of sensorless AC drives** using *H* control design. Adaptivity is introduced to account for the system uncertainties and for identification.

The **identification of electrical parameters** of a three-phase permanent magnet synchronous machine is approached in section 16 by means of a robust estimation strategy.

Chapter 17 considers the effect of uncertainties in pulse with modulation (PVM) DC-DC converters. A **robust controller** is proposed **subject to linear matrix inequality (LMI) constraints**.

The **compensation of harmonic currents** in electrical mains by means of active filtering is addressed in chapter 18. A nonlinear robust controller is designed and tested.

**Passivity** is a crucial property of non-linear control systems. This is discussed in chapter 19 for the control **of permanent magnet synchronous drives**.

Given the wide spectrum covered by this monograph the editor and the authors are confident that the two volumes of 'Robust Control' will be a valuable and stimulating reference for researchers from different disciplines.

> **Andreas Mueller**  Chair of Mechanics and Robotics University Duisburg-Essen Germany

**Part 1** 

**Theoretical Aspects of Robust Control** 

## **Part 1**

## **Theoretical Aspects of Robust Control**

**0**

**1**

*México*

**Parametric Robust Stability**

Robust stability of LTI systems with parametric uncertainty is a very interesting topic to study, industrial world is contained in parametric uncertainty. In industrial reality, there is not a particular system to analyze, there is a family of systems to be analyzed because the values of physical parameters are not known, we know only the lower and upper bounds of each parameter involved in the process, this is known as *Parametric Uncertainty* (Ackermann et al., 1993; Barmish, 1994; Bhattacharyya et al., 1995). The set of parameters involved in a system makes a *Parametric Vector*, the set of all vectors that can exists such that each parameter is kept

The system we are studying is now composed of an infinite number of systems, each system corresponds to a parameter vector contained in the parametric uncertainty box. So as to test the stability of the LTI system with parametric uncertainty we have to prove that all the infinite number of systems are stable, this is called *Parametric Robust Stability*. The parametric robust stability problem is considerably more complicated than determine the stability of an LTI system with fixed parameters. The stability of a LTI system can be analyzed in different ways, this chapter will be analyzed by means of its characteristic polynomial, in the case of parametric uncertainty now exists a set with an infinite number of characteristic polynomials, this is known as a *Family of Polynomials*, and we have to test the stability of the whole family. The parametric robust stability problem in LTI systems with parametric uncertainty is solved in this chapter by means of two tools, the first is a recent stability criterion for LTI systems (Elizondo, 2001B) and the second is the mathematical tool "Sign Decomposition" (Elizondo, 1999). The recent stability criterion maps the prametric robust stability problem to a robust positivity problem of multivariable polynomic functions, sign decomposition solves this

By means of the recent stability criterion (Elizondo, 2001B) is possible to analyze the characteristic polynomial and determine the number of unstable roots on the right side in the complex plane. This criterion is similar to the Routh criterion although without using the traditional division of the Routh criterion. This small difference makes a big advantage when it is analized the robust stability in LTI systems with parametric uncertainty, the elements of the first column of the table (Elizondo, 2001B) they are multivariable polynomic functions and these must be positive for stability conditions. Robust positivity of a multivariable polynomial function is more easier to prove that in the case of quotients of this class of functions, therefore, the recent criterion (Elizondo, 2001B) is easier to use than Routh criterion. There are other

within its lower and upper bounds is called a *Parametric Uncertainty Box*.

problem in necessary and sufficient conditions.

**1. Introduction**

*Facultad de Ingeniería Mecánica y Eléctrica Universidad Autónoma de Nuevo León*

César Elizondo-González

## **Parametric Robust Stability**

### César Elizondo-González

*Facultad de Ingeniería Mecánica y Eléctrica Universidad Autónoma de Nuevo León México*

### **1. Introduction**

Robust stability of LTI systems with parametric uncertainty is a very interesting topic to study, industrial world is contained in parametric uncertainty. In industrial reality, there is not a particular system to analyze, there is a family of systems to be analyzed because the values of physical parameters are not known, we know only the lower and upper bounds of each parameter involved in the process, this is known as *Parametric Uncertainty* (Ackermann et al., 1993; Barmish, 1994; Bhattacharyya et al., 1995). The set of parameters involved in a system makes a *Parametric Vector*, the set of all vectors that can exists such that each parameter is kept within its lower and upper bounds is called a *Parametric Uncertainty Box*.

The system we are studying is now composed of an infinite number of systems, each system corresponds to a parameter vector contained in the parametric uncertainty box. So as to test the stability of the LTI system with parametric uncertainty we have to prove that all the infinite number of systems are stable, this is called *Parametric Robust Stability*. The parametric robust stability problem is considerably more complicated than determine the stability of an LTI system with fixed parameters. The stability of a LTI system can be analyzed in different ways, this chapter will be analyzed by means of its characteristic polynomial, in the case of parametric uncertainty now exists a set with an infinite number of characteristic polynomials, this is known as a *Family of Polynomials*, and we have to test the stability of the whole family.

The parametric robust stability problem in LTI systems with parametric uncertainty is solved in this chapter by means of two tools, the first is a recent stability criterion for LTI systems (Elizondo, 2001B) and the second is the mathematical tool "Sign Decomposition" (Elizondo, 1999). The recent stability criterion maps the prametric robust stability problem to a robust positivity problem of multivariable polynomic functions, sign decomposition solves this problem in necessary and sufficient conditions.

By means of the recent stability criterion (Elizondo, 2001B) is possible to analyze the characteristic polynomial and determine the number of unstable roots on the right side in the complex plane. This criterion is similar to the Routh criterion although without using the traditional division of the Routh criterion. This small difference makes a big advantage when it is analized the robust stability in LTI systems with parametric uncertainty, the elements of the first column of the table (Elizondo, 2001B) they are multivariable polynomic functions and these must be positive for stability conditions. Robust positivity of a multivariable polynomial function is more easier to prove that in the case of quotients of this class of functions, therefore, the recent criterion (Elizondo, 2001B) is easier to use than Routh criterion. There are other

obsevability or stability analysis. In this chapter sign decomposition is applied to parametric

Parametric Robust Stability 5

In this chapter the following topics are studied: recent stability criterion, linear time invariant systems with parametric uncertainty, brief description of sign decomposition and finally a solution for the parametric robust stability problem. All demonstrations of the criterions, theorems, corollaries, lemmas, etc, will be omitted because they are results previously

The study of stability of the LTI systems begun approximately one and a half century ago with three important criterions: *Hermite* in 1856 (Ackermann et al., 1993), 1854 (Bhattacharyya et al., 1995); *Routh* in 1875 (Ackermann et al., 1993), 1877 (Gantmacher, 1990) and *Hurwitz* in 1895 (Gantmacher, 1990). Routh, using Sturm's theorem and Cauchy Index theory of a real rational function, set up a theorem to determine the number *k* of roots of

**Theorem 1.** *(Routh) (Gantmacher, 1990) The number of roots of the real polynomial p*(*s*) = *<sup>c</sup>*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*1*<sup>s</sup>* <sup>+</sup> *<sup>c</sup>*2*s*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *cns<sup>n</sup> in the right half of the complex plane is equal to the number of variations of sign in the first column of the Routh's table with coefficients: ai*,*<sup>j</sup>* = (*ai*−1,1*ai*−2,*j*+<sup>1</sup> −

There are several results related to the Routh criterion, for example (Fuller, 1977; Meinsma, 1995), but they are not appropriate to use in the parametric uncertainty case and they use

In this chapter a recent criterion, an arrange similar to the Rouht table, it is presented. The stability in this recent criterion depends on the positivity of a *sign column*. The recent criterion has two advantages: 1) the numerical operations are reduced with respect to above mentioned criterions; 2) the coefficients are multivariable polynomic functions in the case of parametric uncertainty and robust positivity is easier to test than Routh criterion. The criterion is as

**Theorem 2.** *(Elizondo, 2001B) Given a polynomial p*(*s*) = *<sup>c</sup>*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*1*<sup>s</sup>* <sup>+</sup> *<sup>c</sup>*2*s*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *cn*−1*sn*−<sup>1</sup> <sup>+</sup> *cns<sup>n</sup> with real coefficients, the number of roots on the right half of the complex plane is equal to the number*

> *σ*<sup>1</sup> *cn cn*−<sup>2</sup> *cn*−<sup>4</sup> ··· *σ*<sup>2</sup> *cn*−<sup>1</sup> *cn*−<sup>3</sup> *cn*−<sup>5</sup> ··· *σ*<sup>3</sup> *e*3,1 *e*3,2 ···

The procedure for calculating the elements (*ei*,*j*) is similar to the Routh table but without using the division. On the other hand, the calculation of an element *σ<sup>i</sup>* is more easier than it looks mathematical expression. We can get the sign *σi*, multiplying the sign of the

(*i*+1−*m*)/2 ∏ *j*=1

*Sign*(*em*<sup>+</sup>2(*j*−1),1) <sup>∀</sup>*<sup>i</sup>* <sup>≥</sup> <sup>3</sup>

polynomial with real coefficients on the right half plane of the complex numbers.

robust stability.

published.

described below.

**2. A recent stability criterion for LTI systems**

*ai*−2,1*ai*−1,*j*+1)/*ai*−1,1 <sup>∀</sup>*<sup>i</sup>* <sup>≥</sup> <sup>3</sup>*, ai*,*<sup>j</sup>* <sup>=</sup> *cn*<sup>+</sup>1−*i*−2(*j*−1) <sup>∀</sup>*<sup>i</sup>* <sup>≤</sup> <sup>2</sup>

more mathematical calculations than the Routh criterion.

*of variations of sign in the sign σ column on the follow arrange.*

*σ<sup>i</sup>* = *Sign*(*ei*,1) ∀*i* ≤ 2, *σ<sup>i</sup>* = *Sign*(*ei*,1)

. . . . . . . . . *ei*,*<sup>j</sup>* = (*ei*−1,1*ei*−2,*j*+<sup>1</sup> − *ei*−2,1*ei*−1,*j*+1), ∀<sup>3</sup> ≤ *<sup>i</sup>* ≤ *<sup>n</sup>* + <sup>1</sup> *ei*,*<sup>j</sup>* <sup>=</sup> *cn*<sup>+</sup>1−*i*−2(*j*−1) <sup>∀</sup>*<sup>i</sup>* <sup>≤</sup> <sup>2</sup>

criterions whose its elements are multivariable polynomic functions, such as the Hurwitz criterion and Lienard-Chipart criterion (Gantmacher, 1990), but both use a huge amount of mathematical operations in comparison with the recently stablished stability criterion Elizondo et al. (2005). When industrial cases are analyzed, the difference of mathematical operations is paramount, if the recently stability criterion takes several hours to determine the robust stability, the other criterions take several days. For these reasons the recently stability criterion is used in this chapter instead of other criterions.

*Sign Decomposition* (Elizondo, 1999) also called by some authors as *Sign definite Decomposition* is a mathematical tool able to determine in necessary and sufficient conditions the robust positivity of multivariable polynomic functions by means of extreme points analysis. Sign Decomposition begun as incipient orthogonal ideas of the author in his PhD research. It was not easy to develop this tool as thus it happens in orthogonal works with respect to the contemporary research line, the orthogonal ideas normally are not well seen. This is a very difficult situation on any research work, there may be many opinions, but we must accept that the world keeps working by the aligned but it changes by the orthogonals.

In LTI systems with parametric uncertainty applications, the multivariable polynomic functions to be analyzed depend on bounded physical parameters and some bounds could be negative. So sign decomposition begins with a coordinates transformation from the physical parameters to a set of mathematical parameters such that all the vectors of the new parameters are contained in a positive convex cone; in other words, all the new parameters are non-negatives. In this way, the multivariable polynomic function is made by non-decreasing terms, some of them are preceded by a positive sign and some by a negative sign. Grouping all the positive terms and grouping all the negative terms, then factorizing the negative sign and defining a "positive part" and a "negative part" of the function we obtain two non-decreasing functions. Now the function can be expressed as the positive part minus the negative part. It is obvious that both parts are independent functions, so they can be taken as a basis in with a graphical representation using two axis, the axis of the negative part and the axis of the positive part. Now, suppose that we have a particular vector contained in the parametric uncertainty box , then evaluating the negative part and the positive part a point on the "negative part, positive part plane" is obtained, this point represents the function evaluated in the particular vector in . The forty five degree line crossing at the origin on the "negative part, positive part plane" represents the set of functions with zero value, a point above this line represents a function with positive value and a point below this line represents a function with negative value.

The decomposition of the function in its negative and positive parts may look very simple and non-transcendent but taking into acount that the negative and positive parts are made by the addition of non-decreasing terms, then the negative and positive parts are nondecreasing functions in a vector space, this implies that the positive part and the negative part are bonded. So, geometrically, any point representing the function evaluated at any parameter vector is contained in a rectangle on the "negative part, positive part plane" and if the lowest right vertex is above the forty five degree line then the function is robust positive, obtaining in this way the basis of the "rectangle theorem". By means of this theorem upper and lower bounds of the multivariable polynomic function in the parametric uncertainty box are obtained. Sign decomposition contains a set of definitions, propositions, facts, lemmas, theorems and corollaries, sign decomposition can be applied to several disciplines; in the case of LTI systems with parametric uncertainty, this mathematical tool can be applied to robust controllability, 2 Will-be-set-by-IN-TECH

criterions whose its elements are multivariable polynomic functions, such as the Hurwitz criterion and Lienard-Chipart criterion (Gantmacher, 1990), but both use a huge amount of mathematical operations in comparison with the recently stablished stability criterion Elizondo et al. (2005). When industrial cases are analyzed, the difference of mathematical operations is paramount, if the recently stability criterion takes several hours to determine the robust stability, the other criterions take several days. For these reasons the recently stability

*Sign Decomposition* (Elizondo, 1999) also called by some authors as *Sign definite Decomposition* is a mathematical tool able to determine in necessary and sufficient conditions the robust positivity of multivariable polynomic functions by means of extreme points analysis. Sign Decomposition begun as incipient orthogonal ideas of the author in his PhD research. It was not easy to develop this tool as thus it happens in orthogonal works with respect to the contemporary research line, the orthogonal ideas normally are not well seen. This is a very difficult situation on any research work, there may be many opinions, but we must accept that

In LTI systems with parametric uncertainty applications, the multivariable polynomic functions to be analyzed depend on bounded physical parameters and some bounds could be negative. So sign decomposition begins with a coordinates transformation from the physical parameters to a set of mathematical parameters such that all the vectors of the new parameters are contained in a positive convex cone; in other words, all the new parameters are non-negatives. In this way, the multivariable polynomic function is made by non-decreasing terms, some of them are preceded by a positive sign and some by a negative sign. Grouping all the positive terms and grouping all the negative terms, then factorizing the negative sign and defining a "positive part" and a "negative part" of the function we obtain two non-decreasing functions. Now the function can be expressed as the positive part minus the negative part. It is obvious that both parts are independent functions, so they can be taken as a basis in with a graphical representation using two axis, the axis of the negative part and the axis of the positive part. Now, suppose that we have a particular vector contained in the parametric uncertainty box , then evaluating the negative part and the positive part a point on the "negative part, positive part plane" is obtained, this point represents the function evaluated in the particular vector in . The forty five degree line crossing at the origin on the "negative part, positive part plane" represents the set of functions with zero value, a point above this line represents a function with positive value and a point below this line represents a function

The decomposition of the function in its negative and positive parts may look very simple and non-transcendent but taking into acount that the negative and positive parts are made by the addition of non-decreasing terms, then the negative and positive parts are nondecreasing functions in a vector space, this implies that the positive part and the negative part are bonded. So, geometrically, any point representing the function evaluated at any parameter vector is contained in a rectangle on the "negative part, positive part plane" and if the lowest right vertex is above the forty five degree line then the function is robust positive, obtaining in this way the basis of the "rectangle theorem". By means of this theorem upper and lower bounds of the multivariable polynomic function in the parametric uncertainty box are obtained. Sign decomposition contains a set of definitions, propositions, facts, lemmas, theorems and corollaries, sign decomposition can be applied to several disciplines; in the case of LTI systems with parametric uncertainty, this mathematical tool can be applied to robust controllability,

criterion is used in this chapter instead of other criterions.

with negative value.

the world keeps working by the aligned but it changes by the orthogonals.

obsevability or stability analysis. In this chapter sign decomposition is applied to parametric robust stability.

In this chapter the following topics are studied: recent stability criterion, linear time invariant systems with parametric uncertainty, brief description of sign decomposition and finally a solution for the parametric robust stability problem. All demonstrations of the criterions, theorems, corollaries, lemmas, etc, will be omitted because they are results previously published.

### **2. A recent stability criterion for LTI systems**

The study of stability of the LTI systems begun approximately one and a half century ago with three important criterions: *Hermite* in 1856 (Ackermann et al., 1993), 1854 (Bhattacharyya et al., 1995); *Routh* in 1875 (Ackermann et al., 1993), 1877 (Gantmacher, 1990) and *Hurwitz* in 1895 (Gantmacher, 1990). Routh, using Sturm's theorem and Cauchy Index theory of a real rational function, set up a theorem to determine the number *k* of roots of polynomial with real coefficients on the right half plane of the complex numbers.

**Theorem 1.** *(Routh) (Gantmacher, 1990) The number of roots of the real polynomial p*(*s*) = *<sup>c</sup>*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*1*<sup>s</sup>* <sup>+</sup> *<sup>c</sup>*2*s*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *cns<sup>n</sup> in the right half of the complex plane is equal to the number of variations of sign in the first column of the Routh's table with coefficients: ai*,*<sup>j</sup>* = (*ai*−1,1*ai*−2,*j*+<sup>1</sup> − *ai*−2,1*ai*−1,*j*+1)/*ai*−1,1 <sup>∀</sup>*<sup>i</sup>* <sup>≥</sup> <sup>3</sup>*, ai*,*<sup>j</sup>* <sup>=</sup> *cn*<sup>+</sup>1−*i*−2(*j*−1) <sup>∀</sup>*<sup>i</sup>* <sup>≤</sup> <sup>2</sup>

There are several results related to the Routh criterion, for example (Fuller, 1977; Meinsma, 1995), but they are not appropriate to use in the parametric uncertainty case and they use more mathematical calculations than the Routh criterion.

In this chapter a recent criterion, an arrange similar to the Rouht table, it is presented. The stability in this recent criterion depends on the positivity of a *sign column*. The recent criterion has two advantages: 1) the numerical operations are reduced with respect to above mentioned criterions; 2) the coefficients are multivariable polynomic functions in the case of parametric uncertainty and robust positivity is easier to test than Routh criterion. The criterion is as described below.

**Theorem 2.** *(Elizondo, 2001B) Given a polynomial p*(*s*) = *<sup>c</sup>*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*1*<sup>s</sup>* <sup>+</sup> *<sup>c</sup>*2*s*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *cn*−1*sn*−<sup>1</sup> <sup>+</sup> *cns<sup>n</sup> with real coefficients, the number of roots on the right half of the complex plane is equal to the number of variations of sign in the sign σ column on the follow arrange.*

$$\begin{array}{ccccc}\sigma\_1 & c\_n & c\_{n-2} & c\_{n-4} & \cdots\\\sigma\_2 & c\_{n-1} & c\_{n-3} & c\_{n-5} & \cdots\\\sigma\_3 & c\_{3,1} & c\_{3,2} & \cdots\\\vdots & \vdots & \vdots\\\vdots & \vdots & \vdots\\\sigma\_{i,j} = (e\_{i-1,1}e\_{i-2,j+1} - e\_{i-2,1}e\_{i-1,j+1}), \forall 3 \le i \le n+1\\\ & e\_{i,j} = c\_{n+1-i-2(j-1)} \,\forall i \le 2\\\ \sigma\_i = \operatorname{Sign}(e\_{i,1}) \,\forall i \le 2, \qquad \sigma\_i = \operatorname{Sign}(e\_{i,1}) \prod\_{j=1}^{(i+1-m)/2} \operatorname{Sign}(e\_{m+2(j-1),1}) \,\forall i \ge 3\end{array}$$

The procedure for calculating the elements (*ei*,*j*) is similar to the Routh table but without using the division. On the other hand, the calculation of an element *σ<sup>i</sup>* is more easier than it looks mathematical expression. We can get the sign *σi*, multiplying the sign of the

Example 2. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 2*s*<sup>4</sup> + 2*s*<sup>3</sup> + 2*s*<sup>2</sup> + *s* + 3 by means of criterion 2 determine the number of roots in the right half of the complex plane and compare the results

Parametric Robust Stability 7

It is easy to see by means of two criterions that the polynomial has two roots on the right half

Example 3. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 1*s*<sup>4</sup> + 2*s*<sup>3</sup> + 2*s*<sup>2</sup> + 2*s* + 1 by means of criterion 2

When we try to make the table by means of Elizondo-González 2001 criterion or Routh

Since the element *e*3,1 is equal zero (see table (4)) then this element is replaced by by an *�* > 0,

*σ*<sup>1</sup> 1 2 2 *σ*<sup>2</sup> 1 2 1 *σ*<sup>3</sup> *�* 1 *σ*<sup>4</sup> 2*�* − 1 *� <sup>σ</sup>*<sup>5</sup> <sup>2</sup>*�* <sup>−</sup> <sup>1</sup> <sup>−</sup> *�*<sup>2</sup> *<sup>σ</sup>*<sup>6</sup> (2*�* <sup>−</sup> <sup>1</sup> <sup>−</sup> *�*2)*�*

*σ*<sup>1</sup> = + 1 2 2 *σ*<sup>2</sup> = + 1 2 1 *σ*<sup>3</sup> = + *�* 1 *σ*<sup>4</sup> = − −1 *� σ*<sup>5</sup> = + −1 *σ*<sup>6</sup> = + −*�* Table 6. Example 3. Final result to the solution of the problem of zero in the first column.

From the table (6) is easy to see that the polynomial has two roots on the right half of the

Elizondo-González 2001 *σ*<sup>1</sup> = + 1 2 1 *σ*<sup>2</sup> = + 2 2 3 *σ*<sup>3</sup> = + 2 −1 *σ*<sup>4</sup> = + 6 6 *σ*<sup>5</sup> = − −18 *σ*<sup>6</sup> = + −

Table 3. Example 2. Comparison of stability criterions.

determine the number of roots in the right half of the complex plane.

Table 4. Example 3. Presence of a zero in the first column of elements.

Table 5. Example 3. Solution of the problem of zero in the first column.

Applying the limit *�* → 0 in table (5) is obtained the table (6).

of the complex plane in accordance to the table (3).

criterion, it is truncated because *e*3,1 = 0

thus obtaining the following arrangement.

complex plane.

with the Routh criterion.

element (*ei*,1) by the sign of the immediate superior element (*ei*−1,1) and then jumping in pairs. For example *σ*<sup>6</sup> = *Sign*(*e*6,1)*Sign*(*e*5,1)*Sign*(*e*3,1)*Sign*(*e*1,1). Also *σ*<sup>1</sup> = *Sign*(*cn*) and *σ*<sup>2</sup> = *Sign*(*cn*−2). So also it is not necessary to calculate the last element (*en*+1,1), only its sign is necessary to calculate. Each row of (*ei*,*j*) elements is obtained by means of (*ei*−1,*j*) and (*ei*−2,*j*) elements previously calculated and in Hurwitz criterion a principal minor is not calculated from previous, then the Elizondo-González criterion is more advantageous than Hurwitz criterion as shown in table (1)

**Remark 3.** *a) Given the relation of the above criterion with the Routh criterion, the cases in that one element ei*,*<sup>j</sup> is equal to cero or all the elements of a row are cero, they are treated as so as it is done in the Routh criterion. b) The last element en*+1,1 *is not necessary to calculate, but it is necessary to obtain only its sign*


Mathematical operations in polynomials *n* degree

Table 1. A comparison of stability criterions.

### **2.1 Examples**

Example 1. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 2*s*<sup>4</sup> + 1*s*<sup>3</sup> + 5*s*<sup>2</sup> + 2*s* + 2 by means of criterion 2 determine the number of roots in the right half of the complex plane and compare the results with the Routh criterion.

Applying 2 criterion we obtain the left table. As an example of the procedure to obtain the elements *ei*,*<sup>j</sup>* and *σi*, we have: *e*3,1 = 2 × 1 − 1 × 5, *e*3,2 = 2 × 2 − 1 × 2, *σ*<sup>6</sup> = *Sign*(+) × *Sign*(−56) × *Sign*(−3) × *Sign*(1), *σ*<sup>5</sup> = *Sign*(−56) × *Sign*(−19) × *Sign*(2).


Table 2. Example 1. Comparison of stability criterions.

The left arrengment shows two sign changes in *σ* column so the polynomial has two roots on the right half of the complex plane. By means of Routh criterion is obtained the right table, it shows too two sign changes in the first column which is the same previous result. An interesting observation (see table (2)) is that the left table presents a minus sign in the third row of the *σ* column and the right table presents a minus sign in the same third row but in the first column.

4 Will-be-set-by-IN-TECH

element (*ei*,1) by the sign of the immediate superior element (*ei*−1,1) and then jumping in pairs. For example *σ*<sup>6</sup> = *Sign*(*e*6,1)*Sign*(*e*5,1)*Sign*(*e*3,1)*Sign*(*e*1,1). Also *σ*<sup>1</sup> = *Sign*(*cn*) and *σ*<sup>2</sup> = *Sign*(*cn*−2). So also it is not necessary to calculate the last element (*en*+1,1), only its sign is necessary to calculate. Each row of (*ei*,*j*) elements is obtained by means of (*ei*−1,*j*) and (*ei*−2,*j*) elements previously calculated and in Hurwitz criterion a principal minor is not calculated from previous, then the Elizondo-González criterion is more advantageous than

**Remark 3.** *a) Given the relation of the above criterion with the Routh criterion, the cases in that one element ei*,*<sup>j</sup> is equal to cero or all the elements of a row are cero, they are treated as so as it is done in the Routh criterion. b) The last element en*+1,1 *is not necessary to calculate, but it is necessary to obtain*

> Mathematical operations in polynomials *n* degree grado Hurwitz C. Elizondo *n* × + *o* − × + *o* − 3 4 1 2 1 4 9 2 5 2 5 66 18 9 4 6 193 45 14 6 7 780 145 20 9

Example 1. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 2*s*<sup>4</sup> + 1*s*<sup>3</sup> + 5*s*<sup>2</sup> + 2*s* + 2 by means of criterion 2 determine the number of roots in the right half of the complex plane and compare the results

Applying 2 criterion we obtain the left table. As an example of the procedure to obtain the elements *ei*,*<sup>j</sup>* and *σi*, we have: *e*3,1 = 2 × 1 − 1 × 5, *e*3,2 = 2 × 2 − 1 × 2, *σ*<sup>6</sup> = *Sign*(+) ×

The left arrengment shows two sign changes in *σ* column so the polynomial has two roots on the right half of the complex plane. By means of Routh criterion is obtained the right table, it shows too two sign changes in the first column which is the same previous result. An interesting observation (see table (2)) is that the left table presents a minus sign in the third row of the *σ* column and the right table presents a minus sign in the same third row but in the

*Sign*(−56) × *Sign*(−3) × *Sign*(1), *σ*<sup>5</sup> = *Sign*(−56) × *Sign*(−19) × *Sign*(2). Elizondo-González 2001 *σ*<sup>1</sup> = + 1 1 2 *σ*<sup>2</sup> = + 2 5 2 *σ*<sup>3</sup> = − −3 2 *σ*<sup>4</sup> = + −19 −6 *σ*<sup>5</sup> = + −56 *σ*<sup>6</sup> = + +

Hurwitz criterion as shown in table (1)

Table 1. A comparison of stability criterions.

Table 2. Example 1. Comparison of stability criterions.

*only its sign*

**2.1 Examples**

first column.

with the Routh criterion.

Example 2. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 2*s*<sup>4</sup> + 2*s*<sup>3</sup> + 2*s*<sup>2</sup> + *s* + 3 by means of criterion 2 determine the number of roots in the right half of the complex plane and compare the results with the Routh criterion.


Table 3. Example 2. Comparison of stability criterions.

It is easy to see by means of two criterions that the polynomial has two roots on the right half of the complex plane in accordance to the table (3).

Example 3. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 1*s*<sup>4</sup> + 2*s*<sup>3</sup> + 2*s*<sup>2</sup> + 2*s* + 1 by means of criterion 2 determine the number of roots in the right half of the complex plane.

When we try to make the table by means of Elizondo-González 2001 criterion or Routh criterion, it is truncated because *e*3,1 = 0


Table 4. Example 3. Presence of a zero in the first column of elements.

Since the element *e*3,1 is equal zero (see table (4)) then this element is replaced by by an *�* > 0, thus obtaining the following arrangement.


Table 5. Example 3. Solution of the problem of zero in the first column.

Applying the limit *�* → 0 in table (5) is obtained the table (6).


Table 6. Example 3. Final result to the solution of the problem of zero in the first column.

From the table (6) is easy to see that the polynomial has two roots on the right half of the complex plane.

Example 4. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 1*s*<sup>4</sup> + 2*s*<sup>3</sup> + 2*s*<sup>2</sup> + 1*s* + 1 by means of criterion 2 determine the number of roots in the right half of the complex plane. Applying this criterion we get as following.

maximum *v*max Euclidean vertices of *Q* are defined as so as

In the LTI systems with parametric uncertainty, the characteristic polynomial has coefficients dependent on physical parameters, *<sup>p</sup>*(*s*, *<sup>q</sup>*) = *<sup>c</sup>*0(*q*) + *<sup>c</sup>*1(*q*)*<sup>s</sup>* <sup>+</sup> *<sup>c</sup>*2(*q*)*s*<sup>2</sup> <sup>+</sup> ··· *cn*(*q*)*sn*; so Routh criterion is very difficult to use because it is necessary to test the robust positivity of rational functions dependent on physical parameters. By means of Hurwitz criterion is possible to solve the problem of parametric robust stability by means of robust positivity of principal minors of a matrix dependent on physical parameters, this procedure uses a lot of mathematical calculations. The robust positivity of rational function dependent on physical parameters can be considered as so as a very much difficult problem since only the robust positive test of multivariable polynomic function is very difficult problem (Ackermann et al., 1993) (page 93). So the parametric robust stability problem in LTI systems with parametric uncertainty in the general case is not an easy problem to solve, however in this chapter is

Parametric Robust Stability 9

The characteristic polynomials are classified according to its coefficient of maximum complexity; from the simplest structure coefficient to the most complex are: *Interval*, *Affine, Multilinear and Polynomic*. For example, the coefficients: *ci*(*q*) = *qi*, *ci*(*q*) = 2*q*<sup>1</sup> + 3*q*<sup>2</sup> + 5*q*<sup>3</sup> +

Interval, Affine, Multilinear and Polynomic respectively. The number of polynomials *p*(*s*, *q*) that can exist is infinite since the number of vectors that exist is infinite, the collection of all

The families of polynomials interval and afin are convex sets and these families have subsetting test. This concept, *subsetting test*, means that a family of polynomials is robustly

Kharitonov in (Kharitonov, 1978), by means of his theorem demonstrates that a family of interval polynomials is robust stable if and only if a set of four polynomials are stable. In (Bartlett et al., 1988) by means of their edge theorem, demonstrated that a family of afin polynomials is robustly stable if and only if all the polynomials corresponding to the edges of the parametric uncertainty box are stable. The multilinear an polynomic families are not convex set and they do not have subsetting test. So parametric robust stability of these families can not be resolved by tools based on convexity. In (Elizondo, 1999) was presented a solution for parametric robust stability of any kind of family: Interval, Affine, Multilinear or Polynomic. The solution is based on sign decomposition, and by means of this tool can also

The parametric robust stability problem of LTI systems can be mapped to a problem of robust

The first two are: the Hurwitz and Lienard-Chipart criterions, the other is the recently stability criterion (2). By Hurwitz or Lienard-Chipart criterions can do the mapping but as explained these require making a lot of mathematical calculations. The criterion (2) requires much less mathematical calculations that the criterions mentioned as was shown in table (1),

polynomials that exist is a *Family of Polynomials P*(*s*, *Q*) = {*p*(*s*, *q*)|*q* ∈ *Q*}.

solve the problem of robust controllability or robust observability.

**3.3 Robust stability mapped to robust positivity**

positivity of polynomial functions for at least three ways.

stable if and only if all polynomials contained in the subsetting test are stable.

<sup>1</sup>*q*<sup>2</sup> <sup>+</sup> <sup>2</sup>*q*<sup>2</sup>

2*q*5

**3.2 Parametric robust stability in LTI systems**

*q*4, *ci*(*q*) = 5*q*1*q*<sup>2</sup> + 2*q*2*q*<sup>4</sup> + 5*q*<sup>3</sup> + *q*4, *ci*(*q*) = 2*q*<sup>3</sup>

max *<sup>q</sup>*∈*<sup>Q</sup>* �*q*�2.

presented a solution.

(Elizondo et al., 2005)

*<sup>v</sup>*min 

<sup>2</sup> = min

<sup>4</sup> + *q*3, correspond to classification:

*<sup>q</sup>*∈*<sup>Q</sup>* �*q*�2, �*v*max�<sup>2</sup> <sup>=</sup>


Table 7. Example 4. A row equal zero.

The table (7) generated, it shows the third row equal zero. Then obtaining the derivative of the polynomial "corresponding" to the immediately superiory row *p*(*s*) = *s*<sup>4</sup> + 2*s*<sup>2</sup> + 1 is obtained *p*(*s*) = 4*s*<sup>3</sup> + 4*s*. Now the coefficients of this polynomial replace the zeros of the third row and the procedure continues, obtaining in this way the follow arrangement.


Table 8. Example 4. Solution to the problem of a row equal zero.

We can see in table (8) that there is no sign change in *sigma* column, then there are not roots in the right half complex plane.

### **3. Linear time invariant systems with parametric uncertainty**

#### **3.1 Parametric uncertainty**

All phisical systems are dependent on parameters *qi* and in the physical world does not know the value of the parameters, only know the lower *q*− *<sup>i</sup>* and upper *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* bounds of each parameter, so that *q*− *<sup>i</sup>* <sup>≤</sup> *qi* <sup>≤</sup> *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* , this expression is also written as *qi* ∈ [*q*<sup>−</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *i* ].

For example if we have several electrical resistances with color code of 1,000 ohm, if one measures one of them, the measurement can be: 938, 1,024, or a value close to 1,000 ohm but it is rather difficult that it is exactly 1,000 ohm. By means of tolerance code can be deduced that the resistance will be greater than 900 and less than 1,100 ohm. Another example is the mass of a commercial aircraft, it can fly with few passengers and little baggage or with with many passengers and much baggage, then the mass of the plane is not known until the last passenger to be registered, but not when the plane was designed, however the plane is designed to fly from a minimum mass to a maximum mass.

The set of � parameters involved in a system makes a *Parametric Vector q* = [*q*1, *q*2, ··· , *q*�] *T*, *<sup>q</sup>* ∈ �� and the set of all the possible parameter vectors that may exist makes a *Parametric Uncertainty Box Q* = { *q* = [*q*1, *q*2, ··· , *q*�] *T qi* ∈ [*q*<sup>−</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* ] ∀*i*}. In the case of *qi* > 0 ∀*i* then *Q* = { *q* = [*q*1, *q*2, ··· , *q*�] *T qi* > 0, *qi* ∈ [*q*<sup>−</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* ] ∀*i* } and *Q* is contained in a positive convex cone *<sup>P</sup>*, *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ ��.

For the study of cases involving parametric uncertainty is necessary to define the minimum and maximum vertices of the parametric uncertainty box, so the minimum *v*min and maximum *v*max Euclidean vertices of *Q* are defined as so as *<sup>v</sup>*min <sup>2</sup> = min *<sup>q</sup>*∈*<sup>Q</sup>* �*q*�2, �*v*max�<sup>2</sup> <sup>=</sup>

max *<sup>q</sup>*∈*<sup>Q</sup>* �*q*�2.

6 Will-be-set-by-IN-TECH

Example 4. Given the polynomial *p*(*s*) = *s*<sup>5</sup> + 1*s*<sup>4</sup> + 2*s*<sup>3</sup> + 2*s*<sup>2</sup> + 1*s* + 1 by means of criterion 2 determine the number of roots in the right half of the complex plane. Applying this criterion

The table (7) generated, it shows the third row equal zero. Then obtaining the derivative of the polynomial "corresponding" to the immediately superiory row *p*(*s*) = *s*<sup>4</sup> + 2*s*<sup>2</sup> + 1 is obtained *p*(*s*) = 4*s*<sup>3</sup> + 4*s*. Now the coefficients of this polynomial replace the zeros of the third row and

> *σ*<sup>1</sup> = + 1 2 1 *σ*<sup>2</sup> = + 1 2 1 *σ*<sup>3</sup> = + 4 4 *σ*<sup>4</sup> = + 4 4 *σ*<sup>5</sup> = + *� σ*<sup>6</sup> = + 4*�*

We can see in table (8) that there is no sign change in *sigma* column, then there are not roots

All phisical systems are dependent on parameters *qi* and in the physical world does not know

For example if we have several electrical resistances with color code of 1,000 ohm, if one measures one of them, the measurement can be: 938, 1,024, or a value close to 1,000 ohm but it is rather difficult that it is exactly 1,000 ohm. By means of tolerance code can be deduced that the resistance will be greater than 900 and less than 1,100 ohm. Another example is the mass of a commercial aircraft, it can fly with few passengers and little baggage or with with many passengers and much baggage, then the mass of the plane is not known until the last passenger to be registered, but not when the plane was designed, however the plane is

The set of � parameters involved in a system makes a *Parametric Vector q* = [*q*1, *q*2, ··· , *q*�]

*<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup>

*T*

*<sup>q</sup>* ∈ �� and the set of all the possible parameter vectors that may exist makes a *Parametric*

For the study of cases involving parametric uncertainty is necessary to define the minimum and maximum vertices of the parametric uncertainty box, so the minimum *v*min and

*qi* ∈ [*q*<sup>−</sup>

*<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup>

*<sup>i</sup>* , this expression is also written as *qi* ∈ [*q*<sup>−</sup>

*<sup>i</sup>* and upper *<sup>q</sup>*<sup>+</sup>

*<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *i* ].

*<sup>i</sup>* bounds of each parameter,

*<sup>i</sup>* ] ∀*i*}. In the case of *qi* > 0 ∀*i* then

*<sup>i</sup>* ] ∀*i* } and *Q* is contained in a positive convex

*T*,

the procedure continues, obtaining in this way the follow arrangement.

Table 8. Example 4. Solution to the problem of a row equal zero.

**3. Linear time invariant systems with parametric uncertainty**

the value of the parameters, only know the lower *q*−

designed to fly from a minimum mass to a maximum mass.

*qi* > 0, *qi* ∈ [*q*<sup>−</sup>

*T*

*Uncertainty Box Q* = { *q* = [*q*1, *q*2, ··· , *q*�]

we get as following.

Table 7. Example 4. A row equal zero.

in the right half complex plane.

**3.1 Parametric uncertainty**

*<sup>i</sup>* <sup>≤</sup> *qi* <sup>≤</sup> *<sup>q</sup>*<sup>+</sup>

*Q* = { *q* = [*q*1, *q*2, ··· , *q*�]

cone *<sup>P</sup>*, *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ ��.

so that *q*−

#### **3.2 Parametric robust stability in LTI systems**

In the LTI systems with parametric uncertainty, the characteristic polynomial has coefficients dependent on physical parameters, *<sup>p</sup>*(*s*, *<sup>q</sup>*) = *<sup>c</sup>*0(*q*) + *<sup>c</sup>*1(*q*)*<sup>s</sup>* <sup>+</sup> *<sup>c</sup>*2(*q*)*s*<sup>2</sup> <sup>+</sup> ··· *cn*(*q*)*sn*; so Routh criterion is very difficult to use because it is necessary to test the robust positivity of rational functions dependent on physical parameters. By means of Hurwitz criterion is possible to solve the problem of parametric robust stability by means of robust positivity of principal minors of a matrix dependent on physical parameters, this procedure uses a lot of mathematical calculations. The robust positivity of rational function dependent on physical parameters can be considered as so as a very much difficult problem since only the robust positive test of multivariable polynomic function is very difficult problem (Ackermann et al., 1993) (page 93). So the parametric robust stability problem in LTI systems with parametric uncertainty in the general case is not an easy problem to solve, however in this chapter is presented a solution.

The characteristic polynomials are classified according to its coefficient of maximum complexity; from the simplest structure coefficient to the most complex are: *Interval*, *Affine, Multilinear and Polynomic*. For example, the coefficients: *ci*(*q*) = *qi*, *ci*(*q*) = 2*q*<sup>1</sup> + 3*q*<sup>2</sup> + 5*q*<sup>3</sup> + *q*4, *ci*(*q*) = 5*q*1*q*<sup>2</sup> + 2*q*2*q*<sup>4</sup> + 5*q*<sup>3</sup> + *q*4, *ci*(*q*) = 2*q*<sup>3</sup> <sup>1</sup>*q*<sup>2</sup> <sup>+</sup> <sup>2</sup>*q*<sup>2</sup> 2*q*5 <sup>4</sup> + *q*3, correspond to classification: Interval, Affine, Multilinear and Polynomic respectively. The number of polynomials *p*(*s*, *q*) that can exist is infinite since the number of vectors that exist is infinite, the collection of all polynomials that exist is a *Family of Polynomials P*(*s*, *Q*) = {*p*(*s*, *q*)|*q* ∈ *Q*}.

The families of polynomials interval and afin are convex sets and these families have subsetting test. This concept, *subsetting test*, means that a family of polynomials is robustly stable if and only if all polynomials contained in the subsetting test are stable.

Kharitonov in (Kharitonov, 1978), by means of his theorem demonstrates that a family of interval polynomials is robust stable if and only if a set of four polynomials are stable. In (Bartlett et al., 1988) by means of their edge theorem, demonstrated that a family of afin polynomials is robustly stable if and only if all the polynomials corresponding to the edges of the parametric uncertainty box are stable. The multilinear an polynomic families are not convex set and they do not have subsetting test. So parametric robust stability of these families can not be resolved by tools based on convexity. In (Elizondo, 1999) was presented a solution for parametric robust stability of any kind of family: Interval, Affine, Multilinear or Polynomic. The solution is based on sign decomposition, and by means of this tool can also solve the problem of robust controllability or robust observability.

#### **3.3 Robust stability mapped to robust positivity**

The parametric robust stability problem of LTI systems can be mapped to a problem of robust positivity of polynomial functions for at least three ways.

The first two are: the Hurwitz and Lienard-Chipart criterions, the other is the recently stability criterion (2). By Hurwitz or Lienard-Chipart criterions can do the mapping but as explained these require making a lot of mathematical calculations. The criterion (2) requires much less mathematical calculations that the criterions mentioned as was shown in table (1), (Elizondo et al., 2005)

If we take a particular vector *q* ∈ *Q* and evaluated the *fn*(*q*) and *fp*(*q*) parts, we obtain the coordinates (*fn*(*q*), *fp*(*q*)) of the function in the (*fn*, *fp*) plane. The 45*<sup>o</sup>* line is the set of points where the function is equal zero because *fp*(*q*) = *fn*(*q*) so *f*(*q*) = *fp*(*q*) − *fn*(*q*) = 0 . If a point is above the 45*<sup>o</sup>* line means that *fp*(*q*) > *fn*(*q*) then *f*(*q*) > 0. If a point is below the 45*<sup>o</sup>* line

Parametric Robust Stability 11

It should be noted that independently of the number of parameters in which the function depends on, the function will always be represented in �<sup>2</sup> via (*fn*(*q*), *fp*(*q*)). For example,

decomposed in two bounded continuous nondecreasing and nonnegative functions *f <sup>p</sup>*(*q*) =

by plotting a hundred lines blue color, (one hundred fifty points per line) of variable *q*<sup>3</sup> holding (*q*1, *q*2) constant uniformly distributed in different positions. The process was repeated varying *q*<sup>2</sup> in green color and finally varying *q*<sup>1</sup> in red color. According to the position shown in the graph of the function with respect to the 45*<sup>o</sup>* line, it appears that the function is robustly

2*q*3

<sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*1*q*<sup>2</sup>

2*q*3

*qi* ∈ [0, 1] }. The function has sign decomposition because it is

<sup>3</sup> and *f*(*q*) = *fp*(*q*) − *fn*(*q*). The figure ( 2) was obtained

<sup>3</sup> such that *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �3,

means that *fp*(*q*) < *fn*(*q*) then *f*(*q*) < 0.

the function *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup>

*T*

<sup>1</sup>*q*2, *fn*(*q*) = *<sup>q</sup>*<sup>2</sup> <sup>+</sup> <sup>9</sup>*q*1*q*<sup>2</sup>

positive. But it must be demonstrated mathematically.

Fig. 1. (*fn*, *fp*) plane

*Q* = { *q* = [*q*1, *q*2, *q*3]

Fig. 2. Function in (*fn*, *fp*) plane

4 + *q*1*q*<sup>3</sup> + 8*q*<sup>2</sup>

### **4. Brief description of sign decomposition**

In different areas of sciences the fundamental problem can be mapped to a problem of robust positivity of multivariable polynomic functions. For example the no singularity of a matrix can be analyzed by mean of the robust positivity of its determinant, so it is very useful to have a mathematical tool that solves the problem of robust positivity of multivariable polynomic functions. Practically there are three tools for this purpose: Interval Arithmetic (Moor, 1966); Bernstein Polynomials (Zettler, et all 1998) and Sign Decomposition ((Elizondo, 1999)) whose complete version is developed in (Elizondo, 1999) and its partial versions are presented in (Elizondo, 2000; 2001A;B; 2002A;B), for simplicity only will be mentioned (Elizondo, 1999). Interval arithmetic is very difficult to use because it requires much more calculations than other methods. When robust positivity is analyzed in a very simple function, Bernstein polynomials have advantages over sign decomposition, but when the function is not simple, sign decomposition has advantages over Bernstein polynomials (Graziano et al., 2004). There are several works using sign decomposition instead of Bernstein polynomials, some of them are: (Bhattacharyya et al., 2009; Guerrero, 2006; Keel et al., 2008; 2009; Keel, 2011; Knap et al., 2010; 2011)

### **4.1 Definition of sign decomposition**

The following is a brief description of the more relevant results of Sign Decomposition (Elizondo, 1999). By means of this tool it is possible to determine, in necessary and sufficient conditions, the robust positivity of a multivariable polynomic function depending on � parameters, employing extreme points analysis.

Since mathematically exist the possibility that a parameter *q*ˆ*<sup>i</sup>* has negative value , then this tool begins by a "coordinates transformation" from *q*ˆ*<sup>i</sup>* to *qi* such that the new parameters will be positive *qi* > 0, then an uncertainty box *Q* = { *q* = [*q*1, *q*2, ··· , *q*�] *T qi* > 0, *qi* ∈ [*q*<sup>−</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* ] } is makes, in other words, *<sup>Q</sup>* is in a positive convex cone *<sup>P</sup>*, *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� with minimum *<sup>v</sup>*min and maximum *v*max Euclidean vertices. The transformation is very easy as shown in the equation (1)

$$q\_i = q\_i^- + \frac{\mathfrak{q}\_i - \mathfrak{q}\_i^-}{\mathfrak{q}\_i^+ - \mathfrak{q}\_i^-} (q\_i^+ - q\_i^-) \tag{1}$$

From here on we will assume that if necessary, the transformation was made and work with parameters *qi* > 0. Under this consideration will continue with the rest of this topic.

**Definition 4.** *(Elizondo, 1999) Let f* : �� → � *be a continuous function and let Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� *be a box. It is said that f*(*q*) *has Sign Decomposition in Q if there exist two bounded continuous nondecreasing and nonnegative functions fn*(·) ≥ 0, *fp*(·) ≥ 0, *such that f*(*q*) = *fp*(*q*) − *fn*(*q*) ∀ *q* ∈ *Q. In this way there are defined the Positive Part fp*(*q*) *and Negative Part fn*(*q*) *of the function.*

Negative Part is only a name since Negative Part and Positive Part are nonnegative.

### **4.2** (*fn*, *fp*) **representation**

Is obvious that for the general case, *fn*(·) and *fp*(·) are independent functions then they make a basis in �<sup>2</sup> with graphical representation in the (*fn*(·), *fp*(·)) plane in accordance with figure ( 1).

8 Will-be-set-by-IN-TECH

In different areas of sciences the fundamental problem can be mapped to a problem of robust positivity of multivariable polynomic functions. For example the no singularity of a matrix can be analyzed by mean of the robust positivity of its determinant, so it is very useful to have a mathematical tool that solves the problem of robust positivity of multivariable polynomic functions. Practically there are three tools for this purpose: Interval Arithmetic (Moor, 1966); Bernstein Polynomials (Zettler, et all 1998) and Sign Decomposition ((Elizondo, 1999)) whose complete version is developed in (Elizondo, 1999) and its partial versions are presented in (Elizondo, 2000; 2001A;B; 2002A;B), for simplicity only will be mentioned (Elizondo, 1999). Interval arithmetic is very difficult to use because it requires much more calculations than other methods. When robust positivity is analyzed in a very simple function, Bernstein polynomials have advantages over sign decomposition, but when the function is not simple, sign decomposition has advantages over Bernstein polynomials (Graziano et al., 2004). There are several works using sign decomposition instead of Bernstein polynomials, some of them are: (Bhattacharyya et al., 2009; Guerrero, 2006; Keel et al., 2008; 2009; Keel, 2011; Knap et al.,

The following is a brief description of the more relevant results of Sign Decomposition (Elizondo, 1999). By means of this tool it is possible to determine, in necessary and sufficient conditions, the robust positivity of a multivariable polynomic function depending on �

Since mathematically exist the possibility that a parameter *q*ˆ*<sup>i</sup>* has negative value , then this tool begins by a "coordinates transformation" from *q*ˆ*<sup>i</sup>* to *qi* such that the new parameters will be

makes, in other words, *<sup>Q</sup>* is in a positive convex cone *<sup>P</sup>*, *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� with minimum *<sup>v</sup>*min and maximum *v*max Euclidean vertices. The transformation is very easy as shown in the equation

From here on we will assume that if necessary, the transformation was made and work with

**Definition 4.** *(Elizondo, 1999) Let f* : �� → � *be a continuous function and let Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� *be a box. It is said that f*(*q*) *has Sign Decomposition in Q if there exist two bounded continuous nondecreasing and nonnegative functions fn*(·) ≥ 0, *fp*(·) ≥ 0, *such that f*(*q*) = *fp*(*q*) − *fn*(*q*) ∀ *q* ∈ *Q. In this way there are defined the Positive Part fp*(*q*) *and Negative Part fn*(*q*) *of the*

Is obvious that for the general case, *fn*(·) and *fp*(·) are independent functions then they make a basis in �<sup>2</sup> with graphical representation in the (*fn*(·), *fp*(·)) plane in accordance with figure

− *i*

*<sup>i</sup>* <sup>+</sup> *<sup>q</sup>*ˆ*<sup>i</sup>* <sup>−</sup> *<sup>q</sup>*<sup>ˆ</sup>

*q*ˆ + *<sup>i</sup>* − *q*ˆ − *i* (*q*<sup>+</sup> *<sup>i</sup>* − *q*<sup>−</sup>

parameters *qi* > 0. Under this consideration will continue with the rest of this topic.

Negative Part is only a name since Negative Part and Positive Part are nonnegative.

*T*

*qi* > 0, *qi* ∈ [*q*<sup>−</sup>

*<sup>i</sup>* ) (1)

*<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* ] } is

**4. Brief description of sign decomposition**

2010; 2011)

(1)

*function.*

( 1).

**4.2** (*fn*, *fp*) **representation**

**4.1 Definition of sign decomposition**

parameters, employing extreme points analysis.

positive *qi* > 0, then an uncertainty box *Q* = { *q* = [*q*1, *q*2, ··· , *q*�]

*qi* = *q*<sup>−</sup>

If we take a particular vector *q* ∈ *Q* and evaluated the *fn*(*q*) and *fp*(*q*) parts, we obtain the coordinates (*fn*(*q*), *fp*(*q*)) of the function in the (*fn*, *fp*) plane. The 45*<sup>o</sup>* line is the set of points where the function is equal zero because *fp*(*q*) = *fn*(*q*) so *f*(*q*) = *fp*(*q*) − *fn*(*q*) = 0 . If a point is above the 45*<sup>o</sup>* line means that *fp*(*q*) > *fn*(*q*) then *f*(*q*) > 0. If a point is below the 45*<sup>o</sup>* line means that *fp*(*q*) < *fn*(*q*) then *f*(*q*) < 0.

Fig. 1. (*fn*, *fp*) plane

It should be noted that independently of the number of parameters in which the function depends on, the function will always be represented in �<sup>2</sup> via (*fn*(*q*), *fp*(*q*)). For example, the function *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup> <sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*1*q*<sup>2</sup> 2*q*3 <sup>3</sup> such that *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �3, *Q* = { *q* = [*q*1, *q*2, *q*3] *T qi* ∈ [0, 1] }. The function has sign decomposition because it is decomposed in two bounded continuous nondecreasing and nonnegative functions *f <sup>p</sup>*(*q*) = 4 + *q*1*q*<sup>3</sup> + 8*q*<sup>2</sup> <sup>1</sup>*q*2, *fn*(*q*) = *<sup>q</sup>*<sup>2</sup> <sup>+</sup> <sup>9</sup>*q*1*q*<sup>2</sup> 2*q*3 <sup>3</sup> and *f*(*q*) = *fp*(*q*) − *fn*(*q*). The figure ( 2) was obtained by plotting a hundred lines blue color, (one hundred fifty points per line) of variable *q*<sup>3</sup> holding (*q*1, *q*2) constant uniformly distributed in different positions. The process was repeated varying *q*<sup>2</sup> in green color and finally varying *q*<sup>1</sup> in red color. According to the position shown in the graph of the function with respect to the 45*<sup>o</sup>* line, it appears that the function is robustly positive. But it must be demonstrated mathematically.

Fig. 2. Function in (*fn*, *fp*) plane

Some preliminary properties of the continuous functions *f*(*q*), *g*(*q*), *h*(*q*) with sign decomposition in *Q* and for all *u*(*q*) nondecreasing function in *Q*, are proved in (Elizondo, 1999) as so facts, lemmas and theorems. This properties are employed on the following theorems.

**a)** (*fn*(*q*) + *u*(*q*), *fp*(*q*) + *u*(*q*)) is a (*fn*, *fp*) representation of the function *f*(*q*) ∀*q* ∈ *Q*; **b)** the representation (*fn*(*q*) + *u*(*q*), *fp*(*q*) + *u*(*q*)) of the function is reduced to its minimum expression: (*fp*(*q*), *fn*(*q*)); **c)** *f*(*q*) + *g*(*q*); **d)** *f*(*q*) − *g*(*q*) and **e)** *f*(*q*)*g*(*q*) are functions with sign decomposition in *Q*; **f)** if *f*(*q*) = *g*(*q*) + *h*(*q*), then the positive and negative parts of *f*(*q*) − *g*(*q*) are reduced to their minimum expressions, as follows: *f*(*q*) − *g*(*q*)=(*f*(*q*) − *g*(*q*))*<sup>p</sup>* − (*f*(*q*) − *g*(*q*))*n*, (*f*(*q*) − *g*(*q*))*<sup>n</sup>* = *fn*(*q*) − *gn*(*q*), (*f*(*q*) − *g*(*q*))*<sup>p</sup>* = *fp*(*q*) − *gp*(*q*).

### **4.3 The rectangle theorem**

Since negative part and positive part are bounded continuous nondecreasing functions, then the following inequalities ( 2) are fulfilled.

$$\begin{array}{l} f\_{\boldsymbol{\eta}}(\boldsymbol{\nu}^{\min}) \le f\_{\boldsymbol{\eta}}(\boldsymbol{q}) \le f\_{\boldsymbol{\eta}}(\boldsymbol{\nu}^{\max})\\ f\_{\boldsymbol{p}}(\boldsymbol{\nu}^{\min}) \le f\_{\boldsymbol{p}}(\boldsymbol{q}) \le f\_{\boldsymbol{p}}(\boldsymbol{\nu}^{\max}) \end{array} \tag{2}$$

*The graph of the function always "touches " the rectangle in lower left vertice and upper right vertice.*

Parametric Robust Stability 13

For the purpose of improving the results shown up to this point, the following proposition is necessary. In some cases it is necessary to analyze the function in a Γ box contained in *Q*, <sup>Γ</sup> <sup>⊂</sup> *<sup>Q</sup>*. The <sup>Γ</sup> box has Euclidean Vertices *<sup>μ</sup>*min and *<sup>μ</sup>*max. So, a vector in <sup>Γ</sup> is expressed as so as

**Proposition 7.** *(Elizondo, 1999) Let f* : � → � *be a continuous function in Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �, *let*

*<sup>i</sup>* <sup>−</sup> *<sup>μ</sup>*min

} *with minimum and maximum Euclidean vertices* **<sup>0</sup>**, *<sup>δ</sup>*max <sup>=</sup> *<sup>μ</sup>*max <sup>−</sup> *<sup>μ</sup>*min, *and let q* <sup>∈</sup> <sup>Γ</sup>*<sup>j</sup> a vector such that q* <sup>=</sup> *<sup>μ</sup>*min <sup>+</sup> *<sup>δ</sup> where <sup>δ</sup>* <sup>∈</sup> <sup>Δ</sup>. *Then the function f*(*q*) *is expressed by its: linear, nonlinear and*

*δ*<sup>2</sup> + ··· +

Must be noted that *f* min = *f*(*μ*min). On other hand, it is clear that we can use the concepts of

*fLp*(*δ*) − *fLn*(*δ*) + *fN p*(*δ*) − *fNn*(*δ*) obtaining the following equations (3) where the relation

*<sup>p</sup>* + *fLp*(*δ*) + *fN p*(*δ*)

*.*

*∂ f*(*q*) *q*

 *μ*min *δ*

*<sup>n</sup>* <sup>+</sup> *fLn*(*δ*) + *fNn*(*δ*) (3)

*<sup>i</sup>* <sup>=</sup> *<sup>μ</sup>*max

} *with minimum and maximum Euclidean vertices <sup>μ</sup>*min,

*<sup>i</sup>* } ⊂ *<sup>P</sup>* ⊂ � *be a box with its vertices set*

*<sup>p</sup>* <sup>−</sup> *<sup>f</sup>* min

*<sup>n</sup>* +

*The graph of the function is not necessarily convex.*

Fig. 3. Rectangle theorem

**4.4 The polygon theorem**

<sup>Γ</sup>*<sup>j</sup>* <sup>⊂</sup> *Q be a box with its vertices set* {*μ<sup>i</sup>*

*<sup>f</sup>*(*q*) = *<sup>f</sup>* min <sup>+</sup> *fL*(*δ*) + *fN*(*δ*) <sup>|</sup> *<sup>δ</sup>* <sup>∈</sup> <sup>Δ</sup>∀*<sup>q</sup>* <sup>∈</sup> <sup>Γ</sup>*<sup>j</sup>*

*fL*(*δ*) *Linear Part* = ∇ *f*(*q*)|*μ*min · *δ* ∀*δ* ∈ Δ

 *μ*min

between *δ* and *q* can be appreciated in the figure (4).

*q*1

*<sup>μ</sup>*max, *let* <sup>Δ</sup> <sup>=</sup> {*<sup>δ</sup>* <sup>|</sup> *<sup>δ</sup><sup>i</sup>* <sup>∈</sup> [0, *<sup>δ</sup>*max

*f* min *Indepent Part* = *f*(*μ*min)

<sup>∇</sup> *<sup>f</sup>*(*q*)|*μ*min · *<sup>δ</sup>* <sup>=</sup> *<sup>∂</sup> <sup>f</sup>*(*q*)

{*δi*

*q* = *μ*min + *δ*, where *δ* is a vector in Γ, with origins in *μ*min.

*independent parts, in its minimum expression for all q* <sup>∈</sup> <sup>Γ</sup>*<sup>j</sup>*

*<sup>i</sup>* ], *<sup>δ</sup>*max

*fN*(*δ*) *Nonlinear Part* <sup>=</sup> *<sup>f</sup>*(*μ*min <sup>+</sup> *<sup>δ</sup>*) <sup>−</sup> *<sup>f</sup>* min <sup>−</sup> *fL*(*δ*) <sup>∀</sup>*<sup>δ</sup>* <sup>∈</sup> <sup>Δ</sup>

*∂ f*(*q*) *q*2

*fp*(*q*) = *f* min

*fn*(*q*) = *f* min

 *μ*min

positive part and negative part in the above proposition, So, *<sup>f</sup> <sup>p</sup>*(*q*) <sup>−</sup> *fn*(*q*) = *<sup>f</sup>* min

*δ*<sup>1</sup> +

This means that a function *f*(*q*) with sign decomposition, evaluated at any vector *q* ∈ *Q*, its negative part is contained in a segment and also the positive part is contained in another segment. So, on (*fn*, *fp*) plane the function is contained in a rectangle as expressed by the following theorem according to figure ( 3 ).

**Theorem 5.** *(Elizondo, 1999) Rectangle Theorem. Let f* : �� → � *be a continuous function with sign decomposition in a box Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� *with minimum and maximum Euclidean vertices <sup>v</sup>*min, *<sup>v</sup>*max, *then: a) <sup>f</sup>*(*q*) *is lower and upper bounded by f <sup>p</sup>*(*v*min) <sup>−</sup> *fn*(*v*max) *and fp*(*v*max) <sup>−</sup> *fn*(*v*min) *respectively; b) The graphical representation of the function f*(*q*), <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *Q in* (*fn*, *fp*) *plane is contained in the rectangle with vertices* (*fn*(*v*min), *fp*(*v*min)), (*fn*(*v*max), *fp*(*v*max)), (*fn*(*v*min), *fp*(*v*max)) *and* (*fn*(*v*max), *fp*(*v*min)); *c) if the lower right vertex* (*fn*(*v*max), *fp*(*v*min)) *is over the* <sup>45</sup>*<sup>o</sup> line then f*(*q*) <sup>&</sup>gt; <sup>0</sup> <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>*; *d) if the upper left vertex* (*fn*(*v*min), *fp*(*v*max)) *is below the* <sup>45</sup>*<sup>o</sup> line then f*(*q*) <sup>&</sup>lt; <sup>0</sup> <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *Q. In accordance with figure ( 3 ).*

The above result seems to be very useful, we can say that the rectangle is the "house" where the multivariable function lives in �2. We can know the robust positivity of a function analyzing only one point. It is important to note that this is only sufficient conditions, the lower right vertex can be below the 45*<sup>o</sup>* line and the function could be robustly positive or not be. But if the lower right vertex is above the 45*<sup>o</sup>* line then the function is robustly positive.

For example, the function *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup> <sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*<sup>3</sup> 3*q*1*q*<sup>2</sup> <sup>2</sup> such that *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �3, *Q* = { *q* = [*q*1, *q*2, *q*3] *T qi* ∈ [0, 1] }, has sign decomposition, its minimum and maximum Euclidean vertices are *νmin* = [0, 0] *<sup>T</sup>*, *νmax* = [1, 1] *<sup>T</sup>*, their positive and negative psrtes are: *fp*(*q*) = 4 + *q*1*q*<sup>3</sup> + 8*q*<sup>2</sup> <sup>1</sup>*q*2, *fn*(*q*) = *<sup>q</sup>*<sup>2</sup> <sup>+</sup> <sup>9</sup>*q*<sup>3</sup> 3*q*1*q*<sup>2</sup> <sup>2</sup>. Then the lower bound is *<sup>f</sup> <sup>p</sup>*(*v*min) <sup>−</sup> *fn*(*v*max), *fp*(*v*min) = 4 + (0)(0) + 8(0)(0) = 4, *fn*(*v*max) = 1 + 9(1)(1)(1) = 10, the lower bound is 4 − 10 = −9. The function could be robustly positive, but for now we do not know, It is necessary see more signs of decomposition items.

#### **Remark 6.** *Should be noted three important concepts:*

*The graph of the function does not "fills" the whole rectangle, but it is contained in.*

10 Will-be-set-by-IN-TECH

Some preliminary properties of the continuous functions *f*(*q*), *g*(*q*), *h*(*q*) with sign decomposition in *Q* and for all *u*(*q*) nondecreasing function in *Q*, are proved in (Elizondo, 1999) as so facts, lemmas and theorems. This properties are employed on the following

**a)** (*fn*(*q*) + *u*(*q*), *fp*(*q*) + *u*(*q*)) is a (*fn*, *fp*) representation of the function *f*(*q*) ∀*q* ∈ *Q*; **b)** the representation (*fn*(*q*) + *u*(*q*), *fp*(*q*) + *u*(*q*)) of the function is reduced to its minimum expression: (*fp*(*q*), *fn*(*q*)); **c)** *f*(*q*) + *g*(*q*); **d)** *f*(*q*) − *g*(*q*) and **e)** *f*(*q*)*g*(*q*) are functions with sign decomposition in *Q*; **f)** if *f*(*q*) = *g*(*q*) + *h*(*q*), then the positive and negative parts of *f*(*q*) − *g*(*q*) are reduced to their minimum expressions, as follows: *f*(*q*) − *g*(*q*)=(*f*(*q*) − *g*(*q*))*<sup>p</sup>* − (*f*(*q*) − *g*(*q*))*n*, (*f*(*q*) − *g*(*q*))*<sup>n</sup>* = *fn*(*q*) − *gn*(*q*), (*f*(*q*) − *g*(*q*))*<sup>p</sup>* = *fp*(*q*) − *gp*(*q*).

Since negative part and positive part are bounded continuous nondecreasing functions, then

*fn*(*νmin*) <sup>≤</sup> *fn*(*q*) <sup>≤</sup> *fn*(*νmax*)

This means that a function *f*(*q*) with sign decomposition, evaluated at any vector *q* ∈ *Q*, its negative part is contained in a segment and also the positive part is contained in another segment. So, on (*fn*, *fp*) plane the function is contained in a rectangle as expressed by the

**Theorem 5.** *(Elizondo, 1999) Rectangle Theorem. Let f* : �� → � *be a continuous function with sign decomposition in a box Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� *with minimum and maximum Euclidean vertices <sup>v</sup>*min, *<sup>v</sup>*max, *then: a) <sup>f</sup>*(*q*) *is lower and upper bounded by f <sup>p</sup>*(*v*min) <sup>−</sup> *fn*(*v*max) *and fp*(*v*max) <sup>−</sup> *fn*(*v*min) *respectively; b) The graphical representation of the function f*(*q*), <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *Q in* (*fn*, *fp*) *plane is contained in the rectangle with vertices* (*fn*(*v*min), *fp*(*v*min)), (*fn*(*v*max), *fp*(*v*max)), (*fn*(*v*min), *fp*(*v*max)) *and* (*fn*(*v*max), *fp*(*v*min)); *c) if the lower right vertex* (*fn*(*v*max), *fp*(*v*min)) *is over the* <sup>45</sup>*<sup>o</sup> line then f*(*q*) <sup>&</sup>gt; <sup>0</sup> <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>*; *d) if the upper left vertex* (*fn*(*v*min), *fp*(*v*max)) *is below*

The above result seems to be very useful, we can say that the rectangle is the "house" where the multivariable function lives in �2. We can know the robust positivity of a function analyzing only one point. It is important to note that this is only sufficient conditions, the lower right vertex can be below the 45*<sup>o</sup>* line and the function could be robustly positive or not be. But if the lower right vertex is above the 45*<sup>o</sup>* line then the function is robustly positive.

*<sup>T</sup>*, *νmax* = [1, 1]

3*q*1*q*<sup>2</sup>

*The graph of the function does not "fills" the whole rectangle, but it is contained in.*

*fp*(*v*min) = 4 + (0)(0) + 8(0)(0) = 4, *fn*(*v*max) = 1 + 9(1)(1)(1) = 10, the lower bound is 4 − 10 = −9. The function could be robustly positive, but for now we do not know, It is

<sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*<sup>3</sup>

*qi* ∈ [0, 1] }, has sign decomposition, its minimum and maximum

3*q*1*q*<sup>2</sup>

<sup>2</sup> such that *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �3,

*<sup>T</sup>*, their positive and negative psrtes are:

<sup>2</sup>. Then the lower bound is *<sup>f</sup> <sup>p</sup>*(*v*min) <sup>−</sup> *fn*(*v*max),

*fp*(*νmin*) <sup>≤</sup> *fp*(*q*) <sup>≤</sup> *fp*(*νmax*) (2)

theorems.

**4.3 The rectangle theorem**

the following inequalities ( 2) are fulfilled.

following theorem according to figure ( 3 ).

*the* <sup>45</sup>*<sup>o</sup> line then f*(*q*) <sup>&</sup>lt; <sup>0</sup> <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *Q. In accordance with figure ( 3 ).*

<sup>1</sup>*q*2, *fn*(*q*) = *<sup>q</sup>*<sup>2</sup> <sup>+</sup> <sup>9</sup>*q*<sup>3</sup>

For example, the function *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup>

*T*

necessary see more signs of decomposition items. **Remark 6.** *Should be noted three important concepts:*

Euclidean vertices are *νmin* = [0, 0]

*Q* = { *q* = [*q*1, *q*2, *q*3]

*fp*(*q*) = 4 + *q*1*q*<sup>3</sup> + 8*q*<sup>2</sup>

*The graph of the function always "touches " the rectangle in lower left vertice and upper right vertice. The graph of the function is not necessarily convex.*

Fig. 3. Rectangle theorem

#### **4.4 The polygon theorem**

For the purpose of improving the results shown up to this point, the following proposition is necessary. In some cases it is necessary to analyze the function in a Γ box contained in *Q*, <sup>Γ</sup> <sup>⊂</sup> *<sup>Q</sup>*. The <sup>Γ</sup> box has Euclidean Vertices *<sup>μ</sup>*min and *<sup>μ</sup>*max. So, a vector in <sup>Γ</sup> is expressed as so as *q* = *μ*min + *δ*, where *δ* is a vector in Γ, with origins in *μ*min.

**Proposition 7.** *(Elizondo, 1999) Let f* : � → � *be a continuous function in Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �, *let* <sup>Γ</sup>*<sup>j</sup>* <sup>⊂</sup> *Q be a box with its vertices set* {*μ<sup>i</sup>* } *with minimum and maximum Euclidean vertices <sup>μ</sup>*min, *<sup>μ</sup>*max, *let* <sup>Δ</sup> <sup>=</sup> {*<sup>δ</sup>* <sup>|</sup> *<sup>δ</sup><sup>i</sup>* <sup>∈</sup> [0, *<sup>δ</sup>*max *<sup>i</sup>* ], *<sup>δ</sup>*max *<sup>i</sup>* <sup>=</sup> *<sup>μ</sup>*max *<sup>i</sup>* <sup>−</sup> *<sup>μ</sup>*min *<sup>i</sup>* } ⊂ *<sup>P</sup>* ⊂ � *be a box with its vertices set* {*δi* } *with minimum and maximum Euclidean vertices* **<sup>0</sup>**, *<sup>δ</sup>*max <sup>=</sup> *<sup>μ</sup>*max <sup>−</sup> *<sup>μ</sup>*min, *and let q* <sup>∈</sup> <sup>Γ</sup>*<sup>j</sup> a vector such that q* <sup>=</sup> *<sup>μ</sup>*min <sup>+</sup> *<sup>δ</sup> where <sup>δ</sup>* <sup>∈</sup> <sup>Δ</sup>. *Then the function f*(*q*) *is expressed by its: linear, nonlinear and independent parts, in its minimum expression for all q* <sup>∈</sup> <sup>Γ</sup>*<sup>j</sup> .*

$$\begin{array}{l} f(q) = f^{\min} + f\_{\mathcal{L}}(\delta) + f\_{N}(\delta) \mid \delta \in \Delta \forall q \in \Gamma^{j} \\ f^{\min} \triangleq \text{Independent} & \delta = f(\mu^{\min}) \\ f\_{\mathcal{L}}(\delta) \triangleq \text{Linear Part} = \nabla f(q)|\_{\mu^{\min}} \cdot \delta \quad \forall \delta \in \Delta \\ f\_{N}(\delta) \triangleq \text{Nonlinear Part} = f(\mu^{\min} + \delta) - f^{\min} - f\_{\mathcal{L}}(\delta) \cdot \forall \delta \in \Delta \\ |\nabla f(q)|\_{\mu^{\min}} \cdot \delta = \left. \frac{\partial f(q)}{q\_{1}} \right|\_{\mu^{\min}} \delta\_{1} + \left. \frac{\partial f(q)}{q\_{2}} \right|\_{\mu^{\min}} \delta\_{2} + \dots + \left. \frac{\partial f(q)}{q\_{\ell}} \right|\_{\mu^{\min}} \delta\_{\ell} \end{array}$$

Must be noted that *f* min = *f*(*μ*min). On other hand, it is clear that we can use the concepts of positive part and negative part in the above proposition, So, *<sup>f</sup> <sup>p</sup>*(*q*) <sup>−</sup> *fn*(*q*) = *<sup>f</sup>* min *<sup>p</sup>* <sup>−</sup> *<sup>f</sup>* min *<sup>n</sup>* + *fLp*(*δ*) − *fLn*(*δ*) + *fN p*(*δ*) − *fNn*(*δ*) obtaining the following equations (3) where the relation between *δ* and *q* can be appreciated in the figure (4).

$$\begin{array}{l} f\_p(q) = f\_p^{\min} + f\_{Lp}(\delta) + f\_{Np}(\delta) \\ f\_n(q) = f\_n^{\min} + f\_{Ln}(\delta) + f\_{Nn}(\delta) \end{array} \tag{3}$$

*fN p*(*δ*) = *fp*(*q*) <sup>−</sup> *<sup>f</sup> min*

*fNn*(*δ*) = *fn*(*q*) <sup>−</sup> *<sup>f</sup> min*

the figure ( 2), *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup>

*T*

the function is robustly positive in the Γ box.

� , *<sup>q</sup>*<sup>+</sup>

*<sup>i</sup>* , *<sup>q</sup>*<sup>1</sup> *<sup>i</sup>* ], [*q*<sup>1</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>2</sup>

*q*<sup>2</sup> + 9*q*1*q*<sup>2</sup> 2 

*k* new interval, giving cause to the generation of *k*� new boxes Γ*<sup>i</sup>* = [*γ*<sup>−</sup>

. Through these concepts, the following theorem is obtained.

**4.5 The box partition theorem**

<sup>2</sup> ] ×···× [*q*<sup>−</sup>

generating *k* new intervals: [*q*−

*j*

4 + *q*<sup>1</sup> + 8*q*<sup>2</sup>

<sup>1</sup>*q*<sup>2</sup> −

to obtain new results.

� , *<sup>γ</sup>*<sup>+</sup>

×[*q*<sup>−</sup> <sup>2</sup> , *<sup>q</sup>*<sup>+</sup>

···× [*γ*<sup>−</sup>

*such that Q* =

*Q* = *i* Γ*i*

Γ*<sup>j</sup> box).*

or not.

*f*(*q*) =

box <sup>Γ</sup> <sup>⊂</sup> *<sup>Q</sup>*, with Euclidean vertices *<sup>μ</sup>*min = [0.2 0.2 0.2 ]

*Q* = { *q* = [*q*1, *q*2, *q*3]

*fLp*(*δ*) = ∇ *fp*(*q*)

*fLn*(*δ*) = ∇ *fn*(*q*)|*μ*min · *δ*

Parametric Robust Stability 15

As an illustration of this theme, by means of rectangle theorem and polygon, we will analyze the lower bound of a function in a gama box. Consider the function corresponding to

In accordance with the Rectangle Theorem (3) the lower bound is *fp*(*v*min) <sup>−</sup> *fn*(*v*max) = <sup>−</sup>0.1403. Applying the Polygon Theorem (8) the lower bound is *<sup>f</sup>* min <sup>+</sup> *fL* min <sup>−</sup> *fNn*(*δ*max), so it is necessary to obtain each of these expressions, the results are as follows: *f* min = *<sup>f</sup>*(*μ*min) = 3.9034, *fL* min <sup>=</sup> <sup>−</sup>0.4457, *fNn*(*δ*max) = 3.3825. The last value is obtained of ecuations (4), thus the lower bound is 0.0752. By means of the Rectangle Theorem is obtained *f*(*q*) > −0.1403 ∀*q* ∈ Γ, following the Polygon Theorem is obtained *f*(*q*) > 0.0752 ∀*q* ∈ Γ, so

By means of Rectangle Theorem (3) and Polygon Theorem (8) are obtained sufficient conditions of robust positivity, so to obtain necessary and sufficient conditions is necessary

*<sup>i</sup>* ], ··· , [*q*

**Theorem 9.** *Box Partition Theorem (Elizondo, 1999). Let f* : �� → � *be a continuous function with sign decomposition in Q such that Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� *is a box with minimum and maximum Euclidean vertices v*min, *v*max. *Then the function f*(*q*)*is positive (negative) in Q if and only if a* Γ *boxes set exists,*

This theorem can be applied in two ways, one of them we call " **Analytical Partition**" and the other one "**Constant Partition**". In analytical partition, the box where the function has a negative lower bound is subdivided iteratively. In the case of the function is robustly positive is also obtained information about where the function is close to losing positivity. By means of constant partition is only obtained information on whether the function is robustly positive

To illustrate both procedures, we analyze the robust positivity of the function (Elizondo, 1999)

robust positivity is analized by means of the rectangle theorem because it is more easier to

� ]. In this case it is possible to divide each variable [*q*<sup>−</sup>

*j i* , *q j*+1

� ] with *<sup>μ</sup>*min, *<sup>μ</sup>*max <sup>∈</sup> <sup>Γ</sup>*<sup>i</sup>* minimum and maximum Euclidean vertices of <sup>Γ</sup>*<sup>i</sup>* and

<sup>Γ</sup>*<sup>j</sup> and Lower Bound* <sup>≥</sup> *<sup>c</sup>* <sup>&</sup>gt; <sup>0</sup> *for each* <sup>Γ</sup>*<sup>j</sup> box (U pper Bound* <sup>≤</sup> *<sup>c</sup>* <sup>&</sup>lt; <sup>0</sup> *for each one*

, such that *Q* = { *q* = [*q*1, *q*2]

*<sup>i</sup>* ], ··· [*qk*−<sup>1</sup>

*<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup>

*T*

When it is not possible to know whether the function is positive or not in *Q* = [*q*−

<sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*<sup>3</sup>

*<sup>p</sup>* − *fLp*(*δ*)

*<sup>n</sup>* − *fLn*(*δ*)

3*q*1*q*<sup>2</sup>

*qi* ∈ [0, 1] }. Suppose that the function is analyzed into a gamma

(4)

*T*.

<sup>1</sup> , *<sup>q</sup>*<sup>+</sup> 1 ]

*<sup>i</sup>* ] be a

*<sup>i</sup>* ] in *k* parts,

<sup>2</sup> , *<sup>γ</sup>*<sup>+</sup> <sup>2</sup> ] ×

*<sup>i</sup>* , *<sup>γ</sup>*<sup>+</sup>

*<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup>

<sup>1</sup> ] × [*γ*<sup>−</sup>

*qi* ∈ [0, 1] ∀*i* } . The

*<sup>i</sup>* ], let [*γ*<sup>−</sup>

<sup>1</sup> , *<sup>γ</sup>*<sup>+</sup>

<sup>2</sup> such that *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �3,

*<sup>T</sup>* and *μ*max = [0.85 0.85 0.85 ]

 *<sup>μ</sup>*min · *δ*

Fig. 4. Gamma box

**Theorem 8.** *Polygon Theorem (Elizondo, 1999). Let f* : � → � *be a continuous function with sign decomposition in Q*, *let q*, *δ*, Γ*<sup>j</sup> and* Δ *in accordance with the proposition (7). Then, a) the lower and upper bounds of the function f*(*q*) *are: Lower Bound* <sup>=</sup> *<sup>f</sup>* min <sup>+</sup> *fL* min <sup>−</sup> *fNn*(*δ*max) *and Upper Bound* <sup>=</sup> *<sup>f</sup>* min <sup>+</sup> *fL* max <sup>+</sup> *fN p*(*δ*max) <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>*, *b) the bounds of incise "a", are contained in the interval defined by the bounds of the rectangle theorem 3. f <sup>p</sup>*(*μ*min) <sup>−</sup> *fn*(*μ*max) <sup>≤</sup> *Lower Bound* <sup>≤</sup> *Upper Bound* <sup>≤</sup> *fp*(*μ*max) <sup>−</sup> *fn*(*μ*min), *c) The graphical representation of the function f*(*q*) ∀*q* ∈ Γ *in the* (*fn*, *fp*) *plane is contained in the polygon defined by the intersection of the rectangle of the rectangle theorem (5) and the space between the two* 45*<sup>o</sup> lines separated from the origin by the Lower Bound and U pper Bound in accordance with figure (5).*

Fig. 5. Bounding of the function

The symbolic expression of the nonlinear part used in the above theorem is not necessary to obtain, because we will use only its numerical value. So, from the equations (3), the nonlinear parts are obtained as so as equations ( 4).

12 Will-be-set-by-IN-TECH

**Theorem 8.** *Polygon Theorem (Elizondo, 1999). Let f* : � → � *be a continuous function with sign decomposition in Q*, *let q*, *δ*, Γ*<sup>j</sup> and* Δ *in accordance with the proposition (7). Then, a) the lower and upper bounds of the function f*(*q*) *are: Lower Bound* <sup>=</sup> *<sup>f</sup>* min <sup>+</sup> *fL* min <sup>−</sup> *fNn*(*δ*max) *and Upper Bound* <sup>=</sup> *<sup>f</sup>* min <sup>+</sup> *fL* max <sup>+</sup> *fN p*(*δ*max) <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>*, *b) the bounds of incise "a", are contained in the interval defined by the bounds of the rectangle theorem 3. f <sup>p</sup>*(*μ*min) <sup>−</sup> *fn*(*μ*max) <sup>≤</sup> *Lower Bound* <sup>≤</sup> *Upper Bound* <sup>≤</sup> *fp*(*μ*max) <sup>−</sup> *fn*(*μ*min), *c) The graphical representation of the function f*(*q*) ∀*q* ∈ Γ *in the* (*fn*, *fp*) *plane is contained in the polygon defined by the intersection of the rectangle of the rectangle theorem (5) and the space between the two* 45*<sup>o</sup> lines separated from the origin*

The symbolic expression of the nonlinear part used in the above theorem is not necessary to obtain, because we will use only its numerical value. So, from the equations (3), the nonlinear

*by the Lower Bound and U pper Bound in accordance with figure (5).*

Fig. 4. Gamma box

Fig. 5. Bounding of the function

parts are obtained as so as equations ( 4).

$$\begin{array}{l} f\_{Np}(\delta) = f\_p(q) - f\_p^{\min} - f\_{\mathcal{L}p}(\delta) \\ f\_{N\mathcal{n}}(\delta) = f\_{\mathcal{n}}(q) - f\_{\mathcal{n}}^{\min} - f\_{\mathcal{L}\mathcal{n}}(\delta) \\ f\_{\mathcal{L}p}(\delta) = \nabla f\_p(q)|\_{\mu^{\min}} \cdot \delta \\ f\_{\mathcal{L}\mathcal{n}}(\delta) = \nabla f\_{\mathcal{N}}(q)|\_{\mu^{\min}} \cdot \delta \end{array} \tag{4}$$

As an illustration of this theme, by means of rectangle theorem and polygon, we will analyze the lower bound of a function in a gama box. Consider the function corresponding to the figure ( 2), *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup> <sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*<sup>3</sup> 3*q*1*q*<sup>2</sup> <sup>2</sup> such that *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>* ⊂ �3, *Q* = { *q* = [*q*1, *q*2, *q*3] *T qi* ∈ [0, 1] }. Suppose that the function is analyzed into a gamma box <sup>Γ</sup> <sup>⊂</sup> *<sup>Q</sup>*, with Euclidean vertices *<sup>μ</sup>*min = [0.2 0.2 0.2 ] *<sup>T</sup>* and *μ*max = [0.85 0.85 0.85 ] *T*.

In accordance with the Rectangle Theorem (3) the lower bound is *fp*(*v*min) <sup>−</sup> *fn*(*v*max) = <sup>−</sup>0.1403. Applying the Polygon Theorem (8) the lower bound is *<sup>f</sup>* min <sup>+</sup> *fL* min <sup>−</sup> *fNn*(*δ*max), so it is necessary to obtain each of these expressions, the results are as follows: *f* min = *<sup>f</sup>*(*μ*min) = 3.9034, *fL* min <sup>=</sup> <sup>−</sup>0.4457, *fNn*(*δ*max) = 3.3825. The last value is obtained of ecuations (4), thus the lower bound is 0.0752. By means of the Rectangle Theorem is obtained *f*(*q*) > −0.1403 ∀*q* ∈ Γ, following the Polygon Theorem is obtained *f*(*q*) > 0.0752 ∀*q* ∈ Γ, so the function is robustly positive in the Γ box.

### **4.5 The box partition theorem**

By means of Rectangle Theorem (3) and Polygon Theorem (8) are obtained sufficient conditions of robust positivity, so to obtain necessary and sufficient conditions is necessary to obtain new results.

When it is not possible to know whether the function is positive or not in *Q* = [*q*− <sup>1</sup> , *<sup>q</sup>*<sup>+</sup> 1 ] ×[*q*<sup>−</sup> <sup>2</sup> , *<sup>q</sup>*<sup>+</sup> <sup>2</sup> ] ×···× [*q*<sup>−</sup> � , *<sup>q</sup>*<sup>+</sup> � ]. In this case it is possible to divide each variable [*q*<sup>−</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* ] in *k* parts, generating *k* new intervals: [*q*− *<sup>i</sup>* , *<sup>q</sup>*<sup>1</sup> *<sup>i</sup>* ], [*q*<sup>1</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>2</sup> *<sup>i</sup>* ], ··· , [*q j i* , *q j*+1 *<sup>i</sup>* ], ··· [*qk*−<sup>1</sup> *<sup>i</sup>* , *<sup>q</sup>*<sup>+</sup> *<sup>i</sup>* ], let [*γ*<sup>−</sup> *<sup>i</sup>* , *<sup>γ</sup>*<sup>+</sup> *<sup>i</sup>* ] be a *k* new interval, giving cause to the generation of *k*� new boxes Γ*<sup>i</sup>* = [*γ*<sup>−</sup> <sup>1</sup> , *<sup>γ</sup>*<sup>+</sup> <sup>1</sup> ] × [*γ*<sup>−</sup> <sup>2</sup> , *<sup>γ</sup>*<sup>+</sup> <sup>2</sup> ] × ···× [*γ*<sup>−</sup> � , *<sup>γ</sup>*<sup>+</sup> � ] with *<sup>μ</sup>*min, *<sup>μ</sup>*max <sup>∈</sup> <sup>Γ</sup>*<sup>i</sup>* minimum and maximum Euclidean vertices of <sup>Γ</sup>*<sup>i</sup>* and *Q* = *i* Γ*i* . Through these concepts, the following theorem is obtained.

**Theorem 9.** *Box Partition Theorem (Elizondo, 1999). Let f* : �� → � *be a continuous function with sign decomposition in Q such that Q* <sup>⊂</sup> *<sup>P</sup>* ⊂ �� *is a box with minimum and maximum Euclidean vertices v*min, *v*max. *Then the function f*(*q*)*is positive (negative) in Q if and only if a* Γ *boxes set exists, such that Q* = *j* <sup>Γ</sup>*<sup>j</sup> and Lower Bound* <sup>≥</sup> *<sup>c</sup>* <sup>&</sup>gt; <sup>0</sup> *for each* <sup>Γ</sup>*<sup>j</sup> box (U pper Bound* <sup>≤</sup> *<sup>c</sup>* <sup>&</sup>lt; <sup>0</sup> *for each one* Γ*<sup>j</sup> box).*

This theorem can be applied in two ways, one of them we call " **Analytical Partition**" and the other one "**Constant Partition**". In analytical partition, the box where the function has a negative lower bound is subdivided iteratively. In the case of the function is robustly positive is also obtained information about where the function is close to losing positivity. By means of constant partition is only obtained information on whether the function is robustly positive or not.

To illustrate both procedures, we analyze the robust positivity of the function (Elizondo, 1999) *f*(*q*) = 4 + *q*<sup>1</sup> + 8*q*<sup>2</sup> <sup>1</sup>*q*<sup>2</sup> − *q*<sup>2</sup> + 9*q*1*q*<sup>2</sup> 2 , such that *Q* = { *q* = [*q*1, *q*2] *T qi* ∈ [0, 1] ∀*i* } . The robust positivity is analized by means of the rectangle theorem because it is more easier to apply, although it must be said that the bounds of the polygon theorem are better than the rectangle theorem.

**Analytical Partition** (Elizondo, 1999). In the subfigure 1 of figure (6) shows that the function is robustly positive in boxes Γ<sup>1</sup> and Γ<sup>3</sup> but not in the boxes Γ<sup>2</sup> and Γ4. So it is necessary apply iteratively the partition box to the boxes where the function is not robust positive, in this way is obtained the subfigure 2 of figure (6). Since there is a set of boxes such that *Q* = *j* Γ*j* <sup>|</sup> *<sup>f</sup>*(*q*) <sup>&</sup>gt; <sup>0</sup> <sup>∀</sup>Γ*<sup>j</sup>* , then the function is robustly positive in *Q*. The graphs were made to show

the procedure in visual way, but for more than two dimensions, using software we can get the coordinates and dimensions of sub boxes where the function is close to losing positivity.

Fig. 7. Function in (*fn*, *fp*) plane

negative and positive representation.

In some cases as so as figure (7) it is not easy to determine in graphic way whether a point close to the 45*<sup>o</sup>* line is over this line or not. So in (Elizondo, 1999) the (*α*, *β*) representation was developed, *<sup>α</sup>*(*q*) = *fp*(*q*) + *fn*(*q*), *<sup>β</sup>*(*q*) = *fp*(*q*) <sup>−</sup> *fn*(*q*) , it is similar to rotated 45*<sup>o</sup>* the axis with respect to (*fn*, *fp*) representation implying some graphical and algebraic advantages over the

Parametric Robust Stability 17

**Definition 10.** *(Elizondo, 1999) Let fn*(*q*) *and fp*(*q*) *be the negative and positive parts of a continuous function f*(*q*) *with sign decomposition in Q*. *Let T be the linear transformation described below such that T* <sup>−</sup><sup>1</sup> *exists, then it is called a representation of the function f*(*q*), *in* (*α*, *β*) *coordinates, to the linear transformation* (*α*(*q*), *β*(*q*)) = *T*(*fn*(*q*), *fp*(*q*)) *and the inverse transformation of an*

> *fn*(*q*) *fp*(*q*)

With the purpose to show the advantages of the (*α*, *β*) representation, by means of the rectangle theorem we analyze the same function in the previous subsection *f*(*q*) =

positive implying that all the bounds are positives and consequently the function is robustly

in the figure (9) in (*α*, *β*) representation. We can see that beta axis scale is positive implying

The original idea to develop the representation (*α* , *β*) (Elizondo, 1999) was to solve a visual geometric problem, but this representation has interesting algebraic properties on continuous functions *f*(*q*), *g*(*q*), *h*(*q*) with sign decomposition in *Q* and for all *u*(*q*) nondecreasing

3*q*1*q*<sup>2</sup>

<sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*<sup>3</sup>

*T*−<sup>1</sup> = <sup>1</sup> 2 1 −1 1 1

= *T*−<sup>1</sup> *α*(*q*) *β*(*q*) 

<sup>2</sup> (*α*(*q*) + *β*(*q*))

<sup>2</sup> (*α*(*q*) − *β*(*q*))

applying *k* = 13. We can see in the figure (8) beta axis scale is

<sup>2</sup> corresponding to the figure (2) is shown

(*α*(*q*), *β*(*q*)) *representation is a* (*fn*(*q*), *fp*(*q*)) *representation of the function f*(*q*).

 *fn*(*q*) *fp*(*q*)

*α*(*q*) = *fp*(*q*) + *fn*(*q*) *fp*(*q*) = <sup>1</sup>

*<sup>β</sup>*(*q*) = *fp*(*q*) <sup>−</sup> *fn*(*q*) *fn*(*q*) = <sup>1</sup>

 1 1 <sup>−</sup>1 1

*T* =

 *α*(*q*) *β*(*q*) = *T*

*q*<sup>2</sup> + 9*q*1*q*<sup>2</sup> 2 

function in *Q*, (Elizondo, 1999) as the following:

The function *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup>

the function is robustly positive.

**4.6** (*α*, *β*) **Representation**

4 + *q*<sup>1</sup> + 8*q*<sup>2</sup>

positive.

<sup>1</sup>*q*<sup>2</sup> −

Fig. 6. Partition box

**Constant Partition** (Elizondo, 1999). In this procedure the domaine of each one of the � parameters is divide in *k* equal parts (not necessarily equal), in this way, it is generated a boxes set of *k*� sub boxes Γ*<sup>i</sup>* such that *Q* = *j* Γ*j* . The robust positivity of each Γ*<sup>i</sup>* box can be

analyzed by a computer program so that the computer give us the final result about the robust positivity of the function.

Another way is through a software which plot a × (blue) mark in the (*fn*, *fp*) plane in each (*fn*(*μ*min), *fp*(*μ*min)) and (*fn*(*μ*max), *fp*(*μ*max)) coordinates corresponding to the minimum and maximum vertices of each Γ*<sup>i</sup>* box, and plot too a + (red) mark corresponding to the lower bound of each Γ*<sup>i</sup>* box, as can be appreciated in figure (7) that it was obtained with *k* = 13.

If a <sup>×</sup> (blue) mark is below the 45*<sup>o</sup>* line, means that there is at least one vector for which the function is negative and therefore the function is not robustly positive. If all the × (blue) marks are above the 45*<sup>o</sup>* line, and a + (red) mark is below the 45*<sup>o</sup>* line means that it is necessary to increase the *<sup>k</sup>* number of partitions up to all the <sup>+</sup> (red) and <sup>×</sup> (blue) marks are above the 45*<sup>o</sup>* line. If this is achieved then the function is robustly positive, as shown in figure (7).

In the figure (7) we can see that it is difficult to see that all + (red) marks are above the 45*<sup>o</sup>* line, then with purpose to resolve this difficulty is proposed the following representation.

14 Will-be-set-by-IN-TECH

apply, although it must be said that the bounds of the polygon theorem are better than the

**Analytical Partition** (Elizondo, 1999). In the subfigure 1 of figure (6) shows that the function is robustly positive in boxes Γ<sup>1</sup> and Γ<sup>3</sup> but not in the boxes Γ<sup>2</sup> and Γ4. So it is necessary apply iteratively the partition box to the boxes where the function is not robust positive, in this way is obtained the subfigure 2 of figure (6). Since there is a set of boxes such that *Q* =

the procedure in visual way, but for more than two dimensions, using software we can get the coordinates and dimensions of sub boxes where the function is close to losing positivity.

(a) Subfigure 1 (b) Subfigure 2

**Constant Partition** (Elizondo, 1999). In this procedure the domaine of each one of the � parameters is divide in *k* equal parts (not necessarily equal), in this way, it is generated a

. The robust positivity of each Γ*<sup>i</sup>* box can be

*j* Γ*j*

analyzed by a computer program so that the computer give us the final result about the robust

Another way is through a software which plot a × (blue) mark in the (*fn*, *fp*) plane in each (*fn*(*μ*min), *fp*(*μ*min)) and (*fn*(*μ*max), *fp*(*μ*max)) coordinates corresponding to the minimum and maximum vertices of each Γ*<sup>i</sup>* box, and plot too a + (red) mark corresponding to the lower bound of each Γ*<sup>i</sup>* box, as can be appreciated in figure (7) that it was obtained with *k* = 13. If a <sup>×</sup> (blue) mark is below the 45*<sup>o</sup>* line, means that there is at least one vector for which the function is negative and therefore the function is not robustly positive. If all the × (blue) marks are above the 45*<sup>o</sup>* line, and a + (red) mark is below the 45*<sup>o</sup>* line means that it is necessary to increase the *<sup>k</sup>* number of partitions up to all the <sup>+</sup> (red) and <sup>×</sup> (blue) marks are above the 45*<sup>o</sup>*

In the figure (7) we can see that it is difficult to see that all + (red) marks are above the 45*<sup>o</sup>* line, then with purpose to resolve this difficulty is proposed the following representation.

line. If this is achieved then the function is robustly positive, as shown in figure (7).

, then the function is robustly positive in *Q*. The graphs were made to show

rectangle theorem.

<sup>|</sup> *<sup>f</sup>*(*q*) <sup>&</sup>gt; <sup>0</sup> <sup>∀</sup>Γ*<sup>j</sup>*

Fig. 6. Partition box

positivity of the function.

boxes set of *k*� sub boxes Γ*<sup>i</sup>* such that *Q* =

 *j* Γ*j*

Fig. 7. Function in (*fn*, *fp*) plane

#### **4.6** (*α*, *β*) **Representation**

In some cases as so as figure (7) it is not easy to determine in graphic way whether a point close to the 45*<sup>o</sup>* line is over this line or not. So in (Elizondo, 1999) the (*α*, *β*) representation was developed, *<sup>α</sup>*(*q*) = *fp*(*q*) + *fn*(*q*), *<sup>β</sup>*(*q*) = *fp*(*q*) <sup>−</sup> *fn*(*q*) , it is similar to rotated 45*<sup>o</sup>* the axis with respect to (*fn*, *fp*) representation implying some graphical and algebraic advantages over the negative and positive representation.

**Definition 10.** *(Elizondo, 1999) Let fn*(*q*) *and fp*(*q*) *be the negative and positive parts of a continuous function f*(*q*) *with sign decomposition in Q*. *Let T be the linear transformation described below such that T* <sup>−</sup><sup>1</sup> *exists, then it is called a representation of the function f*(*q*), *in* (*α*, *β*) *coordinates, to the linear transformation* (*α*(*q*), *β*(*q*)) = *T*(*fn*(*q*), *fp*(*q*)) *and the inverse transformation of an* (*α*(*q*), *β*(*q*)) *representation is a* (*fn*(*q*), *fp*(*q*)) *representation of the function f*(*q*).


With the purpose to show the advantages of the (*α*, *β*) representation, by means of the rectangle theorem we analyze the same function in the previous subsection *f*(*q*) = 4 + *q*<sup>1</sup> + 8*q*<sup>2</sup> <sup>1</sup>*q*<sup>2</sup> − *q*<sup>2</sup> + 9*q*1*q*<sup>2</sup> 2 applying *k* = 13. We can see in the figure (8) beta axis scale is positive implying that all the bounds are positives and consequently the function is robustly positive.

The function *<sup>f</sup>*(*q*) = <sup>4</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> <sup>+</sup> *<sup>q</sup>*1*q*<sup>3</sup> <sup>+</sup> <sup>8</sup>*q*<sup>2</sup> <sup>1</sup>*q*<sup>2</sup> <sup>−</sup> <sup>9</sup>*q*<sup>3</sup> 3*q*1*q*<sup>2</sup> <sup>2</sup> corresponding to the figure (2) is shown in the figure (9) in (*α*, *β*) representation. We can see that beta axis scale is positive implying the function is robustly positive.

The original idea to develop the representation (*α* , *β*) (Elizondo, 1999) was to solve a visual geometric problem, but this representation has interesting algebraic properties on continuous functions *f*(*q*), *g*(*q*), *h*(*q*) with sign decomposition in *Q* and for all *u*(*q*) nondecreasing function in *Q*, (Elizondo, 1999) as the following:

This is an usful and inetresting property, but above all properties there are three outstanding properties, it would be very useful if they were fulfilled in complex numbers, they are as

Parametric Robust Stability 19

*Addition f*(*q*) + *g*(*q*) *α*(*q*) = *αf*(*q*) + *αg*(*q*) *β*(*q*) = *β <sup>f</sup>*(*q*) + *βg*(*q*) *Subtraction f*(*q*) − *g*(*q*) *α*(*q*) = *αf*(*q*) + *αg*(*q*) *β*(*q*) = *β <sup>f</sup>*(*q*) − *βg*(*q*) *Product f*(*q*)*g*(*q*) *α*(*q*) = *αf*(*q*)*αg*(*q*) *β*(*q*) = *β <sup>f</sup>*(*q*)*βg*(*q*)

Most be noted that the alpha componet of subtraction is correct with *α*(*q*) = *αf*(*q*) + *αg*(*q*), it is an "addition" of alphas. It is also important to highlight the simplicity with which made

Sign decomposition of the determinant was developed in (Elizondo, 1999) and it was presented an application in (Elizondo, 2001A; 2002B), by simplicity only will mention (Elizondo, 1999). In parametric robust stability is not very useful the sign decomposition of the determinant, but it is a part of sign decomposition. We can analyze robust stability by means of the Hurwitz criterion means the robust positivity of determinants, but it is so much easier by means of criterion (2), see table (1). Taking account that the reader could work in other areas where the nonsingularity of a matrix dependent in parameters is important, then

In order to achieve the procedure to determine the robust positivity in necessary and sufficient conditions of a determinant with real coefficients depending on parameters *qi*, the following fact is presented. By means of the (*α*, *β*) properties (5) is obtained the following fact, in the development of the determinant appears the alpha part and beta part, as shown in the

**Fact 1.** *(Elizondo, 1999) Let M*(*q*) *be a* (2 × 2) *matrix with elements mi*,*j*(*q*) ∈ � *with representation*

*mi*,*j*(*q*) 

*the matrix M*(*q*), *and the determinant* det*α*(*M*(*q*)) = |*M*(*q*)|*<sup>α</sup>* = |*Mα*(*q*)|*<sup>α</sup> will be called the alpha part of the determinant* |*M*(*q*)| , *which is symilar to the usual determinant changing all the subtractions*

Most be noted that: **a)** *βi*,*j*(*q*) = *mi*,*j*(*q*), then, *Mβ*(*q*) = *M*(*q*) and det*β*(*M*(*q*)) = det(*M*(*q*)), **b)** In accordance with the above fact, for a (2 × 2) matrix, the (*α*, *β*) representation of the

det*α*(*M*(*q*)), det*β*(*M*(*q*))

*be a matrix with elements mi*,*j*(*q*) ∈ � *with*

*will be called the alpha part of*

. In the following lemma a

*βi*,*j*(*q*) *will*

> *Mβ*(*q*)

*αi*,*j*(*q*) 

(*αi*,*j*(*q*) , *βi*,*j*(*q*)). *Then the* (*α*, *β*) *representation of the determinant of the matrix M*(*q*) *is:*

*by additions including the sign rule of Cramer. In a similar way, the matrix Mβ*(*q*) =

generalization of the last expression for a (*n* × *n*) matrix is stablished.

*be called the beta part of the matrix M*(*q*), *and the determinant* det*β*(*M*(*q*)) = |*M*(*q*)|*<sup>β</sup>* =

the addition, subtraction and product in alpha beta representation.

sign decomposition of the determinant is included in this chapter.

**4.7.1 The** (*α*, *β*) **representation of the determinant**

(det(*M*(*q*)))*<sup>α</sup>* = (*α*1,1(*q*)*α*2,2(*q*) + *α*2,1(*q*)*α*1,2(*q*)) (det(*M*(*q*)))*<sup>β</sup>* = (*β*1,1(*q*)*β*2,2(*q*) − *β*2,1(*q*)*β*1,2(*q*)).

*will be called the beta part of the determinant* |*M*(*q*)| .

determinant of the matrix *<sup>M</sup>*(*q*) is

(*αi*,*j*(*q*) , *<sup>β</sup>i*,*j*(*q*)) *representation. Then the matrix Mα*(*q*) =

**Definition 11.** *(Elizondo, 1999) Let M*(*q*) =

**4.7 Sign decomposition of the determinant**

(5)

follows:

following fact.

Fig. 8. Function in (*α*, *β*) representation

Fig. 9. Function in (*α*, *β*) representation

**a)** *α*(*q*) is a non-decreasing and non-negative function in *Q*; **b)** *α*(*q*) ≥ *β*(*q*); **c)** *β*(*q*) = *f*(*q*) ∀ *f*(*q*), ∀*q* ∈ *Q*; **d)** the (*α*(*q*) + *u*(*q*), *β*(*q*) + *u*(*q*)) is a *α*, *β* representation of *f*(*q*); **e)** the (*α*(*q*) + *u*(*q*), *β*(*q*)) representation is reduced to its minimum expression (*α*(*q*), *β*(*q*)); **f)** *Addition f*(*q*) + *g*(*q*) : *α*(*q*) = *αf*(*q*) + *αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*) + *βg*(*q*); **g)** *Subtraction f*(*q*) − *g*(*q*) : *α*(*q*) = *αf*(*q*) + *αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*) − *βg*(*q*); **h)** *Product f*(*q*)*g*(*q*), *α*(*q*) = *αf*(*q*)*αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*)*βg*(*q*); **i)** the (*α*, *β*) representation of −*g*(*q*) is as follows: (*αg*(*q*), −*βg*(*q*)); **j)** if *f*(*q*) = *g*(*q*) + *h*(*q*) then the alpha an beta parts of *f*(*q*) − *g*(*q*) are reduced to its minimum expression as follows *α*(*q*) = *αf*(*q*) − *αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*) − *βg*(*q*).

Computationally the (*α*, *β*) representation is better than (*fn*, *fp*) because if the computer does not generate the negative scale in the *β* axis it is implying that all "marks" are positives. 16 Will-be-set-by-IN-TECH

**a)** *α*(*q*) is a non-decreasing and non-negative function in *Q*; **b)** *α*(*q*) ≥ *β*(*q*); **c)** *β*(*q*) = *f*(*q*) ∀ *f*(*q*), ∀*q* ∈ *Q*; **d)** the (*α*(*q*) + *u*(*q*), *β*(*q*) + *u*(*q*)) is a *α*, *β* representation of *f*(*q*); **e)** the (*α*(*q*) + *u*(*q*), *β*(*q*)) representation is reduced to its minimum expression (*α*(*q*), *β*(*q*)); **f)** *Addition f*(*q*) + *g*(*q*) : *α*(*q*) = *αf*(*q*) + *αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*) + *βg*(*q*); **g)** *Subtraction f*(*q*) − *g*(*q*) : *α*(*q*) = *αf*(*q*) + *αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*) − *βg*(*q*); **h)** *Product f*(*q*)*g*(*q*), *α*(*q*) = *αf*(*q*)*αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*)*βg*(*q*); **i)** the (*α*, *β*) representation of −*g*(*q*) is as follows: (*αg*(*q*), −*βg*(*q*)); **j)** if *f*(*q*) = *g*(*q*) + *h*(*q*) then the alpha an beta parts of *f*(*q*) − *g*(*q*) are reduced to its minimum

Computationally the (*α*, *β*) representation is better than (*fn*, *fp*) because if the computer does not generate the negative scale in the *β* axis it is implying that all "marks" are positives.

expression as follows *α*(*q*) = *αf*(*q*) − *αg*(*q*), *β*(*q*) = *β <sup>f</sup>*(*q*) − *βg*(*q*).

Fig. 8. Function in (*α*, *β*) representation

Fig. 9. Function in (*α*, *β*) representation

This is an usful and inetresting property, but above all properties there are three outstanding properties, it would be very useful if they were fulfilled in complex numbers, they are as follows:

$$\begin{array}{llll}\text{Additional} & f(q) + g(q) & \mathfrak{a}(q) = \mathfrak{a}\_f(q) + \mathfrak{a}\_{\mathcal{S}}(q) & \mathfrak{f}(q) = \mathfrak{f}\_f(q) + \mathfrak{f}\_{\mathcal{S}}(q) \\\text{Subtraction } f(q) - \mathfrak{g}(q) & \mathfrak{a}(q) = \mathfrak{a}\_f(q) + \mathfrak{a}\_{\mathcal{S}}(q) & \mathfrak{f}(q) = \mathfrak{f}\_f(q) - \mathfrak{f}\_{\mathcal{S}}(q) \\\text{Product} & f(q)\mathfrak{g}(q) & \mathfrak{a}(q) = \mathfrak{a}\_f(q)\mathfrak{a}\_{\mathcal{S}}(q) & \mathfrak{f}(q) = \mathfrak{f}\_f(q)\mathfrak{f}\_{\mathcal{S}}(q) \end{array} \tag{5}$$

Most be noted that the alpha componet of subtraction is correct with *α*(*q*) = *αf*(*q*) + *αg*(*q*), it is an "addition" of alphas. It is also important to highlight the simplicity with which made the addition, subtraction and product in alpha beta representation.

### **4.7 Sign decomposition of the determinant**

Sign decomposition of the determinant was developed in (Elizondo, 1999) and it was presented an application in (Elizondo, 2001A; 2002B), by simplicity only will mention (Elizondo, 1999). In parametric robust stability is not very useful the sign decomposition of the determinant, but it is a part of sign decomposition. We can analyze robust stability by means of the Hurwitz criterion means the robust positivity of determinants, but it is so much easier by means of criterion (2), see table (1). Taking account that the reader could work in other areas where the nonsingularity of a matrix dependent in parameters is important, then sign decomposition of the determinant is included in this chapter.

### **4.7.1 The** (*α*, *β*) **representation of the determinant**

In order to achieve the procedure to determine the robust positivity in necessary and sufficient conditions of a determinant with real coefficients depending on parameters *qi*, the following fact is presented. By means of the (*α*, *β*) properties (5) is obtained the following fact, in the development of the determinant appears the alpha part and beta part, as shown in the following fact.

**Fact 1.** *(Elizondo, 1999) Let M*(*q*) *be a* (2 × 2) *matrix with elements mi*,*j*(*q*) ∈ � *with representation* (*αi*,*j*(*q*) , *βi*,*j*(*q*)). *Then the* (*α*, *β*) *representation of the determinant of the matrix M*(*q*) *is:*

(det(*M*(*q*)))*<sup>α</sup>* = (*α*1,1(*q*)*α*2,2(*q*) + *α*2,1(*q*)*α*1,2(*q*)) (det(*M*(*q*)))*<sup>β</sup>* = (*β*1,1(*q*)*β*2,2(*q*) − *β*2,1(*q*)*β*1,2(*q*)).

**Definition 11.** *(Elizondo, 1999) Let M*(*q*) = *mi*,*j*(*q*) *be a matrix with elements mi*,*j*(*q*) ∈ � *with* (*αi*,*j*(*q*) , *<sup>β</sup>i*,*j*(*q*)) *representation. Then the matrix Mα*(*q*) = *αi*,*j*(*q*) *will be called the alpha part of the matrix M*(*q*), *and the determinant* det*α*(*M*(*q*)) = |*M*(*q*)|*<sup>α</sup>* = |*Mα*(*q*)|*<sup>α</sup> will be called the alpha part of the determinant* |*M*(*q*)| , *which is symilar to the usual determinant changing all the subtractions by additions including the sign rule of Cramer. In a similar way, the matrix Mβ*(*q*) = *βi*,*j*(*q*) *will be called the beta part of the matrix M*(*q*), *and the determinant* det*β*(*M*(*q*)) = |*M*(*q*)|*<sup>β</sup>* = *Mβ*(*q*) 

*will be called the beta part of the determinant* |*M*(*q*)| .

Most be noted that: **a)** *βi*,*j*(*q*) = *mi*,*j*(*q*), then, *Mβ*(*q*) = *M*(*q*) and det*β*(*M*(*q*)) = det(*M*(*q*)), **b)** In accordance with the above fact, for a (2 × 2) matrix, the (*α*, *β*) representation of the determinant of the matrix *<sup>M</sup>*(*q*) is det*α*(*M*(*q*)), det*β*(*M*(*q*)) . In the following lemma a generalization of the last expression for a (*n* × *n*) matrix is stablished.

**Lemma 12.** *(Elizondo, 1999) Let M*(*q*) *be a* (*n* × *n*) *matrix with elements mi*,*j*(*q*) ∈ � *with representation* (*αi*,*j*(*q*) , *βi*,*j*(*q*) ). *Then the* (*α*, *β*) *representation of the determinant of the matrix <sup>M</sup>*(*q*) *is* det*α*(*M*(*q*)), det*β*(*M*(*q*)) *. In accordance with definition (11)*

The Hurwitz matrix *H*(*q*) is obtained, it is proved that the polynomial *p*(*s*, *q*ˆ) is stable for

Parametric Robust Stability 21

condition of the Frazer and Duncan theorem satisfied, and proving that the determinant is robust positive in *Q*, the second condition of the Frazer and Duncan theorem will be satisfied

The robust positivity of the determinant problem is solved by means of: the box partition theorem 9, the polygon theorem 8 in (*α*, *β*) representation and the sign decomposition of the determinant theorem (13). Taking the partition in 9 equal parts in each one of the two variables *qi* and applying sign decomposition in constant partition way, the function values in minimum and maximum vertices "×" and lower bound "+" are plotted for each <sup>Γ</sup>*<sup>i</sup>* box, as it appears in the figure (10). All lower bound marks "+" are above the alpha axis, then all of bounds are positive, therefore the determinant of the Hurwitz matrix *H*(*q*) is robust positive

In control area, the robust stability of LTI systems with parametric uncertainty problem has been studied in different interesting ways. The problem can be divided in two parts. One of them is that it is not possible to be obtained roots of a polynomial by analytical means for the general case. The second is that we have now a family of polynomials to study instead of a

Since to obtain roots of polynomials for the general case is a difficult problem. Then the extraction of roots of polynomials went mapped firstly to a "position" of roots problem in the complex plane, Routh never tried to extract the roots, his work begun studying the position of the roots. This problem was subjected to a second mapping, it was transferred to mathematical problems of smaller level for example to a positivity problem, as it is the case of: Routh,

*H*(*q*) =

implying that the polynomials family is robust stable.

**5. A solution for the parametric robust stability problem**

Hurwitz, Lienard-Chipart and Elizondo-González 2001 criterions.

Fig. 10. Positivity of the determinant

**5.1 Problem identification**

single polynomial.

⎡ ⎢ ⎢ ⎣

*<sup>T</sup>* and that the determinant of the Hurwitz matrix *H*(*q*ˆ) is positive. Having the first

*c*3(*q*) *c*1(*q*) 0 0 *c*4(*q*) *c*2(*q*) *c*0(*q*) 0 0 *c*3(*q*) *c*1(*q*) 0 0 *c*4(*q*) *c*2(*q*) *c*0(*q*)

⎤ ⎥ ⎥ ⎦

*q*ˆ = [1 1]

too.

#### **4.7.2 Linear, nonlinear and independent parts of the determinant**

When the positivity of the determinant of a matrix with elements *mi*,*j*(*q*) is analyzed via sign decomposition, it is normally necessary to use the box partition and polygon theorems. Then, the independent, linear and nonlinear parts of the determinant need to be obtained. These are obtained in the following theorem.

**Theorem 13.** *(Elizondo, 1999) (Sign Decomposition of the Determinant Theorem) Let q* ∈ Γ ⊆ *Q* | *<sup>q</sup>* <sup>=</sup> *<sup>μ</sup>*min <sup>+</sup> *<sup>δ</sup> be according to the proposition (7 ). Let M*(*q*) ∈ �*n*×*<sup>n</sup> be a matrix with elements mi*,*j*(*q*) *with sign decomposition in Q with representation* (*α*min *<sup>i</sup>*,*<sup>j</sup>* <sup>+</sup> *<sup>α</sup>i*,*j*,*L*(*δ*) + *<sup>α</sup>i*,*j*,*N*(*δ*), *<sup>β</sup>*min *<sup>i</sup>*,*<sup>j</sup>* + *βi*,*j*,*L*(*δ*) + *βi*,*j*,*N*(*δ*)), *then the* (*α*, *β*) *representation of the determinant of the matrix M*(*q*) *is as follows:*

*α*(*q*) = *α*min + *αL*(*δ*) + *αN*(*δ*), *β*(*q*) = *β*min + *βL*(*δ*) + *βN*(*δ*) *α*min = *det<sup>α</sup> <sup>α</sup>*min *i*,*j* , *<sup>β</sup>*min <sup>=</sup> *det <sup>β</sup>*min *i*,*j αL*(*q*) = *k*=*n* ∑ *k*=1 *det<sup>α</sup>* Φ(*k*) *α*min *i*,*j* + [*I* − Φ(*k*)] *αi*,*j*,*L*(*δ*) *βL*(*q*) = *k*=*n* ∑ *k*=1 *det* Φ(*k*) *β*min *i*,*j* + [*I* − Φ(*k*)] *βi*,*j*,*L*(*δ*) Φ(*k*) = *ϕi*,*j*(*k*) | *ϕ*1,1(*k*) = |*sign*(1 − *k*)| *ϕ*2,2(*k*) = |*sign*(2 − *k*)| *. . . ϕn*,*n*(*k*) = |*sign*(*n* − *k*)| *ϕi*,*j*(*k*) = 0 ∀*i* �= *j <sup>α</sup>N*(*δ*) = *<sup>α</sup>*(*q*) <sup>−</sup> *<sup>α</sup>*min <sup>−</sup> *<sup>α</sup>L*(*δ*), *<sup>β</sup>N*(*δ*) = *<sup>β</sup>*(*q*) <sup>−</sup> *<sup>β</sup>*min <sup>−</sup> *<sup>β</sup>L*(*δ*)

#### **4.7.3 Example**

(Elizondo, 1999; 2001A). The Frazer and Duncan Theorem is presented in (Ackermann et al., 1993) in the boundary crossing version as follows. Let *P*(*s*, *Q*) = {*p*(*s*, *q*) | *q* ∈ *Q* ⊂ *P* ⊂ �} be a family of polynomials of invariant degree with parametric uncertainty and real continuous coefficients, then the family *P*(*s*, *Q*) is robust stable if and only if: 1) a stable polynomial *p*(*s*, *q*ˆ) ∈ *P*(*s*, *Q*) exists, 2) det(*H*(*q*)) �= 0 for all *q* ∈ *Q*.

(Ackermann et al., 1993) Given the family of invariant degree polynomials with parametric uncertainty described by: *p*(*s*, *q*) = *c*<sup>0</sup> + *c*1*s* + *c*2*s*<sup>2</sup> + *c*3*s*<sup>3</sup> + *c*4*s*4, with real continuous coefficients: *c*0(*q*) = 3, *c*1(*q*) = 2, *c*2(*q*) = 0.25 + 2*q*<sup>1</sup> + 2*q*2, *c*3(*q*) = 0.5(*q*<sup>1</sup> + *q*2), *c*4(*q*) = *q*1*q*2, such that *qi* ∈ [1, 5]. Determine the robust stability of the family by means of the Frazer and Duncan theorem applying in graphical way the sign decomposition of the determinant theorem (13).

18 Will-be-set-by-IN-TECH

**Lemma 12.** *(Elizondo, 1999) Let M*(*q*) *be a* (*n* × *n*) *matrix with elements mi*,*j*(*q*) ∈ � *with representation* (*αi*,*j*(*q*) , *βi*,*j*(*q*) ). *Then the* (*α*, *β*) *representation of the determinant of the matrix*

When the positivity of the determinant of a matrix with elements *mi*,*j*(*q*) is analyzed via sign decomposition, it is normally necessary to use the box partition and polygon theorems. Then, the independent, linear and nonlinear parts of the determinant need to be obtained. These are

**Theorem 13.** *(Elizondo, 1999) (Sign Decomposition of the Determinant Theorem) Let q* ∈ Γ ⊆ *Q* | *<sup>q</sup>* <sup>=</sup> *<sup>μ</sup>*min <sup>+</sup> *<sup>δ</sup> be according to the proposition (7 ). Let M*(*q*) ∈ �*n*×*<sup>n</sup> be a matrix with elements mi*,*j*(*q*)

, *β*min = *det*

*ϕi*,*j*(*k*) |

*ϕ*1,1(*k*) = |*sign*(1 − *k*)| *ϕ*2,2(*k*) = |*sign*(2 − *k*)| *. . . ϕn*,*n*(*k*) = |*sign*(*n* − *k*)| *ϕi*,*j*(*k*) = 0 ∀*i* �= *j <sup>α</sup>N*(*δ*) = *<sup>α</sup>*(*q*) <sup>−</sup> *<sup>α</sup>*min <sup>−</sup> *<sup>α</sup>L*(*δ*), *<sup>β</sup>N*(*δ*) = *<sup>β</sup>*(*q*) <sup>−</sup> *<sup>β</sup>*min <sup>−</sup> *<sup>β</sup>L*(*δ*)

(Elizondo, 1999; 2001A). The Frazer and Duncan Theorem is presented in (Ackermann et al., 1993) in the boundary crossing version as follows. Let *P*(*s*, *Q*) = {*p*(*s*, *q*) | *q* ∈ *Q* ⊂ *P* ⊂ �} be a family of polynomials of invariant degree with parametric uncertainty and real continuous coefficients, then the family *P*(*s*, *Q*) is robust stable if and only if: 1) a stable

(Ackermann et al., 1993) Given the family of invariant degree polynomials with parametric uncertainty described by: *p*(*s*, *q*) = *c*<sup>0</sup> + *c*1*s* + *c*2*s*<sup>2</sup> + *c*3*s*<sup>3</sup> + *c*4*s*4, with real continuous coefficients: *c*0(*q*) = 3, *c*1(*q*) = 2, *c*2(*q*) = 0.25 + 2*q*<sup>1</sup> + 2*q*2, *c*3(*q*) = 0.5(*q*<sup>1</sup> + *q*2), *c*4(*q*) = *q*1*q*2, such that *qi* ∈ [1, 5]. Determine the robust stability of the family by means of the Frazer and Duncan theorem applying in graphical way the sign decomposition of the determinant

 *β*min *i*,*j* 

+ [*I* − Φ(*k*)]

+ [*I* − Φ(*k*)]

 *αi*,*j*,*L*(*δ*) 

 *βi*,*j*,*L*(*δ*) 

*βi*,*j*,*N*(*δ*)), *then the* (*α*, *β*) *representation of the determinant of the matrix M*(*q*) *is as follows:*

Φ(*k*) =

*α*(*q*) = *α*min + *αL*(*δ*) + *αN*(*δ*), *β*(*q*) = *β*min + *βL*(*δ*) + *βN*(*δ*)

> *α*min *i*,*j*

*. In accordance with definition (11)*

*<sup>i</sup>*,*<sup>j</sup>* <sup>+</sup> *<sup>α</sup>i*,*j*,*L*(*δ*) + *<sup>α</sup>i*,*j*,*N*(*δ*), *<sup>β</sup>*min

*<sup>i</sup>*,*<sup>j</sup>* + *βi*,*j*,*L*(*δ*) +

**4.7.2 Linear, nonlinear and independent parts of the determinant**

*M*(*q*) *is*

**4.7.3 Example**

theorem (13).

det*α*(*M*(*q*)), det*β*(*M*(*q*))

*with sign decomposition in Q with representation* (*α*min

*α*min = *det<sup>α</sup>*

*k*=*n* ∑ *k*=1 *det<sup>α</sup>* Φ(*k*) *α*min *i*,*j* 

*k*=*n* ∑ *k*=1 *det* Φ(*k*) *β*min *i*,*j* 

polynomial *p*(*s*, *q*ˆ) ∈ *P*(*s*, *Q*) exists, 2) det(*H*(*q*)) �= 0 for all *q* ∈ *Q*.

*αL*(*q*) =

*βL*(*q*) =

obtained in the following theorem.

The Hurwitz matrix *H*(*q*) is obtained, it is proved that the polynomial *p*(*s*, *q*ˆ) is stable for *q*ˆ = [1 1] *<sup>T</sup>* and that the determinant of the Hurwitz matrix *H*(*q*ˆ) is positive. Having the first condition of the Frazer and Duncan theorem satisfied, and proving that the determinant is robust positive in *Q*, the second condition of the Frazer and Duncan theorem will be satisfied too.

$$H(q) = \begin{bmatrix} c\_3(q) \ c\_1(q) & 0 & 0\\ c\_4(q) \ c\_2(q) \ c\_0(q) & 0\\ 0 & c\_3(q) \ c\_1(q) & 0\\ 0 & c\_4(q) \ c\_2(q) \ c\_0(q) \end{bmatrix}$$

The robust positivity of the determinant problem is solved by means of: the box partition theorem 9, the polygon theorem 8 in (*α*, *β*) representation and the sign decomposition of the determinant theorem (13). Taking the partition in 9 equal parts in each one of the two variables *qi* and applying sign decomposition in constant partition way, the function values in minimum and maximum vertices "×" and lower bound "+" are plotted for each <sup>Γ</sup>*<sup>i</sup>* box, as it appears in the figure (10). All lower bound marks "+" are above the alpha axis, then all of bounds are positive, therefore the determinant of the Hurwitz matrix *H*(*q*) is robust positive implying that the polynomials family is robust stable.

Fig. 10. Positivity of the determinant

### **5. A solution for the parametric robust stability problem**

### **5.1 Problem identification**

In control area, the robust stability of LTI systems with parametric uncertainty problem has been studied in different interesting ways. The problem can be divided in two parts. One of them is that it is not possible to be obtained roots of a polynomial by analytical means for the general case. The second is that we have now a family of polynomials to study instead of a single polynomial.

Since to obtain roots of polynomials for the general case is a difficult problem. Then the extraction of roots of polynomials went mapped firstly to a "position" of roots problem in the complex plane, Routh never tried to extract the roots, his work begun studying the position of the roots. This problem was subjected to a second mapping, it was transferred to mathematical problems of smaller level for example to a positivity problem, as it is the case of: Routh, Hurwitz, Lienard-Chipart and Elizondo-González 2001 criterions.

**b)** By means of the positive and negative parts, to obtain the components in alpha and beta

Parametric Robust Stability 23

**d)** By means of the rectangle theorem (5) or polygon theorem (8), to analyze the robust positivity in *<sup>Q</sup>* of the coefficients *cn*(*q*) and *cn*−1(*q*). In case of negative bound in a coefficient,

**e)** To make a software to develop the table in accordance to the partition box theorem and to

characteristic polynomial of invariant degree is *p*(*s*, *q*) = *c*4(*q*)*s*<sup>4</sup> + *c*3(*q*)*s*<sup>3</sup> + +*c*2(*q*)*s*<sup>2</sup> +

*c*0*n*(*q*) = *q*2*q*<sup>3</sup> *c*1*n*(*q*) = *q*2*q*<sup>3</sup> *c*2*n*(*q*) = *q*1*q*<sup>2</sup>

*c*3*n*(*q*) = *q*1*q*<sup>2</sup>

*c*4*n*(*q*) = *q*2*q*<sup>3</sup>

*β<sup>i</sup>* = *cpi*(*q*) − *cni*(*q*) *β*<sup>0</sup> = *cp*0(*q*) − *cn*0(*q*) *β*<sup>1</sup> = *cp*1(*q*) − *cn*1(*q*) *β*<sup>2</sup> = *cp*2(*q*) − *cn*2(*q*) *β*<sup>3</sup> = *cp*3(*q*) − *cn*3(*q*) *β*<sup>4</sup> = *cp*4(*q*) − *cn*4(*q*)

<sup>=</sup> <sup>3</sup> + (0)(0)<sup>3</sup> <sup>−</sup> (1)(1) = 2.

<sup>=</sup> <sup>5</sup> + (0)<sup>2</sup> <sup>−</sup> (1)(1)<sup>2</sup> <sup>=</sup> 4.

<sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>3</sup> 2

2

*<sup>T</sup>*|*qi* <sup>∈</sup> [0, 1] <sup>∀</sup>*i*}, its

**Remark 14.** *The sigma column in the criterion (2) is not necessary calculate for robust stability*

Given a LTI system with parametric uncertainty *Q* = {[*q*<sup>1</sup> *q*<sup>2</sup> *q*3]

<sup>3</sup> − *q*2*q*<sup>3</sup>

<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>3</sup> 2

*σ*<sup>1</sup> (*α*4, *β*4) (*α*2, *β*2) (*α*0, *β*0)

<sup>2</sup> − *q*2*q*<sup>3</sup>

<sup>2</sup> − *q*2*q*<sup>3</sup>

3

<sup>2</sup> <sup>−</sup> *<sup>q</sup>*1*q*<sup>2</sup> 2

2

2

**b)** The alpha and beta representation of the coefficients is as follows.

*σ*<sup>2</sup> (*α*3, *β*3) (*α*1, *β*1)

*σ*<sup>3</sup> *α*3,1 = *cα*3*cα*<sup>2</sup> + *cα*4*cα*1, *β*3,1 = *cβ*3*cβ*<sup>2</sup> − *cβ*4*cβ*<sup>1</sup> *α*3,2 = *cα*3*cα*0, *β*3,2 = *cβ*3*cβ*<sup>0</sup>

2

+*c*1(*q*)*s* + *c*0(*q*). To analyze the robust stability of the system.

representation. *α<sup>i</sup>* = *cip*(*q*) + *cin*(*q*), *β<sup>i</sup>* = *cip*(*q*) − *cin*(*q*). **c)** To make a table in accordance to the criterion (2).

include its graph in the following software.

**a)** Positive and negative parts *cpi*(*q*) and *cni*(*q*). *c*0(*q*) = 2 + *q*1*q*2*q*<sup>3</sup>

*c*1(*q*) = 5 + *q*1*q*<sup>3</sup>

*c*3(*q*) = 5 + *q*<sup>2</sup>

*c*4(*q*) = 3 + *q*1*q*<sup>3</sup>

*c*0*p*(*q*) = 2 + *q*1*q*2*q*<sup>3</sup>

*c*2*p*(*q*) = 10 + 4*q*1*q*<sup>3</sup> *c*3*p*(*q*) = 5 + *q*<sup>2</sup>

*c*1*p*(*q*) = 5 + *q*1*q*<sup>3</sup>

*c*4*p*(*q*) = 3 + *q*1*q*<sup>3</sup>

*α<sup>i</sup>* = *cpi*(*q*) + *cni*(*q*), *α*<sup>0</sup> = *cp*0(*q*) + *cn*0(*q*) *α*<sup>1</sup> = *cp*1(*q*) + *cn*1(*q*) *α*<sup>2</sup> = *cp*2(*q*) + *cn*2(*q*) *α*<sup>3</sup> = *cp*3(*q*) + *cn*3(*q*) *α*<sup>4</sup> = *cp*4(*q*) + *cn*4(*q*)

**c)** To make a table in accordance to the criterion (2).

*σ*<sup>4</sup> *α*4,1 = *α*3,1*cα*<sup>1</sup> + *cα*3*α*3,2, *β*4,1 = *β*3,1*cβ*<sup>1</sup> − *cβ*3*β*3,2 *σ*<sup>5</sup> Check robust positivity of *β*4,1 and *β*3,2

**d)** The lower bound of *c*4(*q*) and *c*3(*q*) are as follows.

 [000] *T* − *c*4*<sup>n</sup>* [111] *T*

 [000] *T* − *c*3*<sup>n</sup>* [111] *T*

For *c*4(*q*) is *LB c*<sup>4</sup> = *c*4*<sup>p</sup>*

For *c*3(*q*) is *LB c*<sup>3</sup> = *c*3*<sup>p</sup>*

*<sup>c</sup>*2(*q*) = <sup>10</sup> <sup>+</sup> <sup>4</sup>*q*1*q*<sup>3</sup> <sup>−</sup> *<sup>q</sup>*1*q*<sup>2</sup>

graph the wished *ei*,1 element.

**5.3 Example**

The objective in this chapter is to study the stability of a family of polynomials with invariant degree (the reder can see poles and zeros canellation cases) and real continuous coefficients dependent on parameters with uncertainty. The essence of the problem is that we have now a set of roots in the the complexes plane, and for stability condition all of them must be in the left half of the complex plane for asymptotic stability. How to obtain that the set of roots remains in the left side of the complex plane?

A well known solution is: a) the family *P*(*s*, *Q*) has at least one element *p*(*s*, *q*∗) stable and b) | *H*(*q*)|� 0 ∀*q* ∈ *Q*. The explanation is because the determinant of a Hurwitz matrix is zero when the polynomial has roots in the imaginary axis, so if there is a *q*<sup>∗</sup> ∈ *Q* vector such that *p*(*s*, *q*∗) is stable then its roots are at the left half of the complex plane. On other hand, if a vector *q* slides into *Q* starting from *q*<sup>∗</sup> implies that the coefficients *ci*(*q*) will change in continuous way and the roots of *p*(*s*, *q*∗) will slides too on the complex plane. But if | *H*(*q*)|� 0 ∀*q* ∈ *Q*, it means that does not exist a vector *q* for which *p*(*s*, *q*) has roots in the imaginary axis, implying that the displacement of the roots never cross the imaginary axis. This solution is very difficult to use because to test the robust positivity of a determinant in the general case is a very difficult problem (Ackermann et al., 1993)(page 93).

Another solution was through the subsetting test, the idea worked well in convex families as interval (Kharitonov, 1978) and affine (Bartlett et al., 1988), but it was not in nonconvex families as multilinear and polynomic.

Then it can be concluded that the solution for robust stability of LTI systems with parametric uncertainty problem for the general case: interval, affine, multilinear, polinomic, cannot be sustained in convexity properties nor subsetting test.

### **5.2 A proposed solution**

In (Elizondo, 1999) it was developed a solution for the general case of robust stability of LTI systems with parametric uncertainty without concerning the convexity of the families, the solution consists of two parts.

A part of the solution was the development of a stability criterion, operating with multivariable polynomic functions in parametric uncertainty case, simpler than Hurwitz and Lienard-Chipart criterions (Elizondo et al., 2005). The mentioned criterion is similar to criterion (Elizondo, 2001B) but without the *σ* column, therefore it does not determine the number of unstables roots, it only determines whether the polynomial is stable or not. The amount of mathematical operations required in this criterion is equal to the one of (Elizondo, 2001B) but they are much less that the required ones in Hurwitz and Lienard-Chipart criterions (Elizondo et al., 2005).

The other part of the solution was the development of a mathematical tool capable of solving robust positivity problems of multivariable polynomic functions in necessary and sufficient conditions by means of extreme point analysis.The mathematical tool developed in (Elizondo, 1999) was Sign Decomposition.

Then, the solution proposed for robust stability in LTI systems with parametric uncertainty in the general case is supported in two results: the stability criterion for LTI systems (Elizondo, 2001B) and sign decomposition (Elizondo, 1999). Given a polynomial *p*(*s*, *q*) = *cn*(*q*)*s<sup>n</sup>* + *cn*−1(*q*)*sn*−<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>c</sup>*0(*q*) with real coefficients, where *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>*, *<sup>Q</sup>* <sup>=</sup> {[*q*<sup>1</sup> *<sup>q</sup>*<sup>2</sup> ··· *<sup>q</sup>* ] *<sup>T</sup>*|*qi* <sup>∈</sup> [0, 1] ∀*i*}. The procedure easier to use is by means of the partition box theorem (9) in the modality "Constant Partition", its application could be of the following way.

**a)** Take the equations of the coefficients *ci*(*q*) and decompose them into positive and negative parts *cip*(*q*) and *cin*(*q*). In symbolic way.

20 Will-be-set-by-IN-TECH

The objective in this chapter is to study the stability of a family of polynomials with invariant degree (the reder can see poles and zeros canellation cases) and real continuous coefficients dependent on parameters with uncertainty. The essence of the problem is that we have now a set of roots in the the complexes plane, and for stability condition all of them must be in the left half of the complex plane for asymptotic stability. How to obtain that the set of roots

A well known solution is: a) the family *P*(*s*, *Q*) has at least one element *p*(*s*, *q*∗) stable and b) | *H*(*q*)|� 0 ∀*q* ∈ *Q*. The explanation is because the determinant of a Hurwitz matrix is zero when the polynomial has roots in the imaginary axis, so if there is a *q*<sup>∗</sup> ∈ *Q* vector such that *p*(*s*, *q*∗) is stable then its roots are at the left half of the complex plane. On other hand, if a vector *q* slides into *Q* starting from *q*<sup>∗</sup> implies that the coefficients *ci*(*q*) will change in continuous way and the roots of *p*(*s*, *q*∗) will slides too on the complex plane. But if | *H*(*q*)|� 0 ∀*q* ∈ *Q*, it means that does not exist a vector *q* for which *p*(*s*, *q*) has roots in the imaginary axis, implying that the displacement of the roots never cross the imaginary axis. This solution is very difficult to use because to test the robust positivity of a determinant in the general case

Another solution was through the subsetting test, the idea worked well in convex families as interval (Kharitonov, 1978) and affine (Bartlett et al., 1988), but it was not in nonconvex

Then it can be concluded that the solution for robust stability of LTI systems with parametric uncertainty problem for the general case: interval, affine, multilinear, polinomic, cannot be

In (Elizondo, 1999) it was developed a solution for the general case of robust stability of LTI systems with parametric uncertainty without concerning the convexity of the families, the

A part of the solution was the development of a stability criterion, operating with multivariable polynomic functions in parametric uncertainty case, simpler than Hurwitz and Lienard-Chipart criterions (Elizondo et al., 2005). The mentioned criterion is similar to criterion (Elizondo, 2001B) but without the *σ* column, therefore it does not determine the number of unstables roots, it only determines whether the polynomial is stable or not. The amount of mathematical operations required in this criterion is equal to the one of (Elizondo, 2001B) but they are much less that the required ones in Hurwitz and Lienard-Chipart

The other part of the solution was the development of a mathematical tool capable of solving robust positivity problems of multivariable polynomic functions in necessary and sufficient conditions by means of extreme point analysis.The mathematical tool developed in (Elizondo,

Then, the solution proposed for robust stability in LTI systems with parametric uncertainty in the general case is supported in two results: the stability criterion for LTI systems (Elizondo, 2001B) and sign decomposition (Elizondo, 1999). Given a polynomial *p*(*s*, *q*) = *cn*(*q*)*s<sup>n</sup>* + *cn*−1(*q*)*sn*−<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>c</sup>*0(*q*) with real coefficients, where *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>* <sup>⊂</sup> *<sup>P</sup>*, *<sup>Q</sup>* <sup>=</sup> {[*q*<sup>1</sup> *<sup>q</sup>*<sup>2</sup> ··· *<sup>q</sup>* ]

[0, 1] ∀*i*}. The procedure easier to use is by means of the partition box theorem (9) in the

**a)** Take the equations of the coefficients *ci*(*q*) and decompose them into positive and negative

modality "Constant Partition", its application could be of the following way.

*<sup>T</sup>*|*qi* <sup>∈</sup>

remains in the left side of the complex plane?

families as multilinear and polynomic.

**5.2 A proposed solution**

solution consists of two parts.

criterions (Elizondo et al., 2005).

1999) was Sign Decomposition.

parts *cip*(*q*) and *cin*(*q*). In symbolic way.

is a very difficult problem (Ackermann et al., 1993)(page 93).

sustained in convexity properties nor subsetting test.

**b)** By means of the positive and negative parts, to obtain the components in alpha and beta representation. *α<sup>i</sup>* = *cip*(*q*) + *cin*(*q*), *β<sup>i</sup>* = *cip*(*q*) − *cin*(*q*).

**c)** To make a table in accordance to the criterion (2).

**d)** By means of the rectangle theorem (5) or polygon theorem (8), to analyze the robust positivity in *<sup>Q</sup>* of the coefficients *cn*(*q*) and *cn*−1(*q*). In case of negative bound in a coefficient, include its graph in the following software.

**e)** To make a software to develop the table in accordance to the partition box theorem and to graph the wished *ei*,1 element.

**Remark 14.** *The sigma column in the criterion (2) is not necessary calculate for robust stability*

#### **5.3 Example**

Given a LTI system with parametric uncertainty *Q* = {[*q*<sup>1</sup> *q*<sup>2</sup> *q*3] *<sup>T</sup>*|*qi* <sup>∈</sup> [0, 1] <sup>∀</sup>*i*}, its characteristic polynomial of invariant degree is *p*(*s*, *q*) = *c*4(*q*)*s*<sup>4</sup> + *c*3(*q*)*s*<sup>3</sup> + +*c*2(*q*)*s*<sup>2</sup> + +*c*1(*q*)*s* + *c*0(*q*). To analyze the robust stability of the system.

**a)** Positive and negative parts *cpi*(*q*) and *cni*(*q*).

$$\begin{array}{ll} c\_{0}(q) = 2 + q\_{1}q\_{2}q\_{3}^{3} - q\_{2}q\_{3} \\ c\_{1}(q) = 5 + q\_{1}q\_{2}^{2} - q\_{2}q\_{3} \\ c\_{2}(q) = 10 + 4q\_{1}q\_{3} - q\_{1}q\_{2}^{2} - q\_{2}^{3} \\ c\_{3}(q) = 5 + q\_{2}^{2} - q\_{1}q\_{2}^{2} \\ c\_{4}(q) = 3 + q\_{1}q\_{3}^{2} - q\_{2}q\_{3} \\ c\_{0p}(q) = 2 + q\_{1}q\_{2}q\_{3}^{3} \\ c\_{1p}(q) = 5 + q\_{1}q\_{2}^{2} & c\_{1n}(q) = q\_{2}q\_{3} \\ c\_{2p}(q) = 10 + 4q\_{1}q\_{3} & c\_{2n}(q) = q\_{1}q\_{2}^{2} + q\_{2}^{3} \\ c\_{3p}(q) = 5 + q\_{2}^{2} & c\_{3n}(q) = q\_{1}q\_{2}^{2} \\ c\_{4p}(q) = 3 + q\_{1}q\_{2}^{3} & c\_{4n}(q) = q\_{2}q\_{3} \end{array}$$

**b)** The alpha and beta representation of the coefficients is as follows.


**c)** To make a table in accordance to the criterion (2).


**d)** The lower bound of *c*4(*q*) and *c*3(*q*) are as follows. For *c*4(*q*) is *LB c*<sup>4</sup> = *c*4*<sup>p</sup>* [000] *T* − *c*4*<sup>n</sup>* [111] *T* <sup>=</sup> <sup>3</sup> + (0)(0)<sup>3</sup> <sup>−</sup> (1)(1) = 2. For *c*3(*q*) is *LB c*<sup>3</sup> = *c*3*<sup>p</sup>* [000] *T* − *c*3*<sup>n</sup>* [111] *T* <sup>=</sup> <sup>5</sup> + (0)<sup>2</sup> <sup>−</sup> (1)(1)<sup>2</sup> <sup>=</sup> 4.

Then *c*4(*q*) and *c*3(*q*) are robustly positives in *Q*

**e)** By means of software applying 8 partitions the graphs *e*3,1, *e*3,2, *e*4,1 were obtained as following.

Fig. 13. Element *e*<sup>41</sup> in (*α*, *β*) representation

Springer, ISBN 978-0387198439.

Vol. 1, No. 1, 61-71, DOI: 10.1007/BF02551236.

Prentice Hall, ISBN 0-13-781576-X, NJ, USA.

*Eléctrica de la Universidad Autónoma de Nuevo León*.

978-0023060557.

April, 2000.

2001.

**6. References**

Since *c*4(*q*), *c*3(*q*), *e*31(*q*), *e*32(*q*), *e*41(*q*) are robustly positive, then the system is robustly stable.

Parametric Robust Stability 25

Ackermann, J. & Bartlett, A. (1993). *Robust Control Systems with Uncertain Physical Parameters*,

Barmish, B.R. (1990). *New Tools for Robustness of Linear Systems*, Prentice Hall, ISBN

Bartlett, A.C.; Hollot, C.V. & Lin, H. (1988). Root locations of an entire polytope of

Bhattacharyya, S.P.; Chapellat, H. & Keel, L.H. (1995). *Robust Control the Parametric Approach*,

Bhattacharyya, S.P.;Keel, L.H. & Datta, A. (2009). *Linear Control Theory: Structure, Robustness*

Elizondo-González, C. (1999). Estabilidad y Controlabidad Robusta de Sistemas Lineales con

Elizondo-González, C. (2000). Necessary and Sufficient Conditions for Robust Positivity of

Elizondo-González C. (2001). Robust Positivity of the Determinant Via Sign Decomposition,

Elizondo-González C. (2001). New Stability Criterion on Space Coefficients, *Conferences on*

*and Optimization*, CRC Press,ISBN 978-0-8493-4063-5, Boca Raton.

14-17, ISBN 980-07-7545-5, Orlando, Florida, USA, July, 2001.

polynomials: It suffices to check the edges. *Mathematics of Control Signals Systems*,

Incertidumbre Multilineal. *Programa Doctoral de la Facultad de Ingeniería Mecánica y*

Polynomic Functions Via Sign Decomposition, *3 rd IFAC Symposium on Robust Control Design ROCOND 2000*, pp. 14-17, ISBN-13:9780080432496 , Prague Czech Republic,

*The 5th World Multi-Conference on Systemics Cybernetics and Informatics SCI 2001*, pp.

*Decision and Control IEEE*, SBN 0-7803-7063-5, Orlando, Florida, USA. Diciembre,

Fig. 11. Element *e*<sup>31</sup> in (*α*, *β*) representation

Fig. 12. Element *e*<sup>32</sup> in (*α*, *β*) representation

22 Will-be-set-by-IN-TECH

**e)** By means of software applying 8 partitions the graphs *e*3,1, *e*3,2, *e*4,1 were obtained as

Then *c*4(*q*) and *c*3(*q*) are robustly positives in *Q*

Fig. 11. Element *e*<sup>31</sup> in (*α*, *β*) representation

Fig. 12. Element *e*<sup>32</sup> in (*α*, *β*) representation

following.

Fig. 13. Element *e*<sup>41</sup> in (*α*, *β*) representation

Since *c*4(*q*), *c*3(*q*), *e*31(*q*), *e*32(*q*), *e*41(*q*) are robustly positive, then the system is robustly stable.

#### **6. References**


Elizondo-González, C. (2002). An Application of Recent Reslts on Parametric Robust Stability, *X Congreso Latinoamericano de Control Autmático*, Guadalajara Jalisco, 2002.

**0**

**2**

*USA*

Seddik M. Djouadi

*Knoxville, TN 37996-2100*

**Robustness of Feedback Linear Time-Varying**

*Electrical Engineering & Computer Science Department, University of Tennessee,*

There have been numerous attempts in the literature to generalize results in robust control theory (42; 45) to linear time-varying (LTV) systems (for e.g. (10–13; 30; 33; 37; 39; 40) and references therein). In (12)(13) and (11) the authors studied the optimal weighted sensitivity minimization problem, the two-block problem, and the model-matching problem for LTV systems using inner-outer factorization for positive operators. Abstract solutions involving the computation of induced operators norms of operators are obtained. However, there is no

In (40) the authors rely on state space techniques which lead to algorithms based on infinite dimensional operator inequalities which are difficult to solve. These methods lead to suboptimal controllers and are restricted to finite dimensional systems. An extension of these results to uncertain systems is reported in (41) relying on uniform stability concepts. In (9) both the sensitivity minimization problem in the presence of plant uncertainty, and robust stability for LTV systems in the <sup>∞</sup> induced norm is considered. However, their methods could not be extended to the case of systems operating on finite energy signals. In (37) the standard problem of *H*<sup>∞</sup> control theory for finite-dimensional LTV continuous-time plants is considered. It is shown that a solution to this problem exists if and only if a pair of matrix Riccati differential equations admits positive semidefinite stabilizing solutions. State-space

The gap metric was introduced to study stability robustness of feedback systems. It induces the weakest topology in which feedback stability is robust (6; 7; 31; 32; 38). Extensions of the gap to time-varying systems have been proposed in (33; 34) where a geometric framework was developed. Several results on the gap metric and the gap topology were established, in particular, the concept of a graphable subspace was introduced. In (21) the problem of robust stabilization for LTV systems subject to time-varying normalized coprime factor uncertainty is considered. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, a tight upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure is

Analysis of time-varying control strategies for optimal disturbance rejection for known time-invariant plants has been studied in (2; 16). A robust version of these problems was

clear indication on how to compute optimal linear LTV controllers.

formulae for one solution to the problem are also given.

**1. Introduction**

computed.

**Systems: A Commutant Lifting Approach**


## **Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach**

Seddik M. Djouadi

*Electrical Engineering & Computer Science Department, University of Tennessee, Knoxville, TN 37996-2100 USA*

### **1. Introduction**

24 Will-be-set-by-IN-TECH

26 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

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Keel, L.H. & Bhattacharyya, S.P. (2008). Fixed Order Multivariable Controller Synthesis: A

Keel, L.H. & Bhattacharyya, S.P. (2009). Fixed Order Multivariable Discrete-Time Control,

Keel, L.H. & Bhattacharyya, S.P. (2011). Robust Stability via Sign-Definite Decomposition,

Kharitonov, V. (1978). On a Generalization of a Stability Criterion. *Seria Fizico-matematicheskaia*,

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Meinsma G. (1995). Elementary Proof of the Routh-Hurwitz Test *Systems & Control Letters 25*,

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on Parametric Robust Stability, *X Congreso Latinoamericano de Control Autmático*,

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with applications to performance attainment problems, *American Control Conference.*,

decomposition , *Linear Algebra and its Applications.*, Volume 434, Issue 7, pp 1663-1676,

Dependency Using Bernstein Expansion, *IEEE Transactions on Automatic Control* vol.

There have been numerous attempts in the literature to generalize results in robust control theory (42; 45) to linear time-varying (LTV) systems (for e.g. (10–13; 30; 33; 37; 39; 40) and references therein). In (12)(13) and (11) the authors studied the optimal weighted sensitivity minimization problem, the two-block problem, and the model-matching problem for LTV systems using inner-outer factorization for positive operators. Abstract solutions involving the computation of induced operators norms of operators are obtained. However, there is no clear indication on how to compute optimal linear LTV controllers.

In (40) the authors rely on state space techniques which lead to algorithms based on infinite dimensional operator inequalities which are difficult to solve. These methods lead to suboptimal controllers and are restricted to finite dimensional systems. An extension of these results to uncertain systems is reported in (41) relying on uniform stability concepts. In (9) both the sensitivity minimization problem in the presence of plant uncertainty, and robust stability for LTV systems in the <sup>∞</sup> induced norm is considered. However, their methods could not be extended to the case of systems operating on finite energy signals. In (37) the standard problem of *H*<sup>∞</sup> control theory for finite-dimensional LTV continuous-time plants is considered. It is shown that a solution to this problem exists if and only if a pair of matrix Riccati differential equations admits positive semidefinite stabilizing solutions. State-space formulae for one solution to the problem are also given.

The gap metric was introduced to study stability robustness of feedback systems. It induces the weakest topology in which feedback stability is robust (6; 7; 31; 32; 38). Extensions of the gap to time-varying systems have been proposed in (33; 34) where a geometric framework was developed. Several results on the gap metric and the gap topology were established, in particular, the concept of a graphable subspace was introduced. In (21) the problem of robust stabilization for LTV systems subject to time-varying normalized coprime factor uncertainty is considered. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, a tight upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure is computed.

Analysis of time-varying control strategies for optimal disturbance rejection for known time-invariant plants has been studied in (2; 16). A robust version of these problems was considered in (8; 15) in different induced norm topologies. All these references showed that for *time-invariant nominal* plants and weighting functions, time-varying control laws offer no advantage over time-invariant ones.

• *Pk* the usual truncation operator for some integer *k*, which sets all outputs after time *k* to

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 29

*PkAPk* = *PkA*, ∀*k* positive integers

The subscript "*c*" denotes the restriction of a subspace of operators to its intersection with causal (see (11; 29) for the definition) operators. "⊕" denotes for the direct sum of two spaces.

The commutant lifting theorem has been proposed by Sz.Nagy and Foias (35; 36). It has been used successfully to solve several interpolation problems including *H*<sup>∞</sup> control problems for linear time invariant (LTI) systems (31; 32; 43; 44). In this chapter, we rely on a time-varying version of the commutant lifting theorem which corresponds to nest or triangular algebras. Following (3; 18) a nest <sup>N</sup> of a Hilbert space *<sup>H</sup>*<sup>ˇ</sup> is a family of closed subspaces of *<sup>H</sup>*<sup>ˇ</sup> ordered by inclusion. The triangular or nest algebra T (N ) is the set of all operators *T* such that *TN* ⊆ *N* for every element *N* in N . A representation of T (N ) is an algebra homomorphism *h* from T (N ) into the algebra B(H) of bounded linear operators on a Hilbert space H. A representation is contractive if �*h*(*A*)�≤�*A*�, for all *<sup>A</sup>* ∈ T (<sup>N</sup> ). It is weak continuous if *<sup>h</sup>*(*Ai*) converges to zero in the weak topology of <sup>B</sup>(H) whenever the net {*Ai*} converges to zero in the weak topology of <sup>B</sup>(*H*<sup>ˇ</sup> ). The representation *<sup>h</sup>* is said to be unital if *<sup>h</sup>*(*IH*<sup>ˇ</sup> ) = *<sup>I</sup>*H, where *IH*<sup>ˇ</sup> is the identity operator on *<sup>H</sup>*<sup>ˇ</sup> , and *<sup>I</sup>*<sup>H</sup> the identity operator on <sup>H</sup>. The Sz. Nagy Theorem asserts that any such a representation *h* has a B(*H*ˇ )-dilation, that is, there exists a Hilbert space <sup>K</sup> containing <sup>H</sup>, and a positive representation *<sup>H</sup>* of <sup>B</sup>(*H*<sup>ˇ</sup> ) such that

*<sup>P</sup>*<sup>H</sup> *<sup>H</sup>*(*A*) |H= *<sup>h</sup>*(*A*), where *<sup>P</sup>*<sup>H</sup> is the orthogonal projection from K into H (3; 18).

*be two unital weak continuous contractive representations with* B(*H*ˇ )*-dilations*

*, that is, YH*(*A*) = *H*�

(*A*)*X for all A* ∈ T (N )*, that is, X intertwines h and h*�

We now state the commutant lifting theorem for nest algebras from (3; 18) (see also references

*h* : T (N ) �−→ B(H) *<sup>h</sup>* : <sup>T</sup> (<sup>N</sup> ) �−→ <sup>B</sup>(H�

*<sup>H</sup>* : <sup>B</sup>(*H*<sup>ˇ</sup> ) �−→ <sup>B</sup>(K) *<sup>H</sup>* : <sup>B</sup>(*H*<sup>ˇ</sup> ) �−→ <sup>B</sup>(K�

*respectively. Assume that X* : H �−→ H� *is a linear operator with* �*X*� ≤ 1*, such that Xh*(*A*) =

)

)

(*A*)*Y for all A* <sup>∈</sup> <sup>B</sup>(*H*<sup>ˇ</sup> )*.*

*. Then there exists an operator Y* : K �−→

• An operator *A* ∈ B(*E*, *F*) is said to be causal if it satisfies the operator equation:

zero.

therein).

*h*�

K� *such that* **i)** �*Y*� ≤ 1*.*

**ii)** *Y intertwines H and H*�

**Theorem 1.** *(3; 18) Let*

• *tr*(·) denotes the trace of its argument.

"" stands for the adjoint of an operator.

**2. The commutant lifting theorem**

In this paper, we are interested in optimal disturbance rejection for (*possibly infinite-dimensional*, i.e., systems with an infinite number of states) LTV systems. These systems have been used as models in computational linear algebra and in a variety of computational and communication networks (17). This allows variable number of states which is predominant in networks which can switch on or off certain parts of the system (17), and infinite number of states as in distributed parameter systems.

Using inner-outer factorizations as defined in (3; 11) with respect of the nest algebra of lower triangular (causal) bounded linear operators defined on <sup>2</sup> we show that the problem reduces to a distance minimization between a special operator and the nest algebra. The inner-outer factorization used here holds under weaker assumptions than (12; 13), and in fact, as pointed in ((3) p. 180), is different from the factorization for positive operators used there.

The optimal disturbance attenuation for LTV systems has been addressed using Banach space duality theory in (20; 28). Its robust version which deals with plant uncertainty is addressed in (4; 5; 19) using also duality theory ideas. Furthermore, using the commutant lifting theorem for nest algebras the optimum is shown to be equal to the norm of a compact time-varying Hankel operator defined on the space of causal Hilbert-Schmidt operators. The latter is the "natural" analogous to the Hankel operator used in the LTI case. An operator identity to compute the optimal TV Youla parameter is also provided.

The results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed Hankel-Toeplitz operator generalizing analogous results known to hold in the linear time-invariant (LTI) case (22; 38; 43).

Our approach is purely input-output and does not use any state space realization, therefore the results derived here apply to *infinite dimensional LTV systems*, i.e., TV systems with an infinite number of state variables (33). Although the theory is developed for causal stable system, it can be extended in a straightforward fashion to the unstable case using coprime factorization techniques for LTV systems discussed in (11; 13).

The rest of the chapter is organized as follows. Section 2 the commutant lifting theorem for nest algebras is introduced. In section 3 the optimal disturbance rejection problem is formulated and solved in terms of a TV Hankel operator. A Generalization to the TV mixed sensitivity problem is carried out in section 4. Section 5 contains some concluding remarks.

### **Definitions and notation**

• B(*E*, *F*) denotes the space of bounded linear operators from a Banach space *E* to a Banach space *F*, endowed with the operator norm

$$||A|| := \sup\_{\substack{x \in E\_\prime \|x\| \le 1}} ||Ax||\_\prime \ A \in \mathcal{B}(E\_\prime F).$$

• <sup>2</sup> denotes the usual Hilbert space of square summable sequences with the standard norm

$$\|\|\mathbf{x}\|\|\_2^2 := \sum\_{j=0}^{\infty} |\mathbf{x}\_j|^2, \quad \mathbf{x} := (\mathbf{x}\_{0\prime} \mathbf{x}\_{1\prime} \mathbf{x}\_{2\prime} \cdots) \in \ell^2$$


$$P\_k A P\_k = P\_k A\_\prime \text{ \textquotedbl{}pk\textquotedbl{} positive\textquotedbl{} integers\textquotedbl{}}$$

• *tr*(·) denotes the trace of its argument.

2 Will-be-set-by-IN-TECH

considered in (8; 15) in different induced norm topologies. All these references showed that for *time-invariant nominal* plants and weighting functions, time-varying control laws offer no

In this paper, we are interested in optimal disturbance rejection for (*possibly infinite-dimensional*, i.e., systems with an infinite number of states) LTV systems. These systems have been used as models in computational linear algebra and in a variety of computational and communication networks (17). This allows variable number of states which is predominant in networks which can switch on or off certain parts of the system (17),

Using inner-outer factorizations as defined in (3; 11) with respect of the nest algebra of lower triangular (causal) bounded linear operators defined on <sup>2</sup> we show that the problem reduces to a distance minimization between a special operator and the nest algebra. The inner-outer factorization used here holds under weaker assumptions than (12; 13), and in fact, as pointed

The optimal disturbance attenuation for LTV systems has been addressed using Banach space duality theory in (20; 28). Its robust version which deals with plant uncertainty is addressed in (4; 5; 19) using also duality theory ideas. Furthermore, using the commutant lifting theorem for nest algebras the optimum is shown to be equal to the norm of a compact time-varying Hankel operator defined on the space of causal Hilbert-Schmidt operators. The latter is the "natural" analogous to the Hankel operator used in the LTI case. An operator identity to

The results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed Hankel-Toeplitz operator generalizing analogous results known to hold in the linear time-invariant (LTI) case

Our approach is purely input-output and does not use any state space realization, therefore the results derived here apply to *infinite dimensional LTV systems*, i.e., TV systems with an infinite number of state variables (33). Although the theory is developed for causal stable system, it can be extended in a straightforward fashion to the unstable case using coprime

The rest of the chapter is organized as follows. Section 2 the commutant lifting theorem for nest algebras is introduced. In section 3 the optimal disturbance rejection problem is formulated and solved in terms of a TV Hankel operator. A Generalization to the TV mixed sensitivity problem is carried out in section 4. Section 5 contains some concluding remarks.

• B(*E*, *F*) denotes the space of bounded linear operators from a Banach space *E* to a Banach

• <sup>2</sup> denotes the usual Hilbert space of square summable sequences with the standard norm

2, *x* :=

�*Ax*�, *A* ∈ B(*E*, *F*)

*<sup>x</sup>*0, *<sup>x</sup>*1, *<sup>x</sup>*2, ···

<sup>∈</sup> <sup>2</sup>

in ((3) p. 180), is different from the factorization for positive operators used there.

and infinite number of states as in distributed parameter systems.

compute the optimal TV Youla parameter is also provided.

factorization techniques for LTV systems discussed in (11; 13).

�*A*� := sup

∞ ∑ *j*=0 |*xj*|

*x*∈*E*, �*x*�≤1

space *F*, endowed with the operator norm

�*x*�<sup>2</sup> <sup>2</sup> :=

advantage over time-invariant ones.

(22; 38; 43).

**Definitions and notation**

The subscript "*c*" denotes the restriction of a subspace of operators to its intersection with causal (see (11; 29) for the definition) operators. "⊕" denotes for the direct sum of two spaces. "" stands for the adjoint of an operator.

#### **2. The commutant lifting theorem**

The commutant lifting theorem has been proposed by Sz.Nagy and Foias (35; 36). It has been used successfully to solve several interpolation problems including *H*<sup>∞</sup> control problems for linear time invariant (LTI) systems (31; 32; 43; 44). In this chapter, we rely on a time-varying version of the commutant lifting theorem which corresponds to nest or triangular algebras. Following (3; 18) a nest <sup>N</sup> of a Hilbert space *<sup>H</sup>*<sup>ˇ</sup> is a family of closed subspaces of *<sup>H</sup>*<sup>ˇ</sup> ordered by inclusion. The triangular or nest algebra T (N ) is the set of all operators *T* such that *TN* ⊆ *N* for every element *N* in N . A representation of T (N ) is an algebra homomorphism *h* from T (N ) into the algebra B(H) of bounded linear operators on a Hilbert space H. A representation is contractive if �*h*(*A*)�≤�*A*�, for all *<sup>A</sup>* ∈ T (<sup>N</sup> ). It is weak continuous if *<sup>h</sup>*(*Ai*) converges to zero in the weak topology of <sup>B</sup>(H) whenever the net {*Ai*} converges to zero in the weak topology of <sup>B</sup>(*H*<sup>ˇ</sup> ). The representation *<sup>h</sup>* is said to be unital if *<sup>h</sup>*(*IH*<sup>ˇ</sup> ) = *<sup>I</sup>*H, where *IH*<sup>ˇ</sup> is the identity operator on *<sup>H</sup>*<sup>ˇ</sup> , and *<sup>I</sup>*<sup>H</sup> the identity operator on <sup>H</sup>. The Sz. Nagy Theorem asserts that any such a representation *h* has a B(*H*ˇ )-dilation, that is, there exists a Hilbert space <sup>K</sup> containing <sup>H</sup>, and a positive representation *<sup>H</sup>* of <sup>B</sup>(*H*<sup>ˇ</sup> ) such that *<sup>P</sup>*<sup>H</sup> *<sup>H</sup>*(*A*) |H= *<sup>h</sup>*(*A*), where *<sup>P</sup>*<sup>H</sup> is the orthogonal projection from K into H (3; 18).

We now state the commutant lifting theorem for nest algebras from (3; 18) (see also references therein).

#### **Theorem 1.** *(3; 18) Let*

$$\begin{aligned} h &\colon \mathcal{T}(\mathcal{N}) \longmapsto \mathcal{B}(\mathcal{H}) \\\ h^{\star} &\colon \mathcal{T}(\mathcal{N}) \longmapsto \mathcal{B}(\mathcal{H}') \end{aligned}$$

*be two unital weak continuous contractive representations with* B(*H*ˇ )*-dilations*

$$\begin{aligned} H: &\mathcal{B}(\check{H}) \longmapsto \mathcal{B}(\mathcal{K}) \\ H^\star: &\mathcal{B}(\check{H}) \longmapsto \mathcal{B}(\mathcal{K'}) \end{aligned}$$

*respectively. Assume that X* : H �−→ H� *is a linear operator with* �*X*� ≤ 1*, such that Xh*(*A*) = *h*� (*A*)*X for all A* ∈ T (N )*, that is, X intertwines h and h*� *. Then there exists an operator Y* : K �−→ K� *such that*

**i)** �*Y*� ≤ 1*.*

**ii)** *Y intertwines H and H*� *, that is, YH*(*A*) = *H*� (*A*)*Y for all A* <sup>∈</sup> <sup>B</sup>(*H*<sup>ˇ</sup> )*.* **iii)** *Y dilates X, that is, Y* : M �−→ M� *, and P*H�*<sup>Y</sup>* |M= *XP*<sup>H</sup> |M*, where* H = M�N *is the orthogonal representation of* <sup>H</sup> *as the orthogonal difference of invariant subspaces for H* |T (<sup>N</sup> )*, and similarly for* H� *.*

optimal disturbance rejection can be written in the following form (20)

= inf *Q*∈B*c*(�2,�<sup>2</sup>)

element *N* in N . That is

to the nest <sup>N</sup> , *<sup>N</sup>* has the form *Qn*�<sup>2</sup> for some *<sup>n</sup>*. Define

is outer. The performance index *μ* in (2) can then be written as

*μ* = inf *Q*∈B*c*(�2,�<sup>2</sup>)

Following the classical *H*<sup>∞</sup> control theory (22; 23; 45),we assume

**(A1)** that *T*2*<sup>o</sup>* and *T*3*<sup>o</sup>* are invertible both in B*c*(�2, �2).

*μ* := inf {�*Tzw*� : *K* being robustly stabilizing linear time − varying controller}

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 31

The performance index (2) will be transformed into a shortest distance minimization between a certain bounded linear operator and a subspace to be specified shortly. In order to do so, following (11) define a nest N as a family of closed subspaces of the Hilbert space �<sup>2</sup> containing {0} and �<sup>2</sup> which is closed under intersection and closed span. Let *Qn* :<sup>=</sup> *<sup>I</sup>* − *Pn*, for *<sup>n</sup>* = −1, 0, 1, ··· , where *<sup>P</sup>*−<sup>1</sup> := 0 and *<sup>P</sup>*<sup>∞</sup> := *<sup>I</sup>*. Then *Qn* is a projection, and we can associate to it the following nest <sup>N</sup> :<sup>=</sup> {*Qn*�2, *<sup>n</sup>* <sup>=</sup> <sup>−</sup>1, 0, 1, ···}. In this case the triangular or nest algebra T (N ) is the set of all operators *T* such that *TN* ⊆ *N* for every

<sup>T</sup> (<sup>N</sup> ) = {*<sup>A</sup>* <sup>∈</sup> <sup>B</sup>(�2, �2) : *PnA*(*<sup>I</sup>* <sup>−</sup> *Pn*) = 0, <sup>∀</sup> *<sup>n</sup>*}

Note that the Banach space <sup>B</sup>*c*(�2, �2) is identical to the nest algebra <sup>T</sup> (<sup>N</sup> ). For *<sup>N</sup>* belonging

where *N*� < *N* means *N*� ⊂ *N*, and *N*� > *N* means *N*� ⊃ *N*. The subspaces *N* � *N*<sup>−</sup> are called the atoms of <sup>N</sup> . Since in our case the atoms of <sup>N</sup> span �2, then <sup>N</sup> is said to be atomic (3). The early days of *H*<sup>∞</sup> control theory saw solutions based on the so-called inner-outer factorizations of functions belonging to the Hardy spaces *H*<sup>2</sup> and *H*∞, and their corresponding matrix valued counterparts for multi-input multi-output (MIMO) systems (22; 23). Generalizations in the context of nest algebras have been proposed in (1; 3) as follows: An operator *A* in T (N ) is called *outer* if the range projection *P*(*RA*), *RA* being the range of *A* and *P* the orthogonal projection onto *RA*, commutes with N and *AN* is dense in *N* ∩ *RA* for every *<sup>N</sup>* ∈ N . A partial isometry *<sup>U</sup>* is called *inner* in <sup>T</sup> (<sup>N</sup> ) if *<sup>U</sup>*�*<sup>U</sup>* commutes with <sup>N</sup> (1; 3; 11). In our case, *<sup>A</sup>* ∈ T (<sup>N</sup> ) = <sup>B</sup>*c*(�2, �2) is outer if *<sup>P</sup>* commutes with each *Qn* and *AQn*�<sup>2</sup> is dense in *Qn*�<sup>2</sup> <sup>∩</sup> *<sup>A</sup>*�2. *<sup>U</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2) is inner if *<sup>U</sup>* is a partial isometry and *<sup>U</sup>*�*<sup>U</sup>* commutes with every *Qn*. Applying these notions to the time-varying operator *<sup>T</sup>*<sup>2</sup> <sup>∈</sup> <sup>B</sup>*<sup>c</sup>* (�2, �2), we get *T*<sup>2</sup> = *T*2*iT*2*o*, where *T*2*<sup>i</sup>* and *T*2*<sup>o</sup>* are inner outer operators in B*c*(�2, �2), respectively. Similarly, the operator *<sup>T</sup>*<sup>3</sup> can be factored as *<sup>T</sup>*<sup>3</sup> <sup>=</sup> *<sup>T</sup>*3*oT*3*<sup>i</sup>* where *<sup>T</sup>*3*<sup>i</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2) is inner, *<sup>T</sup>*3*<sup>o</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2)

Assumption (A1) can be relaxed by assuming instead that the outer operators *T*2*<sup>o</sup>* and *T*3*<sup>o</sup>* are bounded below (see Lemma (1) p. 220). Assumption (A1) guarantees that the map *<sup>Q</sup>* −→ *<sup>T</sup>*2*o*B*c*(�2, �2)*T*3*<sup>o</sup>* is bijective. Under this assumption *<sup>T</sup>*2*<sup>i</sup>* becomes an isometry and *<sup>T</sup>*3*<sup>i</sup>*

�*T*<sup>1</sup> − *T*2*QT*3� (2)

<sup>=</sup> {*<sup>A</sup>* <sup>∈</sup> <sup>B</sup>(�2, �2) : (*<sup>I</sup>* <sup>−</sup> *Qn*)*AQn* <sup>=</sup> 0, <sup>∀</sup> *<sup>n</sup>*} (3)

*<sup>N</sup>*<sup>−</sup> <sup>=</sup> {*N*� ∈ N : *<sup>N</sup>*� <sup>&</sup>lt; *<sup>N</sup>*} (4) *<sup>N</sup>*<sup>+</sup> <sup>=</sup> {*N*� ∈ N : *<sup>N</sup>*� <sup>&</sup>gt; *<sup>N</sup>*} (5)

�*T*<sup>1</sup> − *T*2*iT*2*oQT*3*oT*3*i*� (6)

In the next section the optimal disturbance rejection problem is formulated and solved using this Theorem in terms of a TV Hankel operator.

### **3. Time-varying optimal disturbance rejection problem**

In this chapter, we first consider the problem of optimizing performance for causal linear time varying systems by considering the standard block diagram for the optimal disturbance attenuation problem represented in Fig. 1, where *u* represents the control inputs, *y* the measured outputs, *z* is the controlled output, *w* the exogenous perturbations. *P* denotes a causal stable linear time varying plant, and *K* denotes a time varying controller. The

Fig. 1. Block Diagram for Disturbance Rejection

closed-loop transmission from *w* to *z* is denoted by *Tzw*. Using the standard TV Youla parametrization of all stabilizing controllers the closed loop operator *Tzw* can be written as (2; 11; 16),

$$T\_{zw} = T\_1 - T\_2 Q T\_3 \tag{1}$$

where *<sup>T</sup>*1, *<sup>T</sup>*<sup>2</sup> and *<sup>T</sup>*<sup>3</sup> are stable causal LTV operators, that is, *<sup>T</sup>*1, *<sup>T</sup>*<sup>2</sup> and *<sup>T</sup>*<sup>3</sup> <sup>∈</sup> *Bc*(2, <sup>2</sup>). Here it is assumed without loss of generality that *P* is stable, the Youla parameter *Q* := *K*(*I* + *PK*)−<sup>1</sup> is then an operator belonging to *Bc*(2, <sup>2</sup>), and is related in a one-to-one onto fashion to the controller *K* (29). Note that *Q* is allowed to be *time-varying*. If *P* is unstable it suffices to use the coprime factorization techniques in (11; 39) which lead to similar results. The magnitude of the signals *w* and *z* is measured in the 2-norm. The performance index which quantifies optimal disturbance rejection can be written in the following form (20)

4 Will-be-set-by-IN-TECH

*orthogonal representation of* <sup>H</sup> *as the orthogonal difference of invariant subspaces for H* |T (<sup>N</sup> )*, and*

In the next section the optimal disturbance rejection problem is formulated and solved using

In this chapter, we first consider the problem of optimizing performance for causal linear time varying systems by considering the standard block diagram for the optimal disturbance attenuation problem represented in Fig. 1, where *u* represents the control inputs, *y* the measured outputs, *z* is the controlled output, *w* the exogenous perturbations. *P* denotes a causal stable linear time varying plant, and *K* denotes a time varying controller. The

P

K

closed-loop transmission from *w* to *z* is denoted by *Tzw*. Using the standard TV Youla parametrization of all stabilizing controllers the closed loop operator *Tzw* can be written as

where *<sup>T</sup>*1, *<sup>T</sup>*<sup>2</sup> and *<sup>T</sup>*<sup>3</sup> are stable causal LTV operators, that is, *<sup>T</sup>*1, *<sup>T</sup>*<sup>2</sup> and *<sup>T</sup>*<sup>3</sup> <sup>∈</sup> *Bc*(2, <sup>2</sup>). Here it is assumed without loss of generality that *P* is stable, the Youla parameter *Q* := *K*(*I* + *PK*)−<sup>1</sup> is then an operator belonging to *Bc*(2, <sup>2</sup>), and is related in a one-to-one onto fashion to the controller *K* (29). Note that *Q* is allowed to be *time-varying*. If *P* is unstable it suffices to use the coprime factorization techniques in (11; 39) which lead to similar results. The magnitude of the signals *w* and *z* is measured in the 2-norm. The performance index which quantifies

*, and P*H�*<sup>Y</sup>* |M= *XP*<sup>H</sup> |M*, where* H = M�N *is the*

w

u

*Tzw* = *T*<sup>1</sup> − *T*2*QT*<sup>3</sup> (1)

**iii)** *Y dilates X, that is, Y* : M �−→ M�

this Theorem in terms of a TV Hankel operator.

z

y

Fig. 1. Block Diagram for Disturbance Rejection

(2; 11; 16),

**3. Time-varying optimal disturbance rejection problem**

*.*

*similarly for* H�

$$\begin{aligned} \mu &:= \inf \left\{ \|T\_{zw}\| : \text{ } K \text{ being robustly stabilizing linear time} - \text{varying controller} \right\} \\ &= \inf\_{Q \in \mathcal{B}\_{\ell}(\ell^2, \ell^2)} \|T\_1 - T\_2 Q T\_3\| \end{aligned} \tag{2}$$

The performance index (2) will be transformed into a shortest distance minimization between a certain bounded linear operator and a subspace to be specified shortly. In order to do so, following (11) define a nest N as a family of closed subspaces of the Hilbert space �<sup>2</sup> containing {0} and �<sup>2</sup> which is closed under intersection and closed span. Let *Qn* :<sup>=</sup> *<sup>I</sup>* − *Pn*, for *<sup>n</sup>* = −1, 0, 1, ··· , where *<sup>P</sup>*−<sup>1</sup> := 0 and *<sup>P</sup>*<sup>∞</sup> := *<sup>I</sup>*. Then *Qn* is a projection, and we can associate to it the following nest <sup>N</sup> :<sup>=</sup> {*Qn*�2, *<sup>n</sup>* <sup>=</sup> <sup>−</sup>1, 0, 1, ···}. In this case the triangular or nest algebra T (N ) is the set of all operators *T* such that *TN* ⊆ *N* for every element *N* in N . That is

$$\begin{split} \mathcal{T}(\mathcal{N}) &= \{ A \in \mathcal{B}(\ell^2, \ell^2) : P\_\mathbb{R} A (I - P\_\mathbb{N}) = 0, \,\forall \, n \} \\ &= \{ A \in \mathcal{B}(\ell^2, \ell^2) : (I - Q\_\mathbb{N}) A Q\_\mathbb{N} = 0, \,\forall \, n \} \end{split} \tag{3}$$

Note that the Banach space <sup>B</sup>*c*(�2, �2) is identical to the nest algebra <sup>T</sup> (<sup>N</sup> ). For *<sup>N</sup>* belonging to the nest <sup>N</sup> , *<sup>N</sup>* has the form *Qn*�<sup>2</sup> for some *<sup>n</sup>*. Define

$$N^- = \bigvee\{N' \in \mathcal{N} : N' < N\}\tag{4}$$

$$N^{+} = \bigwedge \{ N' \in \mathcal{N} : N' > N \} \tag{5}$$

where *N*� < *N* means *N*� ⊂ *N*, and *N*� > *N* means *N*� ⊃ *N*. The subspaces *N* � *N*<sup>−</sup> are called the atoms of <sup>N</sup> . Since in our case the atoms of <sup>N</sup> span �2, then <sup>N</sup> is said to be atomic (3). The early days of *H*<sup>∞</sup> control theory saw solutions based on the so-called inner-outer factorizations of functions belonging to the Hardy spaces *H*<sup>2</sup> and *H*∞, and their corresponding matrix valued counterparts for multi-input multi-output (MIMO) systems (22; 23). Generalizations in the context of nest algebras have been proposed in (1; 3) as follows: An operator *A* in T (N ) is called *outer* if the range projection *P*(*RA*), *RA* being the range of *A* and *P* the orthogonal projection onto *RA*, commutes with N and *AN* is dense in *N* ∩ *RA* for every *<sup>N</sup>* ∈ N . A partial isometry *<sup>U</sup>* is called *inner* in <sup>T</sup> (<sup>N</sup> ) if *<sup>U</sup>*�*<sup>U</sup>* commutes with <sup>N</sup> (1; 3; 11). In our case, *<sup>A</sup>* ∈ T (<sup>N</sup> ) = <sup>B</sup>*c*(�2, �2) is outer if *<sup>P</sup>* commutes with each *Qn* and *AQn*�<sup>2</sup> is dense in *Qn*�<sup>2</sup> <sup>∩</sup> *<sup>A</sup>*�2. *<sup>U</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2) is inner if *<sup>U</sup>* is a partial isometry and *<sup>U</sup>*�*<sup>U</sup>* commutes with every *Qn*. Applying these notions to the time-varying operator *<sup>T</sup>*<sup>2</sup> <sup>∈</sup> <sup>B</sup>*<sup>c</sup>* (�2, �2), we get *T*<sup>2</sup> = *T*2*iT*2*o*, where *T*2*<sup>i</sup>* and *T*2*<sup>o</sup>* are inner outer operators in B*c*(�2, �2), respectively. Similarly, the operator *<sup>T</sup>*<sup>3</sup> can be factored as *<sup>T</sup>*<sup>3</sup> <sup>=</sup> *<sup>T</sup>*3*oT*3*<sup>i</sup>* where *<sup>T</sup>*3*<sup>i</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2) is inner, *<sup>T</sup>*3*<sup>o</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2) is outer. The performance index *μ* in (2) can then be written as

$$\mu = \inf\_{\mathcal{Q} \in \mathcal{B}\_{\varepsilon}(\ell^2, \ell^2)} \left\| T\_1 - T\_{2i} T\_{2o} \mathcal{Q} T\_{3o} T\_{3i} \right\| \tag{6}$$

Following the classical *H*<sup>∞</sup> control theory (22; 23; 45),we assume **(A1)** that *T*2*<sup>o</sup>* and *T*3*<sup>o</sup>* are invertible both in B*c*(�2, �2).

Assumption (A1) can be relaxed by assuming instead that the outer operators *T*2*<sup>o</sup>* and *T*3*<sup>o</sup>* are bounded below (see Lemma (1) p. 220). Assumption (A1) guarantees that the map *<sup>Q</sup>* −→ *<sup>T</sup>*2*o*B*c*(�2, �2)*T*3*<sup>o</sup>* is bijective. Under this assumption *<sup>T</sup>*2*<sup>i</sup>* becomes an isometry and *<sup>T</sup>*3*<sup>i</sup>*

a co-isometry in which case *T* 2*i T*2*<sup>i</sup>* = *I* and *T*3*iT* <sup>3</sup>*<sup>i</sup>* = *I*. The operators *T*2*<sup>o</sup>* and *T*3*<sup>o</sup>* can be absorbed into the Youla operator *Q*, and expression (6) is then equivalent to

$$\mu = \inf\_{\mathcal{Q} \in \mathcal{B}\_{\ell}(\ell^2, \ell^2)} \|| T\_{2i}^{\star} T\_1 T\_{3i}^{\star} - \mathcal{Q} \|| \tag{7}$$

where now *RA* denotes the right multiplication associated to the operator *A* defined on the

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 33

Let *<sup>M</sup>* :<sup>=</sup> <sup>B</sup>*c*(2, <sup>2</sup>), *<sup>N</sup>* <sup>=</sup> {0}, *<sup>M</sup>*� :<sup>=</sup> <sup>C</sup>2, *<sup>N</sup>*� :<sup>=</sup> <sup>A</sup>2, and *<sup>H</sup>*<sup>1</sup> <sup>=</sup> *<sup>M</sup>* � *<sup>N</sup>*, *<sup>H</sup>*<sup>2</sup> <sup>=</sup> *<sup>M</sup>*� � *<sup>N</sup>*� are orthogonal representations of *H*<sup>1</sup> and *H*<sup>2</sup> of invariant subspaces under *H*|B*c*(2,<sup>2</sup>), that is,

Applying the Commutant Lifting Theorem for representations of nest algebras implies that

By Lemma 4.4. in (18) *<sup>H</sup>*˜ *<sup>B</sup>* is a left multiplication operator acting from <sup>A</sup><sup>2</sup> into <sup>C</sup><sup>2</sup> � A2. That is, *<sup>H</sup>*˜ *<sup>B</sup>* <sup>=</sup> *LK* for some *<sup>K</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>), with �*LK*� <sup>=</sup> �*K*� <sup>=</sup> �*H*˜ *<sup>X</sup>*� <sup>=</sup> �*HB*� by Lemma 4.5. (18).

By Theorem 2.1. (26) the Hankel operator *HB* is a compact operator if and only if *B* belongs to the space <sup>B</sup>*c*(2, <sup>2</sup>) + <sup>K</sup>, where <sup>K</sup> is the space of compact operators on the Hilbert space 2. A basic property of compact operators on Hilbert spaces is that they have maximizing vectors, that is, there exists at least one operator *<sup>A</sup><sup>o</sup>* ∈ A2, �*Ao*�<sup>2</sup> <sup>=</sup> 1 such that *HB* achieves its induced

We can then deduce from (7) and (12) an operator identity for the minimizer, that is, the

In the next section the mixed sensitivity problems for LTV systems is formulated and solved

The mixed sensitivity problem for stable plants (42; 45) involves the sensitivity operator

<sup>2</sup> <sup>+</sup> �*B*2*x*�<sup>2</sup>

2 1 2 , *B* =

<sup>3</sup>*ciA<sup>o</sup>* <sup>−</sup> *HT*

2*i T*1*T* <sup>3</sup>*ci <sup>A</sup><sup>o</sup>*

2*iT*1*<sup>T</sup>*

By Lemma 4.3. (18) *<sup>K</sup>* <sup>=</sup> *<sup>B</sup>* <sup>−</sup> *<sup>Q</sup>*, <sup>∃</sup>*<sup>Q</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) with �*K*� <sup>=</sup> �*HB*� as required.

*QoA<sup>o</sup>* = *T*

, the complementary sensitivity operator *T*<sup>2</sup> =

*Q*∈B*c*(2,<sup>2</sup>)

 �*B*1*x*�<sup>2</sup>

*μ<sup>o</sup>* = inf

where �·� stands for the operator norm in <sup>B</sup>(2, <sup>2</sup> <sup>×</sup> <sup>2</sup>) given by

�*x*�2≤1, *x*∈<sup>2</sup>

assumed to belong to <sup>B</sup>*c*(2, <sup>2</sup> <sup>×</sup> <sup>2</sup>), and is given by the minimization (13; 20; 46)

 *W* 0 − *W V PQ* 

(*A*) = *RA* on <sup>C</sup>2, *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) (15)

(*A*)*HB* <sup>=</sup> *HBh*(*A*) holds for for all *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>),

= (*<sup>I</sup>* − P)*BRA* |A2= (*<sup>I</sup>* − P)*B*P*RA* |A2= *HBh*(*A*) (18)

, i.e., *H*˜ *BH*(*A*) = *H*�

�*HBA<sup>o</sup>*�<sup>2</sup> <sup>=</sup> �*HB*��*Ao*�<sup>2</sup> <sup>=</sup> �*HB*� (19)

 *W V* 

> *B*<sup>1</sup> *B*2

(*A*)*HB* = (*<sup>I</sup>* − P)*RA*(*<sup>I</sup>* − P)*<sup>B</sup>* |A2= (*<sup>I</sup>* − P)*RAB* |A<sup>2</sup> (17)

2*i T*1*T* (·) have

<sup>3</sup>*ci* intertwines *h* and

(*A*)*H*˜ *<sup>B</sup>*, <sup>∀</sup>*<sup>A</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>).

*P* and *T*<sup>3</sup> := *I* which are all

(20)

(21)

(16)

specified Hilbert space, i.e., *RAB* = *BA*, *B* ∈ A2. The representation *h*(·) and *h*�

*H*(*A*) = *H*�

<sup>3</sup>*ci*, then *h*�

*RA*B*c*(2, <sup>2</sup>) <sup>⊂</sup> <sup>B</sup>*c*(2, <sup>2</sup>). Now we have to show that the operator *HT*

dilations *H* = *H*� given by

, that is, if *B* := *T*

norm at *Ao*. That is,

where the unknown is *Qo*.

*T*<sup>1</sup> :=

 *W* 0  2*i T*1*T*

*HB* has a dilation *H*˜ *<sup>B</sup>* that intertwines *H* and *H*�

optimal TV Youla parameter *Qo* as follows

using the commutant lifting theorem.

**4. The time-varying mixed sensitivity problem**

�*B*� = sup

*h*�

*h*�

Expression (7) is the distance from the operator *T* 2*i T*1*T* <sup>3</sup>*<sup>i</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>) to the nest algebra B*<sup>c</sup>* (2, <sup>2</sup>). It is the shortest distance from the bounded linear operator *T* 2*i T*1*T* <sup>3</sup>*<sup>i</sup>* to the space of causal bounded linear operators B*c*(2, <sup>2</sup>), which is a subspace of B(2, <sup>2</sup>). In the sequel, the commutant lifting theorem is used to solve the minimization (7) in terms of a time varying version of Hankel operators.

First, let <sup>C</sup><sup>2</sup> denote the special class of compact operators on <sup>2</sup> called the Hilbert-Schmidt or Schatten 2-class (3; 14) under the norm,

$$\|A\|\_{2} := \left( tr(A^\star A) \right)^{\frac{1}{2}} \tag{8}$$

Note that C<sup>2</sup> is a Hilbert space under the inner product (3)

$$(A, B) = \text{tr}(B^\star A), \quad \forall \ A, \ B \in \mathcal{L}\_2 \tag{9}$$

Define the space

$$\mathcal{A}\_2 := \mathcal{C}\_2 \cap \mathcal{B}\_\mathfrak{c}(\ell^2, \ell^2) \tag{10}$$

Then A<sup>2</sup> is the space of causal Hilbert-Schmidt operators. This space can be viewed as the TV counterpart of the standard Hardy space *H*<sup>2</sup> in the standard *H*<sup>∞</sup> theory. Define the orthogonal projection P of C<sup>2</sup> onto A2. P is the lower triangular truncation, and is analogous to the standard positive Riesz projection (for functions on the unit circle) for the LTI case. Following (27) an operator *<sup>X</sup>* in <sup>B</sup>(2, <sup>2</sup>) determines a Hankel operator *HX* on <sup>A</sup><sup>2</sup> if

$$H\_{\mathbf{X}}A = (I - \mathcal{P})\mathbf{X}A,\quad \text{for } A \in \mathcal{A}\_2 \tag{11}$$

We shall show that the shortest distance *μ* is equal to the norm of a particular LTV Hankel operator using the time varying version of the commutant lifting theorem in Theorem 1, thus generalizing a similar result in the LTI setting. let *HB* be the Hankel operator (*I* − P)*B*P associated with the symbol *B* := *T* 2*i T*1*T* <sup>3</sup>*ci*. The Hankel operator *HB* belongs to the Banach space of bounded linear operators on C2, namely, B(C2, C2). We have then the following Theorem which relates the optimal distance minimization *μ* to the induced norm of the Hankel operator *HT* 2*i T*1*T* 3*ci* .

**Theorem 2.** *Under assumptions (A1) the following holds:*

$$\mu = \left\| H\_{T\_{2i}^\* T\_1 T\_{3i}^\*} \right\| = \left\| (I - \mathcal{P}) T\_{2i}^\star T\_1 T\_{3i}^\star \mathcal{P} \right\| \tag{12}$$

**Proof.** Following (3; 18) let *H*<sup>1</sup> = A<sup>2</sup> and *H*<sup>2</sup> = C<sup>2</sup> � A<sup>2</sup> the orthogonal complement of A<sup>2</sup> in C2. Define the representations *h* and *h*� of A<sup>2</sup> by

$$h(A) = \mathcal{R}\_A \mid\_{H\_{1'}} \quad A \in \mathcal{B}\_{\mathcal{C}}(\ell^2, \ell^2) \tag{13}$$

$$h'(A) = (I - \mathcal{P})R\_{A|H\_2 \prime} \quad A \in \mathcal{B}\_{\mathcal{C}}(\ell^2, \ell^2) \tag{14}$$

where now *RA* denotes the right multiplication associated to the operator *A* defined on the specified Hilbert space, i.e., *RAB* = *BA*, *B* ∈ A2. The representation *h*(·) and *h*� (·) have dilations *H* = *H*� given by

$$H(A) = H'(A) = \mathcal{R}\_A \text{ on } \mathcal{C}\_{2'} \quad A \in \mathcal{B}\_{\mathcal{C}}(\ell^2, \ell^2) \tag{15}$$

(16)

Let *<sup>M</sup>* :<sup>=</sup> <sup>B</sup>*c*(2, <sup>2</sup>), *<sup>N</sup>* <sup>=</sup> {0}, *<sup>M</sup>*� :<sup>=</sup> <sup>C</sup>2, *<sup>N</sup>*� :<sup>=</sup> <sup>A</sup>2, and *<sup>H</sup>*<sup>1</sup> <sup>=</sup> *<sup>M</sup>* � *<sup>N</sup>*, *<sup>H</sup>*<sup>2</sup> <sup>=</sup> *<sup>M</sup>*� � *<sup>N</sup>*� are orthogonal representations of *H*<sup>1</sup> and *H*<sup>2</sup> of invariant subspaces under *H*|B*c*(2,<sup>2</sup>), that is, *RA*B*c*(2, <sup>2</sup>) <sup>⊂</sup> <sup>B</sup>*c*(2, <sup>2</sup>). Now we have to show that the operator *HT* 2*i T*1*T* <sup>3</sup>*ci* intertwines *h* and *h*� , that is, if *B* := *T* 2*i T*1*T* <sup>3</sup>*ci*, then *h*� (*A*)*HB* <sup>=</sup> *HBh*(*A*) holds for for all *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>),

$$\ln'(A)H\_B = (I - \mathcal{P})R\_A(I - \mathcal{P})B \mid\_{\mathcal{A}\_2} = (I - \mathcal{P})R\_AB \mid\_{\mathcal{A}\_2} \tag{17}$$

$$\mathcal{A} = (I - \mathcal{P}) \mathcal{B} \mathcal{R}\_A \mid\_{\mathcal{A}\_2} = (I - \mathcal{P}) \mathcal{B} \mathcal{P} \mathcal{R}\_A \mid\_{\mathcal{A}\_2} = H\_\mathcal{B} h(A) \tag{18}$$

Applying the Commutant Lifting Theorem for representations of nest algebras implies that *HB* has a dilation *H*˜ *<sup>B</sup>* that intertwines *H* and *H*� , i.e., *H*˜ *BH*(*A*) = *H*� (*A*)*H*˜ *<sup>B</sup>*, <sup>∀</sup>*<sup>A</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>). By Lemma 4.4. in (18) *<sup>H</sup>*˜ *<sup>B</sup>* is a left multiplication operator acting from <sup>A</sup><sup>2</sup> into <sup>C</sup><sup>2</sup> � A2. That is, *<sup>H</sup>*˜ *<sup>B</sup>* <sup>=</sup> *LK* for some *<sup>K</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>), with �*LK*� <sup>=</sup> �*K*� <sup>=</sup> �*H*˜ *<sup>X</sup>*� <sup>=</sup> �*HB*� by Lemma 4.5. (18). By Lemma 4.3. (18) *<sup>K</sup>* <sup>=</sup> *<sup>B</sup>* <sup>−</sup> *<sup>Q</sup>*, <sup>∃</sup>*<sup>Q</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) with �*K*� <sup>=</sup> �*HB*� as required.

By Theorem 2.1. (26) the Hankel operator *HB* is a compact operator if and only if *B* belongs to the space <sup>B</sup>*c*(2, <sup>2</sup>) + <sup>K</sup>, where <sup>K</sup> is the space of compact operators on the Hilbert space 2. A basic property of compact operators on Hilbert spaces is that they have maximizing vectors, that is, there exists at least one operator *<sup>A</sup><sup>o</sup>* ∈ A2, �*Ao*�<sup>2</sup> <sup>=</sup> 1 such that *HB* achieves its induced norm at *Ao*. That is,

$$\|\|H\_B A^o\|\|\_2 = \|\|H\_B\|\| \|A^o\|\|\_2 = \|\|H\_B\|\|\tag{19}$$

We can then deduce from (7) and (12) an operator identity for the minimizer, that is, the optimal TV Youla parameter *Qo* as follows

$$Q\_o A^o = T\_{2i}^\star T\_1 T\_{3ci}^\star A^o - H\_{T\_{2i}^\star T\_1 T\_{3ci}^\star} A^o$$

where the unknown is *Qo*.

6 Will-be-set-by-IN-TECH

�*<sup>T</sup>* 2*iT*1*<sup>T</sup>*

of causal bounded linear operators B*c*(2, <sup>2</sup>), which is a subspace of B(2, <sup>2</sup>). In the sequel, the commutant lifting theorem is used to solve the minimization (7) in terms of a time varying

First, let <sup>C</sup><sup>2</sup> denote the special class of compact operators on <sup>2</sup> called the Hilbert-Schmidt or

Then A<sup>2</sup> is the space of causal Hilbert-Schmidt operators. This space can be viewed as the TV counterpart of the standard Hardy space *H*<sup>2</sup> in the standard *H*<sup>∞</sup> theory. Define the orthogonal projection P of C<sup>2</sup> onto A2. P is the lower triangular truncation, and is analogous to the

We shall show that the shortest distance *μ* is equal to the norm of a particular LTV Hankel operator using the time varying version of the commutant lifting theorem in Theorem 1, thus generalizing a similar result in the LTI setting. let *HB* be the Hankel operator (*I* − P)*B*P

space of bounded linear operators on C2, namely, B(C2, C2). We have then the following Theorem which relates the optimal distance minimization *μ* to the induced norm of the Hankel

3*ci*� <sup>=</sup> �(*<sup>I</sup>* − P)*<sup>T</sup>*

**Proof.** Following (3; 18) let *H*<sup>1</sup> = A<sup>2</sup> and *H*<sup>2</sup> = C<sup>2</sup> � A<sup>2</sup> the orthogonal complement of A<sup>2</sup> in

standard positive Riesz projection (for functions on the unit circle) for the LTI case. Following (27) an operator *<sup>X</sup>* in <sup>B</sup>(2, <sup>2</sup>) determines a Hankel operator *HX* on <sup>A</sup><sup>2</sup> if

> 2*i T*1*T*

(*A*)=(*<sup>I</sup>* − P)*RA*|*H*<sup>2</sup>

*tr*(*AA*)

 1 2

(*A*, *<sup>B</sup>*) = *tr*(*<sup>B</sup><sup>A</sup>*), <sup>∀</sup> *<sup>A</sup>*, *<sup>B</sup>* ∈ C<sup>2</sup> (9)

*HX A* = (*I* − P)*XA*, for *A* ∈ A<sup>2</sup> (11)

2*iT*1*<sup>T</sup>*

*<sup>h</sup>*(*A*) = *RA* <sup>|</sup>*H*<sup>1</sup> , *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) (13)

<sup>3</sup>*ci*. The Hankel operator *HB* belongs to the Banach

<sup>3</sup>*ci*P � (12)

, *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) (14)

<sup>A</sup><sup>2</sup> :<sup>=</sup> <sup>C</sup><sup>2</sup> <sup>∩</sup> <sup>B</sup>*c*(2, <sup>2</sup>) (10)

2*i T*1*T*

<sup>3</sup>*<sup>i</sup>* = *I*. The operators *T*2*<sup>o</sup>* and *T*3*<sup>o</sup>* can be

<sup>3</sup>*<sup>i</sup>* − *Q*� (7)

<sup>3</sup>*<sup>i</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>) to the nest algebra

<sup>3</sup>*<sup>i</sup>* to the space

(8)

2*i T*1*T*

*T*2*<sup>i</sup>* = *I* and *T*3*iT*

absorbed into the Youla operator *Q*, and expression (6) is then equivalent to

*μ* = inf *Q*∈B*c*(2,<sup>2</sup>)

B*<sup>c</sup>* (2, <sup>2</sup>). It is the shortest distance from the bounded linear operator *T*

�*A*�<sup>2</sup> :=

a co-isometry in which case *T*

version of Hankel operators.

Define the space

operator *HT*

2*i T*1*T* 3*ci* .

Schatten 2-class (3; 14) under the norm,

associated with the symbol *B* := *T*

**Theorem 2.** *Under assumptions (A1) the following holds:*

*h*�

C2. Define the representations *h* and *h*� of A<sup>2</sup> by

*μ* = �*HT* 2*i T*1*T*

2*i*

Expression (7) is the distance from the operator *T*

Note that C<sup>2</sup> is a Hilbert space under the inner product (3)

In the next section the mixed sensitivity problems for LTV systems is formulated and solved using the commutant lifting theorem.

#### **4. The time-varying mixed sensitivity problem**

The mixed sensitivity problem for stable plants (42; 45) involves the sensitivity operator *T*<sup>1</sup> := *W* 0 , the complementary sensitivity operator *T*<sup>2</sup> = *W V P* and *T*<sup>3</sup> := *I* which are all assumed to belong to <sup>B</sup>*c*(2, <sup>2</sup> <sup>×</sup> <sup>2</sup>), and is given by the minimization (13; 20; 46)

$$\mu\_o = \inf\_{Q \in \mathcal{B}\_\varepsilon(\ell^2, \ell^2)} \left\| \begin{pmatrix} W \\ 0 \end{pmatrix} - \begin{pmatrix} W \\ V \end{pmatrix} P Q \right\|\tag{20}$$

where �·� stands for the operator norm in <sup>B</sup>(2, <sup>2</sup> <sup>×</sup> <sup>2</sup>) given by

$$\|\|B\|\| = \sup\_{\|\mathbf{x}\|\_2 \le 1, \mathbf{x} \in \ell^2} \left( \|\|B\_1 \mathbf{x}\|\|\_2^2 + \|\|B\_2 \mathbf{x}\|\|\_2^2 \right)^{\frac{1}{2}}, \ B = \begin{pmatrix} B\_1 \\ B\_2 \end{pmatrix} \tag{21}$$

The optimization problem (20) can be expressed as a shortest distance problem from the operator *<sup>T</sup>*<sup>1</sup> to the subspace <sup>S</sup> <sup>=</sup> *<sup>T</sup>*2*<sup>P</sup>* <sup>B</sup>*c*(�2, �2) of <sup>B</sup>(�2, �<sup>2</sup> <sup>×</sup> �2).

To ensure closedness of <sup>S</sup>, we assume that *<sup>W</sup>*�*<sup>W</sup>* <sup>+</sup> *<sup>V</sup>*�*<sup>V</sup>* <sup>&</sup>gt; 0, i.e., *<sup>W</sup>*�*<sup>W</sup>* <sup>+</sup> *<sup>V</sup>*�*<sup>V</sup>* as an operator acting on �<sup>2</sup> is a positive operator. In this case, there exists an outer spectral factorization <sup>Λ</sup><sup>1</sup> <sup>∈</sup> <sup>B</sup>*c*(�2, �2), invertible in <sup>B</sup>*c*(�2, �2) such that <sup>Λ</sup>� <sup>1</sup>Λ<sup>1</sup> <sup>=</sup> *<sup>W</sup>*�*<sup>W</sup>* <sup>+</sup> *<sup>V</sup>*�*<sup>V</sup>* (1; 11). Consequently, Λ1*P* as a bounded linear operator in B*c*(�2, �2) has an inner-outer factorization *U*1*G*, where *U*<sup>1</sup> is inner and *G* an outer operator defined on �<sup>2</sup> (3).

Next we assume (**A2**) *G* is invertible, so *U*<sup>1</sup> is unitary, and the operator *G* and its inverse *<sup>G</sup>*−<sup>1</sup> <sup>∈</sup> <sup>B</sup>*c*(�2, �2). The assumption (A2) is satisfied when, for e.g., the outer factor of the plant is invertible. Let *R* := *T*2Λ−<sup>1</sup> <sup>1</sup> *<sup>U</sup>*1, assumption (A2) implies that the operator *<sup>R</sup>*�*<sup>R</sup>* <sup>∈</sup> <sup>B</sup>(�2, �2) has a bounded inverse, this ensures closedness of S. It follows from Corollary 2 (1), that the self-adjoint operator *R*�*R* has a spectral factorization of the form: *R*�*R* = Λ�Λ, where <sup>Λ</sup>, <sup>Λ</sup>−<sup>1</sup> <sup>∈</sup> <sup>B</sup>*c*(�2, �2).

Define the operator *R*<sup>2</sup> := *R*Λ−1, then *R*� <sup>2</sup>*R*<sup>2</sup> = *I*, and S has the equivalent representation, <sup>S</sup> <sup>=</sup> *<sup>R</sup>*2B*c*(�2, �2). After "absorbing" <sup>Λ</sup> into the free parameter *<sup>Q</sup>*, the optimization problem (20) is then equivalent to:

$$\mu\_o = \inf\_{\mathcal{Q} \in \mathcal{B}\_c(\ell^2, \ell^2)} \| |T\_1 - R\_2 \mathcal{Q}| \| \tag{22}$$

where now *RA* denotes the right multiplication associated to the operator *A* defined on the specified Hilbert space. By the Sz. Nagy dilation Theorem there exist dilations *H* (respectively

Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 35

The spaces A<sup>2</sup> and *S* can be written as orthogonal differences of subspaces invariant under *H*

(*A*)<sup>Ξ</sup> = <sup>Π</sup>*RA*Π*T*<sup>1</sup> |A2= <sup>Π</sup>*RA*Π*T*<sup>1</sup> |A<sup>2</sup> = <sup>Π</sup>*RAT*<sup>1</sup> |A2= <sup>Π</sup>*T*1*RA* |A<sup>2</sup>

Applying the commutant lifting theorem for representations of nest algebras implies that Ξ

Lemma 4.4. in (18) Ξ� is a left multiplication operator acting from A<sup>2</sup> into A<sup>2</sup> ⊕ A2, and

Ξ = Π*T*<sup>1</sup> = Π*K*, which implies that Π(*T*<sup>1</sup> − *K*) = 0. Hence, (*T*<sup>1</sup> − *K*)*f* ∈ *R*2A2, for all

3.10 (3) there is a sequence *Fn* of finite rank contractions in B*c*(2, <sup>2</sup>) which converges to the identity operator in the strong \*-topology. By an approximation argument it follows that

<sup>Π</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*2P*<sup>R</sup>*

*T*<sup>1</sup> − *K* = *R*2*Q*, that is, *T*<sup>1</sup> − *R*2*Q* = *K*, with �*K*� = �Ξ�, and the Theorem is proved.

<sup>2</sup>*<sup>Y</sup>* = (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

(*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

= (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*2P*<sup>R</sup>*

, i.e., Ξ�

= Ξ*h*(*A*)

causal. That is, Ξ� = *LK* for some *K* ∈ B*c*(A2, A<sup>2</sup> ⊕ A2), with �*K*� = �Ξ�

*<sup>f</sup>* ∈ A2. That is, (*T*<sup>1</sup> <sup>−</sup> *<sup>K</sup>*)*<sup>f</sup>* <sup>=</sup> *<sup>R</sup>*<sup>2</sup> *<sup>g</sup>*, <sup>∃</sup>*<sup>g</sup>* ∈ A2, which can be written as *<sup>R</sup>*

*<sup>H</sup>*(*A*) = *RA* on <sup>A</sup><sup>2</sup> for *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) (27)

(*A*) = *RA* on <sup>A</sup><sup>2</sup> ⊕ A<sup>2</sup> for *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) (28)

A<sup>2</sup> = A<sup>2</sup> � {0}, *S* = A<sup>2</sup> ⊕ A<sup>2</sup> � *R*2A<sup>2</sup> (29)

*H*(*A*) = *H*�

<sup>2</sup> (*T*<sup>1</sup> <sup>−</sup> *<sup>K</sup>*)*<sup>f</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>), for all *<sup>f</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) of finite rank. By Theorem

<sup>2</sup> )(*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

<sup>2</sup> <sup>−</sup> *<sup>R</sup>*2P*<sup>R</sup>*

<sup>2</sup> ) showing that it is an orthogonal projection. Now we need to show that

<sup>2</sup> ) is *<sup>R</sup>*2A2. Let *<sup>Z</sup>* ∈ A<sup>2</sup> ⊕ A<sup>2</sup> such that (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

<sup>2</sup> )2*<sup>Y</sup>* = (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

<sup>2</sup> ) is a projection. The adjoint of (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

, that is, *h*�

(*A*)Ξ�

<sup>2</sup> (*T*<sup>1</sup> − *K*) we have *g* = *Q f* . We conclude that

<sup>2</sup> (30)

<sup>2</sup> )*Y* (31)

<sup>2</sup> )*Y* (33)

<sup>2</sup>), (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

<sup>2</sup> )*Y* (32)

<sup>2</sup> ), is clearly

<sup>2</sup> )*Z* = 0, so

<sup>2</sup> is the adjoint operator of *R*2. To see that (30)

2P*<sup>R</sup>*

<sup>2</sup>*Z* ∈ A2, implying that *Z* ∈ *R*2A2. We have showed

<sup>2</sup> <sup>+</sup> *<sup>R</sup>*2P*<sup>R</sup>*

(*A*)Ξ = Ξ*h*(*A*) for

, <sup>∀</sup>*<sup>A</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>). By

<sup>2</sup> (*T*<sup>1</sup> − *K*)*f* = *g* ∈ A2.

� = �Ξ�. Then

*H*�

and *H*�

all *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>),

In particular, *R*

*R*

but *R*

) for *h* (respectively *h*�

, respectively, as

) given by

Now we have to show that the operator Ξ intertwines *h* and *h*�

*h*�

has a dilation Ξ� that intertwines *H* and *H*�

<sup>2</sup> (*T*<sup>1</sup> <sup>−</sup> *<sup>K</sup>*) <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>). Letting *<sup>Q</sup>* :<sup>=</sup> *<sup>R</sup>*

The orthogonal projection Π can be computed as

where *<sup>I</sup>* is the identity operator on <sup>A</sup><sup>2</sup> ⊕ A2, *<sup>R</sup>*

holds note that for any *Y* ∈ A<sup>2</sup> ⊕ A2, we have

(*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

<sup>2</sup>*R*<sup>2</sup> <sup>=</sup> *<sup>I</sup>* and <sup>P</sup><sup>2</sup> <sup>=</sup> <sup>P</sup>, therefore

This shows that (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

the null space of (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

<sup>2</sup>*Z*. But *<sup>R</sup>*

equal to (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

*<sup>Z</sup>* <sup>=</sup> *<sup>R</sup>*2P*<sup>R</sup>*

2 )

<sup>2</sup>*<sup>Z</sup>* ∈ C2, then <sup>P</sup>*<sup>R</sup>*

*H*�

The minimization problem (22) gives the optimal mixed sensitivity with respect to controller design (as represented by *Q*). It is solved in terms of a projection of a multiplication operator. If the minimization (22) is achieved by a particular *Qo*, we call it optimal.

**Theorem 3.** *Introduce the orthogonal projection* Π *as follows*

$$
\Pi \,\,\, :\, \mathcal{A}\_2 \oplus \mathcal{A}\_2 \longmapsto (\mathcal{A}\_2 \oplus \mathcal{A}\_2) \odot \mathcal{R}\_2 \mathcal{A}\_2,
$$

*Under assumptions (A2) the following holds:*

$$
\mu\_{\mathcal{o}} = \|\Pi T\_1\|\tag{23}
$$

**Proof.** Denote by *S* := (A<sup>2</sup> ⊕ A2) � *R*2A2. That is, *S* is the orthogonal complement of the subspace *R*2A<sup>2</sup> in A<sup>2</sup> ⊕ A2, and define the operator

$$
\begin{aligned}
\Xi &: \mathcal{A}\_2 \longmapsto \mathcal{S} \\
\Xi &:= \Pi T\_1
\end{aligned}
\tag{24}
$$

We shall show with the help of the commutant lifting theorem that

$$
\mu\_{\mathcal{o}} = \|\Xi\|\tag{25}
$$

To see this we need, as before, a representation of B*c*(�2, �2), that is, an algebra homomorphism, say, *h*(·) (respectively *h*� (·)), from <sup>B</sup>*c*(�2, �2), into the algebra *<sup>B</sup>*(A2, <sup>A</sup>2) (respectively B*c*(*S*, *S*)), of bounded linear operators from A<sup>2</sup> into A<sup>2</sup> respectively from *S* into *S* . Define the representations *h* and *h*� by

$$h: \mathcal{B}\_{\mathfrak{c}}(\ell^2, \ell^2) \longmapsto \mathcal{B}(\mathcal{A}\_2, \mathcal{A}\_2), \ h': \mathcal{B}\_{\mathfrak{c}}(\ell^2, \ell^2) \longmapsto \mathcal{B}\_{\mathfrak{c}}(\mathcal{S}, \mathcal{S}) \tag{26}$$

$$h(A) := \mathcal{R}\_{A\prime} \quad A \in \mathcal{B}\_{\mathfrak{c}}(\ell^2, \ell^2), \ h'(A) := \Pi \mathcal{R}\_{A\prime} \quad A \in \mathcal{B}\_{\mathfrak{c}}(\ell^2, \ell^2)$$

8 Will-be-set-by-IN-TECH

The optimization problem (20) can be expressed as a shortest distance problem from the

To ensure closedness of <sup>S</sup>, we assume that *<sup>W</sup>*�*<sup>W</sup>* <sup>+</sup> *<sup>V</sup>*�*<sup>V</sup>* <sup>&</sup>gt; 0, i.e., *<sup>W</sup>*�*<sup>W</sup>* <sup>+</sup> *<sup>V</sup>*�*<sup>V</sup>* as an operator acting on �<sup>2</sup> is a positive operator. In this case, there exists an outer spectral factorization

Λ1*P* as a bounded linear operator in B*c*(�2, �2) has an inner-outer factorization *U*1*G*, where

Next we assume (**A2**) *G* is invertible, so *U*<sup>1</sup> is unitary, and the operator *G* and its inverse *<sup>G</sup>*−<sup>1</sup> <sup>∈</sup> <sup>B</sup>*c*(�2, �2). The assumption (A2) is satisfied when, for e.g., the outer factor of the plant

has a bounded inverse, this ensures closedness of S. It follows from Corollary 2 (1), that the self-adjoint operator *R*�*R* has a spectral factorization of the form: *R*�*R* = Λ�Λ, where

<sup>S</sup> <sup>=</sup> *<sup>R</sup>*2B*c*(�2, �2). After "absorbing" <sup>Λ</sup> into the free parameter *<sup>Q</sup>*, the optimization problem

The minimization problem (22) gives the optimal mixed sensitivity with respect to controller design (as represented by *Q*). It is solved in terms of a projection of a multiplication operator.

Π : A<sup>2</sup> ⊕ A<sup>2</sup> �−→ (A<sup>2</sup> ⊕ A2) � *R*2A<sup>2</sup>

**Proof.** Denote by *S* := (A<sup>2</sup> ⊕ A2) � *R*2A2. That is, *S* is the orthogonal complement of the

Ξ : A<sup>2</sup> �−→ *S*

To see this we need, as before, a representation of B*c*(�2, �2), that is, an algebra

*μ<sup>o</sup>* = inf

If the minimization (22) is achieved by a particular *Qo*, we call it optimal.

**Theorem 3.** *Introduce the orthogonal projection* Π *as follows*

subspace *R*2A<sup>2</sup> in A<sup>2</sup> ⊕ A2, and define the operator

We shall show with the help of the commutant lifting theorem that

(respectively B*c*(*S*, *S*)), of bounded linear operators from A<sup>2</sup> into A<sup>2</sup>

*<sup>h</sup>*(*A*) :<sup>=</sup> *RA*, *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2), *<sup>h</sup>*�

*<sup>h</sup>* : <sup>B</sup>*c*(�2, �2) �−→ *<sup>B</sup>*(A2, <sup>A</sup>2), *<sup>h</sup>*� : <sup>B</sup>*c*(�2, �2)

*Q*∈B*c*(�2,�<sup>2</sup>)

<sup>1</sup> *<sup>U</sup>*1, assumption (A2) implies that the operator *<sup>R</sup>*�*<sup>R</sup>* <sup>∈</sup> <sup>B</sup>(�2, �2)

<sup>2</sup>*R*<sup>2</sup> = *I*, and S has the equivalent representation,

*μ<sup>o</sup>* = �Π*T*1� (23)

Ξ := Π*T*<sup>1</sup> (24)

*μ<sup>o</sup>* = �Ξ� (25)

(·)), from <sup>B</sup>*c*(�2, �2), into the algebra *<sup>B</sup>*(A2, <sup>A</sup>2)

(*A*) :<sup>=</sup> <sup>Π</sup>*RA*, *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(�2, �2)

respectively from *S*

�−→ B*c*(*S*, *S*) (26)

�*T*<sup>1</sup> − *R*2*Q*� (22)

<sup>1</sup>Λ<sup>1</sup> <sup>=</sup> *<sup>W</sup>*�*<sup>W</sup>* <sup>+</sup> *<sup>V</sup>*�*<sup>V</sup>* (1; 11). Consequently,

operator *<sup>T</sup>*<sup>1</sup> to the subspace <sup>S</sup> <sup>=</sup> *<sup>T</sup>*2*<sup>P</sup>* <sup>B</sup>*c*(�2, �2) of <sup>B</sup>(�2, �<sup>2</sup> <sup>×</sup> �2).

<sup>Λ</sup><sup>1</sup> <sup>∈</sup> <sup>B</sup>*c*(�2, �2), invertible in <sup>B</sup>*c*(�2, �2) such that <sup>Λ</sup>�

*U*<sup>1</sup> is inner and *G* an outer operator defined on �<sup>2</sup> (3).

is invertible. Let *R* := *T*2Λ−<sup>1</sup>

Define the operator *R*<sup>2</sup> := *R*Λ−1, then *R*�

*Under assumptions (A2) the following holds:*

homomorphism, say, *h*(·) (respectively *h*�

. Define the representations *h* and *h*� by

into *S* 

<sup>Λ</sup>, <sup>Λ</sup>−<sup>1</sup> <sup>∈</sup> <sup>B</sup>*c*(�2, �2).

(20) is then equivalent to:

where now *RA* denotes the right multiplication associated to the operator *A* defined on the specified Hilbert space. By the Sz. Nagy dilation Theorem there exist dilations *H* (respectively *H*� ) for *h* (respectively *h*� ) given by

$$H(A) = \mathbb{R}\_A \text{ on } \mathcal{A}\_2 \text{ for } A \in \mathcal{B}\_c(\ell^2, \ell^2) \tag{27}$$

$$H'(A) = \mathbb{R}\_A \text{ on } \mathcal{A}\_2 \oplus \mathcal{A}\_2 \text{ for } A \in \mathcal{B}\_c(\ell^2, \ell^2) \tag{28}$$

The spaces A<sup>2</sup> and *S* can be written as orthogonal differences of subspaces invariant under *H* and *H*� , respectively, as

$$\mathcal{A}\_2 = \mathcal{A}\_2 \ominus \{0\}, \quad \mathcal{S} = \mathcal{A}\_2 \oplus \mathcal{A}\_2 \ominus \mathcal{R}\_2 \mathcal{A}\_2 \tag{29}$$

Now we have to show that the operator Ξ intertwines *h* and *h*� , that is, *h*� (*A*)Ξ = Ξ*h*(*A*) for all *<sup>A</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>),

$$\begin{aligned} \mathcal{H}'(A)\Xi &= \Pi \mathcal{R}\_A \Pi T\_1 \mid\_{\mathcal{A}\_2} = \Pi \mathcal{R}\_A \Pi T\_1 \mid\_{\mathcal{A}\_2} \\ &= \Pi \mathcal{R}\_A T\_1 \mid\_{\mathcal{A}\_2} = \Pi T\_1 \mathcal{R}\_A \mid\_{\mathcal{A}\_2} \\ &= \Xi h(A) \end{aligned}$$

Applying the commutant lifting theorem for representations of nest algebras implies that Ξ has a dilation Ξ� that intertwines *H* and *H*� , i.e., Ξ� *H*(*A*) = *H*� (*A*)Ξ� , <sup>∀</sup>*<sup>A</sup>* <sup>∈</sup> <sup>B</sup>(2, <sup>2</sup>). By Lemma 4.4. in (18) Ξ� is a left multiplication operator acting from A<sup>2</sup> into A<sup>2</sup> ⊕ A2, and causal. That is, Ξ� = *LK* for some *K* ∈ B*c*(A2, A<sup>2</sup> ⊕ A2), with �*K*� = �Ξ� � = �Ξ�. Then Ξ = Π*T*<sup>1</sup> = Π*K*, which implies that Π(*T*<sup>1</sup> − *K*) = 0. Hence, (*T*<sup>1</sup> − *K*)*f* ∈ *R*2A2, for all *<sup>f</sup>* ∈ A2. That is, (*T*<sup>1</sup> <sup>−</sup> *<sup>K</sup>*)*<sup>f</sup>* <sup>=</sup> *<sup>R</sup>*<sup>2</sup> *<sup>g</sup>*, <sup>∃</sup>*<sup>g</sup>* ∈ A2, which can be written as *<sup>R</sup>* <sup>2</sup> (*T*<sup>1</sup> − *K*)*f* = *g* ∈ A2. In particular, *R* <sup>2</sup> (*T*<sup>1</sup> <sup>−</sup> *<sup>K</sup>*)*<sup>f</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>), for all *<sup>f</sup>* <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>) of finite rank. By Theorem 3.10 (3) there is a sequence *Fn* of finite rank contractions in B*c*(2, <sup>2</sup>) which converges to the identity operator in the strong \*-topology. By an approximation argument it follows that *R* <sup>2</sup> (*T*<sup>1</sup> <sup>−</sup> *<sup>K</sup>*) <sup>∈</sup> <sup>B</sup>*c*(2, <sup>2</sup>). Letting *<sup>Q</sup>* :<sup>=</sup> *<sup>R</sup>* <sup>2</sup> (*T*<sup>1</sup> − *K*) we have *g* = *Q f* . We conclude that *T*<sup>1</sup> − *K* = *R*2*Q*, that is, *T*<sup>1</sup> − *R*2*Q* = *K*, with �*K*� = �Ξ�, and the Theorem is proved.

The orthogonal projection Π can be computed as

$$
\Pi = I - \mathsf{R}\_2 \mathsf{\mathcal{P}R}\_2^\star \tag{30}
$$

where *<sup>I</sup>* is the identity operator on <sup>A</sup><sup>2</sup> ⊕ A2, *<sup>R</sup>* <sup>2</sup> is the adjoint operator of *R*2. To see that (30) holds note that for any *Y* ∈ A<sup>2</sup> ⊕ A2, we have

$$(I - R\mathcal{P}\mathcal{R}\_2^\star)^2 Y = (I - R\mathcal{P}\mathcal{R}\_2^\star)(I - R\mathcal{P}\mathcal{R}\_2^\star)Y\tag{31}$$

$$\mathbf{x} = (I - \mathsf{R}\_2 \mathsf{P} \mathsf{R}\_2^\star - \mathsf{R}\_2 \mathsf{P} \mathsf{R}\_2^\star + \mathsf{R}\_2 \mathsf{P} \mathsf{R}\_2^\star \mathsf{P} \mathsf{R}\_2^\star) \mathbf{y} \tag{32}$$

but *R* <sup>2</sup>*R*<sup>2</sup> <sup>=</sup> *<sup>I</sup>* and <sup>P</sup><sup>2</sup> <sup>=</sup> <sup>P</sup>, therefore

$$(I - R\mathcal{P}\mathbb{R}\_2^\star)^2 Y = (I - R\mathcal{P}\mathbb{R}\_2^\star)Y\tag{33}$$

This shows that (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* ) is a projection. The adjoint of (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* ), (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* ), is clearly equal to (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* ) showing that it is an orthogonal projection. Now we need to show that the null space of (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* ) is *<sup>R</sup>*2A2. Let *<sup>Z</sup>* ∈ A<sup>2</sup> ⊕ A<sup>2</sup> such that (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* )*Z* = 0, so *<sup>Z</sup>* <sup>=</sup> *<sup>R</sup>*2P*<sup>R</sup>* <sup>2</sup>*Z*. But *<sup>R</sup> <sup>Z</sup>* ∈ C2, then <sup>P</sup>*<sup>R</sup> Z* ∈ A2, implying that *Z* ∈ *R*2A2. We have showed that the null space of the projection (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* <sup>2</sup> ) is a subset of *R*2A2. Conversely, let *Z* ∈ A2, then

$$(I - R\mathcal{P}\mathcal{R}\_2^\star)\mathcal{R}\_2\mathcal{Z} = R\_2\mathcal{Z} - R\_2\mathcal{P}\mathcal{Z} = R\_2\mathcal{Z} - R\_2\mathcal{Z} = 0\tag{34}$$

hence *<sup>R</sup>*2*<sup>Z</sup>* belongs to the null space of (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>* <sup>2</sup> ), and (30) holds. The operator Ξ has the following explicit form

$$
\Xi = (I - \mathcal{R}\_2 \mathcal{P} \mathcal{R}\_2^\star) T\_1 \tag{35}
$$

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Robustness of Feedback Linear Time-Varying Systems: A Commutant Lifting Approach 37

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which leads to the explicit solution

$$\mu\_0 = \left\| \left( I - \mathcal{R}\_2 \mathcal{P} \mathcal{R}\_2^\star \right) T\_1 \right\| \tag{36}$$

The expression generalizes the solution of the mixed sensitivity problem in the LTI case (25; 43; 46) to the LTV case. This result also applies to solve the robustness problem of feedback systems in the gap metric (38) in the TV case as outlined in (11; 21; 33), since the latter was shown in (11) to be equivalent to a special version of the mixed sensitivity problem (20).

### **5. Conclusion**

The optimal disturbance rejection and the mixed sensitivity problems for LTV systems involve solving shortest distance minimization problems posed in different spaces of bounded linear operators. LTV causal and stable systems form a nest algebras, this allows the commutant lifting theorem for nest algebras to be applied and solve both problems in term of abstract TV Hankel and a TV version generalization of Hankel-Topelitz operators under fairly weak assumptions. Future work includes investigation of numerical solutions based on finite dimensional approximations, and computation of the corresponding controllers.

**Acknowledgement:** This work was partially supported by the National Science Foundation under NSF Award No. CMMI-0825921.

#### **6. References**


10 Will-be-set-by-IN-TECH

<sup>Ξ</sup> = (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*2P*<sup>R</sup>*

*<sup>μ</sup><sup>o</sup>* <sup>=</sup> �(*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*2P*<sup>R</sup>*

The expression generalizes the solution of the mixed sensitivity problem in the LTI case (25; 43; 46) to the LTV case. This result also applies to solve the robustness problem of feedback systems in the gap metric (38) in the TV case as outlined in (11; 21; 33), since the latter was shown in (11) to be equivalent to a special version of the mixed sensitivity problem (20).

The optimal disturbance rejection and the mixed sensitivity problems for LTV systems involve solving shortest distance minimization problems posed in different spaces of bounded linear operators. LTV causal and stable systems form a nest algebras, this allows the commutant lifting theorem for nest algebras to be applied and solve both problems in term of abstract TV Hankel and a TV version generalization of Hankel-Topelitz operators under fairly weak assumptions. Future work includes investigation of numerical solutions based on finite

**Acknowledgement:** This work was partially supported by the National Science Foundation

[1] Arveson W. Interpolation problems in nest algebras, *Journal of Functional Analysis*, 4

[2] Chapellat H., Dahleh M. Analysis of time-varying control strategies for optimal disturbance rejection and robustness, *IEEE Transactions on Automatic Control*, 37 (1992)

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[5] Djouadi S.M. Optimal Robust Disturbance Attenuation for Continuous Time-Varying Systems, International journal of robust and non-linear control, vol. 13, (2003) 1181-1193. [6] G. Zames and A.K. El-Sakkary, Unstable Systems and Feedback: The Gap Metric, in

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Proc. of the Allerton Conference, pp. 380-385, 1980.

<sup>2</sup> ) is a subset of *R*2A2. Conversely, let *Z* ∈ A2,

<sup>2</sup> )*T*<sup>1</sup> (35)

<sup>2</sup> )*T*1� (36)

<sup>2</sup> )*R*2*Z* = *R*2*Z* − *R*2P*Z* = *R*2*Z* − *R*2*Z* = 0 (34)

<sup>2</sup> ), and (30) holds.

that the null space of the projection (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

(*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

hence *<sup>R</sup>*2*<sup>Z</sup>* belongs to the null space of (*<sup>I</sup>* <sup>−</sup> *<sup>R</sup>*P*<sup>R</sup>*

The operator Ξ has the following explicit form

which leads to the explicit solution

under NSF Award No. CMMI-0825921.

**5. Conclusion**

**6. References**

(1975) pp. 67-71.

1734-1746.

1988.

then


[31] T.T. Georgiou and M.C. Smith, Optimal Robustness in the Gap Metric, IEEE Trans. on Automatic Control, vol. 35, No. 6, pp. 673-686, 1990.

**0**

**3**

*Japan*

**A Sum of Squares Optimization Approach to**

Robust control problems for nonlinear systems are usually formulated as *L*2-induced norm minimization problems and those problems are reduced to the solvability of the so-called "Hamilton-Jacobi equation" (see, for example, van der Schaft (1996) and references therein). However, in the case of bilinear systems the usual *L*2-induced norm minimization problem leads to an obvious solution (the zero input is optimal!). To avoid the obvious solution Shimizu et al. (1997) introduced nonlinear weights on the evaluated signal and proposed a design method using linearization of the state-dependent matrix Riccati inequality derived from the Hamilton-Jacobi equation. In contrast to this, the purpose of this paper is to propose a new design method using SOS (Sum-of-Squares) optimization without linearization. It is known that the Hamilton-Jacobi equality coming from the *L*2-induced norm minimization problem is reduced to the solvability of an inequality condition of quadratic form, i.e.,

and this inequality is moreover reduced to the following matrix positive semi-definiteness

where *M*(*x*) is a Riccati-type matrix including the state variables. This matrix inequality is usually called "a state-dependent matrix Riccati inequality" derived from the *L*2-induced norm optimization problem. Most papers have tried to find a solution to the matrix inequality (2) so far. See, for example, Ichihara (2009); Prajna et al. (2004) and the references therein. However, it should be noted that the condition (2) is just a sufficient condition for (1) unless *h*(*x*) is independent of *x*, because *M*(*x*) includes *x*. In most *L*2-induced norm optimization cases, *h*(*x*) includes *x* (in our case *h*(*x*) = *P*−1*x* as shown in the later section) and hence the methods of Ichihara (2009); Prajna et al. (2004) and other papers based upon the condition (2) can have significant conservativeness. Note that Ichihara (2009) proposed a redesign method for reducing the conservativeness; however, it has to find a solution to (2) before applying the redesign method. Hence, the redesign method cannot be applied if the matrix inequality (2)

**1. Introduction**

condition:

does not have a solution.

**Robust Control of Bilinear Systems**

*<sup>h</sup>T*(*x*)*M*(*x*)*h*(*x*) <sup>≥</sup> 0, <sup>∀</sup>*<sup>x</sup>* (1)

*M*(*x*) � 0, ∀*x* (2)

<sup>1</sup>*Kyushu Institute of Technology*

<sup>2</sup>*Akashi National College of Technology*

Eitaku Nobuyama1, Takahiko Aoyagi1 and Yasushi Kami2


## **A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems**

Eitaku Nobuyama1, Takahiko Aoyagi1 and Yasushi Kami2 <sup>1</sup>*Kyushu Institute of Technology* <sup>2</sup>*Akashi National College of Technology Japan*

### **1. Introduction**

12 Will-be-set-by-IN-TECH

38 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

[31] T.T. Georgiou and M.C. Smith, Optimal Robustness in the Gap Metric, IEEE Trans. on

[32] Georgiou T., On the Computation of teh Gap Metric, Systems and Control Letters, vol.

[33] Foias C., Georgiou T. and Smith M.C., Robust Stability of Feedback Systems: A Geometric Approach Using The Gap Metric, SIAM J. Control and Optimization, vol.

[34] Foias C., Georgiou T. and Smith M.C., Geometric Techniques for Robust Stabilization of Linear Time-Varying Systems, Proc. of he 29th CDC, pp. 2868-2873, December 1990. [35] B. Sz. Nagy and C. Foias, Dilatation des commutants des commutateurs d'operateurs,

[36] B. Sz. Nagy, C. Foias, H. Bercovici, and L. Kerchy, Harmonic Analysis of Operators on

[37] Ravi R., Nagpal K.M. and Khargonekar P.P., *H*<sup>∞</sup> Control of Linear Time-Varying Systems: A State Space Approach, SIAM J. Control and Optimization, vol. 29, No.6,

[38] T.T. Georgiou and M.C. Smith, Robust Stabilization in the Gap Metric: Controller Design for Distributed Plants, IEEE Trans. on Automatic Control, vol. 37, No. 8, pp. 1133-1143,

[39] Dale W.N. and Smith M.C., Stabilizability and Existence of System Representation for Discrete-Time-Varying Systems, SIAM J. Control and Optimization, vol. 31, No.6, pp.

[40] Dullerud G.E. and Lall S., A new approach for analysis and synthesis of time-varying systems IEEE Trans. on Automatic Control, Vol.: 44 , Issue: 8 , pp. 1486 - 1497, 1999. [41] C. Pirie and G.E. Dullerud, Robust Controller Synthesis for Uncerttain Time-Varying Systems. SIAM J. Control Optimization., Vol. 40, No. 4, pp. 1312U1331, 2002.

[42] J.C. Doyle, B.A. Francis and A.R. Tannenbaum, Feedback Control Theory, Macmillan,

[43] C. Foias, H. Ozbay and A.R. Tannenbaum, Robust Control of Infinite Dimensional

[44] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, Metric constrained interpolation, commutant lifting and systems. Operator Theory: Advances and Applications, 100.

[46] Djouadi S.M. and Birdwell J.D., On the Optimal Two-Block *H*<sup>∞</sup> Problem, Proceedings of

Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1996.

the American Control Conference, pp. 4289 - 4294, June 2005.

[45] Zhou K. and Doyle J.C., Essentials of Robust Control, Prentice Hall, 1998.

˝

Automatic Control, vol. 35, No. 6, pp. 673-686, 1990.

C.R. Academie Sci. paris, Serie A, 266, pp. 493-495, 1968.

11, pp. 253-257, 1988.

31, No.6, pp. 1518-1537, 1993.

Hilbert Space, Springer, 2010.

Birkhäuser Verlag, Basel, 1998

pp. 1394-1413, 1991.

1538-1557, 1993.

NY, 1990.

1992.

Robust control problems for nonlinear systems are usually formulated as *L*2-induced norm minimization problems and those problems are reduced to the solvability of the so-called "Hamilton-Jacobi equation" (see, for example, van der Schaft (1996) and references therein). However, in the case of bilinear systems the usual *L*2-induced norm minimization problem leads to an obvious solution (the zero input is optimal!). To avoid the obvious solution Shimizu et al. (1997) introduced nonlinear weights on the evaluated signal and proposed a design method using linearization of the state-dependent matrix Riccati inequality derived from the Hamilton-Jacobi equation. In contrast to this, the purpose of this paper is to propose a new design method using SOS (Sum-of-Squares) optimization without linearization.

It is known that the Hamilton-Jacobi equality coming from the *L*2-induced norm minimization problem is reduced to the solvability of an inequality condition of quadratic form, i.e.,

$$h^T(\mathbf{x})M(\mathbf{x})h(\mathbf{x}) \ge \mathbf{0}, \quad \forall \mathbf{x} \tag{1}$$

and this inequality is moreover reduced to the following matrix positive semi-definiteness condition:

$$M(\mathfrak{x}) \succeq \mathbf{0}, \quad \forall \mathfrak{x} \tag{2}$$

where *M*(*x*) is a Riccati-type matrix including the state variables. This matrix inequality is usually called "a state-dependent matrix Riccati inequality" derived from the *L*2-induced norm optimization problem. Most papers have tried to find a solution to the matrix inequality (2) so far. See, for example, Ichihara (2009); Prajna et al. (2004) and the references therein. However, it should be noted that the condition (2) is just a sufficient condition for (1) unless *h*(*x*) is independent of *x*, because *M*(*x*) includes *x*. In most *L*2-induced norm optimization cases, *h*(*x*) includes *x* (in our case *h*(*x*) = *P*−1*x* as shown in the later section) and hence the methods of Ichihara (2009); Prajna et al. (2004) and other papers based upon the condition (2) can have significant conservativeness. Note that Ichihara (2009) proposed a redesign method for reducing the conservativeness; however, it has to find a solution to (2) before applying the redesign method. Hence, the redesign method cannot be applied if the matrix inequality (2) does not have a solution.

In the present paper, to avoid the conservativeness we propose a new method for finding a solution to (1) directly without finding a solution to (2). A key idea of our method is to treat the dependency of *h*(*x*) with *x* as an equality condition and formulate the problem to be concerned as an SOS (Sum of Squares) optimization problem with an equality constraint. After that we apply SOS optimization technique to the problem to propose an iterative algorithm for finding a robust feedback controller.

This paper is organized as follows: In Section 2, the plant to be concerned is described and a robust control problem is formulated after introducing nonlinear weights. Moreover, an inequality condition of quadratic form and the corresponding state-dependent matrix Riccati inequality are derived without using the Hamilton-Jacobi equality. In Section 3, some definitions and basic properties of SOS polynomials and SOS matrices are given. In Section 4, a new iterative method is proposed for finding a solution to the inequality condition of quadratic form. In Section 5, a numerical example is demonstrated to show the efficiency of our method, and in Section 6 this paper is concluded.

*A*

✛

*G*

In this paper, we consider to evaluate the *L*2-induced norm from *w* to *zp* and *u* with the

� *x*˙*z*(*t*) = *Azxz*(*t*) + *Bzzp*(*t*),

� *x*˙*u*(*t*) = *Auxu* + *Buu*(*t*),

*<sup>x</sup>*(*t*) + � <sup>0</sup>

<sup>⎦</sup> , *<sup>z</sup>*(*t*) = �

*D*12 �

where *xz* <sup>∈</sup> **<sup>R</sup>***nz* , *xu* <sup>∈</sup> **<sup>R</sup>***nu* and the matrices in (5) and (6) are real matrices of appropriate size. Here, we assume that the state variable is available for feedback. Then the plant with the frequency weights in Fig. 1 can be represented as a generalized plant *G* in Fig. 2 which is

*w*✲ *u*✲

following frequency weights *Wz*(*s*) and *Wu*(*s*), respectively (see Fig. 1):

*Wz*(*s*) :

*Wu*(*s*) :

*<sup>z</sup>*(*t*) = �

*x*(*t*) =

where *y*(*t*) is the output for feedback and

*C*<sup>11</sup> *C*12 �

⎡ ⎣

*xp*(*t*) *xz*(*t*) *xu*(*t*) ⎤

*Wu*(*s*)

✲

A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 41

✲

✲

*zz*(*t*) = *Czxz*(*t*) + *Dzzp*(*t*), (5)

*zu*(*t*) = *Cuxu*(*t*) + *Duu*(*t*) (6)

*x*˙(*t*) = *Ax*(*t*) + *B*1*w*(*t*) + *B*2(*x*)*u*(*t*) (7)

*y*(*t*) = *x*(*t*) (8)

�

, (9)

*u*(*t*)

*zz*(*t*) *zu*(*t*)

✲

*zz*

*zu*

*xp zp*

✲*z* ✲*y* ✲

*s*

*Bp*2(*x*) ❡ <sup>1</sup>

+ +

✲

*u*

Fig. 1. Plant with frequency weights

Fig. 2. Generalized plant

given by

*w*

✲ ✲ ✲ *<sup>I</sup>* ✲ *Cp*

*Bp*<sup>1</sup> *Wz*(*s*)

✻

+

❄

In this paper, the following notations are used:


**R**[*x*] the set of polynomials in *x*. (**R**[*x*] is also written as **R**[*x*<sup>1</sup> ··· *xn*] for *x* = [*x*<sup>1</sup> ··· *xn*] *T*.)


Σ the set of SOS polynomials. In particular, Σ*<sup>x</sup>* denotes the set of SOS polynomials in *x*.

*I* an identity matrix of appropriate size. In particular, *Ir* denotes the *r* × *r* identity matrix. Moreover, for a square matrix *M*, *M* � 0 and *M* � 0 imply that *M* is positive definite and positive semi-definite, respectively.

### **2. Problem statement**

#### **2.1 Plant and nonlinear weights**

Consider the following bilinear systems:

$$\dot{\mathbf{x}}\_p(t) = A\_p \mathbf{x}\_p(t) + B\_{p1} w(t) + \sum\_{i=1}^{n\_q} B\_{p2i} \mathbf{x}\_p(t) u\_i(t)$$

$$= A\_p \mathbf{x}\_p(t) + B\_{p1} w(t) + B\_{p2} (\mathbf{x}) u(t) \tag{3}$$

$$z\_p(t) = \mathbb{C}\_p \mathfrak{x}\_p(t) \tag{4}$$

where *xp* <sup>∈</sup> **<sup>R</sup>***np* is the state variable, *<sup>u</sup>* := [*u*<sup>1</sup> ··· *unq* ] *<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***nq* is the input, *<sup>w</sup>* <sup>∈</sup> **<sup>R</sup>***nw* is the exogenous input, *zp* <sup>∈</sup> **<sup>R</sup>***<sup>r</sup>* is the output to be evaluated, and the matrices *Ap*, *Bp*1, *Bp*<sup>2</sup>*<sup>i</sup>* (*i* = 1, . . . , *nq*) are real matrices of appropriate sizes with

$$B\_{p2}(\mathfrak{x}) := \sum\_{i=1}^{n\_q} B\_{p2i} \mathfrak{x}\_i$$

Fig. 1. Plant with frequency weights

#### Fig. 2. Generalized plant

2 Will-be-set-by-IN-TECH

In the present paper, to avoid the conservativeness we propose a new method for finding a solution to (1) directly without finding a solution to (2). A key idea of our method is to treat the dependency of *h*(*x*) with *x* as an equality condition and formulate the problem to be concerned as an SOS (Sum of Squares) optimization problem with an equality constraint. After that we apply SOS optimization technique to the problem to propose an iterative

This paper is organized as follows: In Section 2, the plant to be concerned is described and a robust control problem is formulated after introducing nonlinear weights. Moreover, an inequality condition of quadratic form and the corresponding state-dependent matrix Riccati inequality are derived without using the Hamilton-Jacobi equality. In Section 3, some definitions and basic properties of SOS polynomials and SOS matrices are given. In Section 4, a new iterative method is proposed for finding a solution to the inequality condition of quadratic form. In Section 5, a numerical example is demonstrated to show the efficiency of

**R**[*x*] the set of polynomials in *x*. (**R**[*x*] is also written as **R**[*x*<sup>1</sup> ··· *xn*] for *x* = [*x*<sup>1</sup> ··· *xn*]

Σ the set of SOS polynomials. In particular, Σ*<sup>x</sup>* denotes the set of SOS polynomials in *x*. *I* an identity matrix of appropriate size. In particular, *Ir* denotes the *r* × *r* identity matrix. Moreover, for a square matrix *M*, *M* � 0 and *M* � 0 imply that *M* is positive definite and

the exogenous input, *zp* <sup>∈</sup> **<sup>R</sup>***<sup>r</sup>* is the output to be evaluated, and the matrices *Ap*, *Bp*1, *Bp*<sup>2</sup>*<sup>i</sup>*

*nq* ∑ *i*=1

*Bp*2*ix*.

*Bp*2(*x*) :=

*nq* ∑ *i*=1

*Bp*2*ixp*(*t*)*ui*(*t*)

*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***nq* is the input, *<sup>w</sup>* <sup>∈</sup> **<sup>R</sup>***nw* is

= *Apxp*(*t*) + *Bp*1*w*(*t*) + *Bp*2(*x*)*u*(*t*) (3)

*zp*(*t*) = *Cpxp*(*t*) (4)

*x*˙ *<sup>p</sup>*(*t*) = *Apxp*(*t*) + *Bp*1*w*(*t*) +

where *xp* <sup>∈</sup> **<sup>R</sup>***np* is the state variable, *<sup>u</sup>* := [*u*<sup>1</sup> ··· *unq* ]

(*i* = 1, . . . , *nq*) are real matrices of appropriate sizes with

*T*.)

algorithm for finding a robust feedback controller.

our method, and in Section 6 this paper is concluded. In this paper, the following notations are used:

**R** the set of real numbers. **Z** the set of integers.

⊗ the Kroneckar product.

**2. Problem statement**

**Z**+ the set of non-negative integers.

*M<sup>T</sup>* the transpose of the matrix *M*.

positive semi-definite, respectively.

**2.1 Plant and nonlinear weights**

Consider the following bilinear systems:

In this paper, we consider to evaluate the *L*2-induced norm from *w* to *zp* and *u* with the following frequency weights *Wz*(*s*) and *Wu*(*s*), respectively (see Fig. 1):

$$\mathcal{W}\_{\mathbf{z}}(\mathbf{s}) : \begin{cases} \dot{\mathbf{x}}\_{\mathbf{z}}(t) = A\_{\mathbf{z}} \mathbf{x}\_{\mathbf{z}}(t) + B\_{\mathbf{z}} z\_{p}(t),\\ \mathbf{z}\_{\mathbf{z}}(t) = \mathbf{C}\_{\mathbf{z}} \mathbf{x}\_{\mathbf{z}}(t) + D\_{\mathbf{z}} z\_{p}(t), \end{cases} \tag{5}$$

$$\mathcal{W}\_{\boldsymbol{\upmu}}(\mathbf{s}) : \begin{cases} \dot{\boldsymbol{\upmu}}\_{\boldsymbol{\upmu}}(t) = \boldsymbol{A}\_{\boldsymbol{\upmu}} \boldsymbol{\upchi}\_{\boldsymbol{\upmu}} + \boldsymbol{B}\_{\boldsymbol{\upmu}} \boldsymbol{\upmu}(t),\\ \boldsymbol{z}\_{\boldsymbol{\upmu}}(t) = \boldsymbol{C}\_{\boldsymbol{\upmu}} \boldsymbol{x}\_{\boldsymbol{\upmu}}(t) + \boldsymbol{D}\_{\boldsymbol{\upmu}} \boldsymbol{u}(t) \end{cases} \tag{6}$$

where *xz* <sup>∈</sup> **<sup>R</sup>***nz* , *xu* <sup>∈</sup> **<sup>R</sup>***nu* and the matrices in (5) and (6) are real matrices of appropriate size. Here, we assume that the state variable is available for feedback. Then the plant with the frequency weights in Fig. 1 can be represented as a generalized plant *G* in Fig. 2 which is given by

$$\dot{x}(t) = Ax(t) + B\_1 w(t) + B\_2(x)u(t)\tag{7}$$

$$\begin{aligned} z(t) &= \begin{bmatrix} \mathbf{C}\_{11} \\ \mathbf{C}\_{12} \end{bmatrix} x(t) + \begin{bmatrix} 0 \\ D\_{12} \end{bmatrix} u(t) \\ y(t) &= x(t) \end{aligned} \tag{8}$$

where *y*(*t*) is the output for feedback and

$$\mathbf{x}(t) = \begin{bmatrix} \mathbf{x}\_p(t) \\ \mathbf{x}\_z(t) \\ \mathbf{x}\_
u(t) \end{bmatrix}, z(t) = \begin{bmatrix} z\_z(t) \\ z\_
u(t) \end{bmatrix}, \tag{9}$$

$$A = \begin{bmatrix} A\_p & 0 & 0 \\ B\_2 \mathbb{C}\_p \ A\_z & 0 \\ 0 & 0 & A\_\mu \end{bmatrix}, B\_1 = \begin{bmatrix} B\_{p1} \\ 0 \\ 0 \end{bmatrix}, B\_2(\mathbf{x}) = \begin{bmatrix} B\_{p2}(\mathbf{x}) \\ 0 \\ B\_\mu \end{bmatrix} \tag{10}$$

$$\mathbf{C}\_{11} = \begin{bmatrix} D\_{\overline{z}} \mathbf{C}\_{p} \ \mathbf{C}\_{\overline{z}} \ \mathbf{0} \end{bmatrix}, \begin{array}{c} \mathbf{C}\_{12} = \begin{bmatrix} \mathbf{0} \ \mathbf{0} \ \mathbf{C}\_{\mathbf{u}} \end{bmatrix}, \ D\_{12} = D\_{\mathbf{u}}.\tag{11}$$

Let *n* (:= *np* + *nz* + *nu*) denote the dimension of *x*.

The purpose of this paper is to find a feedback controller which reduces the effect of *w* on *z*. For this purpose, the problem to minimize the *L*2-induced norm from *w* to *z* defined by

$$\sup\_{w \neq 0} \frac{||z||\_2}{||w||\_2} \tag{12}$$

With the introduction of the nonlinear weights the new generalized plant *G*¯ shown in Fig. 4 is

A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 43

 *z*¯*z*(*t*) *z*¯*u*(*t*)

Then the problem to be considered in this paper is formulated as the one of finding the

�*z*¯�<sup>2</sup> �*w*�<sup>2</sup>

<sup>1</sup> *<sup>P</sup>* <sup>+</sup> *<sup>C</sup><sup>T</sup>*

*<sup>z</sup>* (*x*))*xTCT*

*<sup>x</sup>*(*t*) + <sup>0</sup>

*az*(*x*)*zz*(*t*)

*au*(*x*)*zu*(*t*)

*au*(*x*)*D*<sup>12</sup>

 . *u*(*t*), (13)

. (14)

<sup>11</sup>*C*<sup>11</sup> < 0 (15)

<sup>2</sup> (*x*)*Px*(*t*). (17)

<sup>11</sup>*C*11*x* ≥ 0. (16)

*x*˙(*t*) = *Ax*(*t*) + *B*1*w*(*t*) + *B*2(*x*)*u*(*t*),

*au*(*x*)*C*<sup>12</sup>

*z*¯(*t*) : =

=

feedback controller which minimizes the *L*2-induced norm from *w* to *z*¯ defined by

sup *w*�=0

1 *<sup>γ</sup>*<sup>2</sup> *PB*1*B<sup>T</sup>*

The next theorem is shown by Ohsaku et al. (1998); Shimizu et al. (1997) using linealization. **Theorem 1.** *Consider the bilinear system* (13) *with C*<sup>12</sup> = 0*. For given γ* > 0 *suppose that there*

<sup>2</sup> (*x*)*Px* + (<sup>1</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

This theorem gives a method for choosing the nonlinear weights after *P* is obtained; however, this means that the nonlinear weights cannot be chosen before obtaining *P* and the condition (16) restricts the choice of the nonlinear weights. In contract to this, in our method given below the nonlinear weights can be chosen a priori and the condition which they have to satisfy is

*Then the L*2*-induced norm from w to z is less than or equal to* ¯ *γ via the feedback control*

*<sup>u</sup>*(*t*) = <sup>−</sup> <sup>1</sup> *a*2 *<sup>u</sup>*(*x*) *BT*

*<sup>u</sup>*(*x*) ∈ **R**[*x*].

*<sup>z</sup>*¯(*t*) = *az*(*x*)*C*<sup>11</sup>

*y*(*t*) = *x*(*t*)

*exits a positive definite symmetric matrix P which satisfies*

*and the nonlinear weights az*(*x*) *and au*(*x*) *satisfy the condition*

*xTPB*2(*x*)*B<sup>T</sup>*

1 *a*2 *<sup>u</sup>*(*x*)

*<sup>z</sup>* (*x*) <sup>∈</sup> **<sup>R</sup>**[*x*] and 1/*a*<sup>2</sup>

*PA* + *ATP* +

represented as

where

just that *a*<sup>2</sup>

is usually considered where �·�<sup>2</sup> denotes the *L*<sup>2</sup> norm. However, the bilinear system (3) is uncontrollable for *x* = 0 because of *B*2(0) = 0, so that the effect of *w* cannot be reduced around *x* = 0. Moreover, it is known that the zero input *u* = 0 is optimal for the problem of minimizing the *L*2-induced norm (12) when the evaluated variable *z* is affine in *x* and *u*. Hence, the minimization problem with respect to the *L*2-induced norm (12) is no use for our purpose.

Although the system is uncontrollable at *x* = 0, the system behavior can be improved by some proper controllers except at *x* = 0. To formulate the problem of finding such controllers Shimizu et al. (1997) introduce nonlinear weights on *z*. It is shown by them that the obvious solution (the zero input) can be avoided by introducing the nonlinear weights.

Fig. 3. Plant with nonlinear weights

$$
\begin{array}{ccc}
\stackrel{w}{\longrightarrow} & \stackrel{\bar{\phantom{w}}}{\to} & \stackrel{\bar{\phantom{z}}}{\longrightarrow} \\
\end{array}
$$

Fig. 4. Generalized plant including nonlinear weights

As in the paper of Shimizu et al. (1997) we will also introduce nonlinear weights on *z* as shown in Fig. 3 where *az*(*x*) and *au*(*x*) are the nonlinear weights which are functions of *x*.

With the introduction of the nonlinear weights the new generalized plant *G*¯ shown in Fig. 4 is represented as

$$\begin{aligned} \dot{x}(t) &= Ax(t) + B\_1 w(t) + B\_2(x)u(t), \\ \dot{z}(t) &= \begin{bmatrix} a\_z(x) \mathbb{C}\_{11} \\ a\_{\boldsymbol{\mu}}(x) \mathbb{C}\_{12} \end{bmatrix} x(t) + \begin{bmatrix} 0 \\ a\_{\boldsymbol{\mu}}(x) D\_{12} \end{bmatrix} u(t), \\ y(t) &= x(t) \end{aligned} \tag{13}$$

where

4 Will-be-set-by-IN-TECH

⎡ ⎢ ⎢ ⎣

, *C*<sup>12</sup> = �

The purpose of this paper is to find a feedback controller which reduces the effect of *w* on *z*. For this purpose, the problem to minimize the *L*2-induced norm from *w* to *z* defined by

is usually considered where �·�<sup>2</sup> denotes the *L*<sup>2</sup> norm. However, the bilinear system (3) is uncontrollable for *x* = 0 because of *B*2(0) = 0, so that the effect of *w* cannot be reduced around *x* = 0. Moreover, it is known that the zero input *u* = 0 is optimal for the problem of minimizing the *L*2-induced norm (12) when the evaluated variable *z* is affine in *x* and *u*. Hence, the minimization problem with respect to the *L*2-induced norm (12) is no use for our

Although the system is uncontrollable at *x* = 0, the system behavior can be improved by some proper controllers except at *x* = 0. To formulate the problem of finding such controllers Shimizu et al. (1997) introduce nonlinear weights on *z*. It is shown by them that the obvious

solution (the zero input) can be avoided by introducing the nonlinear weights.

*Bp*2(*x*) ❡ <sup>1</sup>

+ +

Fig. 4. Generalized plant including nonlinear weights

✲

*u*

Fig. 3. Plant with nonlinear weights

*w*

✲ ✲ ✲ *<sup>I</sup>* ✲ *Cp*

✻

+

❄

*A*

*w*✲ *u*✲

in Fig. 3 where *az*(*x*) and *au*(*x*) are the nonlinear weights which are functions of *x*.

✛

*G*¯

As in the paper of Shimizu et al. (1997) we will also introduce nonlinear weights on *z* as shown

✲

✲ ✲

*z*¯*z*

*z*¯*u*

✲ ✲

*Wu*(*s*) *au*(*x*)*I*

*xp zp*

✲*z*¯ ✲*y*

*s*

*Bp*<sup>1</sup> *Wz*(*s*) *az*(*x*)*I*

✲

✲

�*z*�<sup>2</sup> �*w*�<sup>2</sup>

sup *w*� 0 *Bp*<sup>1</sup> 0 0

⎤ ⎥ ⎥

0 0 *Cu* �

<sup>⎦</sup> , *<sup>B</sup>*2(*x*) =

⎡ ⎢ ⎢ ⎣

*Bp*2(*x*) 0 *Bu*

⎤ ⎥ ⎥

, *D*<sup>12</sup> = *Du*. (11)

<sup>⎦</sup> , (10)

(12)

*A* =

purpose.

⎡ ⎢ ⎢ ⎣

*C*<sup>11</sup> = �

Let *n* (:= *np* + *nz* + *nu*) denote the dimension of *x*.

*Ap* 0 0 *BzCp Az* 0 0 0 *Au* ⎤ ⎥ ⎥ <sup>⎦</sup> , *<sup>B</sup>*<sup>1</sup> <sup>=</sup>

�

*DzCp Cz* 0

$$\begin{aligned} \bar{z}(t) &:= \begin{bmatrix} \bar{z}\_z(t) \\\\ \bar{z}\_\mathcal{u}(t) \end{bmatrix} \\\\ &= \begin{bmatrix} a\_z(\boldsymbol{x}) z\_z(t) \\\\ a\_\mathcal{u}(\boldsymbol{x}) z\_\mathcal{u}(t) \end{bmatrix} .\end{aligned}$$

Then the problem to be considered in this paper is formulated as the one of finding the feedback controller which minimizes the *L*2-induced norm from *w* to *z*¯ defined by

$$\sup\_{w \neq 0} \frac{\|z\|\_2}{\|w\|\_2}.\tag{14}$$

The next theorem is shown by Ohsaku et al. (1998); Shimizu et al. (1997) using linealization.

**Theorem 1.** *Consider the bilinear system* (13) *with C*<sup>12</sup> = 0*. For given γ* > 0 *suppose that there exits a positive definite symmetric matrix P which satisfies*

$$PA + A^T P + \frac{1}{\gamma^2} P B\_1 B\_1^T P + \mathcal{C}\_{11}^T \mathcal{C}\_{11} < 0 \tag{15}$$

*and the nonlinear weights az*(*x*) *and au*(*x*) *satisfy the condition*

$$\frac{1}{a\_{\mu}^{2}(\mathbf{x})}\mathbf{x}^{T}P B\_{2}(\mathbf{x})B\_{2}^{T}(\mathbf{x})P\mathbf{x} + (1 - a\_{z}^{2}(\mathbf{x}))\mathbf{x}^{T}\mathbb{C}\_{11}^{T}\mathbb{C}\_{11}\mathbf{x} \geq 0. \tag{16}$$

*Then the L*2*-induced norm from w to z is less than or equal to* ¯ *γ via the feedback control*

$$u(t) = -\frac{1}{a\_{\mathcal{U}}^2(\mathbf{x})} B\_2^T(\mathbf{x}) P \mathbf{x}(t). \tag{17}$$

This theorem gives a method for choosing the nonlinear weights after *P* is obtained; however, this means that the nonlinear weights cannot be chosen before obtaining *P* and the condition (16) restricts the choice of the nonlinear weights. In contract to this, in our method given below the nonlinear weights can be chosen a priori and the condition which they have to satisfy is just that *a*<sup>2</sup> *<sup>z</sup>* (*x*) <sup>∈</sup> **<sup>R</sup>**[*x*] and 1/*a*<sup>2</sup> *<sup>u</sup>*(*x*) ∈ **R**[*x*].

A typical choice of the nonlinear weights is as follows:

$$a\_{\overline{z}}(\mathbf{x}) = \sqrt{1 + \mathbf{x}^T \mathbf{R}\_{\overline{z}} \mathbf{x}}$$

$$a\_{\mu}(\mathbf{x}) = \frac{1}{\sqrt{1 + \mathbf{x}^T \mathbf{R}\_{\mu} \mathbf{x}}} \tag{18}$$

Proof: First, we will show that the closed-loop system via the feedback (20) with *w*(*t*) = 0 is stable when (19) holds. To show this we adopt *V*(*t*) = *xT*(*t*)*P*−1*x*(*t*) as a Lyapunov function

A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 45

<sup>2</sup> (*x*)*P*−<sup>1</sup> <sup>+</sup> *<sup>C</sup>*<sup>12</sup>

 1 *a*2 *<sup>u</sup>*(*x*) *BT*

*<sup>P</sup>*−1(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)+(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)*TP*−<sup>1</sup> <sup>−</sup> <sup>2</sup>

*zC<sup>T</sup>* <sup>11</sup>*C*<sup>11</sup> +

<sup>11</sup>*C*11*<sup>x</sup>* <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

<sup>1</sup> *<sup>P</sup>*−<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

*x T P*−1*x*

1 *a*2 *u*

≤0, for *x* � 0. (22)

*uxTCT*

<sup>=</sup> <sup>2</sup>*xTP*−1*x*˙ <sup>−</sup> <sup>2</sup>*xTP*−1*Ax* <sup>−</sup> <sup>2</sup>*xTP*−1*B*1*<sup>w</sup>* <sup>−</sup> <sup>2</sup>*xTP*−1*B*2(*x*)*u*. (24)

*xTP*−1*B*2(*x*) + *xTCT*

*uC<sup>T</sup>* 12*C*12 *x*

<sup>1</sup> *<sup>P</sup>*−1*<sup>x</sup>*

12 <sup>1</sup> *a*2 *u BT*

*uu*˜*Tu*˜ + *<sup>φ</sup>*(*x*, *<sup>P</sup>*) (25)

<sup>11</sup>*C*<sup>11</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

<sup>11</sup>*C*<sup>11</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

*<sup>γ</sup>*<sup>2</sup> *<sup>x</sup>TP*−1*B*1*B<sup>T</sup>*

*P*−1*B*2(*x*) + *C<sup>T</sup>*

*uC<sup>T</sup>* 12*C*12 *x*

<sup>2</sup> (*x*)*P*−<sup>1</sup> <sup>+</sup> *<sup>C</sup>*<sup>12</sup>

*x* 

*a*2 *u*

*P*−1*B*2(*x*)*B<sup>T</sup>*

<sup>12</sup>*C*12*<sup>x</sup>* <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

12 *u*  from (20)

<sup>2</sup> (*x*)*P*−<sup>1</sup>

*x* from (19)

*uxTCT*

<sup>2</sup> (*x*)*P*−<sup>1</sup> <sup>+</sup> *<sup>C</sup>*<sup>12</sup>

*x*

 *x*

<sup>12</sup>*u* (23)

*P*−1*B*2(*x*)*B<sup>T</sup>*

<sup>2</sup> (*x*)*P*−<sup>1</sup>

*uuTu* <sup>−</sup> <sup>2</sup>*a*<sup>2</sup>

candidate. Then we have

= 

=*x<sup>T</sup>* 

<sup>&</sup>lt; <sup>−</sup> *<sup>x</sup><sup>T</sup>*

Next, from (13) we have

*<sup>γ</sup>*2|*w*<sup>|</sup>

*<sup>γ</sup>*2|*w*<sup>|</sup>

<sup>2</sup> − |*z*<sup>|</sup>

and the following identity holds:

<sup>2</sup> − |*z*<sup>|</sup>

*TP*−1*x* + *xTP*−1*x*˙

*Ax* − *B*2(*x*)

+ *xTP*−<sup>1</sup>

1

=(*Ax* + *B*2(*x*)*u*)*TP*−1*x* + *xTP*−1(*Ax* + *B*2(*x*)*u*)

 1 *a*2 *<sup>u</sup>*(*x*) *BT*

*Ax* − *B*2(*x*)

*<sup>γ</sup>*<sup>2</sup> *<sup>P</sup>*−1*B*1*B<sup>T</sup>*

<sup>2</sup> <sup>=</sup> *<sup>γ</sup>*2|*w*<sup>|</sup>

This shows the closed-loop system is asymptotically stable.

<sup>0</sup> <sup>=</sup> <sup>2</sup>*xTP*−1*x*˙ <sup>−</sup> <sup>2</sup>*xTP*−1*x*˙

<sup>−</sup> *<sup>a</sup>*<sup>2</sup> *u* 

<sup>−</sup> *<sup>x</sup><sup>T</sup>*

<sup>−</sup> *<sup>a</sup>*<sup>2</sup>

<sup>−</sup> *<sup>x</sup><sup>T</sup>*

<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

*zxTCT*

By adding the both-sides of (24) to (23) and completing the square we have

 1 *a*2 *u*

*zC<sup>T</sup>*

*zC<sup>T</sup>*

<sup>2</sup> <sup>=</sup>2*xTP*−1*x*˙ <sup>+</sup> *<sup>γ</sup>*2*wTw* <sup>−</sup> <sup>2</sup>*xTP*−1*B*1*<sup>w</sup>*

2*P*−1*A* + *a*<sup>2</sup>

*ux<sup>T</sup>* 1 *a*2 *u*

2*P*−1*A* + *a*<sup>2</sup>

*uTu* + 2

<sup>=</sup>2*xTP*−1*x*˙ <sup>+</sup> *<sup>γ</sup>*2*w*˜ *Tw*˜ <sup>−</sup> <sup>1</sup>

*uu*˜*Tu*˜ + *<sup>a</sup>*<sup>2</sup>

<sup>=</sup>2*xTP*−1*x*˙ <sup>+</sup> *<sup>γ</sup>*2*w*˜ *Tw*˜ <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

*dt <sup>V</sup>*(*t*) =*x*˙

*d*

where *Rx* � 0 and *Ru* � 0. Fig. 5 shows an example in the case of *x* ∈ **R**. The weight *az*(*x*) shown in Fig. 5 is utilized for suppressing the effect of *w* on *zz* and *au*(*x*) is for allowing large input values except at *x* = 0.

Fig. 5. Example of nonlinear weights

#### **2.2 Derivation of state-dependent inequalities**

In the sequel, we assume *D*<sup>12</sup> = *I* for simplicity. Then we have the next theorem.

**Theorem 2.** *Suppose that for given γ* > 0 *there exists a positive definite symmetric matrix P which satisfies the following state-dependent inequality:*

$$\phi(\mathbf{x}, P) := \mathbf{x}^T \left[ -P^{-1} (A + B\_2(\mathbf{x}) \mathbb{C}\_{12}) - (A + B\_2(\mathbf{x}) \mathbb{C}\_{12})^T P^{-1} \right. $$

$$ -P^{-1} \left( \frac{1}{\gamma^2} B\_1 B\_1^T - \frac{1}{a\_u^2(\mathbf{x})} B\_2(\mathbf{x}) B\_2^T(\mathbf{x}) \right) P^{-1} - a\_z^2(\mathbf{x}) \mathbb{C}\_{11}^T \mathbb{C}\_{11} \right] \mathbf{x} > 0, \tag{19} $$

$$ \forall \mathbf{x} (\neq 0) \in \mathbf{R}^n \tag{19} $$

*Then by the feedback*

$$u(t) = -\left(\frac{1}{a\_{\mu}^2(\mathbf{x})} B\_2^T(\mathbf{x}) P^{-1} + \mathbf{C}\_{12}\right) \mathbf{x}(t) \tag{20}$$

*the closed-loop system is asymptotically stable and the L*2*-induced norm* (14) *is less than or equal to γ, i.e.,*

$$\sup\_{w \neq 0} \frac{\|\bar{z}\|\_2}{\|w\|\_2} \le \gamma. \tag{21}$$

Proof: First, we will show that the closed-loop system via the feedback (20) with *w*(*t*) = 0 is stable when (19) holds. To show this we adopt *V*(*t*) = *xT*(*t*)*P*−1*x*(*t*) as a Lyapunov function candidate. Then we have

$$\begin{split} \frac{d}{dt}V(t) &= \mathbf{x}^T P^{-1} \mathbf{x} + \mathbf{x}^T P^{-1} \mathbf{x} \\ &= (A\mathbf{x} + B\_2(\mathbf{x})\mathbf{u})^T P^{-1} \mathbf{x} + \mathbf{x}^T P^{-1} (A\mathbf{x} + B\_2(\mathbf{x})\mathbf{u}) \\ &= \left[A\mathbf{x} - B\_2(\mathbf{x}) \left(\frac{1}{a\_u^2(\mathbf{x})} B\_2^T(\mathbf{x}) P^{-1} + \mathbb{C}\_{12}\right) \mathbf{x}\right]^T P^{-1} \mathbf{x} \\ &\quad + \mathbf{x}^T P^{-1} \left[A\mathbf{x} - B\_2(\mathbf{x}) \left(\frac{1}{a\_u^2(\mathbf{x})} B\_2^T(\mathbf{x}) P^{-1} + \mathbb{C}\_{12}\right) \mathbf{x}\right] \quad \text{from (20)} \\ &= \mathbf{x}^T \left[P^{-1} (A + B\_2(\mathbf{x}) \mathbb{C}\_{12}) + (A + B\_2(\mathbf{x}) \mathbb{C}\_{12})^T P^{-1} - \frac{2}{a\_u^2} P^{-1} B\_2(\mathbf{x}) B\_2^T(\mathbf{x}) P^{-1}\right] \mathbf{x} \\ &\quad < -\mathbf{x}^T \left[\frac{1}{\gamma^2} P^{-1} B\_1 B\_1^T P^{-1} + a\_u^2 \mathbb{C}\_{11}^T \mathbb{C}\_{11} + \frac{1}{a\_u^2} P^{-1} B\_2(\mathbf{x}) B\_2^T(\mathbf{x}) P^{-1}\right] \mathbf{x} \quad \text{from (19)} \\ &\le 0, \quad \text{for } \mathbf{x} \neq \mathbf{0}. \end{split} \tag{22}$$

This shows the closed-loop system is asymptotically stable. Next, from (13) we have

$$\gamma^2|w|^2 - |z|^2 = \gamma^2|w|^2 - a\_z^2 \mathbf{x}^T \mathbf{C}\_{11}^T \mathbf{C}\_{11} \mathbf{x} - a\_{\mathbf{u}}^2 \mathbf{x}^T \mathbf{C}\_{12}^T \mathbf{C}\_{12} \mathbf{x} - a\_{\mathbf{u}}^2 \mathbf{u}^T \mathbf{u} - 2a\_{\mathbf{u}}^2 \mathbf{x}^T \mathbf{C}\_{12}^T \mathbf{u} \tag{23}$$

and the following identity holds:

6 Will-be-set-by-IN-TECH

*az*(*x*) = <sup>1</sup> <sup>+</sup> *<sup>x</sup>TRzx*,

where *Rx* � 0 and *Ru* � 0. Fig. 5 shows an example in the case of *x* ∈ **R**. The weight *az*(*x*) shown in Fig. 5 is utilized for suppressing the effect of *w* on *zz* and *au*(*x*) is for allowing large

<sup>1</sup> <sup>+</sup> *<sup>x</sup>TRux*

0.6

0.8

1

−2 −1 <sup>0</sup> <sup>1</sup> <sup>2</sup> 0.4

(b) *au*(*x*)

(18)

*au*(*x*) = <sup>1</sup>

In the sequel, we assume *D*<sup>12</sup> = *I* for simplicity. Then we have the next theorem.

<sup>1</sup> <sup>−</sup> <sup>1</sup> *a*2 *<sup>u</sup>*(*x*)

> 1 *a*2 *<sup>u</sup>*(*x*) *BT*

> > sup *w*�=0

**Theorem 2.** *Suppose that for given γ* > 0 *there exists a positive definite symmetric matrix P which*

<sup>−</sup>*P*−1(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12) <sup>−</sup> (*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)*TP*−<sup>1</sup>

*B*2(*x*)*B<sup>T</sup>*

*the closed-loop system is asymptotically stable and the L*2*-induced norm* (14) *is less than or equal to γ,*

�*z*¯�<sup>2</sup> �*w*�<sup>2</sup> <sup>2</sup> (*x*) 

<sup>2</sup> (*x*)*P*−<sup>1</sup> + *<sup>C</sup>*<sup>12</sup>

<sup>∀</sup>*x*(�<sup>=</sup> <sup>0</sup>) <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* (19)

*<sup>P</sup>*−<sup>1</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

*<sup>z</sup>* (*x*)*C<sup>T</sup>*

≤ *γ*. (21)

<sup>11</sup>*C*<sup>11</sup> *x* > 0,

*x*(*t*) (20)

A typical choice of the nonlinear weights is as follows:

−2 −1 <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>1</sup>

(a) *az* (*x*)

**2.2 Derivation of state-dependent inequalities**

*satisfies the following state-dependent inequality:*

<sup>−</sup> *<sup>P</sup>*−<sup>1</sup>

 1 *<sup>γ</sup>*<sup>2</sup> *<sup>B</sup>*1*B<sup>T</sup>*

*u*(*t*) = −

Fig. 5. Example of nonlinear weights

*φ*(*x*, *P*) :=*x<sup>T</sup>*

*Then by the feedback*

*i.e.,*

input values except at *x* = 0.

1.5

2

2.5

$$\begin{split} 0 &= 2\mathbf{x}^T P^{-1} \dot{\mathbf{x}} - 2\mathbf{x}^T P^{-1} \dot{\mathbf{x}} \\ &= 2\mathbf{x}^T P^{-1} \dot{\mathbf{x}} - 2\mathbf{x}^T P^{-1} A \mathbf{x} - 2\mathbf{x}^T P^{-1} B\_1 w - 2\mathbf{x}^T P^{-1} B\_2(\mathbf{x}) u. \end{split} \tag{24}$$

By adding the both-sides of (24) to (23) and completing the square we have

*<sup>γ</sup>*2|*w*<sup>|</sup> <sup>2</sup> − |*z*<sup>|</sup> <sup>2</sup> <sup>=</sup>2*xTP*−1*x*˙ <sup>+</sup> *<sup>γ</sup>*2*wTw* <sup>−</sup> <sup>2</sup>*xTP*−1*B*1*<sup>w</sup>* <sup>−</sup> *<sup>a</sup>*<sup>2</sup> *u uTu* + 2 1 *a*2 *u xTP*−1*B*2(*x*) + *xTCT* 12 *u* <sup>−</sup> *<sup>x</sup><sup>T</sup>* 2*P*−1*A* + *a*<sup>2</sup> *zC<sup>T</sup>* <sup>11</sup>*C*<sup>11</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> *uC<sup>T</sup>* 12*C*12 *x* <sup>=</sup>2*xTP*−1*x*˙ <sup>+</sup> *<sup>γ</sup>*2*w*˜ *Tw*˜ <sup>−</sup> <sup>1</sup> *<sup>γ</sup>*<sup>2</sup> *<sup>x</sup>TP*−1*B*1*B<sup>T</sup>* <sup>1</sup> *<sup>P</sup>*−1*<sup>x</sup>* <sup>−</sup> *<sup>a</sup>*<sup>2</sup> *uu*˜*Tu*˜ + *<sup>a</sup>*<sup>2</sup> *ux<sup>T</sup>* 1 *a*2 *u P*−1*B*2(*x*) + *C<sup>T</sup>* 12 <sup>1</sup> *a*2 *u BT* <sup>2</sup> (*x*)*P*−<sup>1</sup> <sup>+</sup> *<sup>C</sup>*<sup>12</sup> *x* <sup>−</sup> *<sup>x</sup><sup>T</sup>* 2*P*−1*A* + *a*<sup>2</sup> *zC<sup>T</sup>* <sup>11</sup>*C*<sup>11</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> *uC<sup>T</sup>* 12*C*12 *x* <sup>=</sup>2*xTP*−1*x*˙ <sup>+</sup> *<sup>γ</sup>*2*w*˜ *Tw*˜ <sup>−</sup> *<sup>a</sup>*<sup>2</sup> *uu*˜*Tu*˜ + *<sup>φ</sup>*(*x*, *<sup>P</sup>*) (25)

where

$$\begin{aligned} \tilde{w} &= w - \frac{1}{\gamma^2} B\_1^T P^{-1} \mathbf{x}\_\prime \\\\ \tilde{u} &= u + \left( \frac{1}{a\_u^2} B\_2^T(\mathbf{x}) P^{-1} + \mathcal{C}\_{12} \right) \mathbf{x}\_\prime \end{aligned}$$

Then from (19) and (20)

$$\gamma^2 |w|^2 - |z|^2 \ge 2\varkappa^T P^{-1} \dot{\varkappa} + \gamma^2 \tilde{w}^T \tilde{w}\_{\varkappa}$$

and hence

$$\int\_0^\tau (\gamma^2 |w|^2 - |z|^2) dt \ge \int\_0^\tau (2\mathbf{x}^T P^{-1} \dot{\mathbf{x}} + \gamma^2 \dot{\mathbf{w}}^T \ddot{\mathbf{w}}) dt$$

$$= \mathbf{x}^T(\tau) P^{-1} \mathbf{x}(\tau) - \mathbf{x}^T(0) P^{-1} \mathbf{x}(0) + \int\_0^\tau (\gamma^2 \dot{\mathbf{w}}^T \ddot{\mathbf{w}}) dt.$$

Here, let *x*(0) = 0 and *τ* → ∞ then

$$\gamma^2 \|w\|^2 - \|z\|^2 = \int\_0^\infty (\gamma^2 |w|^2 - |z|^2) dt = \int\_0^\infty (\gamma^2 \tilde{w}^T \tilde{w}) dt \ge 0,\tag{26}$$

which leads to (21). Note that to derive (26) we use lim*τ*→<sup>∞</sup> *x*(*τ*) = 0 which holds because the closed-loop is asymptotically stable. Q.E.D.

Note that *φ*(*x*, *p*) in (19) can be represented as

$$\phi(\mathbf{x}, P) = (P^{-1}\mathbf{x})^T M(\mathbf{x}, P)(P^{-1}\mathbf{x}) \tag{27}$$

The inequality (29) is called "a state-dependent matrix Riccati inequality" and equivalent to

A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 47

*B*2(*x*)*B<sup>T</sup>*

Note that *h* is independent of *x* in (30), whereas there is a relationship of *h* = *P*−1*x* between *h* and *x* in (19). This means that the condition (30) can be very conservative compared with the condition (19). As mentioned in Introduction, most papers have tried to find *P* which satisfies a matrix state-dependent inequality like (29) (or (30)). In contrary to this, we will try to find *P*

In this section, we briefly survey the so-called "SOS (Sum of Squares) optimization."

*<sup>T</sup>* is represented as *xα*<sup>1</sup>

deg(*f*), is defined by the degree of the monomial which has the highest degree among all the monomials included in *<sup>f</sup>*(*x*). For a polynomial matrix *<sup>F</sup>*(*x*) <sup>∈</sup> **<sup>R</sup>***q*×*<sup>r</sup>* the degree of *<sup>F</sup>*(*x*), denoted by deg(*F*), is defined by deg(*F*) := *maxij* deg(*Fij*) where *Fij* denotes the (*i*, *j*) element

A real polynomial *f*(*x*) ∈ **R**[*x*] is said to be an *SOS (Sum of Squares) polynomial* if it can be represented as a sum of squares of some polynomials, i.e., there exist some polynomials

> *p* ∑ *i*=1 *g*2

Moreover, a polynomial symmetric matrix *<sup>F</sup>*(*x*) <sup>∈</sup> **<sup>R</sup>***r*×*r*[*x*] is said to be an *SOS matrix* if it can

*F*(*x*) = *LT*(*x*)*L*(*x*) for some polynomial matrix *L*(*x*) of appropriate size. In this paper, we denote the set of SOS

Here, for a positive integer *d* let *vd*(*x*) be a polynomial vector in *x* of size *<sup>n</sup>*+*d*C*<sup>d</sup>* defined by

<sup>1</sup> *<sup>x</sup>*1*x*<sup>2</sup> ··· *<sup>x</sup>*<sup>2</sup>

<sup>1</sup> *<sup>x</sup>*<sup>1</sup> ··· *xn <sup>x</sup>*<sup>2</sup>

*f*(*x*) =

<sup>2</sup> (*x*) <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

<sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *<sup>h</sup>*(�<sup>=</sup> <sup>0</sup>) <sup>∈</sup> **<sup>R</sup>***n*. (30)

*<sup>z</sup>* (*x*)*PC<sup>T</sup>*

<sup>1</sup> ··· *<sup>x</sup>α<sup>n</sup>*

*<sup>i</sup>*=<sup>1</sup> *α<sup>i</sup>* and the degree of a polynomial *f*(*x*) ∈ **R**[*x*], denoted by

*<sup>f</sup>*(*x*) <sup>∈</sup> <sup>Σ</sup> <sup>⇒</sup> *<sup>f</sup>*(*x*) <sup>≥</sup> <sup>0</sup> (∀*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*), (32) *<sup>F</sup>*(*x*) <sup>∈</sup> <sup>Σ</sup>*r*×*<sup>r</sup>* <sup>⇒</sup> *<sup>F</sup>*(*x*) � <sup>0</sup> (∀*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*). (33)

*<sup>n</sup>* ··· *<sup>x</sup><sup>d</sup>*

<sup>1</sup> ··· *<sup>x</sup><sup>d</sup> n* *<sup>T</sup>* , (34)

*<sup>n</sup>* . The degree of a monomial *xα*, denoted by deg(*xα*),

<sup>11</sup>*C*11*P h* > 0,

*<sup>n</sup>* with *<sup>α</sup>* = (*α*1,..., *<sup>α</sup>n*) <sup>∈</sup> **<sup>Z</sup>***<sup>n</sup>*

*<sup>i</sup>* (*x*). (31)

+.

*<sup>h</sup><sup>T</sup>* [−(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)*<sup>P</sup>* <sup>−</sup> *<sup>P</sup>*(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)

<sup>1</sup> <sup>−</sup> <sup>1</sup> *a*2 *<sup>u</sup>*(*x*)

which satisfies (19) (instead of (30)) to reduce the conservativeness.

<sup>1</sup> ··· *<sup>x</sup>α<sup>n</sup>*

polynomials by <sup>Σ</sup>, and the set of *<sup>r</sup>* <sup>×</sup> *<sup>r</sup>* SOS matrices by <sup>Σ</sup>*r*×*r*.

*vd*(*x*) :=

− 1 *<sup>γ</sup>*<sup>2</sup> *<sup>B</sup>*1*B<sup>T</sup>*

**3. Sum of squares**

of *F*(*x*).

be represented as

**3.1 Definitions and basic properties** A monomial in *x* = [*x*<sup>1</sup> ··· *xn*]

*gi*(*x*) ∈ **R**[*x*] (*i* = 1, . . . , *p*) such that

From the definitions it is obvious that

This is also written as *x<sup>α</sup>* = *xα*<sup>1</sup>

is defined by deg(*xα*) := ∑*<sup>n</sup>*

where

$$M(\mathbf{x}, P) := -\left(A + B\_2(\mathbf{x})\mathbf{C}\_{12}\right)P - P(A + B\_2(\mathbf{x})\mathbf{C}\_{12})^T$$

$$-\left(\frac{1}{\gamma^2}B\_1B\_1^T - \frac{1}{a\_{\rm il}^2(\mathbf{x})}B\_2(\mathbf{x})B\_2^T(\mathbf{x})\right) - a\_z^2(\mathbf{x})P\mathbf{C}\_{11}^T\mathbf{C}\_{11}P.\tag{28}$$

From this we have the next corollary.

**Corollary 1.** *Suppose that for given γ* > 0 *there exists a positive definite symmetric matrix P which satisfies the following state-dependent inequality:*

$$M(\mathbf{x}, P) \succ 0, \quad \forall \mathbf{x} \in \mathbf{R}^n \tag{29}$$

*Then by the feedback* (20) *the closed-loop system is asymptotically stable and the L*2*-induced norm* (14) *is less than or equal to γ.*

Proof: It is obvious from (27) that (19) is satisfied if (29) holds. Hence we obtain this corollary from Theorem 2. Q.E.D. The inequality (29) is called "a state-dependent matrix Riccati inequality" and equivalent to

$$\begin{aligned} &h^T \left[ -(A + B\_2(\mathbf{x}) \mathbf{C}\_{12})P - P(A + B\_2(\mathbf{x}) \mathbf{C}\_{12}) \right. \\ &\left. - \left( \frac{1}{\gamma^2} B\_1 B\_1^T - \frac{1}{a\_{\mathcal{U}}^2(\mathbf{x})} B\_2(\mathbf{x}) B\_2^T(\mathbf{x}) \right) - a\_z^2(\mathbf{x}) P \mathbf{C}\_{11}^T \mathbf{C}\_{11} P \right] h > 0, \\ &\forall \mathbf{x} \in \mathbf{R}^n, h(\neq 0) \in \mathbf{R}^n. \end{aligned} \tag{30}$$

Note that *h* is independent of *x* in (30), whereas there is a relationship of *h* = *P*−1*x* between *h* and *x* in (19). This means that the condition (30) can be very conservative compared with the condition (19). As mentioned in Introduction, most papers have tried to find *P* which satisfies a matrix state-dependent inequality like (29) (or (30)). In contrary to this, we will try to find *P* which satisfies (19) (instead of (30)) to reduce the conservativeness.

### **3. Sum of squares**

8 Will-be-set-by-IN-TECH

<sup>1</sup> *<sup>P</sup>*−1*x*,

<sup>2</sup> (*x*)*P*−<sup>1</sup> + *<sup>C</sup>*<sup>12</sup>

<sup>2</sup> <sup>≥</sup> <sup>2</sup>*xTP*−1*x*˙ <sup>+</sup> *<sup>γ</sup>*2*w*˜ *Tw*˜,

(2*xTP*−1*x*˙ + *γ*2*w*˜ *Tw*˜)*dt*

<sup>=</sup>*xT*(*τ*)*P*−1*x*(*τ*) <sup>−</sup> *<sup>x</sup>T*(0)*P*−1*x*(0) +

<sup>2</sup> − |*z*<sup>|</sup>

which leads to (21). Note that to derive (26) we use lim*τ*→<sup>∞</sup> *x*(*τ*) = 0 which holds because the closed-loop is asymptotically stable. Q.E.D.

<sup>2</sup>)*dt* =

*B*2(*x*)*B<sup>T</sup>*

**Corollary 1.** *Suppose that for given γ* > 0 *there exists a positive definite symmetric matrix P which*

*Then by the feedback* (20) *the closed-loop system is asymptotically stable and the L*2*-induced norm* (14)

Proof: It is obvious from (27) that (19) is satisfied if (29) holds. Hence we obtain this corollary from Theorem 2. Q.E.D.

<sup>2</sup> (*x*) <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

 ∞ 0

*φ*(*x*, *P*)=(*P*−1*x*)*TM*(*x*, *P*)(*P*−1*x*) (27)

 *x*.

> *τ* 0

*<sup>z</sup>* (*x*)*PC<sup>T</sup>*

*<sup>M</sup>*(*x*, *<sup>P</sup>*) � 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* (29)

(*γ*2*w*˜ *Tw*˜)*dt*.

(*γ*2*w*˜ *Tw*˜)*dt* <sup>≥</sup> 0, (26)

<sup>11</sup>*C*11*P*. (28)

*<sup>w</sup>*˜ <sup>=</sup> *<sup>w</sup>* <sup>−</sup> <sup>1</sup>

*u*˜ = *u* +

<sup>2</sup> − |*z*<sup>|</sup>

 *τ* 0

 ∞ 0

(*γ*2|*w*<sup>|</sup>

*<sup>M</sup>*(*x*, *<sup>P</sup>*) :<sup>=</sup> <sup>−</sup> (*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)*<sup>P</sup>* <sup>−</sup> *<sup>P</sup>*(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)*<sup>T</sup>*

<sup>1</sup> <sup>−</sup> <sup>1</sup> *a*2 *<sup>u</sup>*(*x*)

*<sup>γ</sup>*2|*w*<sup>|</sup>

<sup>2</sup>)*dt* <sup>≥</sup>

*<sup>γ</sup>*<sup>2</sup> *<sup>B</sup><sup>T</sup>*

 1 *a*2 *u BT*

where

and hence

where

Then from (19) and (20)

 *τ* 0

(*γ*2|*w*<sup>|</sup>

Here, let *x*(0) = 0 and *τ* → ∞ then

<sup>2</sup> − |*z*<sup>|</sup>

*<sup>γ</sup>*2�*w*�<sup>2</sup> − �*z*�<sup>2</sup> <sup>=</sup>

Note that *φ*(*x*, *p*) in (19) can be represented as

− 1 *<sup>γ</sup>*<sup>2</sup> *<sup>B</sup>*1*B<sup>T</sup>*

From this we have the next corollary.

*is less than or equal to γ.*

*satisfies the following state-dependent inequality:*

In this section, we briefly survey the so-called "SOS (Sum of Squares) optimization."

#### **3.1 Definitions and basic properties**

A monomial in *x* = [*x*<sup>1</sup> ··· *xn*] *<sup>T</sup>* is represented as *xα*<sup>1</sup> <sup>1</sup> ··· *<sup>x</sup>α<sup>n</sup> <sup>n</sup>* with *<sup>α</sup>* = (*α*1,..., *<sup>α</sup>n*) <sup>∈</sup> **<sup>Z</sup>***<sup>n</sup>* +. This is also written as *x<sup>α</sup>* = *xα*<sup>1</sup> <sup>1</sup> ··· *<sup>x</sup>α<sup>n</sup> <sup>n</sup>* . The degree of a monomial *xα*, denoted by deg(*xα*), is defined by deg(*xα*) := ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *α<sup>i</sup>* and the degree of a polynomial *f*(*x*) ∈ **R**[*x*], denoted by deg(*f*), is defined by the degree of the monomial which has the highest degree among all the monomials included in *<sup>f</sup>*(*x*). For a polynomial matrix *<sup>F</sup>*(*x*) <sup>∈</sup> **<sup>R</sup>***q*×*<sup>r</sup>* the degree of *<sup>F</sup>*(*x*), denoted by deg(*F*), is defined by deg(*F*) := *maxij* deg(*Fij*) where *Fij* denotes the (*i*, *j*) element of *F*(*x*).

A real polynomial *f*(*x*) ∈ **R**[*x*] is said to be an *SOS (Sum of Squares) polynomial* if it can be represented as a sum of squares of some polynomials, i.e., there exist some polynomials *gi*(*x*) ∈ **R**[*x*] (*i* = 1, . . . , *p*) such that

$$f(\mathbf{x}) = \sum\_{i=1}^{p} g\_i^2(\mathbf{x}). \tag{31}$$

Moreover, a polynomial symmetric matrix *<sup>F</sup>*(*x*) <sup>∈</sup> **<sup>R</sup>***r*×*r*[*x*] is said to be an *SOS matrix* if it can be represented as

$$F(\mathbf{x}) = L^T(\mathbf{x}) L(\mathbf{x})$$

for some polynomial matrix *L*(*x*) of appropriate size. In this paper, we denote the set of SOS polynomials by <sup>Σ</sup>, and the set of *<sup>r</sup>* <sup>×</sup> *<sup>r</sup>* SOS matrices by <sup>Σ</sup>*r*×*r*. From the definitions it is obvious that

$$f(\mathbf{x}) \in \Sigma \quad \Rightarrow \ f(\mathbf{x}) \ge 0 \quad (\forall \mathbf{x} \in \mathbf{R}^n), \tag{32}$$

$$F(\mathbf{x}) \in \Sigma^{r \times r} \Rightarrow F(\mathbf{x}) \succeq \mathbf{0} \quad (\forall \mathbf{x} \in \mathbf{R}^n). \tag{33}$$

Here, for a positive integer *d* let *vd*(*x*) be a polynomial vector in *x* of size *<sup>n</sup>*+*d*C*<sup>d</sup>* defined by

$$v\_d(\mathbf{x}) := \begin{bmatrix} 1 \ \mathbf{x}\_1 \ \cdots \ \mathbf{x}\_n \ \mathbf{x}\_1^2 \ \mathbf{x}\_1 \mathbf{x}\_2 \ \cdots \ \mathbf{x}\_n^2 \ \cdots \ \mathbf{x}\_1^d \ \cdots \ \mathbf{x}\_n^d \end{bmatrix}^T,\tag{34}$$

which contains all monomials whose degrees are less than or equal to *d* where *<sup>α</sup>*C*<sup>β</sup>* = *α*!/(*β*!(*α* − *β*)!) for positive integers *α* and *β* with *α* ≥ *β*. Then the next lemmas are known.

**Lemma 1.** *(Parrilo (2003)) Let* deg(*f*) = 2*d where f*(*x*) ∈ **R**[*x*]*. Then the following (i) and (ii) are equivalent:*


**Lemma 2.** *(Scherer & Hol (2006)) Let* deg(*F*) = 2*d where F*(*x*) ∈ **R**[*x*] *<sup>r</sup>*×*r. Then the following (i) and (ii) are equivalent:*


Using these lemmas, the problem of determining whether a polynomial *f*(*x*) (or a polynomial matrix *F*(*x*)) is an SOS polynomial (or an SOS matrix) or not is reduced to an SDP (Semi-Definite Programming) problem, which can be solved numerically, of checking the positive semi-definiteness of the corresponding matrix *Q*.

#### **3.2 SOS polynomials with equality constraints**

Let us consider the following equality constraints:

$$f\_j(\mathbf{x}) = 0, \quad j = 1, \ldots, p \tag{35}$$

holds, the condition (37) is satisfied. In fact, (39) implies

summarized in the following lemma.

*(i) There exists λ*(*x*) ∈ **R**[*x*]

*(ii) There exists λ*(*x*) ∈ **R**[*x*]

**4. Proposed method**

Then it is easy to see

*f*0(*x*) ≥

**Lemma 3.** *If the following (i) or (ii) holds, the condition* (37) *holds.*

we propose an SOS optimization method to find such *P*. To this end, let us introduce sufficiently small *�* > 0 and define

From (42), the condition (45) can be written as

and moreover this can be written as

where

as follows:

*p* ∑ *j*=1

*<sup>p</sup> such that L*(*x*, *<sup>λ</sup>*) <sup>∈</sup> <sup>Σ</sup>*.*

Hence, for obtaining the feedback (20) it suffices to find *P* � 0 such that

Moreover, if *L*(*x*, *λ*) ∈ Σ then (39) holds from Lemma 1 and hence (37) holds. These facts are

A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 49

Theorem 2 implies that the state feedback (20) will stabilize the closed-loop system and (25) is satisfied if we can obtain a positive definite symmetric matrix *P* satisfying (19). In this section,

*<sup>p</sup> such that L*(*x*, *<sup>λ</sup>*) <sup>≥</sup> <sup>0</sup> (∀*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*)*.*

*λj*(*x*)*fj*(*x*) = 0, ∀*x* ∈ *S*. (40)

*<sup>M</sup>*˜ (*x*, *<sup>P</sup>*) :<sup>=</sup> *<sup>M</sup>*(*x*, *<sup>P</sup>*) <sup>−</sup> *�I*, (41)

*φ*˜(*x*, *P*) := (*P*−1*x*)*TM*˜ (*x*, *P*)(*P*−1*x*). (42)

*<sup>φ</sup>*˜(*x*, *<sup>P</sup>*) <sup>≥</sup> 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*. (45)

*<sup>h</sup>TM*˜ (*x*, *<sup>P</sup>*)*<sup>h</sup>* <sup>≥</sup> 0, <sup>∀</sup>(*x*, *<sup>h</sup>*) <sup>∈</sup> *<sup>S</sup>*˜ (47)

*<sup>S</sup>*˜ :<sup>=</sup> {(*x*, *<sup>h</sup>*) <sup>∈</sup> **<sup>R</sup>**2*<sup>n</sup>* <sup>|</sup> *<sup>x</sup>* <sup>−</sup> *Ph* <sup>=</sup> <sup>0</sup>}. (48)

*<sup>M</sup>*˜ (*x*, *<sup>P</sup>*) � 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* <sup>⇒</sup> *<sup>M</sup>*(*x*, *<sup>P</sup>*) � 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*, (43)

*<sup>φ</sup>*˜(*x*, *<sup>P</sup>*) <sup>≥</sup> 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* <sup>⇒</sup> *<sup>φ</sup>*(*x*, *<sup>P</sup>*) <sup>&</sup>gt; 0, <sup>∀</sup>*x*(�<sup>=</sup> <sup>0</sup>) <sup>∈</sup> **<sup>R</sup>***n*. (44)

*<sup>h</sup>TM*˜ (*x*, *<sup>P</sup>*)*<sup>h</sup>* <sup>≥</sup> 0, <sup>∀</sup>(*x*, *<sup>h</sup>*) <sup>∈</sup> **<sup>R</sup>**2*<sup>n</sup>* such that *<sup>h</sup>* <sup>=</sup> *<sup>P</sup>*−1*<sup>x</sup>* (46)

By this, the condition (45) is represented as the condition (47) including the equality constraint *x* − *Ph* = 0. For the condition (47) we define a generalized Lagrange function as in Section 3.2

where *fj*(*x*) ∈ **R**[*x*] and their feasible set is defined by

$$S := \{ \mathbf{x} \in \mathbf{R}^n \mid f\_j(\mathbf{x}) = 0, \ j = 1, \dots, p \}. \tag{36}$$

Here we consider the problem of determining whether a given polynomial *f*0(*x*) ∈ **R** is non-negative or not for all *x* ∈ **R** in the feasible set, i.e., the following condition holds or not:

$$f\_0(\mathbf{x}) \ge \mathbf{0}, \quad \forall \mathbf{x} \in \mathbf{S}. \tag{37}$$

For this problem, define a generalized Lagrange function *L*(*x*, *λ*) by

$$L(\mathbf{x}, \lambda) := f\_0(\mathbf{x}) - \sum\_{j=1}^{p} \lambda\_j(\mathbf{x}) f\_j(\mathbf{x}) \tag{38}$$

where *λj*(*x*) ∈ **R**[*x*] and let

$$\lambda(\mathfrak{x}) := \left[ \lambda\_1(\mathfrak{x}) \cdot \dots \cdot \lambda\_p(\mathfrak{x}) \right]^T \in \mathbf{R}[\mathfrak{x}]^p.$$

Then if for given *λ*[*x*] ∈ **R**[*x*] *p*

$$L(\mathbf{x}, \lambda) \ge 0, \quad \forall \mathbf{x} \in \mathbf{R}^n \tag{39}$$

holds, the condition (37) is satisfied. In fact, (39) implies

$$f\_0(\mathbf{x}) \ge \sum\_{j=1}^p \lambda\_j(\mathbf{x}) f\_j(\mathbf{x}) = \mathbf{0}, \quad \forall \mathbf{x} \in \mathcal{S}. \tag{40}$$

Moreover, if *L*(*x*, *λ*) ∈ Σ then (39) holds from Lemma 1 and hence (37) holds. These facts are summarized in the following lemma.

**Lemma 3.** *If the following (i) or (ii) holds, the condition* (37) *holds.*

*(i) There exists λ*(*x*) ∈ **R**[*x*] *<sup>p</sup> such that L*(*x*, *<sup>λ</sup>*) <sup>≥</sup> <sup>0</sup> (∀*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*)*. (ii) There exists λ*(*x*) ∈ **R**[*x*] *<sup>p</sup> such that L*(*x*, *<sup>λ</sup>*) <sup>∈</sup> <sup>Σ</sup>*.*

### **4. Proposed method**

10 Will-be-set-by-IN-TECH

which contains all monomials whose degrees are less than or equal to *d* where *<sup>α</sup>*C*<sup>β</sup>* = *α*!/(*β*!(*α* − *β*)!) for positive integers *α* and *β* with *α* ≥ *β*. Then the next lemmas are known. **Lemma 1.** *(Parrilo (2003)) Let* deg(*f*) = 2*d where f*(*x*) ∈ **R**[*x*]*. Then the following (i) and (ii) are*

*(ii) There exists a positive semi-definite symmetric matrix of appropriate size such that f*(*x*) =

*(ii) There exists a positive semi-definite symmetric matrix of appropriate size Q such that F*(*x*) =

Using these lemmas, the problem of determining whether a polynomial *f*(*x*) (or a polynomial matrix *F*(*x*)) is an SOS polynomial (or an SOS matrix) or not is reduced to an SDP (Semi-Definite Programming) problem, which can be solved numerically, of checking the

Here we consider the problem of determining whether a given polynomial *f*0(*x*) ∈ **R** is non-negative or not for all *x* ∈ **R** in the feasible set, i.e., the following condition holds or

*λ*1(*x*) ··· *λp*(*x*)

*p* ∑ *j*=1

*T*

∈ **R**[*x*] *p*.

*<sup>L</sup>*(*x*, *<sup>λ</sup>*) <sup>≥</sup> 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* (39)

*fj*(*x*) = 0, *j* = 1, . . . , *p* (35)

*f*0(*x*) ≥ 0, ∀*x* ∈ *S*. (37)

*λj*(*x*)*fj*(*x*) (38)

*<sup>S</sup>* :<sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* <sup>|</sup> *fj*(*x*) = 0, *<sup>j</sup>* <sup>=</sup> 1, . . . , *<sup>p</sup>*}. (36)

*<sup>r</sup>*×*r. Then the following (i)*

**Lemma 2.** *(Scherer & Hol (2006)) Let* deg(*F*) = 2*d where F*(*x*) ∈ **R**[*x*]

positive semi-definiteness of the corresponding matrix *Q*.

where *fj*(*x*) ∈ **R**[*x*] and their feasible set is defined by

For this problem, define a generalized Lagrange function *L*(*x*, *λ*) by

*λ*(*x*) :=

*p*

*L*(*x*, *λ*) := *f*0(*x*) −

**3.2 SOS polynomials with equality constraints** Let us consider the following equality constraints:

*equivalent: (i) f*(*x*) ∈ Σ.

*vT*

not:

where *λj*(*x*) ∈ **R**[*x*] and let

Then if for given *λ*[*x*] ∈ **R**[*x*]

*<sup>d</sup>* (*x*)*Qvd*(*x*)*.*

*and (ii) are equivalent: (i) F*(*x*) <sup>∈</sup> <sup>Σ</sup>*r*×*r.*

(*vd*(*x*) <sup>⊗</sup> *Ir*)*TQ*(*vd*(*x*) <sup>⊗</sup> *Ir*)*.*

Theorem 2 implies that the state feedback (20) will stabilize the closed-loop system and (25) is satisfied if we can obtain a positive definite symmetric matrix *P* satisfying (19). In this section, we propose an SOS optimization method to find such *P*.

To this end, let us introduce sufficiently small *�* > 0 and define

$$
\tilde{M}(\mathbf{x}, P) := M(\mathbf{x}, P) - \varepsilon I,\tag{41}
$$

$$\tilde{\Phi}(\mathbf{x}, P) := (P^{-1}\mathbf{x})^T \tilde{M}(\mathbf{x}, P)(P^{-1}\mathbf{x}). \tag{42}$$

Then it is easy to see

$$\tilde{M}(\mathbf{x}, P) \succeq \mathbf{0}, \forall \mathbf{x} \in \mathbf{R}^n \quad \Rightarrow \quad M(\mathbf{x}, P) \succ \mathbf{0}, \forall \mathbf{x} \in \mathbf{R}^n,\tag{43}$$

$$
\tilde{\phi}(\mathbf{x}, P) \ge 0, \,\forall \mathbf{x} \in \mathbf{R}^n \quad \Rightarrow \quad \phi(\mathbf{x}, P) > 0, \,\forall \mathbf{x} (\neq 0) \in \mathbf{R}^n. \tag{44}
$$

Hence, for obtaining the feedback (20) it suffices to find *P* � 0 such that

$$
\tilde{\Phi}(\mathbf{x}, P) \ge \mathbf{0}, \,\forall \mathbf{x} \in \mathbf{R}^n. \tag{45}
$$

From (42), the condition (45) can be written as

$$h^T \tilde{M}(\mathbf{x}, P)h \ge 0, \quad \forall (\mathbf{x}, h) \in \mathbf{R}^{2n} \quad \text{such that} \quad h = P^{-1}\mathbf{x} \tag{46}$$

and moreover this can be written as

$$h^T \tilde{\mathcal{M}}(\mathbf{x}, P)h \ge 0, \quad \forall (\mathbf{x}, h) \in \tilde{\mathcal{S}} \tag{47}$$

where

$$\tilde{S} := \{ (\mathbf{x}, \hbar) \in \mathbf{R}^{2n} \mid \mathbf{x} - \hbar \hbar = 0 \}. \tag{48}$$

By this, the condition (45) is represented as the condition (47) including the equality constraint *x* − *Ph* = 0. For the condition (47) we define a generalized Lagrange function as in Section 3.2 as follows:

$$L(\mathbf{x}, h, \lambda; P) := h^T \mathbf{M}(\mathbf{x}, P)h - \lambda^T(\mathbf{x}, h)(\mathbf{x} - Ph) \tag{49}$$

where *λ*(*x*, *h*) ∈ **R**[*x*, *h*] *<sup>n</sup>*. Then, from Lemma 3, (47) is satisfied if there exit *<sup>λ</sup>* and *<sup>P</sup>*(� <sup>0</sup>) which satisfies

$$L(\mathbf{x}, h, \lambda; P) \ge 0, \quad \forall (\mathbf{x}, h) \in \mathbf{R}^{2n} \tag{50}$$

Since *G* is afffine in *P*, it can be written as

(51) is satisfied if *<sup>G</sup>* <sup>∈</sup> <sup>Σ</sup>(*nz*+1)×(*nz*+1)

Now, let us summarize our method as an algorithm.

Lemma 2.

minimize *γ*).

*SDP problem*

*and let Hk* := *H*∗*.*

*the next SDP problem*

**5. Numerical example**

al. (1999).

where

*and let γ*¯ *<sup>k</sup>*+<sup>1</sup> := *γ*¯ <sup>∗</sup> *and Pk*<sup>+</sup><sup>1</sup> = *P*∗*.*

*exit; otherwise, let k* := *k* + 1 *and go to Step 1.*

The motion equation of the suspension system is given by

*xb* is the displacement of the car body, *xw* is the displacement of the car wheel,

*xr* is the displacement of the road,

*<sup>G</sup>*(*x*, *<sup>h</sup>*; *<sup>H</sup>*, *<sup>P</sup>*)=(*vd*<sup>2</sup> (*x*, *<sup>h</sup>*) <sup>⊗</sup> *Inz*<sup>+</sup>1)*TQ*2(*P*)(*vd*<sup>2</sup> (*x*, *<sup>h</sup>*) <sup>⊗</sup> *Inz*<sup>+</sup>1) (56)

(*x*,*h*) which is equivalent to the existence of *Q*2(*H*) � 0 by

*<sup>γ</sup>*¯,*<sup>H</sup> <sup>γ</sup>*¯ *s.t. Q*1(*H*) � 0, (57)

*γ*¯ *s.t. Q*2(*P*) � 0, (58)

where *d*<sup>2</sup> = deg(*G*), the degree of *G* as a polynomial matrix in *x* and *h*, *vd*<sup>2</sup> (*x*, *h*) is a vector of size <sup>2</sup>*n*+*d*2*Cd*<sup>2</sup> which contains all monomials in *x* and *h* whose degrees are less than or equal to *d*2, and *Q*2(*P*) is a real symmetric matrix of appropriate size. Hence, in the case of fixed *H*, the problem to be concerned is reduced to an SDP problem to find *P* such that *Q*2(*P*) � 0, because

A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 51

Note that *Lm* and *G* are also affine in *γ*¯ and hence we can consider to maximize *γ*¯ = 1/*γ*<sup>2</sup> (i.e.,

**Step 1** *Let P* = *Pk and get the optimal value γ*¯ <sup>∗</sup> *and its optimizer H*<sup>∗</sup> *by solving numerically the next*

**Step 2** *Let H* = *Hk and get the optimal value γ*¯ <sup>∗</sup> *and its optimizer P*<sup>∗</sup> � 0 *by solving numerically*

**Step 3** *If* |*γ*¯ *<sup>k</sup>*+<sup>1</sup> − *γ*¯ *<sup>k</sup>*| *is sufficiently small (i.e., γ*¯ *<sup>k</sup> is convergent), then return Pk*<sup>+</sup><sup>1</sup> *as a solution and*

Note that a feasible solution to *<sup>M</sup>*˜ (0, *<sup>P</sup>*) � 0 for large *<sup>γ</sup>* can be used as an initial value *<sup>P</sup>*0, because *<sup>M</sup>*˜ (0, *<sup>P</sup>*) � 0 is a usual Riccati inequality and has a feasible solution for large *<sup>γ</sup>*.

In this section, we give a numerical example. The bilinear system to be concerned is the semi-active suspension system for automobiles introduced by Ohsaku et al. (1998); Sampei et

> *Mbx*¨*<sup>b</sup>* = *Cs*(*x*˙*<sup>w</sup>* − *x*˙*b*) + *Cv*(*x*˙*<sup>w</sup>* − *x*˙*b*) + *Ks*(*xw* − *xb*) (59) *Mwx*¨*<sup>w</sup>* = −*Cs*(*x*˙*<sup>w</sup>* − *x*˙*b*) − *Cv*(*x*˙*<sup>w</sup>* − *x*˙*b*) − *Ks*(*xw* − *xb*) + *Kt*(*xr* − *xw*) (60)

**Algorithm 1. Step. 0** *Choose an initial value P*<sup>0</sup> � 0 *and small �. Let k* := 0 *and γ*¯0 = 0*.*

max

max *γ*¯,*P*�0

Here, suppose the degree of *λ* is given, say *m*, then *λ* can be written as

$$
\lambda(\mathbf{x}, h) = H \upsilon\_m(\mathbf{x}, h)
$$

where *vm*(*x*, *h*) is a vector of size <sup>2</sup>*n*+*mCm* which contains all monomials in *x* and *h* whose degrees are less than or equal to *m*, and *H* is an *n* × (2*n*+*mCm*) real matrix. From this, (50) is reduced to

$$L\_{\mathfrak{m}}(\mathbf{x}, h; H, P) := h^T \tilde{M}(\mathbf{x}, P)h - v\_{\mathfrak{m}}^T(\mathbf{x}, h)H^T(\mathbf{x} - Ph) \ge 0, \quad \forall (\mathbf{x}, h) \in \mathbf{R}^{2n}, \tag{51}$$

and the problem to be concerned becomes the one for finding matrices *P* and *H* which satisfies (51).

Note that *Lm*(*x*, *h*; *H*, *P*) includes the product of *H* and *P* in the last term. Hence, we consider an iterative algorithm which repeats a step of finding *H* for fixed *P* and a step of finding *P* for fixed *H*.

First, suppose *P* is fixed. In this case, *Lm*(*x*, *h*; *H*, *P*) can be written as

$$L\_m(\mathbf{x}, h; H, P) = v\_{d\_1}^T(\mathbf{x}, h) Q\_1(H) v\_{d\_1}(\mathbf{x}, h) \tag{52}$$

where *d*<sup>1</sup> = deg(*Lm*), the degree of *Lm* as a polynomial in *x* and *h*, *vd*<sup>1</sup> (*x*, *h*) is a vector of size <sup>2</sup>*n*+*d*1*Cd*<sup>1</sup> which contains all monomials in *x* and *h* whose degrees are less than or equal to *d*1, and *Q*1(*H*) is a (2*n*+*d*1*Cd*<sup>1</sup> ) × (2*n*+*d*1*Cd*<sup>1</sup> ) symmetric matrix. Then *Q*1(*H*) is affine in *H* because so is *Lm*. Hence, in the case of fixed *P*, the problem to be concerned is reduced to an SDP problem to find *H* such that *Q*1(*H*) � 0, because (51) is satisfied if *Lm* ∈ Σ(*x*,*h*) which is equivalent to the existence of *Q*1(*H*) � 0 by Lemma 1.

Next, suppose *H* is fixed. In this case, *Lm* is not affine in *P*, but by Schur complement (51) is equivalent to

$$G(\mathbf{x}, h; H, P) := \begin{bmatrix} \mathbf{G}\_{11}(\mathbf{x}, h; H, P) \ a\_z^2(\mathbf{x}) h^T P \mathbf{C}\_{11}^T \\ a\_z^2(\mathbf{x}) \mathbf{C}\_{11} P h & a\_z^2(\mathbf{x}) I\_{h\_z} \end{bmatrix} \succeq \mathbf{0}, \quad \forall (\mathbf{x}, h) \in \mathbf{R}^{2n}, \tag{53}$$

which is affine in *P*, where

$$\begin{aligned} \mathbb{G}\_{11}(\mathbf{x}, h; H, P) &:= h^{T} \left[ -(A + B\_{2}(\mathbf{x}) \mathbb{C}\_{12})P - P(A + B\_{2}(\mathbf{x}) \mathbb{C}\_{12})^{T} - \gamma B\_{1} B\_{1}^{T} \\ &+ \bar{a}\_{\mu}^{2}(\mathbf{x}) B\_{2}(\mathbf{x}) B\_{2}^{T}(\mathbf{x}) \right] h - v\_{m}^{T}(\mathbf{x}, h) H^{T}(\mathbf{x} - Ph) \end{aligned} \tag{54}$$

and

$$\bar{\gamma} := \frac{1}{\gamma^2}, \; \bar{a}\_{\iota}(\mathbf{x}) := \frac{1}{a\_{\iota}(\mathbf{x})}.\tag{55}$$

Since *G* is afffine in *P*, it can be written as

12 Will-be-set-by-IN-TECH

*λ*(*x*, *h*) = *Hvm*(*x*, *h*)

where *vm*(*x*, *h*) is a vector of size <sup>2</sup>*n*+*mCm* which contains all monomials in *x* and *h* whose degrees are less than or equal to *m*, and *H* is an *n* × (2*n*+*mCm*) real matrix. From this, (50) is

and the problem to be concerned becomes the one for finding matrices *P* and *H* which satisfies

Note that *Lm*(*x*, *h*; *H*, *P*) includes the product of *H* and *P* in the last term. Hence, we consider an iterative algorithm which repeats a step of finding *H* for fixed *P* and a step of finding *P* for

*d*1

where *d*<sup>1</sup> = deg(*Lm*), the degree of *Lm* as a polynomial in *x* and *h*, *vd*<sup>1</sup> (*x*, *h*) is a vector of size <sup>2</sup>*n*+*d*1*Cd*<sup>1</sup> which contains all monomials in *x* and *h* whose degrees are less than or equal to *d*1, and *Q*1(*H*) is a (2*n*+*d*1*Cd*<sup>1</sup> ) × (2*n*+*d*1*Cd*<sup>1</sup> ) symmetric matrix. Then *Q*1(*H*) is affine in *H* because so is *Lm*. Hence, in the case of fixed *P*, the problem to be concerned is reduced to an SDP problem to find *H* such that *Q*1(*H*) � 0, because (51) is satisfied if *Lm* ∈ Σ(*x*,*h*) which is

Next, suppose *H* is fixed. In this case, *Lm* is not affine in *P*, but by Schur complement (51) is

<sup>2</sup> (*x*) *<sup>h</sup>* <sup>−</sup> *<sup>v</sup><sup>T</sup>*

*<sup>γ</sup>*<sup>2</sup> , *<sup>a</sup>*¯*u*(*x*) :<sup>=</sup> <sup>1</sup>

*<sup>z</sup>* (*x*)*hTPCT*

*<sup>z</sup>* (*x*)*Inz*

11

<sup>−</sup>(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)*<sup>P</sup>* <sup>−</sup> *<sup>P</sup>*(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2(*x*)*C*12)*<sup>T</sup>* <sup>−</sup> *<sup>γ</sup>*¯*B*1*B<sup>T</sup>*

*au*(*x*)

Here, suppose the degree of *λ* is given, say *m*, then *λ* can be written as

First, suppose *P* is fixed. In this case, *Lm*(*x*, *h*; *H*, *P*) can be written as

*Lm*(*x*, *h*; *H*, *P*) = *v<sup>T</sup>*

*G*11(*x*, *h*; *H*, *P*) *a*<sup>2</sup>

*u*(*x*)*B*2(*x*)*B<sup>T</sup>*

*<sup>γ</sup>*¯ :<sup>=</sup> <sup>1</sup>

*<sup>z</sup>* (*x*)*C*11*Ph a*<sup>2</sup>

*Lm*(*x*, *<sup>h</sup>*; *<sup>H</sup>*, *<sup>P</sup>*) :<sup>=</sup> *<sup>h</sup>TM*˜ (*x*, *<sup>P</sup>*)*<sup>h</sup>* <sup>−</sup> *<sup>v</sup><sup>T</sup>*

equivalent to the existence of *Q*1(*H*) � 0 by Lemma 1.

*a*2

+ *a*¯ 2

*G*(*x*, *h*; *H*, *P*) :=

*<sup>G</sup>*11(*x*, *<sup>h</sup>*; *<sup>H</sup>*, *<sup>P</sup>*) :=*h<sup>T</sup>*

which is affine in *P*, where

where *λ*(*x*, *h*) ∈ **R**[*x*, *h*]

which satisfies

reduced to

(51).

fixed *H*.

equivalent to

and

*<sup>L</sup>*(*x*, *<sup>h</sup>*, *<sup>λ</sup>*; *<sup>P</sup>*) :<sup>=</sup> *<sup>h</sup>TM*˜ (*x*, *<sup>P</sup>*)*<sup>h</sup>* <sup>−</sup> *<sup>λ</sup>T*(*x*, *<sup>h</sup>*)(*<sup>x</sup>* <sup>−</sup> *Ph*) (49)

*<sup>n</sup>*. Then, from Lemma 3, (47) is satisfied if there exit *<sup>λ</sup>* and *<sup>P</sup>*(� <sup>0</sup>)

*<sup>L</sup>*(*x*, *<sup>h</sup>*, *<sup>λ</sup>*; *<sup>P</sup>*) <sup>≥</sup> 0, <sup>∀</sup>(*x*, *<sup>h</sup>*) <sup>∈</sup> **<sup>R</sup>**2*<sup>n</sup>* (50)

*<sup>m</sup>*(*x*, *<sup>h</sup>*)*HT*(*<sup>x</sup>* <sup>−</sup> *Ph*) <sup>≥</sup> 0, <sup>∀</sup>(*x*, *<sup>h</sup>*) <sup>∈</sup> **<sup>R</sup>**2*n*, (51)

(*x*, *h*)*Q*1(*H*)*vd*<sup>1</sup> (*x*, *h*) (52)

� 0, <sup>∀</sup>(*x*, *<sup>h</sup>*) <sup>∈</sup> **<sup>R</sup>**2*n*, (53)

*<sup>m</sup>*(*x*, *<sup>h</sup>*)*HT*(*<sup>x</sup>* <sup>−</sup> *Ph*) (54)

. (55)

1

$$\mathbf{G}(\mathbf{x}, h; H, P) = (\upsilon\_{d\_2}(\mathbf{x}, h) \otimes I\_{n\_2 + 1})^T \mathbf{Q}\_2(P) (\upsilon\_{d\_2}(\mathbf{x}, h) \otimes I\_{n\_2 + 1}) \tag{56}$$

where *d*<sup>2</sup> = deg(*G*), the degree of *G* as a polynomial matrix in *x* and *h*, *vd*<sup>2</sup> (*x*, *h*) is a vector of size <sup>2</sup>*n*+*d*2*Cd*<sup>2</sup> which contains all monomials in *x* and *h* whose degrees are less than or equal to *d*2, and *Q*2(*P*) is a real symmetric matrix of appropriate size. Hence, in the case of fixed *H*, the problem to be concerned is reduced to an SDP problem to find *P* such that *Q*2(*P*) � 0, because (51) is satisfied if *<sup>G</sup>* <sup>∈</sup> <sup>Σ</sup>(*nz*+1)×(*nz*+1) (*x*,*h*) which is equivalent to the existence of *Q*2(*H*) � 0 by Lemma 2.

Note that *Lm* and *G* are also affine in *γ*¯ and hence we can consider to maximize *γ*¯ = 1/*γ*<sup>2</sup> (i.e., minimize *γ*).

Now, let us summarize our method as an algorithm.

**Algorithm 1. Step. 0** *Choose an initial value P*<sup>0</sup> � 0 *and small �. Let k* := 0 *and γ*¯0 = 0*.*

**Step 1** *Let P* = *Pk and get the optimal value γ*¯ <sup>∗</sup> *and its optimizer H*<sup>∗</sup> *by solving numerically the next SDP problem*

$$\max\_{\tilde{\gamma},H} \tilde{\gamma} \quad \text{s.t.} \quad Q\_1(H) \succeq 0,\tag{57}$$

*and let Hk* := *H*∗*.*

**Step 2** *Let H* = *Hk and get the optimal value γ*¯ <sup>∗</sup> *and its optimizer P*<sup>∗</sup> � 0 *by solving numerically the next SDP problem*

$$\max\_{\tau\_{\gamma}P \succ 0} \bar{\gamma} \quad \text{s.t.} \quad Q\_2(P) \succeq 0,\tag{58}$$

*and let γ*¯ *<sup>k</sup>*+<sup>1</sup> := *γ*¯ <sup>∗</sup> *and Pk*<sup>+</sup><sup>1</sup> = *P*∗*.*

**Step 3** *If* |*γ*¯ *<sup>k</sup>*+<sup>1</sup> − *γ*¯ *<sup>k</sup>*| *is sufficiently small (i.e., γ*¯ *<sup>k</sup> is convergent), then return Pk*<sup>+</sup><sup>1</sup> *as a solution and exit; otherwise, let k* := *k* + 1 *and go to Step 1.*

Note that a feasible solution to *<sup>M</sup>*˜ (0, *<sup>P</sup>*) � 0 for large *<sup>γ</sup>* can be used as an initial value *<sup>P</sup>*0, because *<sup>M</sup>*˜ (0, *<sup>P</sup>*) � 0 is a usual Riccati inequality and has a feasible solution for large *<sup>γ</sup>*.

### **5. Numerical example**

In this section, we give a numerical example. The bilinear system to be concerned is the semi-active suspension system for automobiles introduced by Ohsaku et al. (1998); Sampei et al. (1999).

The motion equation of the suspension system is given by

$$M\_b \ddot{\mathbf{x}}\_b = \mathbb{C}\_s(\dot{\mathbf{x}}\_w - \dot{\mathbf{x}}\_b) + \mathbb{C}\_v(\dot{\mathbf{x}}\_w - \dot{\mathbf{x}}\_b) + \mathbf{K}\_s(\mathbf{x}\_w - \mathbf{x}\_b) \tag{59}$$

$$M\_{\overline{w}}\ddot{x}\_{\overline{w}} = -\mathsf{C}\_{\mathsf{s}}(\dot{\mathsf{x}}\_{\overline{w}} - \dot{\mathsf{x}}\_{\overline{b}}) - \mathsf{C}\_{\mathsf{v}}(\dot{\mathsf{x}}\_{\overline{w}} - \dot{\mathsf{x}}\_{\overline{b}}) - \mathsf{K}\_{\mathsf{s}}(\mathsf{x}\_{\overline{w}} - \mathsf{x}\_{\overline{b}}) + \mathsf{K}\_{\mathsf{t}}(\mathsf{x}\_{\mathsf{r}} - \mathsf{x}\_{\overline{w}}) \tag{60}$$

where

*xb* is the displacement of the car body,

*xw* is the displacement of the car wheel,

*xr* is the displacement of the road,

and the nonlinear weights are given by

minimizing the following *L*2-induced norm:

Fig. 7. Disturbance from the road surface

*az*(*x*) = <sup>1</sup> <sup>+</sup> *<sup>x</sup>TRzx*,

A Sum of Squares Optimization Approach to Robust Control of Bilinear Systems 53

with some positive numbers *mz*, *mu*. Note that this nonlinear weights do not satisfy the condition (16) in Theorem 1 in general. Hence, the method by Ohsaku et al. (1998); Shimizu et

The objective of the robust control to be concerned is to minimize the effect of the disturbance (the road roughness) on the velocity of the car body, which is formulated as the problem of

> *z*¯<sup>2</sup> *w*<sup>2</sup> .

Fig. 7 shows the disturbance *w*(*t*) = *x*˙*r*(*t*) and Fig. 8 shows the simulation results where the dashed red line shows the velocity of the car body without feedback control and the solid blue line shows the one with feedback control designed by our method. It can be seen that the amplitude of the body velocity by our method is suppressed compared with that without feedback control. This means that the effect of the disturbance on the body velocity is reduced

sup *w*=0

<sup>1</sup> <sup>+</sup> *<sup>x</sup>TRux*

, (63)

*au*(*x*) = <sup>1</sup>

*Rz* = *mz I*, *Ru* = *mu I*,

al. (1997) cannot be applied to this example with this weights.

by our method, which shows the efficiency of our method.

*Mb* is the mass of the car body,

*Mw* is the mass of the car wheel,

*Ks* is the spring constant of the suspension,

*Kt* is the elastic coefficient of the tire,

*Cs* is the fixed damping coefficient of the suspension,

*Cv* is the variable damping coefficient of the suspension,

and *Cv* is the input and *x*˙*<sup>r</sup>* is the disturbance.

Fig. 6. Semi-active suspension system

Then the state-space representation of the generalized system with *Wz* = *I*, *Wu* = 1 and nonlinear weights *az*(*x*), *au*(*x*) is given by

$$
\dot{x}(t) = Ax(t) + B\_1 w(t) + B\_2(x)u(t) \tag{61}
$$

$$\overline{z}(t) = a\_{\overline{z}}(\mathbf{x})\mathbb{C}\_{11}\mathbf{x}(t) + a\_{\boldsymbol{\mu}}(\mathbf{x})u(t) \tag{62}$$

where

$$\begin{aligned} \mathbf{x}(t) &= \begin{bmatrix} \mathbf{x}\_{\overline{w}}(t) - \mathbf{x}\_{b}(t) \\ \dot{\mathbf{x}}\_{r}(t) \\ \mathbf{x}\_{r}(t) - \mathbf{x}\_{w}(t) \\ \dot{\mathbf{x}}\_{w}(t) \end{bmatrix}, \mathbf{u}(t) = \mathbf{C}\_{v}(t), \; \mathbf{w}(t) = \dot{\mathbf{x}}\_{r}(t) \\\\ \mathbf{A} &= \begin{bmatrix} 0 & -1 & 0 & 1 \\ \frac{\dot{\mathbf{x}}\_{r}}{M\_{b}} & -\frac{\dot{\mathbf{C}}\_{r}}{M\_{b}} & 0 & \frac{\dot{\mathbf{C}}\_{r}}{M\_{b}} \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}, \; \mathbf{B}\_{1} = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \; \mathbf{B}\_{2}(\mathbf{x}) = \begin{bmatrix} 0 \\ \frac{\dot{\mathbf{x}}\_{\overline{w}} - \dot{\mathbf{x}}\_{b}}{M\_{b}} \\ 0 \\ -\frac{\dot{\mathbf{x}}\_{\overline{w}} - \dot{\mathbf{x}}\_{b}}{M\_{\overline{w}}} \end{bmatrix}, \\\\ \mathbf{C}\_{11} = \begin{bmatrix} 0 \ \mathbf{1} \ \mathbf{0} \ \mathbf{0} \end{bmatrix} \end{aligned}$$

and the nonlinear weights are given by

14 Will-be-set-by-IN-TECH

✟✟ *<sup>u</sup>* <sup>=</sup> *Cv* ✟✟✯

✻*xr*

, *u*(*t*) = *Cv*(*t*), *w*(*t*) = *x*˙*r*(*t*)

⎡ ⎢ ⎢ ⎣

⎤ ⎥ ⎥

<sup>⎦</sup> , *<sup>B</sup>*2(*x*) =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 *x*˙*w*−*x*˙*<sup>b</sup> Mb* 0 <sup>−</sup> *<sup>x</sup>*˙*w*−*x*˙*<sup>b</sup> Mw*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

*x*˙(*t*) = *Ax*(*t*) + *B*1*w*(*t*) + *B*2(*x*)*u*(*t*) (61)

*z*¯(*t*) = *az*(*x*)*C*11*x*(*t*) + *au*(*x*)*u*(*t*) (62)

Then the state-space representation of the generalized system with *Wz* = *I*, *Wu* = 1 and

✻*xw*

✻*xb*

*Mb* is the mass of the car body, *Mw* is the mass of the car wheel,

*Ks* is the spring constant of the suspension,

*Cs* is the fixed damping coefficient of the suspension, *Cv* is the variable damping coefficient of the suspension,

*Mb*

*Cs*

*Mw*

❳❳✘✘❳ ✘ ❳❳✘✘❳ ✘✘

*Kt*

✁✁ ✁ ✁ ✁ ✁ ✁ ✁✁ ✁ ✁✁ ✁✁ ✁✁

❡

*Kt* is the elastic coefficient of the tire,

and *Cv* is the input and *x*˙*<sup>r</sup>* is the disturbance.

❳❳✘✘❳ ✘ ❳❳✘✘❳ ✘✘

*Ks*

Fig. 6. Semi-active suspension system

*x*(*t*) =

*A* =

*C*<sup>11</sup> = �

where

nonlinear weights *az*(*x*), *au*(*x*) is given by

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*Ks Mb* <sup>−</sup> *Cs*

<sup>−</sup> *Ks Mw*

0100 �

*xw*(*t*) − *xb*(*t*) *x*˙*b*(*t*) *xr*(*t*) − *xw*(*t*) *x*˙*w*(*t*)

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

0 −10 1

0 00 −1

*Cs Mw*

*Mb* <sup>0</sup> *Cs*

*Kt Mw* <sup>−</sup> *Cs Mw*

*Mb*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, *B*<sup>1</sup> =

$$\begin{aligned} a\_{\overline{z}}(\mathbf{x}) &= \sqrt{1 + \mathbf{x}^T \mathbf{R}\_{\overline{z}} \mathbf{x}}, \\ a\_{\mu}(\mathbf{x}) &= \frac{1}{\sqrt{1 + \mathbf{x}^T \mathbf{R}\_{\mu} \mathbf{x}}} , \\ R\_{\overline{z}} &= m\_{\overline{z}} I\_{\prime} \\ R\_{\mu} &= m\_{\mu} I\_{\prime} \end{aligned} \tag{63}$$

with some positive numbers *mz*, *mu*. Note that this nonlinear weights do not satisfy the condition (16) in Theorem 1 in general. Hence, the method by Ohsaku et al. (1998); Shimizu et al. (1997) cannot be applied to this example with this weights.

The objective of the robust control to be concerned is to minimize the effect of the disturbance (the road roughness) on the velocity of the car body, which is formulated as the problem of minimizing the following *L*2-induced norm:

$$\sup\_{w \neq 0} \frac{||\tilde{z}||\_2}{||w||\_2}.$$

Fig. 7 shows the disturbance *w*(*t*) = *x*˙*r*(*t*) and Fig. 8 shows the simulation results where the dashed red line shows the velocity of the car body without feedback control and the solid blue line shows the one with feedback control designed by our method. It can be seen that the amplitude of the body velocity by our method is suppressed compared with that without feedback control. This means that the effect of the disturbance on the body velocity is reduced by our method, which shows the efficiency of our method.

Fig. 7. Disturbance from the road surface

**4** 

**Spatially Sampled Robust Repetitive Control** 

Repetitive control is one control algorithm based on the Internal Model Principle (Francis & Wonham, 1976) and has been widely implemented in various applications. A repetitive control based system has been shown to work well for tracking periodic reference commands or for rejecting periodic disturbances. Although the idea has been verified as early as 1981 (Inoue et al., 1981), a rigorous analysis and synthesis of repetitive controllers for continuous-time systems was not proposed until 1989, by Hara et al. (Hara et al., 1988). Tomizuka et al. (Tomizuka et al., 1989) addressed the analysis and synthesis of discrete-time repetitive controller, considering the fact that digital implementation of a repetitive controller is simpler and more straightforward. Since then, repetitive control has gained popularity in applications where periodic disturbances rejection or repetitive tracking are required, see (Wang et al., 2009; Cuiyan et al., 2004) and the references therein. These include controls of disk drive servo (Tomizuka et al., 1989; Guo, 1997; Moon et al., 1998), hydraulic closed-loop servo for material testing (Srinivasan & Shaw, 1993), vibration suppression (Hillerstrom, 1996), rejection of load disturbances in steel casting process (Manayathara et al., 1996), servo control for a positioning table (Yamada et al., 1999), X-Y table (Tung et al., 1993), noncircular turning process (Alter & Tsao, 1994), motor speed ripple reduction (Godler et al., 1995; Rodriguez et al., 2000), and eccentricity compensation

In literatures, repetitive controllers are synthesized and operate in time domain, which is in accordance with the fact that models or differential equations of physical systems are mostly derived using time as the independent variable. One of the key steps for designing a repetitive controller is to determine the period, or equivalently, the number of delay taps (q−1, q is the one step advance operator). This can usually be done by analyzing the periodic tracking or disturbance signal using techniques such as fast Fourier transform (FFT). To ensure effectiveness of the design, an underlying assumption is that the frequency constitutions of the periodic tracking or disturbance signal do not vary with respect to time, which corresponds to a stationary or time-invariant frequency spectrum of the signal. This assumption can be satisfied when the design objective is to track a prespecified periodic trajectory. However, it might be violated for disturbance rejection problems where the frequency constitutions of the disturbance are time-varying. For a motion system with rotary components such as gear-train, the disturbances due to gear

**1. Introduction** 

(Garimella & Srinivasan, 1996).

Cheng-Lun Chen1 and George T.-C. Chiu2

*1National Chung Hsing University, 2Purdue University, West Lafayette,* 

*1Taiwan, R.O.C.* 

*2USA* 

Fig. 8. Velocities of the car body (dashed red: open-loop, solid blue: by our method)

### **6. Conclusions**

In this paper, first we have derived an inequality condition of quadratic form for the robust control problem of bilinear systems with nonlinear weights, and then proposed an iterative method for finding a solution to the inequality condition. Finally, we have given a numerical example to show the effectiveness of our method.

### **7. References**


van der Schaft, A. J. (1996). *L*2*-gain and passivity techniques in nonlinear control*, Springer-Verlag.

## **Spatially Sampled Robust Repetitive Control**

Cheng-Lun Chen1 and George T.-C. Chiu2

*1National Chung Hsing University, 2Purdue University, West Lafayette, 1Taiwan, R.O.C. 2USA* 

### **1. Introduction**

16 Will-be-set-by-IN-TECH

54 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Fig. 8. Velocities of the car body (dashed red: open-loop, solid blue: by our method)

In this paper, first we have derived an inequality condition of quadratic form for the robust control problem of bilinear systems with nonlinear weights, and then proposed an iterative method for finding a solution to the inequality condition. Finally, we have given a numerical

Huang, J. & Lin, C.-F. (1995). Numerical approach to computing nonlinear *H*∞ control laws.

Ichihara, H. (2009). Optimal control for polynomial systems using matrix sum of squares relaxations. *IEEE Transactions on Automatic Control*, Vol. 54, No. 5, pp. 1048-1053. Ohsaku S.; Nakayama T.; Kamimura I. & Motozono, Y. (1998). Nonlinear *H*∞ state feedback controller for semi-active controlled suspension. *Proceedings of AVEC '98*, pp. 63-68. Parrilo, P. A. (2003). Semidefinite programming relaxations for semialgebraic problems.

Prajna, S.; Papachristodoulou, A. & Wu, F. (2004). Nonlinear control synthesis by sum of squares optimization. *Proceedings of the Asian Control Conference 2004*, pp. 157-165. Sampei, M.; Ohsaku S. & Kamimura, I. (1999). Nonlinear H-infinity Control Theory, its

Scherer, C. W. & Hol, C. W. (2006). Matrix sum-of-squares relaxations for robust semi-definite

Shimizu, E.; Sampei M. & Koga M. (1997). Design of a nonlinear *H*∞ state feedback controller

van der Schaft, A. J. (1996). *L*2*-gain and passivity techniques in nonlinear control*, Springer-Verlag.

Limitation and Possibility - Its Application to Semiactive Suspension. *Systems, Control*

for bilinear systems with nonlinear weight. *Proceedings of the 36th IEEE Conference on*

*Journal of Guidance, Control and Dynamics*, Vol. 18, pp. 989-994.

*Mathematical Programming*, Vol. 96, pp. 293-320.

*Decision and Control*, pp. 2323-2324.

*and Information*, Vol. 43, No. 10, pp. 544-552 (in Japanese).

programs. *Mathematical Programming*, Vol. 107, pp. 189-211.

**6. Conclusions**

**7. References**

example to show the effectiveness of our method.

Repetitive control is one control algorithm based on the Internal Model Principle (Francis & Wonham, 1976) and has been widely implemented in various applications. A repetitive control based system has been shown to work well for tracking periodic reference commands or for rejecting periodic disturbances. Although the idea has been verified as early as 1981 (Inoue et al., 1981), a rigorous analysis and synthesis of repetitive controllers for continuous-time systems was not proposed until 1989, by Hara et al. (Hara et al., 1988). Tomizuka et al. (Tomizuka et al., 1989) addressed the analysis and synthesis of discrete-time repetitive controller, considering the fact that digital implementation of a repetitive controller is simpler and more straightforward. Since then, repetitive control has gained popularity in applications where periodic disturbances rejection or repetitive tracking are required, see (Wang et al., 2009; Cuiyan et al., 2004) and the references therein. These include controls of disk drive servo (Tomizuka et al., 1989; Guo, 1997; Moon et al., 1998), hydraulic closed-loop servo for material testing (Srinivasan & Shaw, 1993), vibration suppression (Hillerstrom, 1996), rejection of load disturbances in steel casting process (Manayathara et al., 1996), servo control for a positioning table (Yamada et al., 1999), X-Y table (Tung et al., 1993), noncircular turning process (Alter & Tsao, 1994), motor speed ripple reduction (Godler et al., 1995; Rodriguez et al., 2000), and eccentricity compensation (Garimella & Srinivasan, 1996).

In literatures, repetitive controllers are synthesized and operate in time domain, which is in accordance with the fact that models or differential equations of physical systems are mostly derived using time as the independent variable. One of the key steps for designing a repetitive controller is to determine the period, or equivalently, the number of delay taps (q−1, q is the one step advance operator). This can usually be done by analyzing the periodic tracking or disturbance signal using techniques such as fast Fourier transform (FFT). To ensure effectiveness of the design, an underlying assumption is that the frequency constitutions of the periodic tracking or disturbance signal do not vary with respect to time, which corresponds to a stationary or time-invariant frequency spectrum of the signal. This assumption can be satisfied when the design objective is to track a prespecified periodic trajectory. However, it might be violated for disturbance rejection problems where the frequency constitutions of the disturbance are time-varying. For a motion system with rotary components such as gear-train, the disturbances due to gear

Spatially Sampled Robust Repetitive Control 57

in spatial domain. Specifically, the variable of time is rendered implicit for the transformed system in spatial domain with angular displacement being the new independent variable. This is attained by using the relationship between angular displacement and velocity along with imposing an assumption of bijective mapping between time and angular displacement. The resulting nonlinear system was linearized at a fixed angular velocity and a stabilizing controller with built-in repetitive control action was synthesized. In (Chen et al., 2006), robust control methods were employed to address issues associated with using a linearized plant model in the controller synthesis and actuator saturation. Although effective for small angular velocity fluctuations, the effectiveness of a linearized approach is limited when the application requires a large variation in operating speed. (Mahawan & Luo, 2000) demonstrated the feasibility of augmenting a spatially sampled repetitive controller to a time-sampled stabilizing controller, where no reformulation and linearization of the openloop plant model is required. However, the complexity of the method lies in the need to solve an optimization problem in real-time to synchronize the hardware and software interrupts associated with time and spatial sampling, respectively. In addition, although reasonable for trajectory tracking, the assumption of a known mapping between time and angular displacement is rarely applicable for disturbance rejection applications. The lack of considerations to modeling uncertainty is another area that can be improved from the methods proposed in (Nakano et al., 1996) and (Mahawan & Luo, 2000). Instead of linearizing the resulting nonlinear plant model, (Chen and Chiu, 2008) shows that the nonlinear plant model can be formulated into a quasi-linear parameter varying (quasi-LPV) system, where the angular speed is one of the measurable varying parameters. Leveraging existing results in controller synthesis for LPV systems (Becker & Packard, 1994; Apkarian et al., 1995; Gahinet, 1996; Gahinet & Apkarian, 1994, 1995) and the robust repetitive design formulation outlined in (Chen et al., 2006; Hanson & Tsao, 2000), an LPV gain-scheduling controller can be obtained that addresses bounded modeling uncertainties, actuator

This book chapter will provide the reader with a review and summary of recent advances in design of spatially sampled repetitive control systems. Specifically, we will elaborate on a few designs which account for the robustness property of the system, i.e., capability in tackling modeling uncertainties and actuator saturation. Current issues and future research

Section 2 demonstrates how to transform a generic time-domain system into its counterpart in spatial domain. It is also shown that nonlinearity such as actuator saturation may be

Section 3 presents a design of spatially sampled robust repetitive control. A well-known approach for designing controllers for nonlinear systems with a well defined operating point is to first linearize the system around the nominal operating point. Once the linear system is extracted, linear robust design paradigm can be applied to establish a design

Section 4 presents another design of spatially sampled robust repetitive control. By reformulating the transformed spatial-domain system as a quasi-linear parameter varying (quasi-LPV) system, we gain access to the LPV design framework for gain-scheduling controllers. Hence, an LPV gain-scheduling repetitive control (LPVRC) system can be synthesized by augmenting the repetitive controller with the LPV controller. The LPVRC design is superior to others in the sense that 1) It requires less computation effort when compared to nonlinear design; 2) It is robust to spatially periodic disturbances when

directions will also be discussed. The outline of this chapter is as follows:

properly modeled and incorporated into the spatial-domain open-loop system.

saturation and spatially periodic disturbances.

framework with embedded repetitive controller.

eccentricity or tooth profile error are inherently angular displacement dependent or spatially periodic. They are periodic with respect to angular displacement, but not necessarily periodic with respect to time. Gear eccentricity induces disturbances with period equal to one revolution and tooth profile error induces disturbances with fundamental frequency equal to the number of teeth per revolution. The spatial periods for these two types of disturbances do not change with the angular velocity. However, the corresponding temporal frequencies will be proportional to the angular velocity and vary accordingly when the system operates at variable speeds. As an example, for a single stage motor/gear transmission system operating an output speed of v revolution per second, the eccentricity error of the final gear will show up as a periodic disturbance with temporal frequency of v Hz. As the operating speed changes, a proportional changes to the temporal frequency of the disturbance will occur whereas its spatial frequency is fixed at 1 cycle per revolution. Suppose that a repetitive controller is implemented using a constant-angular-displacement sampling period (spatially sampled) approach, e.g., m samples per revolution of the final gear, to tackle this disturbance. The number of required delay taps, which reflects the period of the disturbance, will be a constant m regardless of the angular velocity. On the other hand, a repetitive controller synthesized using the conventional approach, i.e., based on the temporal frequency of the disturbance (v Hz), and implemented with constant-time sampling period will not be effective if the number of delay taps for the repetitive controller is not tuned/adapted in real-time in accordance with the angular velocity. If the period fluctuation is small, methods have been shown to improve the robustness of the repetitive controller by increasing the notch width in the frequency domain of the repetitive controller at the cost of reduced attenuation for the periodic disturbance (Onuki & Ishioka, 2001). When the period variation is large, there are two approaches to address the varying period in a repetitive control framework. For situation where the period variation can not be measured or unknown, adaptive control approaches have been shown to be effective in adapting the period of the repetitive controller (Hillerstrom, 1996; Manayathara et al., 1996; Wit & Praly, 2000) at the expense of response time and transient response. When the period variation is known or can be measured, such as the case in gear noise induced disturbance, better trade-off between period adaptability and effectiveness of repetitive control can be made.

Recent researches started studying control problems of rejecting/tracking spatially periodic disturbances/references in spatial domain, i.e., using spatially sampled repetitive controllers. As explained earlier, a spatially sampled repetitive controller has its repetitive kernel (i.e., e-st or z-N with positive feedback) synthesized and operate with respect to angular displacement. Hence its capability for rejecting/tracking spatially periodic disturbances/references will not degrade when the controlled system operates at varying speed. Note that a typical repetitive control system consists of repetitive (i.e., a repetitive kernel) and non-repetitive (e.g., a stabilizing controller) portions. Given a time-domain open-loop system and with the repetitive kernel to be synthesized and implemented in spatial domain, design of the non-repetitive portion that properly interfaces the repetitive kernel and the open-loop system actually poses a challenge. (Nakano et al., 1996) initiated a fundamental design of spatially sampled repetitive controller in 1996. Although the proposed design is rudimentary due to its focus on simple linear time-invariant systems, it has recently motivated several more advanced designs (Mahawan & Luo, 2000; Chen et al., 2006). The design started by transforming a given open-loop system in time domain into one 56 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

eccentricity or tooth profile error are inherently angular displacement dependent or spatially periodic. They are periodic with respect to angular displacement, but not necessarily periodic with respect to time. Gear eccentricity induces disturbances with period equal to one revolution and tooth profile error induces disturbances with fundamental frequency equal to the number of teeth per revolution. The spatial periods for these two types of disturbances do not change with the angular velocity. However, the corresponding temporal frequencies will be proportional to the angular velocity and vary accordingly when the system operates at variable speeds. As an example, for a single stage motor/gear transmission system operating an output speed of v revolution per second, the eccentricity error of the final gear will show up as a periodic disturbance with temporal frequency of v Hz. As the operating speed changes, a proportional changes to the temporal frequency of the disturbance will occur whereas its spatial frequency is fixed at 1 cycle per revolution. Suppose that a repetitive controller is implemented using a constant-angular-displacement sampling period (spatially sampled) approach, e.g., m samples per revolution of the final gear, to tackle this disturbance. The number of required delay taps, which reflects the period of the disturbance, will be a constant m regardless of the angular velocity. On the other hand, a repetitive controller synthesized using the conventional approach, i.e., based on the temporal frequency of the disturbance (v Hz), and implemented with constant-time sampling period will not be effective if the number of delay taps for the repetitive controller is not tuned/adapted in real-time in accordance with the angular velocity. If the period fluctuation is small, methods have been shown to improve the robustness of the repetitive controller by increasing the notch width in the frequency domain of the repetitive controller at the cost of reduced attenuation for the periodic disturbance (Onuki & Ishioka, 2001). When the period variation is large, there are two approaches to address the varying period in a repetitive control framework. For situation where the period variation can not be measured or unknown, adaptive control approaches have been shown to be effective in adapting the period of the repetitive controller (Hillerstrom, 1996; Manayathara et al., 1996; Wit & Praly, 2000) at the expense of response time and transient response. When the period variation is known or can be measured, such as the case in gear noise induced disturbance, better trade-off between period adaptability and effectiveness of repetitive

Recent researches started studying control problems of rejecting/tracking spatially periodic disturbances/references in spatial domain, i.e., using spatially sampled repetitive controllers. As explained earlier, a spatially sampled repetitive controller has its repetitive kernel (i.e., e-st or z-N with positive feedback) synthesized and operate with respect to angular displacement. Hence its capability for rejecting/tracking spatially periodic disturbances/references will not degrade when the controlled system operates at varying speed. Note that a typical repetitive control system consists of repetitive (i.e., a repetitive kernel) and non-repetitive (e.g., a stabilizing controller) portions. Given a time-domain open-loop system and with the repetitive kernel to be synthesized and implemented in spatial domain, design of the non-repetitive portion that properly interfaces the repetitive kernel and the open-loop system actually poses a challenge. (Nakano et al., 1996) initiated a fundamental design of spatially sampled repetitive controller in 1996. Although the proposed design is rudimentary due to its focus on simple linear time-invariant systems, it has recently motivated several more advanced designs (Mahawan & Luo, 2000; Chen et al., 2006). The design started by transforming a given open-loop system in time domain into one

control can be made.

in spatial domain. Specifically, the variable of time is rendered implicit for the transformed system in spatial domain with angular displacement being the new independent variable. This is attained by using the relationship between angular displacement and velocity along with imposing an assumption of bijective mapping between time and angular displacement. The resulting nonlinear system was linearized at a fixed angular velocity and a stabilizing controller with built-in repetitive control action was synthesized. In (Chen et al., 2006), robust control methods were employed to address issues associated with using a linearized plant model in the controller synthesis and actuator saturation. Although effective for small angular velocity fluctuations, the effectiveness of a linearized approach is limited when the application requires a large variation in operating speed. (Mahawan & Luo, 2000) demonstrated the feasibility of augmenting a spatially sampled repetitive controller to a time-sampled stabilizing controller, where no reformulation and linearization of the openloop plant model is required. However, the complexity of the method lies in the need to solve an optimization problem in real-time to synchronize the hardware and software interrupts associated with time and spatial sampling, respectively. In addition, although reasonable for trajectory tracking, the assumption of a known mapping between time and angular displacement is rarely applicable for disturbance rejection applications. The lack of considerations to modeling uncertainty is another area that can be improved from the methods proposed in (Nakano et al., 1996) and (Mahawan & Luo, 2000). Instead of linearizing the resulting nonlinear plant model, (Chen and Chiu, 2008) shows that the nonlinear plant model can be formulated into a quasi-linear parameter varying (quasi-LPV) system, where the angular speed is one of the measurable varying parameters. Leveraging existing results in controller synthesis for LPV systems (Becker & Packard, 1994; Apkarian et al., 1995; Gahinet, 1996; Gahinet & Apkarian, 1994, 1995) and the robust repetitive design formulation outlined in (Chen et al., 2006; Hanson & Tsao, 2000), an LPV gain-scheduling controller can be obtained that addresses bounded modeling uncertainties, actuator saturation and spatially periodic disturbances.

This book chapter will provide the reader with a review and summary of recent advances in design of spatially sampled repetitive control systems. Specifically, we will elaborate on a few designs which account for the robustness property of the system, i.e., capability in tackling modeling uncertainties and actuator saturation. Current issues and future research directions will also be discussed. The outline of this chapter is as follows:

Section 2 demonstrates how to transform a generic time-domain system into its counterpart in spatial domain. It is also shown that nonlinearity such as actuator saturation may be properly modeled and incorporated into the spatial-domain open-loop system.

Section 3 presents a design of spatially sampled robust repetitive control. A well-known approach for designing controllers for nonlinear systems with a well defined operating point is to first linearize the system around the nominal operating point. Once the linear system is extracted, linear robust design paradigm can be applied to establish a design framework with embedded repetitive controller.

Section 4 presents another design of spatially sampled robust repetitive control. By reformulating the transformed spatial-domain system as a quasi-linear parameter varying (quasi-LPV) system, we gain access to the LPV design framework for gain-scheduling controllers. Hence, an LPV gain-scheduling repetitive control (LPVRC) system can be synthesized by augmenting the repetitive controller with the LPV controller. The LPVRC design is superior to others in the sense that 1) It requires less computation effort when compared to nonlinear design; 2) It is robust to spatially periodic disturbances when

Spatially Sampled Robust Repetitive Control 59

ˆ ˆ ( ) ( ( )), ( ) ( ( )), ˆ ˆ( ) ( ( )), ( ) ( ( )),

usually be satisfied for most rotary motion system where the rotary component rotates only

() ( ) ( ) ˆ ˆ ˆ( ) *dx t d dx dx dt dt d d*

ˆ( ) ˆ ˆ ˆ ˆ ˆˆ ( ) , , , ,

*dx fx fx gx gx u <sup>d</sup>*

independent variable. Note that the concept of transfer function is still valid for linear

 <sup>0</sup> ˆ() ( ) ˆ *<sup>s</sup> Gs g e d* .

This definition will be useful for describing the linear portion of the overall control system.

Suppose a state space realization of an LTI model for a typical rotary system can be

( ) ( ) ( ) ( ) ( ) 0 ()

where *x*(*t*) is the system state vector, *x t* ( ) denotes the time derivative of the state vector, *v*(*t*) is the output disturbance vector that contains spatially periodic components, *z*(*t*) denotes the output vector related to system performance, *y*(*t*) is the measurement vector, and *u*(*t*) is the control input vector. Those signals are linearly related by the matrices shown in (4), i.e., *A*, *Bv*, *Cz*, etc. and all of the matrices and vectors are assumed to have compatible dimensions. If (*t*) is a strictly monotonic function of *t* such that its inverse *t* = 1(*t*) exists and does not

*xt A B B xt zt C D D vt y t C D ut* 

*v u z zv zu y yv*

11 1

 

 

> 

( ) ( ( )), ( ) ( ( )), and ( ) ( ( )).

 

 

*t ft f t gt g*

1

> 

 

Equation (3) can be regarded as an NPI system with the angular displacement

position-invariant systems if we define the Laplace transform of a signal *g*ˆ( )

*xx yy uu dd*

 

ˆ( ) ( ( )),

 

**2.1 Transformation of an LTI rotary system with actuator saturation** 

vanish, variables in time domain will have a well defined counterpart in the

1 1

*yy vv uu*

( ) ( ( )), ( ) ( ( )),

*xx zz*

 

 ˆ ˆ ˆˆ ˆ () () () .

 

*y xd x d*

*yn y*

 

where we denote ˆ as the

in one direction. Since

(1) may be rewritten as

 

expressed as

 

angular displacement domain as

1 1 1 1

 

  -domain representation of . Note that, in practice, (2) can

 

     

(4)

  -domain, i.e.,

(3)

as the

in the

 

 

 

compared to temporal-based design; 3) It allows wider operation range when compared to designs using linearization approaches.

Section 5 concludes the chapter and points out issues and future research directions relevant to spatially sampled robust repetitive control.

### **2. Problem formulation – Position-invariant rotary systems**

In this section, we show how a generic nonlinear time-invariant (NTI) model can be transformed into a nonlinear position-invariant (NPI; as opposed to the definition of timeinvariant) model by choosing an alternate independent variable (angular displacement instead of time) and defining a new set of states (or coordinates) with respect to the angular displacement. Note that the transformation described here is equivalent to a nonlinear coordinate transformation or a diffeomorphism. The NPI model will be used for the subsequent design and discussion. In Section 2.1, we further demonstrate this transformation for a typical linear time-invariant (LTI) rotary system with actuator saturation, which will be utilized in subsequent design.

Consider the mathematical model of a single-input single-output (SISO) nth-order NTI system with model uncertainties, and subject to output disturbance, i.e.,

$$\begin{aligned} \mathbf{x}(\mathbf{t}) &= \left[ \mathbf{f}\_{\mathbf{t}}(\mathbf{x}(\mathbf{t}), \mathbf{f}\_{\mathbf{t}}) + \Delta \mathbf{f}\_{\mathbf{t}}(\mathbf{x}(\mathbf{t}), \mathbf{f}\_{\mathbf{t}}) \right] + \left[ \mathbf{g}\_{\mathbf{t}}(\mathbf{x}(\mathbf{t}), \mathbf{f}\_{\mathbf{g}}) + \Delta \mathbf{g}\_{\mathbf{t}}(\mathbf{x}(\mathbf{t}), \mathbf{f}\_{\mathbf{g}}) \right] \mathbf{u}(\mathbf{t}) \\ \mathbf{y} &= \Psi \mathbf{x}(\mathbf{t}) + \mathbf{d}(\mathbf{t}) = \mathbf{x}\_{\mathbf{n}}(\mathbf{t}) + \mathbf{d}\_{\mathbf{y}}(\mathbf{t}) \end{aligned} \tag{1}$$

where <sup>1</sup> *T <sup>n</sup> xt x t x t* , 0 01 , *u t* and *yt* correspond to control input and measured output angular velocity of the system, respectively. ( ) *<sup>y</sup> d t* represents a class of position-dependent disturbances which constitutes bounded spatially periodic and non-periodic components. Here we refer non-periodic disturbances to signals whose Fourier transform or power spectral density is zero above a certain finite frequency. The only available information of the disturbances is the number of distinctive spatial frequencies and the spectrum distribution for non-periodic disturbance components. *f xt <sup>t</sup>* , *<sup>f</sup>* and *g xt <sup>t</sup>* , *<sup>g</sup>* are known vector-valued functions with unknown but bounded system parameters, i.e., *ff f* <sup>1</sup> *<sup>k</sup>* and *gg g* <sup>1</sup> *<sup>l</sup>* ; *f xt <sup>t</sup>* , *<sup>f</sup>* and *g xt <sup>t</sup>* , *<sup>g</sup>* represent unstructured modeling inaccuracy, which are also assumed to be bounded. Instead of using time *t* as the independent variable, consider an alternate independent variable ( )*t* , i.e., the angular displacement. Since by definition

$$\mathcal{A}(t) = \int\_0^t \rho(\tau)d\tau + \mathcal{A}(0),$$

where ( )*t* is the angular velocity, the following condition

$$
\rho a(t) = \frac{d\theta}{dt} > 0, \; \forall \; t \ge 0 \tag{2}
$$

will guarantee that ( )*t* is strictly monotonic such that <sup>1</sup> *t* ( ) exists. Thus all the variables in the time domain can be transformed into their counterparts in the -domain, i.e.,

$$\begin{aligned} \hat{\mathfrak{x}}(\theta) &= \mathfrak{x}(\hat{\mathscr{L}}^{-1}(\theta))\_{\prime}, \hat{\mathfrak{y}}(\theta) = \mathfrak{y}(\hat{\mathscr{L}}^{-1}(\theta))\_{\prime}, \\ \hat{\mathfrak{u}}(\theta) &= \mathfrak{u}(\hat{\mathscr{L}}^{-1}(\theta))\_{\prime}, \hat{\mathfrak{d}}(\theta) = \mathfrak{d}(\hat{\mathscr{L}}^{-1}(\theta))\_{\prime}, \\ \hat{\mathfrak{o}}(\theta) &= \mathfrak{o}(\hat{\mathscr{L}}^{-1}(\theta))\_{\prime} \end{aligned}$$

where we denote ˆ as the -domain representation of . Note that, in practice, (2) can usually be satisfied for most rotary motion system where the rotary component rotates only in one direction. Since

$$\frac{d\mathbf{x}(t)}{dt} = \frac{d\theta}{dt}\frac{d\hat{\mathbf{x}}(\theta)}{d\theta} = \hat{\alpha}(\theta)\frac{d\hat{\mathbf{x}}(\theta)}{d\theta}$$

(1) may be rewritten as

58 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

compared to temporal-based design; 3) It allows wider operation range when compared to

Section 5 concludes the chapter and points out issues and future research directions relevant

In this section, we show how a generic nonlinear time-invariant (NTI) model can be transformed into a nonlinear position-invariant (NPI; as opposed to the definition of timeinvariant) model by choosing an alternate independent variable (angular displacement instead of time) and defining a new set of states (or coordinates) with respect to the angular displacement. Note that the transformation described here is equivalent to a nonlinear coordinate transformation or a diffeomorphism. The NPI model will be used for the subsequent design and discussion. In Section 2.1, we further demonstrate this transformation for a typical linear time-invariant (LTI) rotary system with actuator

Consider the mathematical model of a single-input single-output (SISO) nth-order NTI

x t = f x t ,f + Δf x t ,f + g x t ,f + Δg x t ,f u t

t ft f t g t g

input and measured output angular velocity of the system, respectively. ( ) *<sup>y</sup> d t* represents a class of position-dependent disturbances which constitutes bounded spatially periodic and non-periodic components. Here we refer non-periodic disturbances to signals whose Fourier transform or power spectral density is zero above a certain finite frequency. The only available information of the disturbances is the number of distinctive spatial frequencies and

bounded. Instead of using time *t* as the independent variable, consider an alternate

<sup>0</sup> ( ) ( ) (0), *<sup>t</sup>*

( ) 0, t >0 *<sup>d</sup> <sup>t</sup>*

*t d*

 

( )*t* is strictly monotonic such that

variables in the time domain can be transformed into their counterparts in the

(1)

, 0 01 , *u t* and *yt* correspond to control

*<sup>g</sup>* are known vector-valued functions with unknown but bounded system

*<sup>g</sup>* represent unstructured modeling inaccuracy, which are also assumed to be

( )*t* , i.e., the angular displacement. Since by definition

*gg g* 

 <sup>1</sup> *<sup>l</sup>* ; *f xt <sup>t</sup>* ,

*dt* (2)

<sup>1</sup> *t* ( ) exists. Thus all the


  *<sup>f</sup>* and

*<sup>f</sup>* and

**2. Problem formulation – Position-invariant rotary systems** 

system with model uncertainties, and subject to output disturbance, i.e.,

y = Ψx t +d t =x t +d t

 *ff f* 

 <sup>1</sup> *<sup>k</sup>* and

( )*t* is the angular velocity, the following condition

*T*

where <sup>1</sup>

*g xt <sup>t</sup>* ,

*g xt <sup>t</sup>* ,

where 

i.e.,

parameters, i.e.,

independent variable

will guarantee that

*<sup>n</sup> xt x t x t*

 

n y

the spectrum distribution for non-periodic disturbance components. *f xt <sup>t</sup>* ,

designs using linearization approaches.

to spatially sampled robust repetitive control.

saturation, which will be utilized in subsequent design.

$$\begin{split} \hat{\boldsymbol{\mu}}(\boldsymbol{\theta}) \frac{d\hat{\boldsymbol{\Sigma}}(\boldsymbol{\theta})}{d\boldsymbol{\theta}} &= \left[ \boldsymbol{f}\_{t} \Big( \hat{\boldsymbol{\boldsymbol{x}}}(\boldsymbol{\theta}), \boldsymbol{\phi}\_{f} \Big) + \Delta \boldsymbol{f}\_{t} \Big( \hat{\boldsymbol{\boldsymbol{x}}}(\boldsymbol{\theta}), \boldsymbol{\phi}\_{f} \Big) \right] + \left[ \boldsymbol{\mathcal{g}}\_{t} \Big( \hat{\boldsymbol{\boldsymbol{x}}}(\boldsymbol{\theta}), \boldsymbol{\phi}\_{\boldsymbol{\xi}} \Big) + \Delta \boldsymbol{g}\_{t} \Big( \hat{\boldsymbol{\boldsymbol{x}}}(\boldsymbol{\theta}), \boldsymbol{\phi}\_{\boldsymbol{\xi}} \Big) \right] \hat{\boldsymbol{\boldsymbol{n}}}(\boldsymbol{\theta}) \\ \hat{\boldsymbol{y}}(\boldsymbol{\theta}) = \Psi \hat{\boldsymbol{x}}(\boldsymbol{\theta}) + \hat{\boldsymbol{d}}\_{\boldsymbol{y}}(\boldsymbol{\theta}) = \hat{\boldsymbol{x}}\_{n}(\boldsymbol{\theta}) + \hat{\boldsymbol{d}}\_{\boldsymbol{y}}(\boldsymbol{\theta}). \end{split} \tag{3}$$

Equation (3) can be regarded as an NPI system with the angular displacement as the independent variable. Note that the concept of transfer function is still valid for linear position-invariant systems if we define the Laplace transform of a signal *g*ˆ( ) in the angular displacement domain as

$$
\hat{G}(\tilde{\mathbf{s}}) = \int\_0^\infty \hat{\mathbf{g}}(\theta) e^{-\tilde{\mathbf{s}}\theta} d\theta \ .
$$

This definition will be useful for describing the linear portion of the overall control system.

### **2.1 Transformation of an LTI rotary system with actuator saturation**

Suppose a state space realization of an LTI model for a typical rotary system can be expressed as

$$
\begin{bmatrix}
\dot{\mathbf{x}}(t) \\
\mathbf{z}(t) \\
\mathbf{y}(t)
\end{bmatrix} = \begin{bmatrix}
A & B\_v & B\_u \\
\mathbf{C}\_z & D\_{zv} & D\_{zu} \\
\mathbf{C}\_y & D\_{yv} & \mathbf{0}
\end{bmatrix} \begin{bmatrix}
\mathbf{x}(t) \\
\mathbf{v}(t) \\
\mathbf{u}(t)
\end{bmatrix} \tag{4}
$$

where *x*(*t*) is the system state vector, *x t* ( ) denotes the time derivative of the state vector, *v*(*t*) is the output disturbance vector that contains spatially periodic components, *z*(*t*) denotes the output vector related to system performance, *y*(*t*) is the measurement vector, and *u*(*t*) is the control input vector. Those signals are linearly related by the matrices shown in (4), i.e., *A*, *Bv*, *Cz*, etc. and all of the matrices and vectors are assumed to have compatible dimensions. If (*t*) is a strictly monotonic function of *t* such that its inverse *t* = 1(*t*) exists and does not vanish, variables in time domain will have a well defined counterpart in the -domain, i.e.,

$$\begin{aligned} \tilde{\boldsymbol{x}}(\boldsymbol{\theta}) & \triangleq \boldsymbol{x}(\boldsymbol{\Phi}^{-1}(\boldsymbol{\theta})), \ \tilde{\boldsymbol{z}}(\boldsymbol{\theta}) \triangleq \boldsymbol{z}(\boldsymbol{\Phi}^{-1}(\boldsymbol{\theta})),\\ \tilde{\boldsymbol{y}}(\boldsymbol{\theta}) & \triangleq \boldsymbol{y}(\boldsymbol{\Phi}^{-1}(\boldsymbol{\theta})), \ \tilde{\boldsymbol{v}}(\boldsymbol{\theta}) \triangleq \boldsymbol{v}(\boldsymbol{\Phi}^{-1}(\boldsymbol{\theta})), \ \text{and } \tilde{\boldsymbol{u}}(\boldsymbol{\theta}) \triangleq \boldsymbol{u}(\boldsymbol{\Phi}^{-1}(\boldsymbol{\theta})).\end{aligned}$$

Suppose the angular velocity can be measured in real-time and written as

$$
\tilde{\alpha}(\theta) = \mathbb{C}\_{\alpha} \tilde{\mathfrak{x}}(\theta) + \alpha\_0 \neq 0,\tag{5}
$$

Spatially Sampled Robust Repetitive Control 61

used in the z-transform) includes the plant, and all linear weighting filters whose magnitude responses are used to specify the frequency-wise bounds on the output signals and modeling uncertainty. All mathematical operations within the generalized plant are either addition of two signals or scalar multiplication of signals, which renders *P z*( ) linear. Note that only motor actuators are considered in the framework and driven by the control input *u* calculated by the controller *K z*( ) . Other type of actuators can also be considered. Second, the inputs to the controller *y* are output signal measurement, e.g. velocity error from the rotary component. Third, the variable *w* includes those external signals such as periodic disturbances while the variable *z* includes those physical quantities which are important to system performance. Furthermore, *p* and *q* represent the input and output of the generalized uncertainty which is formed by all the uncertainty blocks from the generalized plant. The uncertainty blocks are usually formed by the modeling error and plant nonlinearity. There exist standard procedures and techniques to 'pull out' uncertainties from the generalized

*A*

*CQ CP*

*Δ*

Generalized *q* uncertainty *p*

*AK*

Stabilizing controller *K(z)*

*CK1 CK2*

Fig. 1. LFT representation of the EP closed-loop control system using motor actuators.

,

*q p*

*IM k k YW k ID k IQ k ZW k*

 

*y x Dw <sup>C</sup> z Cx D w*

*C*

Based on the LFT representation, a discrete-position state space realization of the to-becontrolled system (the generalized plant plus the generalized uncertainty) can be written as

1 12 [],

*k k M M k Wk*

*x Ax B B u B w*

,

,

Generalized plant *P(z)*

*BW*

motor 1 motor 2

*u y*

*BK1 BK2*

*CIQ CIM CID* *z*

(7)

plant (Zhou & Doyle, 1997).

*w BM1 BM2*

where 0 is the nominal angular velocity and *C* is an appropriate output matrix. Applying the aforementioned transformation, substituting (5) into (4), and imposing the saturation function

$$sat(\boldsymbol{\mu}) = \begin{cases} \boldsymbol{\mu}\_{\max \prime} \; \boldsymbol{\mu} \ge \boldsymbol{\mu}\_{\max} \\ \boldsymbol{\mu}\_{\prime} \; \boldsymbol{\mu}\_{\min} < \boldsymbol{\mu} < \boldsymbol{\mu}\_{\max} \\ \boldsymbol{\mu}\_{\min \prime} \; \boldsymbol{\mu} \le \boldsymbol{\mu}\_{\min} \end{cases}$$

on *u*( ) , we have

$$
\begin{bmatrix}
\dot{\tilde{\mathbf{x}}}(\theta) \\
\tilde{z}(\theta) \\
\tilde{y}(\theta)
\end{bmatrix} = \begin{bmatrix}
A/\{\mathbf{C}\_{o}\tilde{\mathbf{x}} + a\_{0}\} & B\_{v}\{\{\mathbf{C}\_{o}\tilde{\mathbf{x}} + a\_{0}\} & B\_{u}/\{\mathbf{C}\_{o}\tilde{\mathbf{x}} + a\_{0}\} \\
\mathbf{C}\_{z} & D\_{zv} & D\_{zu} \\
\mathbf{C}\_{y} & D\_{yv} & 0
\end{bmatrix} \begin{bmatrix}
\tilde{\mathbf{x}}(\theta) \\
\tilde{v}(\theta) \\
\text{sat}(\tilde{u}(\theta))
\end{bmatrix}.
\tag{6}
$$

The system expressed by (6) is an angular displacement reformulated (ADR) system with the angular displacement as the independent variable.

### **3. Linear spatially sampled robust repetitive control**

Linear robust controller design is aiming at synthesizing a feedback controller so that stability and performance of the overall (closed-loop) control system is insensitive (i.e., robust) to external disturbances and model uncertainties. There are four popular terms used to characterize the performance of a linear feedback control system, namely nominal stability, nominal performance, robust stability, and robust performance (Zhou & Doyle, 1997). We say that a feedback control system is stable if its output signals are bounded when subject to bounded input signals. A feedback control system meets (steady-state) performance if it is stable and the ratio of the sizes (measured by a mathematical norm, e.g., 2-norm) of its output to input signals is bounded above by certain frequency dependent number. In most cases, stability comes first, and performance comes next in the priority of the design. Nominal stability/performance is to be satisfied by controller design only for a plant, i.e., the model of the to-be-controlled system (free of parameter uncertainty), while robust stability/performance is to be satisfied by more challenged design for a set of plants, which include the nominal one and those due to plant parameter variation. Linear fractional transformation (LFT) is a popular and effective technique to formulate and pose a robust control design problem as will be demonstrated next.

### **3.1 Synthesis of the robust controller**

Start the design by first looking at the LFT representation of the desired closed loop control system depicted in Fig. 1, which incorporate two motors as the actuators. An LFT representation basically consists of three blocks: generalized plant, generalized uncertainty, and the stabilizing controller. Several variables and components need to be explained here. First, the generalized plant *P z*( ) (i.e., discrete-position system with *z* denoted the variable

60 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

<sup>0</sup> ( ) ( ) 0, *C x*

max max min max min min

> 

> >

(6)

,

*u uu*

,

*u uu*

(5)

is an appropriate output matrix. Applying

 

the aforementioned transformation, substituting (5) into (4), and imposing the saturation

<sup>000</sup> () ( ) ( ) ( ) () ( ) () . ( ) 0 ( ( ))

 

The system expressed by (6) is an angular displacement reformulated (ADR) system with

Linear robust controller design is aiming at synthesizing a feedback controller so that stability and performance of the overall (closed-loop) control system is insensitive (i.e., robust) to external disturbances and model uncertainties. There are four popular terms used to characterize the performance of a linear feedback control system, namely nominal stability, nominal performance, robust stability, and robust performance (Zhou & Doyle, 1997). We say that a feedback control system is stable if its output signals are bounded when subject to bounded input signals. A feedback control system meets (steady-state) performance if it is stable and the ratio of the sizes (measured by a mathematical norm, e.g., 2-norm) of its output to input signals is bounded above by certain frequency dependent number. In most cases, stability comes first, and performance comes next in the priority of the design. Nominal stability/performance is to be satisfied by controller design only for a plant, i.e., the model of the to-be-controlled system (free of parameter uncertainty), while robust stability/performance is to be satisfied by more challenged design for a set of plants, which include the nominal one and those due to plant parameter variation. Linear fractional transformation (LFT) is a popular and effective technique to formulate and pose a robust

Start the design by first looking at the LFT representation of the desired closed loop control system depicted in Fig. 1, which incorporate two motors as the actuators. An LFT representation basically consists of three blocks: generalized plant, generalized uncertainty, and the stabilizing controller. Several variables and components need to be explained here. First, the generalized plant *P z*( ) (i.e., discrete-position system with *z* denoted the variable

*x ACx B Cx B Cx x z C D Dv y C D sat u*

as the independent variable.

*v u z zv zu*

 

Suppose the angular velocity can be measured in real-time and written as

 

( ) ,

*y yv*

 

**3. Linear spatially sampled robust repetitive control** 

control design problem as will be demonstrated next.

**3.1 Synthesis of the robust controller** 

*sat u u u u u*

0 is the nominal angular velocity and *C*

where 

function

on *u*( ) 

, we have

the angular displacement

used in the z-transform) includes the plant, and all linear weighting filters whose magnitude responses are used to specify the frequency-wise bounds on the output signals and modeling uncertainty. All mathematical operations within the generalized plant are either addition of two signals or scalar multiplication of signals, which renders *P z*( ) linear. Note that only motor actuators are considered in the framework and driven by the control input *u* calculated by the controller *K z*( ) . Other type of actuators can also be considered. Second, the inputs to the controller *y* are output signal measurement, e.g. velocity error from the rotary component. Third, the variable *w* includes those external signals such as periodic disturbances while the variable *z* includes those physical quantities which are important to system performance. Furthermore, *p* and *q* represent the input and output of the generalized uncertainty which is formed by all the uncertainty blocks from the generalized plant. The uncertainty blocks are usually formed by the modeling error and plant nonlinearity. There exist standard procedures and techniques to 'pull out' uncertainties from the generalized plant (Zhou & Doyle, 1997).

Fig. 1. LFT representation of the EP closed-loop control system using motor actuators.

Based on the LFT representation, a discrete-position state space realization of the to-becontrolled system (the generalized plant plus the generalized uncertainty) can be written as

$$\begin{aligned} \mathbf{x}\_{k+1} &= A \mathbf{x}\_k + [B\_{M1} \quad B\_{M2}] \boldsymbol{\mu}\_k + B\_{VV} \mathbf{w}\_{k'}\\ \mathbf{y}\_k &= \begin{bmatrix} \mathbf{C}\_{IM} \\ \mathbf{C}\_{ID} \end{bmatrix} \mathbf{x}\_k + D\_{YY} \mathbf{w}\_{k'}\\ \mathbf{z}\_k &= \mathbf{C}\_{IQ} \mathbf{x}\_k + D\_{ZIV} \mathbf{w}\_{k'}\\ \mathbf{q} &= \Delta \mathbf{p}\_{\prime} \end{aligned} \tag{7}$$

and the optimal stabilizing controller *K z*( ) , parameters of which stabilize the system and also minimize the size of the transfer function from *w* to *z* (or the ratio between the sizes of *w* and *z* if induced matrix norm is used) in the presence of the generalized uncertainty, can be represented as

$$K(z) = \begin{bmatrix} C\_{K1} \\ C\_{K2} \end{bmatrix} \begin{bmatrix} zI - A\_K \end{bmatrix}^{-1} \begin{bmatrix} B\_{K1} & B\_{K2} \end{bmatrix} + D\_K. \tag{8}$$

Spatially Sampled Robust Repetitive Control 63

( ) ( ) ( ).

independent variable. Note that this transformation will render those position-dependent

*<sup>B</sup> x e x ed u*

.

The procedures summarized in the literature (Chen & Chiu, 2001) can now be applied to the plant model expressed in Eq. (12) for synthesizing a two degree of freedom (TDOF) discrete-

*C z* <sup>3</sup>

*<sup>N</sup> z q z*

<sup>3</sup> *q* <sup>3</sup> *p* **Repetitive control** <sup>3</sup> *z*

*A A <sup>T</sup> <sup>T</sup> <sup>u</sup> k k <sup>k</sup>*

0 0 <sup>1</sup>

 

> 

<sup>0</sup> <sup>0</sup>

**Model uncertainty**

*y yv*

0 0 ( ) () ()

*x u*

 

*y Cx D v*

*dx A B*

*u*

 

*v* periodic and stationary. The performance of a repetitive controller


0.5*P z W z* <sup>1</sup>

(in number of samples per revolution), we can discretize Eq. (11) and acquire

 , and <sup>1</sup> *v vf* ( ) ( ( )) 

<sup>0</sup> , we have

.

(11)

(12)

*e*

*w*

*W z* <sup>2</sup> *<sup>f</sup> z* <sup>2</sup> *z* <sup>2</sup> *<sup>q</sup>* <sup>2</sup> *<sup>p</sup>*

  ( )*t* as the

*z*

**performance**

 , <sup>1</sup> *y yf* ( ) ( ( )) 

*d*

*y Cx D v*

*k k <sup>k</sup> y yv*

Equation (11) is a linear position invariant (LPI) system with the angular position

where <sup>1</sup> *x xf* ( ) ( ( )) 

disturbances within

a discrete-position model, i.e.

synthesized in the

frequency *T*

4 *u*

position robust repetitive controller.

<sup>1</sup> 0.5 (1 ) *u*

<sup>1</sup> *z*

<sup>1</sup> *q* <sup>1</sup> *p*

*C z* <sup>4</sup>

4 *y*

Fig. 2. The proposed TDOF robust repetitive control system.

<sup>3</sup> *y* <sup>3</sup> *u*

 , <sup>1</sup> *u uf* ( ) ( ( )) 

Linearize the equation around the nominal angular velocity

The corresponding optimization (or robust performance) problem can be formulated as

minimize ( ) subject to ( ) stabilizes the system, *H z wz K z*

where ( ) *H z wz* is the transfer function from *w* to *z*, and is some induced matrix norm. The decision variables to be found are *AK* , *B B K K* 1 2 , 1 2 *<sup>T</sup> C C K K* and *DK* . It has been shown that the above problem is nonconvex and a sophisticated search algorithm (e.g., D-K iteration) needs to be implemented in order to locate the global optimal solution. An alternative way is to consider a suboptimal controller which is the solution to the following problem

$$\begin{array}{l}\text{minimize } \left\| H\_{qw \to pz}(z) \right\| \\\text{subject to } K(z) \text{ stabilizes the system.} \end{array}$$

where ( ) *H z qw pz* is the transfer function from *q w* to *p z* , and is some induced matrix norm. This is the so-called mixed-sensitivity optimization problem and is convex. There have been standard software tools for solving this type of problems (Gahinet & Nemirovski, 1995).

#### **3.2 Discrete-position model of the system**

Suppose that the open-loop LTI system *P*(s) has a state space realization, i.e.,

$$\begin{aligned} \frac{d\mathbf{x}(t)}{dt} &= A\mathbf{x}(t) + B\_u \boldsymbol{\mu}(t) \\ \mathbf{y}(t) &= \mathbf{C}\_y \mathbf{x}(t) + D\_{yv} \boldsymbol{\nu}(t), \end{aligned} \tag{9}$$

where *v t*( ) denotes disturbances at the plant output. Equation (9) is basically a simplified version of (4). Instead of using time *t* as the independent variable, we can pick angular position, ( )*t* , as the independent variable, i.e. ( )*t* . Thus in the -domain Eq. (9) can be expressed as

$$\begin{aligned} \frac{d\phi}{dt} \frac{d\tilde{\mathbf{x}}(\phi)}{d\phi} &= A \tilde{\mathbf{x}}(\phi) + B\_u \tilde{\boldsymbol{u}}(\phi) \\ \tilde{\mathbf{y}}(\phi) &= C\_y \tilde{\mathbf{x}}(\phi) + D\_{yv} \tilde{\mathbf{y}}(\phi) \end{aligned} \tag{10}$$

62 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

and the optimal stabilizing controller *K z*( ) , parameters of which stabilize the system and also minimize the size of the transfer function from *w* to *z* (or the ratio between the sizes of *w* and *z* if induced matrix norm is used) in the presence of the generalized uncertainty, can

() ( ) [ ] . *<sup>K</sup>*

subject to ( ) stabilizes the system,

where ( ) *H z wz* is the transfer function from *w* to *z*, and is some induced matrix norm.

shown that the above problem is nonconvex and a sophisticated search algorithm (e.g., D-K iteration) needs to be implemented in order to locate the global optimal solution. An alternative way is to consider a suboptimal controller which is the solution to the following

> subject to ( ) stabilizes the system, *H z qw pz*

where ( ) *H z qw pz* is the transfer function from *q w* to *p z* , and is some induced matrix norm. This is the so-called mixed-sensitivity optimization problem and is convex. There have been standard software tools for solving this type of problems (Gahinet &

( ) () ()

*dx t Ax t B u t*

*y t C xt D vt*

*u*

( )*t* . Thus in the

( ) ( ) ( ),

where *v t*( ) denotes disturbances at the plant output. Equation (9) is basically a simplified version of (4). Instead of using time *t* as the independent variable, we can pick angular

*y yv*

 

( ) () ()

*u*

  *y Cx D v*

( ) ( ) ( ),

*d dx Ax B u*

*y yv*

*K z*

*K z zI A B B D*

1 2

(8)

*<sup>T</sup> C C K K* and *DK* . It has been

(9)

(10)


*K KK K*

1 1

The corresponding optimization (or robust performance) problem can be formulated as

*H z wz*

2

minimize ( )

The decision variables to be found are *AK* , *B B K K* 1 2 , 1 2

minimize ( )

Suppose that the open-loop LTI system *P*(s) has a state space realization, i.e.,

*dt*

*dt d*

 

*K z*

*K C*

*C*

be represented as

problem

position,

expressed as

Nemirovski, 1995).

**3.2 Discrete-position model of the system** 

( )*t* , as the independent variable, i.e.

where <sup>1</sup> *x xf* ( ) ( ( )) , <sup>1</sup> *u uf* ( ) ( ( )) , <sup>1</sup> *y yf* ( ) ( ( )) , and <sup>1</sup> *v vf* ( ) ( ( )) . Linearize the equation around the nominal angular velocity <sup>0</sup> , we have

$$\begin{aligned} \frac{d\tilde{\mathbf{x}}(\theta)}{d\phi} &= \frac{A}{a\phi\_0} \tilde{\mathbf{x}}(\theta) + \frac{B\_u}{a\phi\_0} \tilde{\mathbf{u}}(\theta) \\ \tilde{\mathbf{y}}(\theta) &= \mathbf{C}\_y \tilde{\mathbf{x}}(\theta) + D\_{yv} \tilde{\mathbf{v}}(\theta) .\end{aligned} \tag{11}$$

Equation (11) is a linear position invariant (LPI) system with the angular position ( )*t* as the independent variable. Note that this transformation will render those position-dependent disturbances within *v* periodic and stationary. The performance of a repetitive controller synthesized in the -domain will not be compromised. Properly choosing spatial sampling frequency *T* (in number of samples per revolution), we can discretize Eq. (11) and acquire a discrete-position model, i.e.

$$\begin{aligned} \tilde{\mathbf{x}}\_{k+1} &= e^{\frac{A}{a\_0}r\_\theta} \tilde{\mathbf{x}}\_k + \left( \int\_0^{r\_\theta} e^{\frac{A}{a\_0}r} d\tau \right) \frac{B\_u}{a\_0} \tilde{\mathbf{u}}\_k \\ \tilde{\mathbf{y}}\_k &= \mathbf{C}\_y \tilde{\mathbf{x}}\_k + \mathbf{D}\_{y\overline{\mathbf{o}}} \tilde{\mathbf{o}}\_k. \end{aligned} \tag{12}$$

The procedures summarized in the literature (Chen & Chiu, 2001) can now be applied to the plant model expressed in Eq. (12) for synthesizing a two degree of freedom (TDOF) discreteposition robust repetitive controller.

Fig. 2. The proposed TDOF robust repetitive control system.

### **3.3 TDOF robust repetitive controller**

To reduce system sensitivity or increase system robustness to unmodeled dynamics or nonlinearity (i.e. actuator saturation), we can formulate the control problem within a unified linear design framework, i.e. using LFT. The proposed TDOF control structure is depicted in Fig. 2. The actual plant is represented as a saturation element 1 0.5(1 ) with | |1 <sup>1</sup> followed by a nominal model *P z*( ) with output multiplicative uncertainties *W*2 2 . *W*2 is the frequency-dependent uncertainty weighting filter such that 2 1 . It can be picked to be any stable filter with its magnitude upper bounding the multiplicative error between the model and the actual plant, i.e.

$$\left| \frac{\hat{P}\left(e^{jw}\right) - P\left(e^{jw}\right)}{P\left(e^{jw}\right)} \right| \le \left| \mathcal{W}\_2\left(e^{jw}\right) \right|\_{\prime} \,\,\forall w \tag{13}$$

Spatially Sampled Robust Repetitive Control 65

angular velocity varies. The velocity variation can be caused by friction, which is usually time-varying and difficult to be taken into account during design of the controller. Based on the proposed TDOF repetitive controller design, Fig. 3 shows the effect of nominal velocity variation on the performance of the sensitivity reduction. Parameters of the repetitive controller were specified to reject a disturbance located at 16 Hz when the system is operating at a nominal angular velocity of 3.14 rad/s. It can be seen that as the nominal velocity deviates from the desired value, the ability of the repetitive controller to reject the disturbance at 16 Hz degrades significantly. As shown in Fig. 3, a 0.2% variation in the nominal speed has an order of magnitude effect in the effectiveness of disturbance rejection. This high sensitivity to operating velocity is the motivation for pursuing the spatial-based


deviation from nominal velocity at 3.14 rad/s (%)

The proposed discrete-position repetitive controller was implemented on a typical 600-dpi laser printing system. An optical encoder was mounted on the main rotary component, i.e., an organic photoconductor (OPC) drum. A spatial sampling scheme that uses the encoder pulses (instead of a master clock) to trigger the interrupt of the control algorithm at intervals of equal angular position was implemented. Instead of counting number of pulses within a sampling period, the angular velocity was determined by monitoring the amount of time elapsed for fixed number of encoder pulses. This method actually enables low-cost encoders to achieve high-resolution velocity measurement. The spatial sampling frequency was set at 2000 samples/rev such that the discrete-position repetitive controller has a period of *N*=2000/16=125. The engine started printing when velocity data of 10 revolutions were collected from the OPC drum for analysis. Fig. 4 shows the measured angular velocity from the OPC drum. Note that as the paper goes through the printing process, it slightly increased the load on the transmission system. This impact decreased the nominal angular velocity from 3.14 rad/s to 3.07 rad/s. However, the frequency spectrums, as shown in Fig. 5, indicated that the performance of the discrete-position repetitive control system was not

Fig. 3. Effect of nominal angular velocity variation on performance of the repetitive

repetitive control.

controller.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

**3.5 Spatial-based repetitive control** 

amplification ratio w.r.t. a disturbance at 32 cycles/rev

Furthermore, the kernel of the repetitive controller *<sup>N</sup> qzz* is replaced by a fictitious uncertainty <sup>3</sup> . Also another fictitious uncertainty *<sup>f</sup>* is connected between the disturbance input and plant output. *W*1 is the frequency-dependent weighting filter that approximates human contrast sensitivity function (Chen et al., 2003). Thus, a TDOF controller is obtained by solving the following mixed-sensitivity optimization problem given by

$$\gamma\_{opt} = \inf\_{\mathbf{K}\text{ stabilizing}} \left\| \begin{array}{l} \mathcal{W}\_1 \left( \mathbf{1} + \mathcal{P} \mathbf{C}\_4 \right)^{-1} \\\\ \mathcal{C}\_4 P \left( \mathbf{1} + \mathcal{C}\_4 P \right)^{-1} \\\\ \left\| \mathcal{W}\_2 \mathcal{P} \mathcal{C}\_4 \left( \mathbf{1} + \mathcal{P} \mathbf{C}\_4 \right)^{-1} \right\|\_{\infty} \\\\ \mathbf{1} - P \left( \mathbf{1} - \mathcal{C}\_4 P \right)^{-1} \mathcal{C}\_3 \end{array} \right\|\_{\infty}^{\prime} \tag{14}$$

where

 1 2 3 4 Motor/Gear transmission system Performance weighting Uncertainty weighting The TDOF controller *P W W KCC* 

With upper and lower LFT denoted by *Fu* , and *Fl*, , respectively (Zhou & Doyle, 1997), the robust performance of the designed control system can further be evaluated by looking at the structure singular value of *F F MR K l u* , , with respect to the uncertainty block 1 2 , , *<sup>f</sup> diag* , i.e. *F F MR K l u* , , . Note that *<sup>N</sup> Rz qzz* is the kernel of the repetitive controller.

#### **3.4 Effect of nominal angular velocity variation on temporal-based repetitive control**

A repetitive control system creates comb-like notches in the system sensitivity function at periodic disturbance frequencies. For a motor/gear rotary system where significant disturbance sources come from gear eccentricity or tooth profile error, temporal frequencies of those disturbances will be proportional to the nominal angular velocity. Thus the performance of temporal-based repetitive control systems will deteriorate as the nominal 64 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

To reduce system sensitivity or increase system robustness to unmodeled dynamics or nonlinearity (i.e. actuator saturation), we can formulate the control problem within a unified linear design framework, i.e. using LFT. The proposed TDOF control structure is depicted in Fig. 2. The actual plant is represented as a saturation element 1 0.5(1 ) with | |1 <sup>1</sup> followed by a nominal model *P z*( ) with output multiplicative uncertainties *W*2 2 . *W*2 is the frequency-dependent uncertainty weighting filter such that 2 1 . It can be picked to be any stable filter with its magnitude upper bounding the multiplicative error between the

*jw jw*

*jw Pe Pe*

*P e*

by solving the following mixed-sensitivity optimization problem given by

inf

*KCC*

3 4

*opt*

1 2

*P W W*

 

<sup>2</sup> <sup>ˆ</sup>

Furthermore, the kernel of the repetitive controller *<sup>N</sup> qzz* is replaced by a fictitious uncertainty <sup>3</sup> . Also another fictitious uncertainty *<sup>f</sup>* is connected between the disturbance input and plant output. *W*1 is the frequency-dependent weighting filter that approximates human contrast sensitivity function (Chen et al., 2003). Thus, a TDOF controller is obtained

,

1

 

1

1 4 3

*F F MR K l u* , , . Note that *<sup>N</sup> Rz qzz* is the kernel of

, (14)

(13)

*jw*

 

1 4

4 4 K stabilizing 1

1 1

 

Motor/Gear transmission system

The TDOF controller

Performance weighting Uncertainty weighting

With upper and lower LFT denoted by *Fu* , and *Fl*, , respectively (Zhou & Doyle, 1997), the robust performance of the designed control system can further be evaluated by looking at the structure singular value of *F F MR K l u* , , with respect to the uncertainty

**3.4 Effect of nominal angular velocity variation on temporal-based repetitive control**  A repetitive control system creates comb-like notches in the system sensitivity function at periodic disturbance frequencies. For a motor/gear rotary system where significant disturbance sources come from gear eccentricity or tooth profile error, temporal frequencies of those disturbances will be proportional to the nominal angular velocity. Thus the performance of temporal-based repetitive control systems will deteriorate as the nominal

*W PC CP CP W PC PC P CP C*

1 1

24 4

1

*We w*

**3.3 TDOF robust repetitive controller** 

model and the actual plant, i.e.

where

block 1 2 , , *<sup>f</sup> diag* , i.e.

the repetitive controller.

angular velocity varies. The velocity variation can be caused by friction, which is usually time-varying and difficult to be taken into account during design of the controller. Based on the proposed TDOF repetitive controller design, Fig. 3 shows the effect of nominal velocity variation on the performance of the sensitivity reduction. Parameters of the repetitive controller were specified to reject a disturbance located at 16 Hz when the system is operating at a nominal angular velocity of 3.14 rad/s. It can be seen that as the nominal velocity deviates from the desired value, the ability of the repetitive controller to reject the disturbance at 16 Hz degrades significantly. As shown in Fig. 3, a 0.2% variation in the nominal speed has an order of magnitude effect in the effectiveness of disturbance rejection. This high sensitivity to operating velocity is the motivation for pursuing the spatial-based repetitive control.

Fig. 3. Effect of nominal angular velocity variation on performance of the repetitive controller.

### **3.5 Spatial-based repetitive control**

The proposed discrete-position repetitive controller was implemented on a typical 600-dpi laser printing system. An optical encoder was mounted on the main rotary component, i.e., an organic photoconductor (OPC) drum. A spatial sampling scheme that uses the encoder pulses (instead of a master clock) to trigger the interrupt of the control algorithm at intervals of equal angular position was implemented. Instead of counting number of pulses within a sampling period, the angular velocity was determined by monitoring the amount of time elapsed for fixed number of encoder pulses. This method actually enables low-cost encoders to achieve high-resolution velocity measurement. The spatial sampling frequency was set at 2000 samples/rev such that the discrete-position repetitive controller has a period of *N*=2000/16=125. The engine started printing when velocity data of 10 revolutions were collected from the OPC drum for analysis. Fig. 4 shows the measured angular velocity from the OPC drum. Note that as the paper goes through the printing process, it slightly increased the load on the transmission system. This impact decreased the nominal angular velocity from 3.14 rad/s to 3.07 rad/s. However, the frequency spectrums, as shown in Fig. 5, indicated that the performance of the discrete-position repetitive control system was not

Spatially Sampled Robust Repetitive Control 67

& Tsao, 2000), can be considered for the ADR system represented in (6). In this section, we first demonstrate that the ADR system with actuator saturation can be formulated into a linear parameter varying (LPV) system. Next, we show that with additional parameterization, LPV gain-scheduling controller synthesis methods (Becker & Packard, 1994; Apkarian et al., 1995) can be applied to the ADR system. Finally, repetitive control and anti-windup (Wu et al., 2000) formulations can be incorporated into the LPV framework to

Assume that the angular velocity described by (5) can be measured in real-time and the input *u* and the output *sat u*( ) of the actuator saturation is accessible. By defining two

1 () and , *sat u*

 

( ) () () (,) ( ) ( ) () () ( ) 0 ()

 

 

( ) , ( , ) , and

 

Equation (15) represents a linear parameter-varying (LPV) system with two varying parameters whose values are accessible in real-time. Strictly speaking, (15) represents a quasi-LPV system since one of the varying parameters (*ρ*) is a function of the system states

> ( ) ( ) ( ) ( ) ( ) 0 ()

 

Affine LPV representation has many desirable properties that can facilitate subsequent controller design. For the quasi-LPV system represented by (15), by defining an augmented

> 

 

*x AB B x z CD D v y CD u*

*v u z zv zu y yv*

 

*x x AB B z CDD v y CD u*

*y yv*

*v vu u*

*B BB B*

 

*v u z zv zu*

 

*u*

 

   

= 1, (15) can be written as an affine LPV

. (16)

(15)

reject spatially periodic disturbances and avoid actuator saturation.

() ,

*A A*

() .

 

> 

*zu zu*

*D D*

 **4.1 State-dependent linear parameter varying (LPV) system** 

Without the actuator saturation constraint, i.e.

varying parameters

we can rewrite (6) as

(Shamma & Athans, 1992).

varying parameter

where

system,

degraded by this variation in the nominal velocity. Fig. 5 also shows that capability of the temporal-based repetitive controller was compromised due to frequency shifting of those periodic disturbances.

Fig. 4. Measured OPC angular velocity during printing.

Fig. 5. Experimental PC velocity variation spectrum.

### **4. Linear parameter varying spatially sampled repetitive control**

Several controller design approaches, e.g., design by linearization as shown previously and design for linear periodic system using the lifting technique (Chen & Francis, 1995; Hanson & Tsao, 2000), can be considered for the ADR system represented in (6). In this section, we first demonstrate that the ADR system with actuator saturation can be formulated into a linear parameter varying (LPV) system. Next, we show that with additional parameterization, LPV gain-scheduling controller synthesis methods (Becker & Packard, 1994; Apkarian et al., 1995) can be applied to the ADR system. Finally, repetitive control and anti-windup (Wu et al., 2000) formulations can be incorporated into the LPV framework to reject spatially periodic disturbances and avoid actuator saturation.

### **4.1 State-dependent linear parameter varying (LPV) system**

Assume that the angular velocity described by (5) can be measured in real-time and the input *u* and the output *sat u*( ) of the actuator saturation is accessible. By defining two varying parameters

$$\rho = \frac{1}{\tilde{\rho}} \text{ and } \phi = \frac{\text{sat}(\tilde{u})}{\tilde{u}} \text{ }$$

we can rewrite (6) as

$$
\begin{bmatrix}
\dot{\tilde{\mathbf{x}}}(\theta) \\
\tilde{z}(\theta) \\
\tilde{y}(\theta)
\end{bmatrix} = \begin{bmatrix}
\tilde{A}(\rho) \\
\hline C\_z \\
 C\_y \\
\end{bmatrix} \begin{bmatrix}
\tilde{B}\_v(\rho) & \tilde{B}\_u(\rho,\theta) \\
D\_{zv} & D\_{zu}(\theta) \\
D\_{yv} & 0
\end{bmatrix} \begin{bmatrix}
\tilde{\mathbf{x}}(\theta) \\
\tilde{v}(\theta) \\
\tilde{u}(\theta)
\end{bmatrix} \tag{15}
$$

where

66 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

degraded by this variation in the nominal velocity. Fig. 5 also shows that capability of the temporal-based repetitive controller was compromised due to frequency shifting of those

> nominal speed

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> 2.7

32

16

**4. Linear parameter varying spatially sampled repetitive control** 

Fig. 4. Measured OPC angular velocity during printing.

After compensation (position-based)

After

compensation (temporal-based)

compensation

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Fig. 5. Experimental PC velocity variation spectrum.

psd

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

psd

psd

angular position (rev)

0 10 20 30 40 50 60 70 80 90 100 0

0 5 10 15 20 25 30 35 40 45 50

48

temporal frequency (Hz)

Several controller design approaches, e.g., design by linearization as shown previously and design for linear periodic system using the lifting technique (Chen & Francis, 1995; Hanson

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>0</sup>

48 Before 96

spatial frequency (cycles/rev)

periodic disturbances.

2.8 2.9 3 3.1 3.2 3.3 3.4 3.5

angular velocity (rad/s)

$$\begin{aligned} \tilde{A}(\rho) &= A\rho, \\ \tilde{B}\_{\upsilon}(\rho) &= B\_{\upsilon}\rho, \; \tilde{B}\_{\iota}(\rho\_{\prime}\phi) = B\_{\iota\iota}\rho\phi\_{\prime} \; \text{and} \\ \tilde{D}\_{zu}(\phi) &= D\_{zu}\phi. \end{aligned}$$

Equation (15) represents a linear parameter-varying (LPV) system with two varying parameters whose values are accessible in real-time. Strictly speaking, (15) represents a quasi-LPV system since one of the varying parameters (*ρ*) is a function of the system states (Shamma & Athans, 1992).

Without the actuator saturation constraint, i.e. = 1, (15) can be written as an affine LPV system,

$$
\begin{bmatrix}
\dot{\tilde{\mathbf{x}}}(\theta) \\
\tilde{z}(\theta) \\
\tilde{y}(\theta)
\end{bmatrix} = \begin{bmatrix}
A\rho & B\_{\upsilon}\rho & B\_{\mu}\rho \\
\hline C\_{z} & D\_{z\upsilon} & D\_{zu} \\
C\_{y} & D\_{y\upsilon} & 0
\end{bmatrix} \begin{bmatrix}
\tilde{\mathbf{x}}(\theta) \\
\tilde{v}(\theta) \\
\tilde{u}(\theta)
\end{bmatrix}. \tag{16}
$$

Affine LPV representation has many desirable properties that can facilitate subsequent controller design. For the quasi-LPV system represented by (15), by defining an augmented varying parameter

> 

such that ( , ) *B BB u uu* , (15) can be represented by a pseudo-affine LPV system with three varying parameters (, , ), i.e.,

$$
\begin{bmatrix}
\dot{\tilde{\mathbf{x}}}(\theta) \\
\tilde{z}(\theta) \\
\tilde{y}(\theta)
\end{bmatrix} = \begin{bmatrix}
\tilde{A}(\rho) & \tilde{B}\_{v}(\rho) & \tilde{B}\_{u}(\eta) \\
\mathbf{C}\_{z} & D\_{zv} & D\_{zu}(\theta) \\
\mathbf{C}\_{y} & D\_{yv} & 0
\end{bmatrix} \begin{bmatrix}
\tilde{\mathbf{x}}(\theta) \\
\tilde{\mathbf{y}}(\theta) \\
\tilde{u}(\theta)
\end{bmatrix}. \tag{17}
$$

Spatially Sampled Robust Repetitive Control 69

() 0 0 0 () ( ) 0 () ( ) 1 0 1 0 () ( ) 1 0 0 0 ()

 

.

(20)

We will briefly summarize the results pertinent to the synthesis of an LPV gain-scheduling controller. Note that these results are originally derived for time-based systems, i.e., using time as the independent variable. However, they are equally applicable for an ADR system

For the LPV system represented by (17), suppose a parameter-dependent output feedback

( ) () () , ( ) () () *K KKK K K*

 

 

 

() () ( ) () ( ) () () ( )

 

*BC A B D*

*C D D CD C D D D D*

 

*u*

 

 

*B*

*A BD C BC B BD D*

*K y K K yv*

*z zu K y zu K zv zu K yv*

0 . <sup>0</sup> () () () 0 0 ()

) forms a parameter vector. Equation (20) is a full-order design in the sense

*<sup>K</sup> x R* . Note that the controller is parameterized by the measurable but

*u K y u K v u K yv*

 

0 () () () 0 0

*A B I*

 

 

*K K*

 

 

in the angular

*y yv K K*

*C D C D*

*x ABx u C D y*

 

varying parameter vector *ψ*, which explains the gain-scheduling characteristics. Define

*cl <sup>K</sup> x xx* , the closed-loop LPV system with (17) and (20) can be expressed as

*x ABx zC D <sup>v</sup>*

 ( ) ( ) ( ) () , ( ) ( ) () () *cl cl cl cl cl cl*

() () ( ) ( ) ( ) () () ( ) () ( ) () ( ) () ( )

*zv zu*

In the above equations, all *I*'s and 0's are identity and zero matrices, respectively, with compatible dimensions for block matrix addition and multiplication. Denote the above LPV

<sup>0</sup> () ( ) *<sup>s</sup> Gs g e d*

The quadratic LPV *γ*-performance problem can be summarized in the following theorem:

.

*D D*

*I*

1 1 2 2

*x x x b a cx z v y u*

 

**4.2 Synthesis of gain-scheduling controller for an affine LPV system** 

using angular displacement as the independent variable.

where  *=* (, , 

where

*<sup>T</sup>*

that *<sup>n</sup> x R* implies *<sup>n</sup>*

*cl cl*

 

 

*A B*

*C D*

*cl cl*

displacement domain to be

dynamic controller is to be designed from *y* to *u* , represented by

() 0 () 000

 

 

*A B*

*z*

*C*

*v*

closed-loop system as *Pcl*. Define the Laplace transform of a signal *g*( )

 

The name pseudo-affine is used since is not an independent parameter but depends on the other two parameters and . The impact of over-parameterizing the parameter space will be discussed in later section. Controller synthesis problem for a pseudo-affine LPV system (17) or an affine LPV system (16) can be reduced to solving a finite set of linear matrix inequalities (LMIs) under conditions satisfied by the parameter variation set and the input/output matrices.

The following example demonstrates the process of reformulating a simple 2nd order motor system model to a pseudo-affine LPV system in the angular displacement domain. Consider a transfer function representation of an LTI model for a permanent magnet brushless dc motor,

$$Z(\mathbf{s}) = Y(\mathbf{s}) + V(\mathbf{s}) = \frac{c}{s^2 + a\mathbf{s} + b} \mathcal{U}(\mathbf{s}) + V(\mathbf{s})\,,\tag{18}$$

where *U s*( ) is the voltage input to the motor, *V s*( ) is the output disturbance, and *Y s*( ) and *Z s*( ) are the undisturbed and disturbed angular position output, respectively. A state space model for (18) can be obtained by defining a set of state variables 1 2 () () () () *T T xt x t <sup>y</sup> <sup>t</sup> <sup>y</sup> <sup>t</sup>* , i.e.,

$$
\begin{array}{c|c|c||c}
\dot{\mathbf{x}}\_1(t) & 0 & 1 & 0 & 0 \\
\dot{\mathbf{x}}\_2(t) & -b & -a & 0 & c \\
z(t) & \begin{bmatrix} -b & -a & 0 & c \\ \hline 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix} & v(t) \\
\mathbf{y}(t) & \mathbf{1} & 0 & 0 & \mathbf{0} \end{array} \tag{19}
$$

Since

$$
\begin{bmatrix}
\dot{\boldsymbol{x}}\_1(t) \\
\dot{\boldsymbol{x}}\_2(t)
\end{bmatrix} = \frac{d\boldsymbol{\theta}(t)}{dt} \begin{bmatrix}
\dot{\tilde{\boldsymbol{x}}}\_1(\boldsymbol{\theta}) \\
\dot{\tilde{\boldsymbol{x}}}\_2(\boldsymbol{\theta})
\end{bmatrix} = o(t) \begin{bmatrix}
\dot{\tilde{\boldsymbol{x}}}\_1(\boldsymbol{\theta}) \\
\dot{\tilde{\boldsymbol{x}}}\_1(\boldsymbol{\theta})
\end{bmatrix},
$$

where (*t*) and (*t*) are the motor angular position and angular velocity, respectively. We can represent (19) as an ADR pseudo-affine LPV system by defining three varying parameters,

$$\begin{array}{c} \varphi = 1/\tilde{\alpha}(\theta) \,, \ \phi = \text{sat}(\tilde{\imath}i)/\tilde{\imath} \,, \ \text{and} \ \eta = \rho\phi \,\ . \end{array}$$

From (17), the associated LPV system can be written as


#### **4.2 Synthesis of gain-scheduling controller for an affine LPV system**

We will briefly summarize the results pertinent to the synthesis of an LPV gain-scheduling controller. Note that these results are originally derived for time-based systems, i.e., using time as the independent variable. However, they are equally applicable for an ADR system using angular displacement as the independent variable.

For the LPV system represented by (17), suppose a parameter-dependent output feedback dynamic controller is to be designed from *y* to *u* , represented by

$$
\begin{bmatrix}
\dot{\tilde{\boldsymbol{x}}}\_{\mathcal{K}}(\boldsymbol{\theta}) \\
\tilde{\boldsymbol{u}}(\boldsymbol{\theta})
\end{bmatrix} = \begin{bmatrix}
\tilde{A}\_{\mathcal{K}}(\boldsymbol{\nu}) & \tilde{B}\_{\mathcal{K}}(\boldsymbol{\nu}) \\
\tilde{\mathcal{C}}\_{\mathcal{K}}(\boldsymbol{\nu}) & \tilde{D}\_{\mathcal{K}}(\boldsymbol{\nu})
\end{bmatrix} \begin{bmatrix}
\tilde{\boldsymbol{x}}\_{\mathcal{K}} \\
\tilde{\boldsymbol{\nu}}
\end{bmatrix} \tag{20}
$$

where  *=* (, , ) forms a parameter vector. Equation (20) is a full-order design in the sense that *<sup>n</sup> x R* implies *<sup>n</sup> <sup>K</sup> x R* . Note that the controller is parameterized by the measurable but varying parameter vector *ψ*, which explains the gain-scheduling characteristics. Define *<sup>T</sup> cl <sup>K</sup> x xx* , the closed-loop LPV system with (17) and (20) can be expressed as

$$
\begin{bmatrix}
\dot{\tilde{\boldsymbol{x}}}\_{cl}(\boldsymbol{\theta}) \\
\tilde{\boldsymbol{z}}(\boldsymbol{\theta})
\end{bmatrix} = \begin{bmatrix}
\tilde{A}\_{cl}(\boldsymbol{\nu}) & \tilde{B}\_{cl}(\boldsymbol{\nu}) \\
\tilde{C}\_{cl}(\boldsymbol{\nu}) & \tilde{D}\_{cl}(\boldsymbol{\nu})
\end{bmatrix} \begin{bmatrix}
\tilde{\boldsymbol{x}}\_{cl}(\boldsymbol{\theta}) \\
\tilde{\boldsymbol{v}}(\boldsymbol{\theta})
\end{bmatrix}'
$$

where

68 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

( ) () () () ( ) ( ) () (). ( ) 0 ()

 

be discussed in later section. Controller synthesis problem for a pseudo-affine LPV system (17) or an affine LPV system (16) can be reduced to solving a finite set of linear matrix inequalities (LMIs) under conditions satisfied by the parameter variation set and the

The following example demonstrates the process of reformulating a simple 2nd order motor system model to a pseudo-affine LPV system in the angular displacement domain. Consider a transfer function representation of an LTI model for a permanent magnet brushless dc

> <sup>2</sup> () () () () () *<sup>c</sup> Zs Ys Vs Us Vs*

where *U s*( ) is the voltage input to the motor, *V s*( ) is the output disturbance, and *Y s*( ) and *Z s*( ) are the undisturbed and disturbed angular position output, respectively. A state space model for (18) can be obtained by defining a set of state variables

1 11 2 2 1 () ( ) ( ) ( ) ( ) ( ) () () *xt x x d t <sup>t</sup> x t dt x x*

 

,

can represent (19) as an ADR pseudo-affine LPV system by defining three varying

*sat u u* ( ) , and

(*t*) are the motor angular position and angular velocity, respectively. We

 .

() 0 1 0 0 () ( ) 0 () () 1 0 1 0 () ( ) 1 0 0 0 ( ( ))

1 1 2 2

*x t x t xt b a c xt z t v t y t sat u t*

*x x AB B z C DD v y CD u*

*y yv*

*v u z zv zu*

, (15) can be represented by a pseudo-affine LPV system with

 

. The impact of over-parameterizing the parameter space will

is not an independent parameter but depends on the

*s as b* , (18)

. (19)

(17)

such that ( , ) *B BB u uu* 

three varying parameters (

other two parameters

input/output matrices.

motor,

Since

where (*t*) and

parameters,

 

The name pseudo-affine is used since

1 2 () () () () *T T xt x t <sup>y</sup> <sup>t</sup> <sup>y</sup> <sup>t</sup>* , i.e.,

 and 

 

 

 1 () , 

From (17), the associated LPV system can be written as

, , ), i.e.,

() () ( ) () ( ) () () ( ) () () ( ) ( ) ( ) () () ( ) () ( ) () ( ) () ( ) () 0 () 000 *u K y u K v u K yv cl cl K y K K yv cl cl z zu K y zu K zv zu K yv v z A BD C BC B BD D A B BC A B D C D C D D CD C D D D D A B C* 0 () () () 0 0 0 . <sup>0</sup> () () () 0 0 () *u K K y yv K K zv zu B A B I I C D C D D D* 

In the above equations, all *I*'s and 0's are identity and zero matrices, respectively, with compatible dimensions for block matrix addition and multiplication. Denote the above LPV closed-loop system as *Pcl*. Define the Laplace transform of a signal *g*( ) in the angular displacement domain to be

$$G(\tilde{\mathbf{s}}) = \int\_0^{\infty} \tilde{\mathbf{g}}(\theta) e^{-\tilde{\mathbf{s}}\theta} d\theta \,.$$

The quadratic LPV *γ*-performance problem can be summarized in the following theorem:

**Theorem 3.1** The LPV closed-loop system Pcl is exponentially stable and the scaled *H* norm of the system is less than a scalar > 0, i.e.,

$$\left\| L^{\mathcal{Y}2} P\_d L^{-\mathcal{Y}2} \right\|\_o = \left\| L^{\mathcal{Y}2} \tilde{\mathcal{C}}\_d(\mathcal{\nu}) (\tilde{\text{s}} I - \tilde{A}\_d(\mathcal{\nu}))^{-1} \tilde{B}\_d(\mathcal{\nu}) + \tilde{D}\_d(\mathcal{\nu}) L^{-\mathcal{Y}2} \right\|\_o < \gamma \tag{21}$$

for all belonging to a parameter variation set , if there exists a symmetric positive definite matrix *X* **R***<sup>n</sup><sup>n</sup>* and a scaling matrix *L* reflecting certain parameter structure such that

$$
\begin{bmatrix}
\tilde{A}\_d^T(\boldsymbol{\nu})\boldsymbol{X} + \boldsymbol{X}\tilde{A}\_d(\boldsymbol{\nu}) & \boldsymbol{X}\tilde{B}\_d(\boldsymbol{\nu}) & \tilde{C}\_d^T(\boldsymbol{\nu}) \\
\tilde{B}\_d^T(\boldsymbol{\nu})\boldsymbol{X} & -\boldsymbol{\gamma}\boldsymbol{L} & \tilde{D}\_d^T(\boldsymbol{\nu}) \\
\tilde{C}\_d(\boldsymbol{\nu}) & \tilde{D}\_d(\boldsymbol{\nu}) & -\boldsymbol{\gamma}\boldsymbol{L}^{-1}
\end{bmatrix} < 0 \tag{22}
$$

Spatially Sampled Robust Repetitive Control 71

**Remark** The scaling matrix *L* in the above theorems takes into account the structural information on the mapping relating input *v* and output *z* in (17), which can include unmodeled dynamics, errors in sensing the varying parameters, and uncertain parameters which can not be measured in real-time. However, the resulting matrix inequalities are nonconvex, mainly due to (26), and computational techniques such as scaling/controller

To simplify the subsequent derivation, we will be conservative and ignore the structural information of the mapping between *v* and *z* . This is equivalent to setting *L* = *J* = *I* and removing the constraint defined by (26) from the above theorem. The advantage of doing so is that (23)-(25) become LMIs in *R* and *S* and the optimization becomes a convex problem that can be solved using numerical solvers based on interior point method, e.g., (Gainet et al., 1995). To check the solvability of the problem for the system given by (17) using

> ( ) , ( ) , ( ) , ( ) , ( ) , ( ) ( ),

 

*A AB B B B C CD D D D*

 

*z z zv zv zu zu*

Since constitutes infinite number of elements, inequalities (23)-(25) pose solvability issue with infinite number of LMI constraints. It was suggested in (Becker & Packard, 1994) that the parameter space be gridded and a controller is synthesized such that it satisfies the solvability conditions at the finite number of parameter values. However, for fixed grid spacing, the number of grid points grows rapidly as the number of parameters increases. Another way to reduce the number of constraints is to take advantage of the properties of

**Definition** An LPV system is polytopic if the state-space matrices of the system depend

1 1

**Proposition** Let *f* : **R** be a convex function where is a convex set with vertices

*i i*

 

*r r*

*v vu u*

 

: 1, 0 ,

 

for all *x* if and only if *f*(

*<sup>i</sup>* is the parameter vector

*i*) < 

*yv* ; that is, those matrices are

for *i*=1, 2,…, *r* (Apkarian et al., 1995). In other

*<sup>i</sup>*'s,

for *i*=1,

*ii i i*

 

, and ( ) *D D yv*

it can be easily shown (using the above proposition) that (23) and (24) in Theorem 3.2 hold if and only if they hold for the matrices corresponding to the vertices of the parameter

> *zv i* ( )

 

( ) , ( ) .

*y y yv yv*

*C CD D*

 

iteration or D/K iteration will be required to solve for matrices *R* and *S*.

Theorem 3.2, the following substitution is used

polytopic LPV systems.

i.e.,

affinely on the varying parameters that lie within a polytope, i.e.,

corresponding to a vertex of the polytope.

*ii i i*

 

: 1, 0

. Then *f*(*x*) <

*v i* ( ) , *C*

 

1 1

, ( ) *C C y y*

*<sup>i</sup>* , *B*

*i i* 

2,…, *r* (Berkovitz, 2002).

i. () 0 *Dyu* 

ii. ( ) *B B u u* 

polytope, i.e., *A*( )

*r r*

where *r* is the number of vertices of the polytope and

For a polytopic LPV system satisfying the following two assumptions:

*z i* ( ) , *D*

constant matrices that are independent of the varying parameters,

; that is, no direct transmission from *u* to *y* ,

, ( ) *D D zu zu*

**Proof:** See (Becker & Packard, 1994) or (Gahinet & Apkarian, 1994).

With the help of the projection lemma and the completion lemma, the following theorem can be derived to provide the necessary and sufficient conditions for the solvability of the (quadratic) LPV *γ*-performance problem stated above.

**Theorem 3.2** For a given , let *NR*() and *NS*() denote orthonormal bases of the null spaces of () () *T T B D u zu* and () () *C D y yv* , respectively. The LPV *γ*-performance problem is solvable if and only if there exist symmetric matrices (*R, S*) **R***<sup>n</sup><sup>n</sup>* and symmetric scaling matrices *L* and *J* such that the following matrix inequalities

$$
\begin{bmatrix} N\_R(\boldsymbol{\nu}) & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{I} \end{bmatrix}^T \begin{bmatrix} \tilde{A}(\boldsymbol{\nu})\boldsymbol{R} + R\tilde{A}^T(\boldsymbol{\nu}) & R\tilde{C}\_z^T(\boldsymbol{\nu}) \\ \tilde{C}\_z(\boldsymbol{\nu})\boldsymbol{R} & -\boldsymbol{\gamma}\boldsymbol{I} \\ \hline \\ \tilde{B}\_v^T(\boldsymbol{\nu}) & \tilde{D}\_{zv}^T(\boldsymbol{\nu}) \end{bmatrix} \begin{bmatrix} \tilde{B}\_v(\boldsymbol{\nu}) & \boldsymbol{0} \\ \tilde{D}\_R(\boldsymbol{\nu}) & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{I} \end{bmatrix} < \boldsymbol{0},\tag{23}
$$

$$
\begin{bmatrix} N\_S(\boldsymbol{\nu}) & \mathbf{0} \\ 0 & I \end{bmatrix}^T \begin{bmatrix} \tilde{A}^T(\boldsymbol{\nu})\mathbf{S} + S\tilde{A}(\boldsymbol{\nu}) & S\tilde{B}\_v(\boldsymbol{\nu}) \\ \tilde{B}\_v^T(\boldsymbol{\nu})\mathbf{S} & -\gamma I & \tilde{D}\_{zv}^T(\boldsymbol{\nu}) \\ \hline \\ \tilde{C}\_z(\boldsymbol{\nu}) & \tilde{D}\_{zv}(\boldsymbol{\nu}) & -\gamma L \end{bmatrix} \begin{bmatrix} N\_S(\boldsymbol{\nu}) & 0 \\ 0 & I \end{bmatrix} < 0,\tag{24}
$$

$$
\begin{bmatrix} R & I \\ I & S \end{bmatrix} \succeq \mathbf{0} \tag{25}
$$

$$\mathbf{L}\mathbf{J} = \mathbf{I} \tag{26}$$

hold for all .

**Proof:** Follows the proof in the appendix of (Gahinet & Apkarian, 1995). The only difference being that most matrices are now parameter dependent.

If the LPV *γ*-performance problem is solvable, the two symmetric matrices *R* and *S* along with the value of and the system matrices ( *A* ( ) , *B <sup>v</sup>* ( ) , *B <sup>u</sup>*( ) , etc.) can be used to synthesize the controller matrices ( *A <sup>K</sup>* ( ) , *B <sup>K</sup>* ( ) , *C <sup>K</sup>* ( ) , and *D <sup>K</sup>* ( ) ) (Becker & Packard, 1994; Gahinet, 1996).

70 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

**Theorem 3.1** The LPV closed-loop system Pcl is exponentially stable and the scaled *H* norm

belonging to a parameter variation set , if there exists a symmetric positive

( ) () 0

 

*<sup>n</sup>* and a scaling matrix *L* reflecting certain parameter structure such

12 12 12 <sup>1</sup> 1 2 ( )( ( )) ( ) ( ) , *L P L L C sI A B D L cl cl cl cl cl* (21)

 

) denote orthonormal bases of the null

*<sup>n</sup>* and symmetric

(23)

(24)

() () *C D y yv* , respectively. The LPV *γ*-performance

 

 

*I S* (25)

*LJ I* (26)

*<sup>u</sup>*( ) 

> *<sup>K</sup>* ( )

, etc.) can be used to

) (Becker & Packard,

(22)

**Proof:** See (Becker & Packard, 1994) or (Gahinet & Apkarian, 1994).

, let *NR*(

scaling matrices *L* and *J* such that the following matrix inequalities

and the system matrices ( *A* ( )

*<sup>K</sup>* ( ) , *B*

being that most matrices are now parameter dependent.

(quadratic) LPV *γ*-performance problem stated above.

 () () *T T B D u zu* and

synthesize the controller matrices ( *A*

hold for all

.

with the value of

1994; Gahinet, 1996).

> 0, i.e.,

> 

*cl cl*

problem is solvable if and only if there exist symmetric matrices (*R, S*) **R***<sup>n</sup>*

 <sup>1</sup>

*A X XA XB C B X LD C DL*

*T T cl cl cl cl T T cl cl*

() () () ()

With the help of the projection lemma and the completion lemma, the following theorem can be derived to provide the necessary and sufficient conditions for the solvability of the

) and *NS*(

() () () () ()0 ()0 ( ) ( ) 0, 0 0

*CR JD I I B DL*

() () () () ()0 ()0 ( ) ( ) 0, 0 0

*B S JD I I C DL*

 

 

0

() ()

*T T T v z S S T T v zv*

*A S SA SB C N N*

 

 

() ()

 

**Proof:** Follows the proof in the appendix of (Gahinet & Apkarian, 1995). The only difference

If the LPV *γ*-performance problem is solvable, the two symmetric matrices *R* and *S* along

*<sup>K</sup>* ( ) , *C*

, *B*

> *<sup>K</sup>* ( )

*<sup>v</sup>* ( ) , *B*

, and *D*

*R I*

*z zv*

*T T v zv*

*T T T z v R R z zv*

*A R RA RC B N N*

 

 

() ()

of the system is less than a scalar

for all

that

definite matrix *X* **R***<sup>n</sup>*

**Theorem 3.2** For a given

spaces of 

**Remark** The scaling matrix *L* in the above theorems takes into account the structural information on the mapping relating input *v* and output *z* in (17), which can include unmodeled dynamics, errors in sensing the varying parameters, and uncertain parameters which can not be measured in real-time. However, the resulting matrix inequalities are nonconvex, mainly due to (26), and computational techniques such as scaling/controller iteration or D/K iteration will be required to solve for matrices *R* and *S*.

To simplify the subsequent derivation, we will be conservative and ignore the structural information of the mapping between *v* and *z* . This is equivalent to setting *L* = *J* = *I* and removing the constraint defined by (26) from the above theorem. The advantage of doing so is that (23)-(25) become LMIs in *R* and *S* and the optimization becomes a convex problem that can be solved using numerical solvers based on interior point method, e.g., (Gainet et al., 1995). To check the solvability of the problem for the system given by (17) using Theorem 3.2, the following substitution is used

$$\begin{split} \tilde{A}(\boldsymbol{\nu}) &= A \, \rho, \, \tilde{B}\_{\boldsymbol{\nu}}(\boldsymbol{\nu}) = B\_{\boldsymbol{\nu}} \rho, \, \tilde{B}\_{\boldsymbol{u}}(\boldsymbol{\nu}) = B\_{\boldsymbol{u}} \boldsymbol{\eta}, \\ \tilde{C}\_{z}(\boldsymbol{\nu}) &= C\_{z'} \, \tilde{D}\_{z\boldsymbol{\nu}}(\boldsymbol{\nu}) = D\_{z\boldsymbol{\nu}'} \, \tilde{D}\_{z\boldsymbol{u}}(\boldsymbol{\nu}) = \tilde{D}\_{z\boldsymbol{u}}(\boldsymbol{\phi}), \\ \tilde{C}\_{y}(\boldsymbol{\nu}) &= C\_{\boldsymbol{y}'} \, \tilde{D}\_{\boldsymbol{y}\boldsymbol{v}}(\boldsymbol{\nu}) = D\_{\boldsymbol{y}\boldsymbol{v}}. \end{split}$$

Since constitutes infinite number of elements, inequalities (23)-(25) pose solvability issue with infinite number of LMI constraints. It was suggested in (Becker & Packard, 1994) that the parameter space be gridded and a controller is synthesized such that it satisfies the solvability conditions at the finite number of parameter values. However, for fixed grid spacing, the number of grid points grows rapidly as the number of parameters increases. Another way to reduce the number of constraints is to take advantage of the properties of polytopic LPV systems.

**Definition** An LPV system is polytopic if the state-space matrices of the system depend affinely on the varying parameters that lie within a polytope, i.e.,

$$\forall \nu \in \Psi = \left\{ \sum\_{i=1}^r \alpha\_i \nu\_i \, : \, \sum\_{i=1}^r \alpha\_i = 1, \, \alpha\_i \ge 0 \right\},$$

where *r* is the number of vertices of the polytope and *<sup>i</sup>* is the parameter vector corresponding to a vertex of the polytope.

**Proposition** Let *f* : **R** be a convex function where is a convex set with vertices *<sup>i</sup>*'s, i.e., 1 1 : 1, 0 *r r ii i i i i* . Then *f*(*x*) < for all *x* if and only if *f*(*i*) < for *i*=1,

2,…, *r* (Berkovitz, 2002).

For a polytopic LPV system satisfying the following two assumptions:


it can be easily shown (using the above proposition) that (23) and (24) in Theorem 3.2 hold if and only if they hold for the matrices corresponding to the vertices of the parameter polytope, i.e., *A*( ) *<sup>i</sup>* , *B v i* ( ) , *C z i* ( ) , *D zv i* ( ) for *i*=1, 2,…, *r* (Apkarian et al., 1995). In other

Spatially Sampled Robust Repetitive Control 73

rad/rev is included to attenuate the controller gain in the high frequency region that is similar to the *q*-filter used in a digital repetitive controller. As shown in Fig. 6, the repetitive controller takes *y* as input and creates a new input <sup>2</sup> *y* to the 'to-be-designed' LPV

To address actuator saturation, an anti-windup scheme as proposed in [36] can be formulated that feeds the difference between the actuator input and output back to the

> *y* <sup>1</sup> ( 1) . *u*

Δ

*v z*

*G*( ) 

*u*

Fig. 6. LPV gain-scheduling control system with repetitive controller.

LPV Controller

2 *y*

*K*( ) 

*RC*

Repetitive Controller

Open-loop LPV System

*y*

 *ni* is

> *r*

*<sup>i</sup>*, and

*<sup>i</sup>* < 1, to ensure

*i* < 

 *ni* rad/rev. The gain of the repetitive controller

*i*. A low-pass filter with roll-off frequency

1 , then 1 *y* 0 and this additional input is

*q*

*W2*

*W1*

, then 1 *y* 0 , which provides additional

where *k* is the number of spatially sinusoidal disturbances that is to be compensated.

the *ith* disturbance frequency in rad/rev. Damping ratios associated with the poles,

*i*, of the repetitive filter need to satisfy the condition 0 <

controller. This corresponds to creating a new input for the LPV controller, i.e.,

*i* and 

zeros, 

controller *K*(

).

sensitivity reduction at spatial frequency

If the control *u* does not saturate, i.e.,

Actuator Saturation

deactivated. If the control *u* saturates, i.e., 1

degree of freedom for manipulating the control *u* .

*p*

*RC*( *s* ) can be adjusted by varying

words, only the 2 1 *r* LMIs corresponding to the vertices of the parameter polytope need to be formed for solving matrices *R* and *S* in Theorem 3.2.

The affine-LPV system represented in (16) qualifies as a polytopic system with matrix 0 *Dyu* and the matrices *A*, *Bv*, and *Bu* depend on the varying parameter . Although the pseudo-affine LPV system represented by (17) has similar structure as (16) with 0 *Dyu* and the matrices *A* , *Bv* , *Bu* , and *Dzu* depend on varying parameters, it is not polytopic due to the dependency of the varying parameter *η* on the two varying parameters and . In such cases, a polytope can usually be found to bound and replace the parameter variation set. To satisfy the assumption (ii), parameter dependency of the *Bu* or the *Bu* matrix can be removed by filtering the input channel, as will be discussed in section 3.3. For systems without direct transmission between *u* and *z* , e.g. the brushless dc motor system demonstrated in section 3.1, it is easy to verify that 0 *D D zu zu* .

### **4.3 Incorporating spatial-sampled repetitive control and actuator anti-windup**

The overall control structure is summarized in Fig. 6. Here *G*( ) along with the actuator saturation block represents the pseudo-affine LPV system, denotes the modeling uncertainty, and *W*1 and *W*2 are weighting filters whose frequency-dependent magnitudes are used to bound the performance specifications and model uncertainty. The repetitive controller is denoted by *RC* and the LPV controller to be designed is denoted by *K*(). The open-loop LPV system (within the dashed-line block in Fig. 6) can be expressed as

$$
\begin{bmatrix}
\dot{\tilde{x}}(\theta) \\
\tilde{q}(\theta) \\
\tilde{z}(\theta) \\
\tilde{y}(\theta) \\
\end{bmatrix} = \begin{bmatrix}
\tilde{A}(\rho) & \tilde{B}\_{p}(\rho) & \tilde{B}\_{v}(\rho) & \tilde{B}\_{u}(\eta) \\
\hline \mathbb{C}\_{q} & D\_{qp} & D\_{qv} & \tilde{D}\_{qu}(\theta) \\
\mathbb{C}\_{z} & D\_{zp} & D\_{zv} & \tilde{D}\_{zu}(\phi) \\
\mathbb{C}\_{y} & D\_{yp} & D\_{yv} & 0 \\
\end{bmatrix} \begin{bmatrix}
\tilde{x}(\theta) \\
\tilde{p}(\theta) \\
\tilde{v}(\theta) \\
\tilde{u}(\theta) \\
\end{bmatrix} \tag{27}
$$

which differs from (17) in that unstructured model uncertainty, connecting output *q* to input *p* , and weighting filters are also incorporated. The current formulation considers two types of perturbations. One is due to the varying parameters, which is bounded and can be measured in real-time. The other is due to modeling error, which is also bounded but can not be measured in real-time. In (27), without actuator saturation constraint, i.e. = 1, we have = and the matrices *Dqu* and *Dzu* become constant matrices and independent of varying parameters.

To account for spatially periodic disturbances, we will consider a low-order and attenuated spatial-based repetitive controller that takes the form of

$$RC(\tilde{\mathbf{s}}) = \frac{1}{\tilde{\mathbf{s}}/o\_r + 1} \prod\_{i=1}^{k} \frac{\tilde{\mathbf{s}}^2 + 2\zeta\_i o\_{ni}\tilde{\mathbf{s}} + o\_{ni}^2}{\tilde{\mathbf{s}}^2 + 2\zeta\_i o\_{ni}\tilde{\mathbf{s}} + o\_{ni}^2}$$

or equivalently in state space representation

$$\begin{aligned} \dot{\tilde{\mathfrak{X}}}\_{rc} &= A\_{rc} \tilde{\mathfrak{X}}\_{rc} + B\_{rc} \tilde{\mathfrak{Y}}\_{rc} \\ \tilde{\mathfrak{Y}}\_2 &= C\_{rc} \tilde{\mathfrak{X}}\_{rc} \end{aligned} \prime$$

72 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

words, only the 2 1 *r* LMIs corresponding to the vertices of the parameter polytope need

The affine-LPV system represented in (16) qualifies as a polytopic system with matrix

pseudo-affine LPV system represented by (17) has similar structure as (16) with 0 *Dyu* and

such cases, a polytope can usually be found to bound and replace the parameter variation

removed by filtering the input channel, as will be discussed in section 3.3. For systems without direct transmission between *u* and *z* , e.g. the brushless dc motor system

saturation block represents the pseudo-affine LPV system, denotes the modeling uncertainty, and *W*1 and *W*2 are weighting filters whose frequency-dependent magnitudes are used to bound the performance specifications and model uncertainty. The repetitive

() () () () ( ) ( )

*AB B B <sup>x</sup> <sup>x</sup> q C D DD p z C D DD v y u CD D*

 

*pv u q qp qv qu z zp zv zu*

( ) ( ) ( ) , ( ) ( ) ( )

( ) ( ) 0

which differs from (17) in that unstructured model uncertainty, connecting output *q* to input *p* , and weighting filters are also incorporated. The current formulation considers two types of perturbations. One is due to the varying parameters, which is bounded and can be measured in real-time. The other is due to modeling error, which is also bounded but can

To account for spatially periodic disturbances, we will consider a low-order and attenuated

1

*rc rc rc rc rc rc x Ax By*

 ,

*s s s*

*r i i ni ni*

<sup>1</sup> <sup>2</sup> ( ) <sup>1</sup> <sup>2</sup> *k*

2

*y Cx*

*s s RC s*

2 2 2 2

 

 

*i ni ni*

 

 

*y yp yv*

not be measured in real-time. In (27), without actuator saturation constraint, i.e.

*Dqu* and

the dependency of the varying parameter *η* on the two varying parameters

**4.3 Incorporating spatial-sampled repetitive control and actuator anti-windup** 

controller is denoted by *RC* and the LPV controller to be designed is denoted by *K*(

open-loop LPV system (within the dashed-line block in Fig. 6) can be expressed as

set. To satisfy the assumption (ii), parameter dependency of the

demonstrated in section 3.1, it is easy to verify that 0 *D D zu zu* .

The overall control structure is summarized in Fig. 6. Here *G*( )

depend on the varying parameter

*Dzu* depend on varying parameters, it is not polytopic due to

*Dzu* become constant matrices and independent of

*Bu* or the *Bu*

 and . In

along with the actuator

(27)

= 1, we

. Although the

matrix can be

). The

to be formed for solving matrices *R* and *S* in Theorem 3.2.

, *Bv*, and *Bu*

*Bu* , and

spatial-based repetitive controller that takes the form of

and the matrices

or equivalently in state space representation

0 *Dyu* and the matrices *A*

*Bv* ,

the matrices *A* ,

have = 

varying parameters.

where *k* is the number of spatially sinusoidal disturbances that is to be compensated.  *ni* is the *ith* disturbance frequency in rad/rev. Damping ratios associated with the poles, *<sup>i</sup>*, and zeros, *i*, of the repetitive filter need to satisfy the condition 0 < *i* < *<sup>i</sup>* < 1, to ensure sensitivity reduction at spatial frequency  *ni* rad/rev. The gain of the repetitive controller *RC*( *s* ) can be adjusted by varying *i* and *i*. A low-pass filter with roll-off frequency *r* rad/rev is included to attenuate the controller gain in the high frequency region that is similar to the *q*-filter used in a digital repetitive controller. As shown in Fig. 6, the repetitive controller takes *y* as input and creates a new input <sup>2</sup> *y* to the 'to-be-designed' LPV controller *K*().

To address actuator saturation, an anti-windup scheme as proposed in [36] can be formulated that feeds the difference between the actuator input and output back to the controller. This corresponds to creating a new input for the LPV controller, i.e.,

$$
\tilde{y}\_1 = (\phi - 1)\tilde{\mu}.
$$

If the control *u* does not saturate, i.e., 1 , then 1 *y* 0 and this additional input is deactivated. If the control *u* saturates, i.e., 1 , then 1 *y* 0 , which provides additional degree of freedom for manipulating the control *u* .

Fig. 6. LPV gain-scheduling control system with repetitive controller.

The open-loop LPV system with repetitive and anti-windup control (within the dashed-line block in Fig. 6) can be shown to have the following state-space representation:

$$
\begin{bmatrix}
\dot{\tilde{x}}(\theta) \\
\dot{\tilde{x}}\_{rc}(\theta) \\
\tilde{y}(\theta) \\
\tilde{z}(\theta) \\
\tilde{y}(\theta) \\
\tilde{y}\_1(\theta) \\
\tilde{y}\_2(\theta)
\end{bmatrix} = \begin{bmatrix}
\tilde{A}(\rho) & 0 & \tilde{B}\_p(\rho) & \tilde{B}\_v(\rho) & \tilde{B}\_u(\eta) \\
B\_{rc}C\_y & A\_{rc} & B\_{rd}D\_{yp} & B\_{rc}D\_{yr} & 0 \\
C\_q & 0 & D\_{qp} & D\_{qr} & D\_{qu}(\theta) \\
C\_z & 0 & D\_{pp} & D\_{zv} & \tilde{D}\_{zu}(\theta) \\
C\_y & 0 & D\_{yp} & D\_{pv} & 0 \\
0 & 0 & 0 & 0 & \phi - 1 \\
\tilde{y}\_2(\theta) \\
\end{bmatrix} \begin{bmatrix}
\tilde{x}(\theta) \\
\tilde{x}\_v(\theta) \\
\tilde{y}(\theta) \\
\tilde{z}(\theta) \\
\tilde{u}(\theta) \\
\end{bmatrix}. \tag{28}
$$

Spatially Sampled Robust Repetitive Control 75

0 0 0 0 0 0 00 0 0 00 0 0 0 0

*z zu i zp zv*

3

*o*

*C*

*x A BC B B x BC A BD BD x BC A BD BD*

( ) () 0 0 0 0 () () () 0 ( ) 000 0 0 ( ) 0 00 0 0

1 1 1 1 1

2

*o*

*C*

Open-loop LPV System

*<sup>v</sup> <sup>z</sup>*

*p q*

Δ

*rc rc y rc rc yp rc yv o o y o o yp o yv*

0 00 0 0 0 0 ( ) 0 0 000 0 0 ( )

*B C A x*

3 3 2

*o rc o o*

0 0 0 0 () 0 ( ) 0 0 0 0 () 0 ( )

*C v*

*y*

,

*T*

, ˆˆ ˆ , , , ˆ

*H3*

2 *y*ˆ

*X xx x x x x*

*rc o o o i*

2

Fig. 7. LPV gain-scheduling control system with repetitive controller, anti-windup scheme,

 123

1 2

*Z qz Y yy y V pv Uu*

*T TT*

*y*ˆ

1 *y*ˆ

*K*( ) 

*H1*

<sup>1</sup> *y RC*

2 *y*

Open-loop LPV System with RC and Anti-windup

0 0 00 0 0 0 0 ( )

*q qu i qp qv i*

*C DC D D x C DC D D p*

 

 

2 1

( 1) 0 0 0 ( )

*C x*

*i i o*

*A B x*

*i o*

( ) ( )

 

 

*x x*

*rc*

.

(29)

3

 

*u*ˆ( )

*ui p v*

 

 

( ) ( ) ( ) ( ) ˆ( ) ˆ ( ) ˆ ( )

 

3

*o i*

*x x q z y y y*

1 2

2 0 2

1

*o*

( ) 0 00 0

*x A B*

*o o*

*u*

*u*ˆ

and sensor/actuator dynamics.

By making the following definitions

*F H2*

Note that the LPV controller *K*() now has three inputs 1 2 (, , ) *y y y* and one output *u* . The parameter dependency of the input and output matrices (e.g., *Bu* and *Cy* ), if any, can be removed by considering the dynamics of the sensors and actuators (Apkarian et al., 1995). Let

$$
\begin{bmatrix}
\dot{\tilde{\mathbf{x}}}\_{o1} \\
\dot{\tilde{\mathbf{x}}}\_{o2} \\
\dot{\tilde{\mathbf{x}}}\_{o3}
\end{bmatrix} = \begin{bmatrix}
A\_{o1} & 0 & 0 \\
0 & A\_{o2} & 0 \\
0 & 0 & A\_{o3}
\end{bmatrix} \begin{bmatrix}
\tilde{\mathbf{x}}\_{o1} \\
\tilde{\mathbf{x}}\_{o2} \\
\tilde{\mathbf{x}}\_{o3}
\end{bmatrix} + \begin{bmatrix}
B\_{o1} & 0 & 0 \\
0 & B\_{o2} & 0 \\
0 & 0 & B\_{o3}
\end{bmatrix} \begin{bmatrix}
\tilde{y} \\
\tilde{y}\_1 \\
\tilde{y}\_2
\end{bmatrix},
$$

$$
\begin{bmatrix}
\dot{\tilde{y}} \\
\hat{y}\_1 \\
\dot{\tilde{y}}\_2
\end{bmatrix} = \begin{bmatrix}
\mathbf{C}\_{o1} & 0 & 0 \\
0 & \mathbf{C}\_{o2} & 0 \\
0 & 0 & \mathbf{C}\_{o3}
\end{bmatrix} \begin{bmatrix}
\tilde{\mathbf{x}}\_{o1} \\
\tilde{\mathbf{x}}\_{o2} \\
\tilde{\mathbf{x}}\_{o3}
\end{bmatrix},
$$

where 1 2 (, , ) *yy y* ˆˆ ˆ represent the new outputs, and

$$\begin{aligned} \dot{\tilde{x}}\_i &= A\_i \tilde{x}\_i + B\_i \hat{u}\_{\prime} \\ \tilde{u} &= \mathbf{C}\_i \tilde{x}\_{i\prime} \end{aligned}$$

where *u*ˆ represent the new input. This action is equivalent to passing each input or output channel of the open-loop LPV system in (28) through a low-pass filter

$$\begin{aligned} H\_j(\tilde{\mathbf{s}}) &= \mathbf{C}\_{\circ j} (\tilde{\mathbf{s}}I - A\_{\circ j})^{-1} B\_{\circ j} \; \; j = \mathbf{1} \text{, } \mathbf{2} \text{, } \mathbf{3} \text{ or } \\ F(\tilde{\mathbf{s}}) &= \mathbf{C}\_i (\tilde{\mathbf{s}}I - A\_i)^{-1} B\_i \end{aligned}$$

respectively, before connecting to the LPV controller *K*(), as depicted in Fig. 7. The bandwidth of the low-pass filters depends on the sensor and actuator dynamics. For negligible senor or actuator dynamics, the bandwidth can be assigned to be much larger than that of the open-loop system to minimize possible interference. With the inclusion of the anti-windup formulation and the input/output filters, the overall open-loop LPV system with parameter-free input-output matrices can be found to be

74 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

The open-loop LPV system with repetitive and anti-windup control (within the dashed-line

<sup>0</sup> ( ) ( )

*BC A BD BD x*

*q qp qv qu z zp zv zu*

removed by considering the dynamics of the sensors and actuators (Apkarian et al., 1995).

2 2 2 21 3 3 3 3 2

> 1 22 2 3 3

*yC x y Cx y C x* 

ˆ 0 0,

, *i ii i i i x Ax Bu u Cx* 

1

,

1 ( ) ( ) , 1,2,3 or

bandwidth of the low-pass filters depends on the sensor and actuator dynamics. For negligible senor or actuator dynamics, the bandwidth can be assigned to be much larger than that of the open-loop system to minimize possible interference. With the inclusion of the anti-windup formulation and the input/output filters, the overall open-loop LPV system

 

where *u*ˆ represent the new input. This action is equivalent to passing each input or output

*x A xB y x A x By x A x B y* 

0 0 0 0

ˆ 0 0

ˆ 0 0

channel of the open-loop LPV system in (28) through a low-pass filter

() ( )

 

respectively, before connecting to the LPV controller *K*(

with parameter-free input-output matrices can be found to be

*F s C sI A B*

*j oj oj oj i ii*

*H s C sI A B j*

0 0 0 0 0 0 0 0,

1 1

ˆ,

 

*o o o o o o*

*pvu*

) now has three inputs 1 2 (, , ) *y y y* and one output *u* . The

( ) ( ) . ( ) ( )

(28)

and *Cy* ), if any, can be

 

), as depicted in Fig. 7. The

*rc p v u*

block in Fig. 6) can be shown to have the following state-space representation:

*rc y rc rc yp rc yv rc*

*z C D DD*

 

*rc*

1 1 1 1

*o o o o o oo o o o o o*

*y C DD*

() 0 () () () ( )

*A BBB*

( ) 0 ( ) ( ) 0 ( ) ( ) 0 0 ( ) 00 0 0 1 ( ) 0 000

*q C D DD x*

 

*y yp yv*

1 2

*<sup>y</sup> <sup>C</sup>*

parameter dependency of the input and output matrices (e.g., *Bu*

*y*

Note that the LPV controller *K*(

Let

 

*x x*

 

where 1 2 (, , ) *yy y* ˆˆ ˆ represent the new outputs, and

 1 1 1 1 1 2 0 2 3 1 2 ( ) () 0 0 0 0 () () () 0 ( ) 000 0 0 ( ) 0 00 0 0 ( ) 0 00 0 ( ) ( ) ( ) ( ) ˆ( ) ˆ ( ) ˆ ( ) *ui p v rc rc y rc rc yp rc yv o o y o o yp o yv o o o i x A BC B B x BC A BD BD x BC A BD BD x A B x x q zy y y* 2 1 3 3 2 3 1 2 3 ( ) ( ) ( 1) 0 0 0 ( ) 0 00 0 0 0 0 ( ) 0 0 000 0 0 ( ) 0 0 0 0 () 0 ( ) 0 0 0 0 () 0 ( ) 0 0 00 0 0 0 0 ( ) 0 0 0 0 0 0 00 0 0 00 0 0 0 0 *rc i o o rc o o i i o q qu i qp qv i z zu i zp zv o o o x x C x B C A x A B x C DC D D x C DC D D p <sup>C</sup> vC C* . *u*ˆ( ) (29)

Fig. 7. LPV gain-scheduling control system with repetitive controller, anti-windup scheme, and sensor/actuator dynamics.

By making the following definitions

$$\begin{aligned} \tilde{X} &= \begin{bmatrix} \tilde{\boldsymbol{x}} & \tilde{\boldsymbol{x}}\_{rc} & \tilde{\boldsymbol{x}}\_{o1} & \tilde{\boldsymbol{x}}\_{o2} & \tilde{\boldsymbol{x}}\_{o3} & \tilde{\boldsymbol{x}}\_{i} \end{bmatrix}^{\boldsymbol{T}}, \\ \tilde{\boldsymbol{Z}} &= \begin{bmatrix} \tilde{\boldsymbol{q}} & \tilde{\boldsymbol{z}} \end{bmatrix}^{\boldsymbol{T}}, \ \tilde{\boldsymbol{Y}} = \begin{bmatrix} \hat{\boldsymbol{y}} & \hat{\boldsymbol{y}}\_{1} & \hat{\boldsymbol{y}}\_{2} \end{bmatrix}^{\boldsymbol{T}}, \ \tilde{\boldsymbol{V}} = \begin{bmatrix} \tilde{\boldsymbol{p}} & \tilde{\boldsymbol{v}} \end{bmatrix}^{\boldsymbol{T}}, \ \tilde{\boldsymbol{U}} = \begin{bmatrix} \boldsymbol{\tilde{u}}\_{r} \end{bmatrix} \end{aligned}$$

we can rewrite the above system as

$$
\begin{bmatrix}
\dot{\tilde{X}} \\
\tilde{Z} \\
\tilde{Z} \\
\tilde{Y}
\end{bmatrix} = \begin{bmatrix}
A(\boldsymbol{\nu}) & B\_{\boldsymbol{\nu}}(\boldsymbol{\nu}) & B\_{\boldsymbol{\nu}} \\
C\_{Z}(\boldsymbol{\nu}) & D\_{Z\boldsymbol{\nu}}(\boldsymbol{\nu}) & D\_{Z\boldsymbol{\nu}} \\
& C\_{\boldsymbol{\nu}} & D\_{\boldsymbol{\nu}\boldsymbol{\nu}} & 0
\end{bmatrix} \begin{bmatrix}
\tilde{X} \\
\tilde{V} \\
\tilde{\mathcal{U}} \\
\tilde{\mathcal{U}}
\end{bmatrix} \tag{30}
$$

Spatially Sampled Robust Repetitive Control 77

Motor Driver

**Motor**

counter register

decoder circuit

Fig. 8. Experimental setup for the closed-loop control of a typical 600-dpi monochrome laser

The output multiplicative modeling errors are obtained by comparing the frequency responses of the plant model and the experimental platform, as shown in Fig. 9. Note that the spatial frequency response shown in Fig. 9 is obtained from the temporal frequency response where the spatial frequency in cycle per revolution is scaled by the nominal angular velocity. A stable 1st order filter that upper bounds the multiplicative model

16 1 ( ) 0.03 . 700 1

m ultiplicative error for 2nd order fit m ultiplicative error for 3rd order fit

*s* 

10<sup>1</sup>

<sup>10</sup><sup>2</sup> <sup>10</sup>-2

Fig. 9. Output multiplicative uncertainties for the experimental platform approximated using a 2nd or 3rd order transfer function. The solid line is the magnitude of a 1st order filter

frequency (cycles/rev)

BLDC

Control Algorithm (firmware)

D/A

D/A

**Encoder**

uncertainty can be found to be

10-1

magnitude

that upper bounds the uncertainties.

10<sup>0</sup>

printer.

ENC ENC OPC

LaserPack

ELECTRONICS

2

10<sup>0</sup>

*<sup>s</sup> W s*

where the system and input/output matrices are of appropriate dimensions and can be identified from (2). Note that the matrices *BU* and *CY* are free of varying parameters.

### **4.4 Discretization of angular displacement reformulated systems**

A spatial sampling scheme that uses the output pulses of an optical shaft encoder (instead of a clock signal) to trigger the interrupt of the control algorithm at intervals of equal angular displacement was implemented. The constant angular displacement based sampling effectively discretized the control system in the angular displacement domain. Note that an ADR system, see (15), without varying parameters can be viewed as an LTI system with angular displacement as the independent variable as compared with time *t*. Theorems or methods used to derive the discrete equivalent of LTI systems, e.g., z-transform (impulse invariant), zero-order hold (step invariant), and bilinear or trapezoid rule, can be applied to ADR systems with slightest modification. What needs to be kept in mind is that the sampling behavior has changed from equal time interval (in sec) to equal angular displacement interval (in revolution).

### **4.5 Experimental setup and validation**

Rotational velocity regulation in a laser printer will be used to verify the effectiveness of the proposed spatially sampled repetitive control in rejecting spatially periodic disturbances. A 600-dpi monochrome laser printer is used as the experimental platform that comprised of one brushless dc motor, with a set of gear and a photosensitive drum. The hardware setup is depicted in Fig. 8. The motor velocity is regulated by adjusting the voltage input to a pulse width modulated (PWM) power drive. A digital encoder with a resolution of 50,000 pulses/rev is mounted on the photosensitive drum to measurement of angular displacement and velocity. To maintain the desired dot placement accuracy, the photosensitive drum is expected to rotate at a nominal angular velocity of 0.5 rev/sec. This corresponds to a motor voltage input of 2.56 volts. The saturation limits for the input voltage are identified to be 0.5 volts around the nominal value, i.e., max *u* 3.06 and min *u* 2.06 . According to the frequency spectrum of the measured speed fluctuations, spatially periodic components at spaitial frequencies of 32, 48, and 96 cycles/rev need to be reduced, since they caused visible bands in printed images. A 2nd order transfer function from the motor voltage input to the drum angular velocity output is obtained to approximate the actual frequency response of the experimental platform, i.e.

$$P\_{yn}(\text{s}) = \frac{4.184 \times 10^5}{\text{s}^2 + 2246 \text{s} + 8.932 \times 10^4} \text{.}$$

76 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

 

where the system and input/output matrices are of appropriate dimensions and can be

A spatial sampling scheme that uses the output pulses of an optical shaft encoder (instead of a clock signal) to trigger the interrupt of the control algorithm at intervals of equal angular displacement was implemented. The constant angular displacement based sampling effectively discretized the control system in the angular displacement domain. Note that an ADR system, see (15), without varying parameters can be viewed as an LTI system with

methods used to derive the discrete equivalent of LTI systems, e.g., z-transform (impulse invariant), zero-order hold (step invariant), and bilinear or trapezoid rule, can be applied to ADR systems with slightest modification. What needs to be kept in mind is that the sampling behavior has changed from equal time interval (in sec) to equal angular

Rotational velocity regulation in a laser printer will be used to verify the effectiveness of the proposed spatially sampled repetitive control in rejecting spatially periodic disturbances. A 600-dpi monochrome laser printer is used as the experimental platform that comprised of one brushless dc motor, with a set of gear and a photosensitive drum. The hardware setup is depicted in Fig. 8. The motor velocity is regulated by adjusting the voltage input to a pulse width modulated (PWM) power drive. A digital encoder with a resolution of 50,000 pulses/rev is mounted on the photosensitive drum to measurement of angular displacement and velocity. To maintain the desired dot placement accuracy, the photosensitive drum is expected to rotate at a nominal angular velocity of 0.5 rev/sec. This corresponds to a motor voltage input of 2.56 volts. The saturation limits for the input voltage are identified to be 0.5 volts around the nominal value, i.e., max *u* 3.06 and min *u* 2.06 . According to the frequency spectrum of the measured speed fluctuations, spatially periodic components at spaitial frequencies of 32, 48, and 96 cycles/rev need to be reduced, since they caused visible bands in printed images. A 2nd order transfer function from the motor voltage input to the drum angular velocity output is obtained to

approximate the actual frequency response of the experimental platform, i.e.

*P s yu s s*

5

2 4 4.184 10 ( ) . 2246 8.932 10

 

*X X AB B ZC D DV Y C D U*

() ()

 

*Z ZV ZU Y YV*

() () ,

*V U*

0

as the independent variable as compared with time *t*. Theorems or

(30)

identified from (2). Note that the matrices *BU* and *CY* are free of varying parameters.

**4.4 Discretization of angular displacement reformulated systems** 

we can rewrite the above system as

angular displacement

displacement interval (in revolution).

**4.5 Experimental setup and validation** 

Fig. 8. Experimental setup for the closed-loop control of a typical 600-dpi monochrome laser printer.

The output multiplicative modeling errors are obtained by comparing the frequency responses of the plant model and the experimental platform, as shown in Fig. 9. Note that the spatial frequency response shown in Fig. 9 is obtained from the temporal frequency response where the spatial frequency in cycle per revolution is scaled by the nominal angular velocity. A stable 1st order filter that upper bounds the multiplicative model uncertainty can be found to be

$$\mathcal{W}\_2(\mathfrak{s}) = 0.03 \frac{\mathfrak{s}/16 + 1}{\mathfrak{s}/700 + 1} \text{.} $$

Fig. 9. Output multiplicative uncertainties for the experimental platform approximated using a 2nd or 3rd order transfer function. The solid line is the magnitude of a 1st order filter that upper bounds the uncertainties.

Spatially Sampled Robust Repetitive Control 79

variation from –80 % to +100% around the nominal value of 0.5 rev/sec and a 10-to-1 saturation limit, respectively. The parameter variation set is not convex but can be

*W*1 can be properly determined to reflect the different performance requirement for the

*b* as 

<sup>1</sup> () () () () (2 1000) 1

10-2 <sup>100</sup> <sup>102</sup> <sup>104</sup> 10-2

frequency (cycles/rev)

Fig. 11. The parameter-dependent performance weighting *W1* and uncertainty weighting *W2*. where the frequency value of 1000 cycles/rev is specified to reflect the negligible sensor and actuator dynamics. The low-order attenuated repetitive controller can be expressed as

> 32,48,96 0.002,0.002,0.001 <sup>1</sup> 2 0.1 ( ) , (2 200) 1 <sup>2</sup> *<sup>n</sup>*

where the periodic disturbances are at 32, 48 and 96 cycles/rev. A feasible LPV controller is determined based on the above parameters, which attains *γ* = 1.1669. The controller can be

*s s*

*n*

*s s RC s*

*W1 W2*

1 *s* 

are empirically determined based on a velocity

(1,0.1,0.1) (see Fig. 10). The polytope will be used

[0.1, 1] in , the parameters of the weighting filter

< 1) system. Fig. 11 shows the magnitude curves of *W*<sup>1</sup>

2 2 2 2

 

*n n nn n*

  [0.1, 1]. The magnitude curve of *W*<sup>2</sup>

(10,1,10) ,

 and 

123

0.1

*s*

written as

*H s H s H s Fs*

shown to lie within a polytope with four vertices located at 1

*b* and *e*=5/3×

is also shown in the figure. The low-pass filters ( ) *H s <sup>j</sup>* and *F s*( ) are selected as

The upper and lower bounds of

10-1

100

101

for the following design. Given that

(1,1,1) , and 4

= 1) and saturated (

*b* = 212, *b* = 0.1

2 

(10,0.1,1) , 3

unsaturated (

with *k* = 0.03,

Note that the affine nature of the open-loop LPV system (29) will be intact after inclusion of the parameter independent filter, *W*2. The selection of *W*1 requires more considerations. First of all, the LPV controller will be independent of the saturation indicator if the performance weighting does not depend on [36]. In other words, the design problem using parameterfree *W*1 will degenerate to one without actuator saturation. Secondly, if a parameter dependent *W*1 filter is chosen, the affine nature of the LPV open-loop system will be preserved after incorporating the filter. Thus, a feasible *W*1 filter can assume the following state space realization

$$\begin{aligned} \dot{\tilde{\mathbf{x}}}\_{w1} &= -(a\flat\_{b}\phi + b)\tilde{\mathbf{x}}\_{w1} + \sqrt{e}\tilde{z}\_{w2}, \\ \hat{z} &= \sqrt{e}\phi \tilde{\mathbf{x}}\_{w1} + k\tilde{z}\_{w2} \end{aligned}$$

which has the transfer function

$$\mathcal{W}\_1(\tilde{\mathbf{s}}) = k + \frac{e\phi}{\tilde{\mathbf{s}} + a\_b \phi + b} \cdot \mathbf{s}$$

Note that the magnitude curve of *W*1 can be specified by tuning the constant values of *<sup>b</sup>* , *b*, *e*, and *k*. Specifically, *k* can be used to specify the lower bound for the *W*1 magnitude at high frequencies (i.e., as *s* ); the coefficient *b* can be used to specify the lower bound of the corner frequencies; coefficients *<sup>b</sup>* and *e* can be used to specify the exact corner frequencies and the *W*1 magnitude at low frequencies (i.e., as 0 *s* ). The parameter variation set is determined to be

$$\overline{\Psi} = \{ (\rho, \phi, \eta) \colon 1 \le \rho \le 10, \, 0.1 \le \phi \le 1, \, \eta = \rho \phi \}\dots$$

Fig. 10. Parameter variation set and the selected (convex) polytope which bounds the set.

78 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Note that the affine nature of the open-loop LPV system (29) will be intact after inclusion of the parameter independent filter, *W*2. The selection of *W*1 requires more considerations. First

free *W*1 will degenerate to one without actuator saturation. Secondly, if a parameter dependent *W*1 filter is chosen, the affine nature of the LPV open-loop system will be preserved after incorporating the filter. Thus, a feasible *W*1 filter can assume the following

1

1( ) .

*e*, and *k*. Specifically, *k* can be used to specify the lower bound for the *W*1 magnitude at high frequencies (i.e., as *s* ); the coefficient *b* can be used to specify the lower bound of the

and the *W*1 magnitude at low frequencies (i.e., as 0 *s* ). The parameter variation set is

Fig. 10. Parameter variation set and the selected (convex) polytope which bounds the

*x b x ez z e x kz* 

ˆ , *w bw w*

*<sup>e</sup> Ws k*

Note that the magnitude curve of *W*1 can be specified by tuning the constant values of

( , , ) : 1 10, 0.1 1,

 

() ,

*b*

 

.

*s b* 

  [36]. In other words, the design problem using parameter-

*<sup>b</sup>* and *e* can be used to specify the exact corner frequencies

  if the performance

*<sup>b</sup>* , *b*,

of all, the LPV controller will be independent of the saturation indicator

 

1 1

weighting does not depend on

which has the transfer function

corner frequencies; coefficients

determined to be

set.

state space realization

The upper and lower bounds of and are empirically determined based on a velocity variation from –80 % to +100% around the nominal value of 0.5 rev/sec and a 10-to-1 saturation limit, respectively. The parameter variation set is not convex but can be shown to lie within a polytope with four vertices located at 1 (10,1,10) , 2 (10,0.1,1) , 3 (1,1,1) , and 4 (1,0.1,0.1) (see Fig. 10). The polytope will be used for the following design. Given that [0.1, 1] in , the parameters of the weighting filter *W*1 can be properly determined to reflect the different performance requirement for the unsaturated ( = 1) and saturated ( < 1) system. Fig. 11 shows the magnitude curves of *W*<sup>1</sup> with *k* = 0.03, *b* = 212, *b* = 0.1*b* and *e*=5/3×*b* as [0.1, 1]. The magnitude curve of *W*<sup>2</sup> is also shown in the figure. The low-pass filters ( ) *H s <sup>j</sup>* and *F s*( ) are selected as

$$H\_1(\tilde{\mathbf{s}}) = H\_2(\tilde{\mathbf{s}}) = H\_3(\tilde{\mathbf{s}}) = F(\tilde{\mathbf{s}}) = \frac{1}{\tilde{s}/(2\pi \times 1000) + 1}$$

Fig. 11. The parameter-dependent performance weighting *W1* and uncertainty weighting *W2*. where the frequency value of 1000 cycles/rev is specified to reflect the negligible sensor and actuator dynamics. The low-order attenuated repetitive controller can be expressed as

2 2 2 2 32,48,96 0.002,0.002,0.001 <sup>1</sup> 2 0.1 ( ) , (2 200) 1 <sup>2</sup> *<sup>n</sup> n n n nn n s s RC s s s s* 

where the periodic disturbances are at 32, 48 and 96 cycles/rev. A feasible LPV controller is determined based on the above parameters, which attains *γ* = 1.1669. The controller can be written as

$$
\begin{bmatrix}
\dot{\tilde{\boldsymbol{x}}}\_{k} \\
\dot{\tilde{\boldsymbol{x}}}\_{\kappa}
\end{bmatrix} = \begin{bmatrix}
A\_{\kappa}(\boldsymbol{\nu}) & \boldsymbol{0} \\
\boldsymbol{0} & A\_{\kappa}
\end{bmatrix} \begin{bmatrix}
\tilde{\boldsymbol{x}}\_{k} \\
\tilde{\boldsymbol{x}}\_{\kappa}
\end{bmatrix} + \begin{bmatrix}
B\_{\kappa1}(\boldsymbol{\nu}) & B\_{\kappa2}(\boldsymbol{\nu}) & B\_{\kappa3}(\boldsymbol{\nu}) & \boldsymbol{0} \\
\boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} & B\_{\kappa}
\end{bmatrix} \begin{bmatrix}
\hat{\boldsymbol{y}}\_{1} \\
\hat{\boldsymbol{y}} \\
\hat{\boldsymbol{y}}\_{2} \\
\hat{\boldsymbol{y}}
\end{bmatrix} \tag{31}
$$

Spatially Sampled Robust Repetitive Control 81

The experiment was performed by activating the LPVRC controller and rotating the photosensitive drum for 40 revolutions with step change in nominal velocity. During the operation, the nominal motor input voltage was changed at the 10th, 20th and 30th revolution, which shifted the nominal drum angular velocity. This can be seen in Fig. 13, which depicts the histories of drum angular velocity, motor input voltage and the three varying parameters with respect to the drum angular position. Fig. 14 compares the spatial frequency spectrum of the velocity signals within each 10-revolution interval to that of the uncompensated system. We can see that the performance of the LPVRC controlled system is insensitive to changes in nominal drum angular velocity. Note that the magnitude increases near dc frequency are due to the transient responses. As a comparison, Fig. 15 shows the responses when the system is under the control of a fixed temporal repetitive controller. As expected, a fixed-period repetitive controller operating in the time domain is unable to effectively compensate for the disturbances whose temporal periods change with the

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> 0.4

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> -1

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> 0.6

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> 1.5

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>1</sup>

drum angular position (rev)

The spatial sampling scheme proposed in this section raises a practical issue when synthesizing digital full-order repetitive controllers. The available sampling frequencies when conducting the scheme depends on the encoder resolution. For example, if the resolution of an encoder is 5000 pulses/rev, the highest sampling frequency achievable using the scheme will be 5000 cycles/rev. Other available sampling frequencies, depending on implementable divide-by-N circuits, might be 2500 (when the pulses are divided by 2), 500 (divided by 10), etc. Due to limited choices of sampling frequencies, the number of delay

*N*), which is the ratio of the sampling frequency and the

Fig. 13. Histories of drum angular velocity, motor input voltage and the three varying

rotational speed of the system.

0.5 0.6 velocity (rev/s)

> 0 1

0.8 1 1.2

2 2.5

> 2 3

parameters with respect to drum angular position.

taps *N* for the repetitive kernel (i.e. *q*

ida

input (volt)

phi

ro

$$
\begin{bmatrix} \hat{\boldsymbol{\mu}} \\ \tilde{\boldsymbol{y}}\_{2} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{\boldsymbol{\kappa}}(\boldsymbol{\nu}) & \mathbf{0} \\ \mathbf{0} & \mathbf{C}\_{\boldsymbol{\kappa}} \end{bmatrix} \begin{bmatrix} \tilde{\boldsymbol{x}}\_{k} \\ \tilde{\boldsymbol{x}}\_{\kappa} \end{bmatrix} + \begin{bmatrix} D\_{\boldsymbol{\kappa}1}(\boldsymbol{\nu}) & D\_{\boldsymbol{\kappa}2}(\boldsymbol{\nu}) & D\_{\boldsymbol{\kappa}3}(\boldsymbol{\nu}) & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \hat{\boldsymbol{y}}\_{1} \\ \hat{\boldsymbol{y}}\_{2} \\ \hat{\boldsymbol{y}}\_{2} \\ \tilde{\boldsymbol{y}} \end{bmatrix} \tag{32}
$$

where

$$\begin{aligned} B\_{\mathbb{K}}(\boldsymbol{\psi}) &= [B\_{\mathbb{K}1}(\boldsymbol{\psi}) \quad B\_{\mathbb{K}2}(\boldsymbol{\psi}) \quad B\_{\mathbb{K}3}(\boldsymbol{\psi})],\\ D\_{\mathbb{K}}(\boldsymbol{\psi}) &= [D\_{\mathbb{K}1}(\boldsymbol{\psi}) \quad D\_{\mathbb{K}2}(\boldsymbol{\psi}) \quad D\_{\mathbb{K}3}(\boldsymbol{\psi})].\end{aligned}$$

$$\boldsymbol{\psi} = \{\sum\_{i=1}^{4} a\_{i} \boldsymbol{\psi}\_{i} : a\_{i} \ge 0, \sum\_{i=1}^{4} a\_{i} = 1\}$$

We can view (31) and (32) as an LPV repetitive controller (LPVRC). For practical implementation, the vertex controllers need to be transformed into their discrete-position invariant counterparts, e.g., using bilinear transformation. The nominal performance (NP), robust stability (RS), and robust performance (RP) curves for the four vertex systems are shown in Fig. 12.

Fig. 12. NP, RS and RP curves for the four vertex closed-loop systems.

80 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

 

> 

ˆ ˆ () 0 () () ()0 , <sup>0</sup> 0 0 00 <sup>ˆ</sup> *K KK K K*

We can view (31) and (32) as an LPV repetitive controller (LPVRC). For practical implementation, the vertex controllers need to be transformed into their discrete-position invariant counterparts, e.g., using bilinear transformation. The nominal performance (NP), robust stability (RS), and robust performance (RP) curves for the four vertex systems are

10-3 10-2 10-1 <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> <sup>0</sup>

10-3 10-2 10-1 <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> <sup>0</sup>

10-3 10-2 10-1 <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> <sup>0</sup>

freqency (cycles/rev)

Fig. 12. NP, RS and RP curves for the four vertex closed-loop systems.

{ : 0, 1}

*ii i i i i*

1 1

123 123 4 4

( ) [ ( ) ( ) ( )], ( ) [ ( ) ( ) ( )].

*DDDD*

*uC x D D D y y C x y*

2 2

*KKKK KKKK*

*BBBB*

 

*xA xB B B y x A x B y*

*rc rc rc rc*

*K K KK K K*

*rc rc*

 

> >

123

123

() 0 () () () 0 <sup>ˆ</sup> , <sup>0</sup> <sup>000</sup> <sup>ˆ</sup>

2

(31)

(32)

1

ˆ

*y*

*y*

1

ˆ

*y*

NP1 NP2 NP3 NP4

*y*

where

shown in Fig. 12.

0.5

0.5

0.5

1

1

RS1 RS2 RS3 RS4

RP1 RP2 RP3 RP4

1

  The experiment was performed by activating the LPVRC controller and rotating the photosensitive drum for 40 revolutions with step change in nominal velocity. During the operation, the nominal motor input voltage was changed at the 10th, 20th and 30th revolution, which shifted the nominal drum angular velocity. This can be seen in Fig. 13, which depicts the histories of drum angular velocity, motor input voltage and the three varying parameters with respect to the drum angular position. Fig. 14 compares the spatial frequency spectrum of the velocity signals within each 10-revolution interval to that of the uncompensated system. We can see that the performance of the LPVRC controlled system is insensitive to changes in nominal drum angular velocity. Note that the magnitude increases near dc frequency are due to the transient responses. As a comparison, Fig. 15 shows the responses when the system is under the control of a fixed temporal repetitive controller. As expected, a fixed-period repetitive controller operating in the time domain is unable to effectively compensate for the disturbances whose temporal periods change with the rotational speed of the system.

Fig. 13. Histories of drum angular velocity, motor input voltage and the three varying parameters with respect to drum angular position.

The spatial sampling scheme proposed in this section raises a practical issue when synthesizing digital full-order repetitive controllers. The available sampling frequencies when conducting the scheme depends on the encoder resolution. For example, if the resolution of an encoder is 5000 pulses/rev, the highest sampling frequency achievable using the scheme will be 5000 cycles/rev. Other available sampling frequencies, depending on implementable divide-by-N circuits, might be 2500 (when the pulses are divided by 2), 500 (divided by 10), etc. Due to limited choices of sampling frequencies, the number of delay taps *N* for the repetitive kernel (i.e. *qN*), which is the ratio of the sampling frequency and the

Spatially Sampled Robust Repetitive Control 83

bound for the initial states can be established or estimated, a bound for the state-dependent varying parameters can be estimated, and the polytope which contains the parameter variation set can be determined more accurately. Since the proposed LPV control system has the property of being robust to unstructured but bounded uncertainty (specified by *W*<sup>2</sup> and ), the issues of measurement noise and uncertainty can be accounted for in the

proposed formulation if they can be incorporated into the *W*2 filter and the block.

provides a bound for the state-dependent varying parameters, e.g.,

0 5 10 15 20 25 30 35 40

time (sec)

frequency (Hz)

Fig. 15. Frequency spectra of the velocity signals for the closed-loop system using fixed

In this chapter, the notion of spatial-based repetitive control system and its historical development were introduced. Two designs, which were experimentally verified on a rotary motion system, representative of recent advancement in this field were presented. The designs, which are applicable to a generic class of LTI systems, address important practical issues such as actuator saturation and modelling uncertainty. However, several drawbacks and limitations are worth notice. First of all, the designs resorted to linear robust control paradigm and account for only unstructured uncertainty. It is well known that such control approach might lead to limited performance if information regarding the uncertainty (e.g., structure) is not properly utilized. Second, the LPVRC design relies on a common Lyapunov function, which also results in conservative design. The design is further degraded if the number of varying parameters increases or the varying parameter space is nonconvex. Finally, both designs along with other exiting ones are applicable only to rotary systems operating unidirectionally. The LPVRC design can improve by employing parameter varying Lyapunov function (Apkarian & Adams, 1998). On the other hand, since the openloop spatial-based system, i.e., (3) or (6), is nonlinear, we may apply nonlinear control

0 50 0 50 0 50

. If a

The ellipsoid

2

0

period temporal repetitive control.

1

16

2

psd

**5. Conclusion** 

3

0 50

24

48

2.5

3

3.5

velocity (rad/s)

4

disturbance frequency, might end up being non-integral when tackling certain disturbance frequencies.

Other nonlinear control design approaches (e.g., sliding mode and adaptive control) can also be employed. However, it is not clear if frequency-wise tradeoff between performance and stability can be easily performed within those nonlinear design frameworks.

It is also worth mentioning that the LPV gain-scheduling design may encounter the following implementation issues:


A feasible solution for the first issue is to setup the parameter variation set more accurately. Note that (22) implies that () () 0 *<sup>T</sup> A X XA cl cl* , and we can pick a Lyapunov function ( ( )) ( ) ( ) *<sup>T</sup> V x x Xx cl cl cl* for the closed-loop system such that *dV/d* < 0. Thus, the state of the closed-loop system starting from 0 ( ) *cl x* will stay within an ellipsoid centered at the equilibrium point and defined by

$$\boldsymbol{\varepsilon} = \left\{ \widetilde{\boldsymbol{\mathfrak{x}}}\_{cl}(\boldsymbol{\theta}) \Big| \widetilde{\boldsymbol{\mathfrak{x}}}\_{cl}^T(\boldsymbol{\theta}) \boldsymbol{\mathfrak{X}} \widetilde{\boldsymbol{\mathfrak{x}}}\_{cl}(\boldsymbol{\theta}) \leq \widetilde{\boldsymbol{\mathfrak{x}}}\_{cl}^T(\boldsymbol{\theta}\_0) \boldsymbol{\mathfrak{X}} \widetilde{\boldsymbol{\mathfrak{x}}}\_{cl}(\boldsymbol{\theta}\_0) \right\}.$$

Fig. 14. Frequency spectra of the velocity signals for the open-loop and closed-loop systems. Spectra for the closed-loop system are divided into four, with each corresponding to signals measured from each 10-revolution interval (psd is abbreviation for power spectrum density)

82 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

disturbance frequency, might end up being non-integral when tackling certain disturbance

Other nonlinear control design approaches (e.g., sliding mode and adaptive control) can also be employed. However, it is not clear if frequency-wise tradeoff between performance

It is also worth mentioning that the LPV gain-scheduling design may encounter the

A feasible solution for the first issue is to setup the parameter variation set more accurately.

 ( ) ( ) ( ) ( ) ( ). 0 0 *T T cl cl cl cl cl*

*x x Xx x Xx*

 

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

Fig. 14. Frequency spectra of the velocity signals for the open-loop and closed-loop systems. Spectra for the closed-loop system are divided into four, with each corresponding to signals measured from each 10-revolution interval (psd is abbreviation for power spectrum density)

frequency (cycles/rev)

*cl* , and we can pick a Lyapunov function

 

will stay within an ellipsoid

  < 0. Thus, the state of the

Open-loop

LPVRC

centered at the

 

for the closed-loop system such that *dV/d*

and stability can be easily performed within those nonlinear design frameworks.

i. The state-dependent varying parameters may leave the parameter variation set. ii. The measurement of the varying parameters may be contaminated by noise. iii. There may be delay induced in the measurement of the varying parameters.

frequencies.

following implementation issues:

( ( )) ( ) ( ) *<sup>T</sup> V x x Xx cl cl cl*

0

0.05

0

0.05

psd

0

0.05

0

0.05

0

0.05

equilibrium point and defined by

Note that (22) implies that () () 0 *<sup>T</sup> A X XA cl*

0 – 10th revolution

10th – 20th revolution

20th – 30th revolution

30th – 40th revolution

 

closed-loop system starting from 0 ( ) *cl x*

  The ellipsoid provides a bound for the state-dependent varying parameters, e.g., . If a bound for the initial states can be established or estimated, a bound for the state-dependent varying parameters can be estimated, and the polytope which contains the parameter variation set can be determined more accurately. Since the proposed LPV control system has the property of being robust to unstructured but bounded uncertainty (specified by *W*<sup>2</sup> and ), the issues of measurement noise and uncertainty can be accounted for in the proposed formulation if they can be incorporated into the *W*2 filter and the block.

Fig. 15. Frequency spectra of the velocity signals for the closed-loop system using fixed period temporal repetitive control.

### **5. Conclusion**

In this chapter, the notion of spatial-based repetitive control system and its historical development were introduced. Two designs, which were experimentally verified on a rotary motion system, representative of recent advancement in this field were presented. The designs, which are applicable to a generic class of LTI systems, address important practical issues such as actuator saturation and modelling uncertainty. However, several drawbacks and limitations are worth notice. First of all, the designs resorted to linear robust control paradigm and account for only unstructured uncertainty. It is well known that such control approach might lead to limited performance if information regarding the uncertainty (e.g., structure) is not properly utilized. Second, the LPVRC design relies on a common Lyapunov function, which also results in conservative design. The design is further degraded if the number of varying parameters increases or the varying parameter space is nonconvex. Finally, both designs along with other exiting ones are applicable only to rotary systems operating unidirectionally. The LPVRC design can improve by employing parameter varying Lyapunov function (Apkarian & Adams, 1998). On the other hand, since the openloop spatial-based system, i.e., (3) or (6), is nonlinear, we may apply nonlinear control

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### **6. References**


84 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

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**5** 

*Japan* 

**An Iterative Approach to the Fixed-Order** 

It is well known that the robust disturbance attenuation against uncertainties can be achieved by the robust *H* controllers and some practical situations make us use the fixedorder controllers. These facts imply that the fixed-order robust *H* controllers are important for practical control problems. However it is difficult to design such robust controllers, because the robust *H* control problems include an infinite number of matrix inequality constraints, in other words, they are described by Robust Semi-Definite Programming (RSDP) problems. For obtaining a feasible solution of the RSDP problems coming from the robust control problems with state feedback controllers or full-order controllers, many numerical methods have been proposed. Classically, the quadratic stability theory, i.e. a common constant Lyapunov function for the entire uncertain set is used for reducing the infinite constraints to the finite ones at the expense of conservatism (Boyd et al. 1994). Recently, parameter dependent Lyapunov functions are used to improve the conservatism (Chesi et al. 2005) - (Ichihara et al. 2003), (Kami et al. 2009) - (Shaked 2001), (Xie 2008) and some one-shot type approaches using extended LMI conditions, which allows to use the affine parameter dependent Lyapunov functions, have been proposed (Pipeleers et al. 2009), (Shaked 2001), (Xie 2008). However these methods can not always produce the robust controller, because common additional variables are required and these methods can not be used for designing fixed-order controllers. In this sense, an iterative type approach may be

useful to the problems such that these one-shot type approaches can not be applied.

for control problems to our knowledge.

In the field of the numerical optimization, there are two types of iterative approaches for finding feasible or locally optimal solutions of the optimization problems: one is an interiorpoint approach which needs an initial feasible solution to be carried out and the other is an exterior-point approach which does not need it. From these facts, exterior-point approach can be efficient for obtaining the solutions of the problems such that feasible solutions are difficult to be found. However, there are no exterior-point approaches except those in (Iwasaki & Skelton 1995), (Kami & Nobuyama 2004), (Kami et al. 2009), (Vanbierviet 2009)

**1. Introduction** 

**Robust** *H* **Control Problem Using** 

Yasushi Kami and Eitaku Nobuyama *Akashi National College of Technology and Kyushu Institute of Technology* 

**a Sequence of Infeasible Controllers** 


## **An Iterative Approach to the Fixed-Order Robust** *H* **Control Problem Using a Sequence of Infeasible Controllers**

Yasushi Kami and Eitaku Nobuyama *Akashi National College of Technology and Kyushu Institute of Technology Japan* 

### **1. Introduction**

86 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

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nonlinear rotary systems with spatially periodic parametric variation. *International* 

It is well known that the robust disturbance attenuation against uncertainties can be achieved by the robust *H* controllers and some practical situations make us use the fixedorder controllers. These facts imply that the fixed-order robust *H* controllers are important for practical control problems. However it is difficult to design such robust controllers, because the robust *H* control problems include an infinite number of matrix inequality constraints, in other words, they are described by Robust Semi-Definite Programming (RSDP) problems. For obtaining a feasible solution of the RSDP problems coming from the robust control problems with state feedback controllers or full-order controllers, many numerical methods have been proposed. Classically, the quadratic stability theory, i.e. a common constant Lyapunov function for the entire uncertain set is used for reducing the infinite constraints to the finite ones at the expense of conservatism (Boyd et al. 1994). Recently, parameter dependent Lyapunov functions are used to improve the conservatism (Chesi et al. 2005) - (Ichihara et al. 2003), (Kami et al. 2009) - (Shaked 2001), (Xie 2008) and some one-shot type approaches using extended LMI conditions, which allows to use the affine parameter dependent Lyapunov functions, have been proposed (Pipeleers et al. 2009), (Shaked 2001), (Xie 2008). However these methods can not always produce the robust controller, because common additional variables are required and these methods can not be used for designing fixed-order controllers. In this sense, an iterative type approach may be useful to the problems such that these one-shot type approaches can not be applied.

In the field of the numerical optimization, there are two types of iterative approaches for finding feasible or locally optimal solutions of the optimization problems: one is an interiorpoint approach which needs an initial feasible solution to be carried out and the other is an exterior-point approach which does not need it. From these facts, exterior-point approach can be efficient for obtaining the solutions of the problems such that feasible solutions are difficult to be found. However, there are no exterior-point approaches except those in (Iwasaki & Skelton 1995), (Kami & Nobuyama 2004), (Kami et al. 2009), (Vanbierviet 2009) for control problems to our knowledge.

An Iterative Approach to the Fixed-Order

1. ( ( ), ) *A* 

2. ( ( ), , ) *A BC* 

where ( ) *<sup>r</sup>*

holds.

follows:

**Robust** *H* **synthesis problem:**  Given an *H* norm bound *<sup>p</sup>*

denotes the *H*¥ norm.

feedback controller *<sup>s</sup>* :

Moreover, we have the following assumptions:

*B* is controllable for all

Robust *H∞* Control Problem Using a Sequence of Infeasible Controllers 89

For this system let us consider the following fixed-order controller *d* or the static state

 

0 *r n* . Note that *d* and *s* become state feedback controllers in the case that *E I*

() ( , ) () (). () ( ) () *cl cl cl clw cl cl*

*x t A K x t B wt*

( , ): ( ) , ( ): *A K A BKE C K C DKE cl*

1 ( ) 0 () , ( ) , ( 0,1, , ), ( ) 0 0

 

 0 0 , , 0, 0, 0 0

*x t A A AA i N*

*B E D C*

( ) ( ), ( ) ( ), , , , , *cl x t xt A A B BE EC CD DK K*

For the closed-loop system (6) the control problem to be solved in this paper is defined as

*I I B A*

 

*N*

*zt C Kx t*

 

0

*B E CC DD K*

 

, find *K* which achieves

For the control problem (11) the following lemma holds (Boyd et al. 1994):

(,) *T K zw p* 

where ( ,) *T K zw q* is the transfer function from *w* to *z* of the closed-loop system (6) and ¥⋅

*x t A*

For the controller *<sup>d</sup>* ( ) *cl x t* and the coefficient matrices in (7) are given by

() () ()

*x t Ax t Byt ut Cx t Dyt*

() () () *c cc c*

*cc c*

*<sup>c</sup> x t* **R** is the controller state and *r* is the prescribed integer which achieves

.

(4)

: () () *<sup>s</sup> u t Kx t* (5)

(6)

*i*

(8)

. (9)

(7)

(10)

*c c c c*

(11)

.

:

Via the controller *d* and *s* the closed-loop system can be described by

*cl ii i cl i*

For the controller *<sup>s</sup>* ( ) *cl x t* and the coefficient matrices are given by

*x t*

*d*

*<sup>w</sup>* is controllable and observable for all

In this paper, we deal with the fixed-order robust *H* controller synthesis problem against time invariant polytopic uncertainties, which can be described by parameter dependent bilinear matrix inequality (PDBMI) problems. The purpose of this paper is to propose an iterative approach which is like an exterior-point one. To do that, we introduce an `axisshifted system' which is obtained by shifting the imaginary axis of the complex plane so that all perturbing closed-poles are included in the LHS of the shifted imaginary axis. Our approach constructs a sequence of infeasible controller variables on which the shifted imaginary axis returns to the original position while the *H* norm of the axis-shifted system is less than the prescribed *H* norm bound. The advantage of our approach is to be able to use any controller variables as an initial point. The efficiency of our approach is shown by a numerical example.

In this paper, the following notations are used. **R** , *n m* **R** and *<sup>n</sup>* **S** are the sets of real scalars, *n m* real matrices and *n n* real symmetric matrices, respectively. He{ *<sup>Z</sup>* }, \* *T A B C* and

( ) denote *<sup>T</sup> Z Z* , the block symmetric matrix *<sup>T</sup> A B B C* and a set of eigenvalues, respectively. Moreover, denotes a hyper-rectangle and vert indicates the set of vertices of .

### **2. Problem formulation**

In this paper, we consider the following plant *P*( ) with a time invariant uncertain parameter : <sup>1</sup> *<sup>N</sup>* :

$$P(\theta) \coloneqq \begin{cases} \dot{\mathbf{x}}(t) = A(\theta)\mathbf{x}(t) + Bu(t) + B\_w w(t) \\ z(t) = \mathbf{C}x(t) + Du(t) \\ y(t) = \mathbf{E}x(t) \end{cases} \tag{1}$$

$$A(\theta) \coloneqq A\_0 + \sum\_{i=1}^{N} \theta\_i A\_i \tag{2}$$

where *x t*( ) is the plant state, *w t*( ) is any exogenous input, *u t*( ) is the control input, *z t*( ) is the performance output, *y*( )*t* is the measurement output and : <sup>1</sup> *<sup>N</sup>* is an uncertain parameter vector whose elements satisfy

$$\theta\_i \in \left[\underline{\theta\_i} \quad \overline{\theta\_i}\right], i = 1, \dots, N. \tag{3}$$

$$\underbrace{\overline{\phantom{\frac{1}{n}\sum\limits\_{i=1}^{n}\sum\limits\_{j=1}^{n}\frac{1}{n}}}\_{\text{ $V$ }} \cdot \text{ $\epsilon$ }^{\text{-}} $$

Fig. 1. Control system.

Moreover, we have the following assumptions:

1. ( ( ), ) *A B* is controllable for all .

88 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

In this paper, we deal with the fixed-order robust *H* controller synthesis problem against time invariant polytopic uncertainties, which can be described by parameter dependent bilinear matrix inequality (PDBMI) problems. The purpose of this paper is to propose an iterative approach which is like an exterior-point one. To do that, we introduce an `axisshifted system' which is obtained by shifting the imaginary axis of the complex plane so that all perturbing closed-poles are included in the LHS of the shifted imaginary axis. Our approach constructs a sequence of infeasible controller variables on which the shifted imaginary axis returns to the original position while the *H* norm of the axis-shifted system is less than the prescribed *H* norm bound. The advantage of our approach is to be able to use any controller variables as an initial point. The efficiency of our approach is shown by a

In this paper, the following notations are used. **R** , *n m* **R** and *<sup>n</sup>* **S** are the sets of real scalars,

respectively. Moreover, denotes a hyper-rectangle and vert indicates the set of

*A B B C* 

() ( )() () ()

1

 

*i AA A*

*i i*

*N*

*<sup>w</sup> x t A x t Bu t B w t*

*T A B C* 

and a set of eigenvalues,

with a time invariant uncertain

(2)

*i N* (3)

(1)

 : <sup>1</sup> *<sup>N</sup>* is an

and

*n m* real matrices and *n n* real symmetric matrices, respectively. He{ *<sup>Z</sup>* }, \*

( ): ( ) ( ) ( ) () ()

0

where *x t*( ) is the plant state, *w t*( ) is any exogenous input, *u t*( ) is the control input, *z t*( ) is

, 1, , . *i ii*

*P*( ) 

*z w*

*K u y*

*P z t Cx t Du t y t Ex t*

( ):

( ) denote *<sup>T</sup> Z Z* , the block symmetric matrix *<sup>T</sup>*

In this paper, we consider the following plant *P*( )

the performance output, *y*( )*t* is the measurement output and

 

uncertain parameter vector whose elements satisfy

 <sup>1</sup> *<sup>N</sup>* :

numerical example.

vertices of .

parameter

**2. Problem formulation** 

 : 

Fig. 1. Control system.

2. ( ( ), , ) *A BC <sup>w</sup>* is controllable and observable for all .

For this system let us consider the following fixed-order controller *d* or the static state feedback controller *<sup>s</sup>* :

$$\begin{aligned} \Sigma\_d &: \begin{cases} \dot{\mathbf{x}}\_c(t) = A\_c \mathbf{x}\_c(t) + B\_c y(t) \\ u(t) = \mathbf{C}\_c \mathbf{x}\_c(t) + D\_c y(t) \end{cases} \end{aligned} \tag{4}$$

$$
\Sigma\_s: \mathfrak{u}(t) = \mathbb{K}\mathfrak{x}(t) \tag{5}
$$

where ( ) *<sup>r</sup> <sup>c</sup> x t* **R** is the controller state and *r* is the prescribed integer which achieves 0 *r n* . Note that *d* and *s* become state feedback controllers in the case that *E I* holds.

Via the controller *d* and *s* the closed-loop system can be described by

$$\begin{cases} \dot{\mathbf{x}}\_{cl}(t) = A\_{cl}(\overline{\mathbf{K}}, \theta) \mathbf{x}\_{cl}(t) + B\_{clw} w(t) \\\ z(t) = \mathbf{C}\_{cl}(\overline{\mathbf{K}}) \mathbf{x}\_{cl}(t) \end{cases} . \tag{6}$$

$$A\_{cl}(\overline{K}, \theta) \coloneqq \overline{A}(\theta) + \overline{B}\overline{K}\overline{E}, \mathbb{C}(\overline{K}) \coloneqq \overline{\mathbb{C}} + \overline{D}\overline{K}\overline{E} \tag{7}$$

For the controller *<sup>d</sup>* ( ) *cl x t* and the coefficient matrices in (7) are given by

$$\mathbf{x}\_{cl}(t) = \begin{bmatrix} \mathbf{x}(t) \\ \mathbf{x}\_{cl}(t) \end{bmatrix}, \overline{A}(\theta) = \overline{A}\_0 + \sum\_{i=1}^{N} \theta\_i \overline{A}\_i, \overline{A}\_i = \begin{bmatrix} A\_i & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \langle i = 0, 1, \cdots, N \rangle\_{\prime} \tag{8}$$

$$
\overline{B} = \begin{bmatrix} B & 0 \\ 0 & I \end{bmatrix}, \overline{E} = \begin{bmatrix} E & 0 \\ 0 & I \end{bmatrix}, \overline{C} = \begin{bmatrix} C & 0 \end{bmatrix}, \overline{D} = \begin{bmatrix} D & 0 \end{bmatrix}, \overline{K} = \begin{bmatrix} D\_c & C\_c \\ B\_c & A\_c \end{bmatrix}. \tag{9}
$$

For the controller *<sup>s</sup>* ( ) *cl x t* and the coefficient matrices are given by

$$\mathbf{x}\_{cl}(t) = \mathbf{x}(t), \overline{A}(\theta) = A(\theta), \overline{B} = B, \overline{E} = E, \overline{C} = C, \overline{D} = D, \overline{K} = K \tag{10}$$

For the closed-loop system (6) the control problem to be solved in this paper is defined as follows:

### **Robust** *H* **synthesis problem:**

Given an *H* norm bound *<sup>p</sup>* , find *K* which achieves

$$\left\| T\_{zw}(\overline{K}, \theta) \right\|\_{\infty} < \gamma\_p \tag{11}$$

where ( ,) *T K zw q* is the transfer function from *w* to *z* of the closed-loop system (6) and ¥⋅ denotes the *H*¥ norm.

For the control problem (11) the following lemma holds (Boyd et al. 1994):

Lemma 1 (,) *T K zw p* holds if there exists a parameter dependent Lyapunov function

$$P^{\mathcal{P}}\left(\theta\right) \coloneqq P\_0^{\mathcal{P}} + \sum\_{i=1}^{N} \theta\_i P\_i^{\mathcal{P}} > 0 \tag{12}$$

An Iterative Approach to the Fixed-Order

where <sup>1</sup> : () *<sup>T</sup>*

feasible solution of (17).

" =*i N* 1, , .

shown in the next subsection.

**3.2 Multi-convex relaxation** 

multi-convex upper bound of *M*(, ) *z*

Fig. 2).

respect to any

*f* ( ) 

**3.1 Multi-convex function** 

**Definition 1:** If the function *f* ( ),

Lemma 2 The next statements hold:

is said as a multi-convex function.

Fig. 2. The concept of the multi-convex functions.

parameter vector whose elements

**3. Multi-convex relaxation method** 

In this section, let us consider the next PDMI problem

0

From the definition the multi-convex function has the next properties:

 *<sup>j</sup>* in the case that ( 1, , 1, 1, , ) *<sup>i</sup>* 

1. The function *<sup>f</sup>* ( ) *<sup>q</sup>* is the multi-convex function if and only if ( ) <sup>0</sup>

2. The maximum of the multi-convex function *f* ( ) *q* on *q* Î W is on the vertex of W (See

Using these properties the relaxation method for obtaining the feasible solution of (17) is

In this subsection, we show a relaxation method with multi-convex function (Ichihara et al. 2003) which is needed to derive our approach. The key idea of this method is to make the

> .

Robust *H∞* Control Problem Using a Sequence of Infeasible Controllers 91

1 1

*i i ji*

*<sup>i</sup>* **R** are in the given range

*N N N*

*i i i j ij*

 

(17)

*i ii* 

<sup>1</sup> *<sup>N</sup>* becomes a multi-convex function with

*i jj N* are fixed then the function

*T*

and 0 *<sup>M</sup>* ( ) *<sup>z</sup>* ,

*i f q q* ¶ <sup>³</sup> ¶

hold

 *<sup>N</sup>* is a

find s.t. ( , ) : ( ) ( ) ( ) 0,

*N i zz z z* **R** is a vector of decision variables, <sup>1</sup> :

( ) *Mi z* and ( ) *Mij z* are symmetric matrices with appropriate sizes. It is well known that feasible solutions of the PDMI problem (17) are difficult to be obtained, because this problem has an infinite number of constraints corresponding to all points on . In this section, we show the multi-convex relaxation method (Ichihara et al. 2003) which is used for reducing the infinitely constrained problem to a finitely constrained one for obtaining a

In this subsection, we review the definition and the properties of the multi-convex function.

 

*z Mz M z M z M z*

which satisfies

$$
\begin{bmatrix}
P^{\circ}(\theta)A\_{cl}(\overline{\mathbf{K}},\theta) + A\_{cl}(\overline{\mathbf{K}},\theta)^{T}P^{\circ}(\theta) & \* & \* \\
B\_{clw}^{T}P^{\circ}(\theta) & -\gamma\_{p}I & \* \\
C\_{cl}(\overline{\mathbf{K}}) & 0 & -\gamma\_{p}I
\end{bmatrix} < 0. \tag{13}
$$

This lemma implies that the robust *H* synthesis problem (11) can be described as PDBMI problem, which has an infinite number of BMI constraints corresponding to all points on . Hence it is difficult to obtain the feasible controller variables *K* achieving (13). One well known classical method for obtaining *s* in the case that *E I* is to use quadratic (parameter independent constant) Lyapunov functions (Boyd et al. 1994). i.e., defining

$$P^{\oplus}(\theta)^{-1} \coloneqq X \ \ \ \ W \coloneqq \text{KX} \tag{14}$$

to get the controller variables from

$$K = \mathcal{V}\mathcal{W}\mathcal{X}^{-1} \tag{15}$$

where *X* and *W* are the solutions of the next inequalities:

$$\begin{bmatrix} A(\theta)X + XA(\theta)^T + B\mathcal{W} + \mathcal{W}^T B^T & \* & \* \\ & B\_w^T & -\mathcal{Y}\_p I & \* \\ & CX + DW & 0 & -\mathcal{Y}\_p I \end{bmatrix} < 0, \forall \theta \in \text{vert } \Omega \tag{16}$$

However the quadratic Lyapunov functions *X* do not always exist and even if they exist the obtained controller includes a high conservatism. Moreover, this method can be only used in the case that *E I* .

Recently, various studies with parameter dependent Lyapunov functions have been reported to reduce the conservatism (Chesi et al. 2005) - (Ichihara et al. 2003), (Kami et al. 2009) - (Shaked 2001), (Xie 2008). Especially, some interesting one-shot type approaches for designing static state feedback controllers or full-order controllers with extended matrix inequality conditions have been proposed (Pipeleers et. al., 2009), (Shaked 2001), (Xie 2008). However these methods do not always produce the feasible controllers in some cases, because some additional common matrix variables are required and this method can not be used in the case that *E I* . In this paper, we propose an iterative approach to the fixedorder robust *H* synthesis problem, which can be used if *E I* . The features of our approach are to constructs a controller sequence from the infeasible region to the feasible one and to be able to use any matrix as an initial point.

### **3. Multi-convex relaxation method**

90 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

0

() ( ,) ( ,) () \* \*

 

<sup>1</sup> *P X* () : 

() () \* \*

*T T T*

*w p*

However the quadratic Lyapunov functions *X* do not always exist and even if they exist the obtained controller includes a high conservatism. Moreover, this method can be only

Recently, various studies with parameter dependent Lyapunov functions have been reported to reduce the conservatism (Chesi et al. 2005) - (Ichihara et al. 2003), (Kami et al. 2009) - (Shaked 2001), (Xie 2008). Especially, some interesting one-shot type approaches for designing static state feedback controllers or full-order controllers with extended matrix inequality conditions have been proposed (Pipeleers et. al., 2009), (Shaked 2001), (Xie 2008). However these methods do not always produce the feasible controllers in some cases, because some additional common matrix variables are required and this method can not be used in the case that *E I* . In this paper, we propose an iterative approach to the fixedorder robust *H* synthesis problem, which can be used if *E I* . The features of our approach are to constructs a controller sequence from the infeasible region to the feasible

*B I CX DW I*

*T*

*clw p*

This lemma implies that the robust *H* synthesis problem (11) can be described as PDBMI problem, which has an infinite number of BMI constraints corresponding to all points on . Hence it is difficult to obtain the feasible controller variables *K* achieving (13). One well known classical method for obtaining *s* in the case that *E I* is to use quadratic (parameter independent constant) Lyapunov functions (Boyd et al. 1994). i.e., defining

*B P I*

*PP P*

*cl cl T*

 

where *X* and *W* are the solutions of the next inequalities:

 

one and to be able to use any matrix as an initial point.

*T*

*A X XA BW W B*

*P AK AK P*

 

1 ( ): 0 *N i i*

 

*i*

( ) 0

*cl p*

0

*p*

*C K I*

holds if there exists a parameter dependent Lyapunov function

( ) \* 0.

 

(12)

(13)

(16)

, : *W KX* (14)

<sup>1</sup> *K WX* (15)

\* 0, vert

 

Lemma 1 (,) *T K zw*

which satisfies

 *p* 

to get the controller variables from

used in the case that *E I* .

In this section, let us consider the next PDMI problem

$$\text{find } z \text{ s.t. } M(z, \theta) \coloneqq M\_0(z) + \sum\_{i=1}^{N} \theta\_i M\_i(z) + \sum\_{i=1}^{N} \sum\_{j=i}^{N} \theta\_i \theta\_j M\_{ij}(z) < 0, \forall \theta \in \Omega \tag{17}$$

where <sup>1</sup> : () *<sup>T</sup> N i zz z z* **R** is a vector of decision variables, <sup>1</sup> : *T <sup>N</sup>* is a parameter vector whose elements *<sup>i</sup>* **R** are in the given range *i ii* and 0 *<sup>M</sup>* ( ) *<sup>z</sup>* , ( ) *Mi z* and ( ) *Mij z* are symmetric matrices with appropriate sizes. It is well known that feasible solutions of the PDMI problem (17) are difficult to be obtained, because this problem has an infinite number of constraints corresponding to all points on . In this section, we show the multi-convex relaxation method (Ichihara et al. 2003) which is used for reducing the infinitely constrained problem to a finitely constrained one for obtaining a feasible solution of (17).

### **3.1 Multi-convex function**

In this subsection, we review the definition and the properties of the multi-convex function. **Definition 1:** If the function *f* ( ), <sup>1</sup> *<sup>N</sup>* becomes a multi-convex function with respect to any *<sup>j</sup>* in the case that ( 1, , 1, 1, , ) *<sup>i</sup> i jj N* are fixed then the function

*f* ( ) is said as a multi-convex function.

From the definition the multi-convex function has the next properties: Lemma 2 The next statements hold:

1. The function *<sup>f</sup>* ( ) *<sup>q</sup>* is the multi-convex function if and only if ( ) <sup>0</sup> *i f q q* ¶ <sup>³</sup> ¶ hold

" =*i N* 1, , .

2. The maximum of the multi-convex function *f* ( ) *q* on *q* Î W is on the vertex of W (See Fig. 2).

Using these properties the relaxation method for obtaining the feasible solution of (17) is shown in the next subsection.

Fig. 2. The concept of the multi-convex functions.

#### **3.2 Multi-convex relaxation**

In this subsection, we show a relaxation method with multi-convex function (Ichihara et al. 2003) which is needed to derive our approach. The key idea of this method is to make the multi-convex upper bound of *M*(, ) *z* .

The multi-convex relaxation method can be described as the next lemma:

Lemma 3 *z* is a feasible solution of the PDMI problem (17) if there exist *z* , *n n Qi* **R** and ( 1, , ) *<sup>n</sup> Ri N <sup>i</sup>* **S** which achieve

$$
\begin{bmatrix} M(z, \theta) & \* & \* & \* \\ Q\_1 - \theta\_1 R\_1 & -R\_1 & \* & \* \\ \vdots & 0 & \ddots & \* \\ Q\_N - \theta\_N R\_N & 0 & 0 & -R\_N \end{bmatrix} < 0, \forall \theta \in \text{vert } \Omega \tag{18}
$$

$$M\_{ii}(z) + R\_i \ge 0 (i = 1, \cdots N). \tag{19}$$

An Iterative Approach to the Fixed-Order

and the inequality (27) can be transformed into

Therefore, *z* is a feasible solution of *M z* (,) 0

into that with a finite number of constraints.

plane, i.e., there always exists a finite scalar

and we introduce the following system using

This system has the next property.

**Proof** It is obvious from (29).

Now, letting (,,) *T K zw*

 

imaginary axis of the complex plane to the right by

(30) is called as `axis-shifted system' in this paper.

 

lemma holds for the *H* norm condition

Lemma 4 The system (30) is robustly stable for the parameter

(28) for all vert 

approach:

*M z*

11 11 1

*R RR*

(, ) \* \* \*

*NN NN* 0 0 *<sup>N</sup>*

Using this lemma the problem (17) with an infinite number of constraints can be reduced

In this section, we propose an iterative approach to the robust *H* control problem (11) using Lemma 3. To do that, we introduce an `axis-shifted system' which is obtained by shifting the imaginary axis so that all perturbing poles are located in the LHS of the imaginary axis. The key idea of our approach is to return the shifted imaginary axis to the

Firstly, we add the practical assumption for the closed-loop system (6) such that the poles of the system (6) do not exist infinitely far from the imaginary axis on the RHS of the complex

> 

() ( ( , ) ) () (). () ( ) () *cl cl cl clw*

*x t A K Ix t B wt*

which achieves:

(30)

be a transfer function of the system (30) from *w* to *z* the next

.

( ,, ) *T K zw <sup>p</sup> qb g* ¥ <sup>&</sup>lt; . (31)

 

(See Fig. 3). In this sense, the system

max Re[ ] ,

and replacing *Ri i* by *Qi* in (28) we have (18).

*RR R*

**4. Iterative approach to the robust** *H* **synthesis problems** 

original position while the *H* norm of the axis-shifted system is less than *<sup>p</sup>*

( ( , ))

 *A K cl*

*cl cl*

 

*zt C Kx t*

Remark 1. In this paper, we interpret the meaning of " (,) *AK I cl*

Lemma 5 (31) holds if there exists a parameter dependent Lyapunov function

our approach is to be able to use any controller variables as an initial point.

Robust *H∞* Control Problem Using a Sequence of Infeasible Controllers 93

\* \* 0, vert . 0 \*

(28)

if there exist *z* , *i* and *Ri* which achieve

(29)

, which is needed to derive our iterative

. The feature of

" as shifting the

**Proof** *M z*( , ) 0, holds iff we have

$$f(z, \theta) \coloneqq \mathbf{x}^T M(z, \theta) \mathbf{x} < 0, \forall \theta \in \Omega, \forall \mathbf{x}(\neq 0) \in \mathbb{R}^n. \tag{20}$$

Now, let us define 1*f z*(, ) and 2*f* ( ) as

$$f\_1(\mathbf{z}, \boldsymbol{\theta}) \coloneqq \mathbf{x}^T M(\mathbf{z}, \boldsymbol{\theta}) \mathbf{x} + \sum\_{i=1}^N \theta\_i^2 \mathbf{x}^T R\_i \mathbf{x}\_i \tag{21}$$

$$\mathbb{E}\, f\_2(\theta) \coloneqq \sum\_{i=1}^{N} \theta\_i^2 \ge ^T \mathbb{R}\_i \mathbf{x}\_{\prime} \tag{22}$$

respectively, where *<sup>n</sup> Ri* **S** achieve

$$\mathbf{x}^T (M\_{ii}(z) + R\_i)\mathbf{x} \ge \mathbf{0}, i = \mathbf{1}, \cdots \mathbf{N} \tag{23}$$

which is the necessary and sufficient condition for 1*f z*(, ) to be multi-convex function with respect to . Then the function *f z*(, ) given by (24) becomes a multi-convex upper bound function of 1 2 *fz f z f* ( , )( ( , ) ( )) :

$$
\overline{f}(z,\theta) \coloneqq f\_1(z,\theta) - \overline{f}\_2(\theta) \, , \tag{24}
$$

$$\overline{f}\_2(\boldsymbol{\theta}) \coloneqq \sum\_{i=1}^{N} \left( \boldsymbol{\theta}\_i \mathbf{B}\_i^T \mathbf{R}\_i + \boldsymbol{\theta}\_i \mathbf{R}\_i \mathbf{B}\_i - \mathbf{B}\_i^T \mathbf{R}\_i \mathbf{B}\_i \right). \tag{25}$$

This is because 2 2 *f f* () () holds from

$$-\boldsymbol{\theta}\_{i}^{2}\boldsymbol{R}\_{i}\leq - \left(\boldsymbol{\theta}\_{i}\mathbf{B}\_{i}^{T}\boldsymbol{R}\_{i} + \boldsymbol{\theta}\_{i}\mathbf{R}\_{i}\mathbf{B}\_{i} - \mathbf{B}\_{i}^{T}\boldsymbol{R}\_{i}\mathbf{B}\_{i}\right) \forall \mathbf{B}\_{i}\in\mathbf{R}^{m\times n}.\tag{26}$$

Then, from the property of the multi-convex functions *f z*(, ) 0 holds iff we have

$$M(z, \theta) + \sum\_{i=1}^{N} (\theta\_i I - \mathbf{B}\_i)^T R\_i (\theta\_i I - \mathbf{B}\_i) < 0, \forall \theta \in \text{vert } \Omega \tag{27}$$

and the inequality (27) can be transformed into

92 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

( , ): ( , ) ,

2

*f z x M z x x Rx*

1 ( ): , *N*

1 2 *fz f z f* ( , ): ( , ) ( )

<sup>2</sup>

*f RR R*

( ): .

<sup>2</sup> ( ), *<sup>T</sup> <sup>T</sup> n n*

( , ) ( ) ( ) 0, vert

*T i i ii i*

*Mz I R I qq q q*

 

*T T ii i iii i ii*

 

 *x Rx*

*i*

\* \* 0, vert 0 \*

(18)

( ) 0( 1, ). *Mii i zR i N* (19)

*x* **R** . (20)

*ii i x M z Rx i N* (23)

given by (24) becomes a multi-convex upper bound

(25)

 

*i i ii i iii i ii i R RR R* **<sup>R</sup>** . (26)

+ -B -B < " Î W å (27)

(21)

(22)

to be multi-convex function with

, (24)

holds iff we have

2

*i i*

1

*i*

*T i i*

*N T T* **R** and

Lemma 3 *z* is a feasible solution of the PDMI problem (17) if there exist *z* , *n n Qi*

The multi-convex relaxation method can be described as the next lemma:

(, ) \* \* \*

*N NN* 0 0 *<sup>N</sup>*

*QR R*

1 11 1

holds iff we have

( , ) : ( , ) 0, , ( 0) *T n f z x Mz x*

2

*f* 

1

*i*

Then, from the property of the multi-convex functions *f z*(, ) 0

*N*

1

*i*

=

holds from

*N*

*QR R*

 and 2*f* ( ) as

1

( ( ) ) 0, : 1, *<sup>T</sup>*

which is the necessary and sufficient condition for 1*f z*(, )

 :

. Then the function *f z*(, )

 

*M z*

( 1, , ) *<sup>n</sup> Ri N <sup>i</sup>* **S** which achieve

 

respectively, where *<sup>n</sup> Ri* **S** achieve

function of 1 2 *fz f z f* ( , )( ( , ) ( )) 

This is because 2 2 *f f* () ()

**Proof** *M z*( , ) 0, 

respect to

Now, let us define 1*f z*(, )

$$
\begin{bmatrix} M(z, \theta) & \* & \* & \* \\ R\_1 \mathbf{B}\_1 - \theta\_1 \mathbf{R}\_1 & -\mathbf{R}\_1 & \* & \* \\ \vdots & 0 & \ddots & \* \\ R\_N \mathbf{B}\_N - \theta\_N \mathbf{R}\_N & 0 & 0 & -\mathbf{R}\_N \end{bmatrix} < 0, \forall \theta \in \text{vert } \Omega. \tag{28}
$$

Therefore, *z* is a feasible solution of *M z* (,) 0 if there exist *z* , *i* and *Ri* which achieve (28) for all vert and replacing *Ri i* by *Qi* in (28) we have (18).

Using this lemma the problem (17) with an infinite number of constraints can be reduced into that with a finite number of constraints.

### **4. Iterative approach to the robust** *H* **synthesis problems**

In this section, we propose an iterative approach to the robust *H* control problem (11) using Lemma 3. To do that, we introduce an `axis-shifted system' which is obtained by shifting the imaginary axis so that all perturbing poles are located in the LHS of the imaginary axis. The key idea of our approach is to return the shifted imaginary axis to the original position while the *H* norm of the axis-shifted system is less than *<sup>p</sup>* . The feature of our approach is to be able to use any controller variables as an initial point.

Firstly, we add the practical assumption for the closed-loop system (6) such that the poles of the system (6) do not exist infinitely far from the imaginary axis on the RHS of the complex plane, i.e., there always exists a finite scalar which achieves:

$$\max\_{\lambda \in \sigma(A\_{cl}(\tilde{k}, \theta))} \text{Re}[\mathcal{A}] < \beta / \forall \theta \in \Omega \tag{29}$$

and we introduce the following system using , which is needed to derive our iterative approach:

$$\begin{cases} \dot{\mathbf{x}}\_{cl}(t) = (A\_{cl}(\overline{\mathbf{K}}, \theta) - \beta I)\mathbf{x}\_{cl}(t) + B\_{clw}w(t) \\ \mathbf{z}(t) = \mathbf{C}\_{cl}(\overline{\mathbf{K}})\mathbf{x}\_{cl}(t) \end{cases} . \tag{30}$$

This system has the next property.

Lemma 4 The system (30) is robustly stable for the parameter .

**Proof** It is obvious from (29).

Remark 1. In this paper, we interpret the meaning of " (,) *AK I cl* " as shifting the imaginary axis of the complex plane to the right by (See Fig. 3). In this sense, the system (30) is called as `axis-shifted system' in this paper.

Now, letting (,,) *T K zw* be a transfer function of the system (30) from *w* to *z* the next lemma holds for the *H* norm condition

$$\left\| \overline{T}\_{zw}(\overline{\mathcal{K}}\_{\prime}\theta\_{\prime}\beta) \right\|\_{\infty} < \gamma\_p \,. \tag{31}$$

Lemma 5 (31) holds if there exists a parameter dependent Lyapunov function

Fig. 3. Concept of complex plane of the axis-shifted system.

$$P(\theta) \coloneqq P\_0^{\circ \circ} + \sum\_{i=1}^{N} \theta\_i P\_i^{\circ \circ} > 0 \tag{32}$$

An Iterative Approach to the Fixed-Order

iterative approach. Lemma 6 *K* and *Pi*

, (,, ) *T K zw* 

 

**Proof:** Let us consider the next matrix:

Then, from (29), we can choose

which implies that there exists

symmetric matrix:

( : 1, , ), vert *Ri N <sup>i</sup>* 

*M*

, which achieve

*p*

 

Robust *H∞* Control Problem Using a Sequence of Infeasible Controllers 95

*T T i j ji j i i j*

*i i ii*

*PA AP*

*PA AP P A AP*

*ij <sup>T</sup>*

0 0 \* 0 0 0

\* \*

achieve the *H* norm constraint (31) if there exist *Qi* and

( ( ), , , , ) \* \*

 

0

*N NN N*

*Q R R*

1 11 1

*p*

 

0 0\* 0 00

Then we can get the next the next lemma with Lemma 3 which is needed to derive our

\* ( ( ), , , , ) : <sup>0</sup>

0, *Mii i R i* . (40)

 

*<sup>T</sup> <sup>T</sup> <sup>p</sup> <sup>T</sup> clw*

*<sup>T</sup> <sup>T</sup> <sup>p</sup> <sup>T</sup> clw*

*<sup>T</sup> <sup>T</sup> <sup>p</sup> clw*

*g*

( ,) ( ,) ( ) <sup>0</sup> ( )

*<sup>B</sup> AK AK B CK*

<sup>1</sup> <sup>0</sup> <sup>1</sup> ( ,) ( ,) ( ) <sup>2</sup> <sup>0</sup> ( )

2 ( ,) ( ,) ( ) , vert 0 ( )


*<sup>B</sup> AK AK B CK*

which achieves

*b qq q*

*<sup>B</sup> I AK AK B CK*

*MP K*

, (,, ) *T K zw* 

*QR R MP K*

Moreover, we have the next lemma with respect to the existence of

 *<sup>p</sup>* for given controller variables *Kk* . Lemma 7 For a given *Kk* achieving the condition (29) there always exists

*cl k cl k clw cl*

 

*cl k cl k clw cl*

*cl k cl k clw cl*

  \* \*

*i j*

(38)

, (39)

which achieves

(42)

, (43)

. (44)

achieving

*i j*

*<sup>p</sup>* (41)

<sup>1</sup> 0

which is larger than the maximum eigenvalue of the next

<sup>1</sup> 0

*g*

*p cl*

*p cl*

*p cl*

*T*

*C K*

*C K*

*C K*

which achieves

$$M^{\circ}(P^{\circ}(\theta), \overline{K}, \beta, \theta, \gamma\_p) < 0 \tag{33}$$

where

$$\mathcal{M}^{\pi}(P^{\pi}(\theta), \overline{\mathcal{K}}, \mathcal{J}, \theta, \mathcal{V}\_{\mathcal{P}}) \coloneqq \begin{bmatrix} P^{\pi}(\theta)(A\_{cl}(\overline{\mathcal{K}}, \theta) - \beta I) + (A\_{cl}(\overline{\mathcal{K}}, \theta) - \beta I)^{T}P^{\pi}(\theta) & \* & \* \\ & B\_{cl\pi}^{\top}P^{\pi}(\theta) & & \neg\_{\mathcal{V}}I & \* \\ & & C\_{cl}(\overline{\mathcal{K}}) & & 0 & \neg\_{\mathcal{V}}I \end{bmatrix} . \tag{34}$$

**Proof** It is obvious from Lemma 1.

Remark 2. If (,) *A K cl* is robustly stable we can let 0 and in this case ( , ,0) ( , ) *TK TK zw zw* holds.

Now, the inequality (33) can be described as

$$\mathcal{M}\_0^{\varphi} + \sum\_{i=1}^{N} \theta\_i \mathcal{M}\_i^{\varphi} + \sum\_{i=1}^{N} \sum\_{j=i}^{N} \theta\_i \theta\_j \mathcal{M}\_{ij}^{\varphi} < 0 \tag{35}$$

$$\begin{array}{c} \begin{bmatrix} P\_0^{\pi} \left( \overline{A}\_0 + \overline{B} \overline{K} \overline{E} \right) + \left( \overline{A}\_0 + \overline{B} \overline{K} \overline{E} \right)^T P\_0^{\pi} & \* & \* \\\\ B\_{clw}^{\top} P\_0^{\pi} & & -\gamma\_p I & \* \\\\ \overline{C} + \overline{D} \overline{K} \overline{E} & 0 & -\gamma\_p I \end{bmatrix} \end{array} \tag{36}$$

$$\boldsymbol{M}\_{i}^{\infty} = \begin{bmatrix} P\_0^{\infty} \overline{A}\_i + \overline{A}\_i^T P\_0^{\infty} + P\_i^{\infty} (\overline{A}\_0 + \overline{B} \overline{K} \overline{E}) + (\overline{A}\_0 + \overline{B} \overline{K} \overline{E})^T P\_i^{\infty} & \* & \* \\ & \boldsymbol{B}\_{\text{clw}}^T P\_i^{\infty} & & \boldsymbol{0} & \* \\ & \boldsymbol{0} & & \boldsymbol{0} & \boldsymbol{0} \end{bmatrix} \tag{37}$$

94 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

0

is robustly stable

0

0 0

*i clw i*

¥ ¥

*M B P*

1 ( ): 0 *N i i*

 

*i PP P*

( ( ), , , , ) 0 *MP K* 

 *p*

( ( ), , , , ) : ( ) \* .

*i i i j ij*

( )( ) \* \*

0 00 0 ( )( ) \*\*

*T T ii i i T*

<sup>é</sup> + + + ++ <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>=</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

*P A A P P A BKE A BKE P*

¥ ¥¥ ¥

 

*clw p*

*T*

*C DKE I*

*P AK I AK I P*

*cl cl T p clw p*

*MP K B P I*

1 1

0 0 0 0

*P A BKE A BKE P M B P I*

*T*

*i i ji MM M* 

*N NN*

 

 

Im Im

O

which achieves

Remark 2. If (,) *A K cl*

( , ,0) ( , ) *TK TK zw*

 

**Proof** It is obvious from Lemma 1.

 *zw* holds. Now, the inequality (33) can be described as

where

Fig. 3. Concept of complex plane of the axis-shifted system.

Re

Imaginary axis

Original Imaginary axis

(32)

(33)

( ) 0

0

0

0 0 0

(35)

*T*

 

0 and in this case

(36)

(37)

(34)

*cl p*

\*

*p*

0 \*

*C K I*

( )( ( , ) ) ( ( , ) ) ( ) \* \*

we can let

  

Perturbing closed-poles

<sup>O</sup> Shifted

$$M\_{ij} = \begin{bmatrix} \begin{bmatrix} P\_i^{\alpha}A\_j + A\_j^T P\_i^{\alpha} + P\_j^{\alpha} A\_i + A\_i^T P\_j^{\alpha} & \* & \* \\ 0 & 0 & \* \\ 0 & 0 & 0 \end{bmatrix} & i = j \end{bmatrix} \quad i = j$$

$$\begin{bmatrix} P\_i^{\alpha}A\_i + A\_i^T P\_i^{\alpha} & \* & \* \\ 0 & 0 & \* \\ 0 & 0 & 0 \end{bmatrix} \quad \quad i \neq j$$

Then we can get the next the next lemma with Lemma 3 which is needed to derive our iterative approach.

Lemma 6 *K* and *Pi* achieve the *H* norm constraint (31) if there exist *Qi* and ( : 1, , ), vert *Ri N <sup>i</sup>* , which achieve

$$
\overline{\mathcal{M}}^{\alpha}(\mathbb{P}^{\upsilon}(\theta), \overline{\mathcal{K}}, \beta, \gamma\_{\mathcal{P}}, \theta) \coloneqq \begin{bmatrix}
\mathcal{M}^{\upsilon}(\mathbb{P}^{\upsilon}(\theta), \overline{\mathcal{K}}, \beta, \gamma\_{\mathcal{P}}, \theta) & \* & \cdots & \* \\
& Q\_{1}^{\upsilon} - \theta\_{1} R\_{1}^{\upsilon} & -R\_{1} & \* \\
& \vdots & & \ddots & \vdots \\
& Q\_{N}^{\upsilon} - \theta\_{N} R\_{N}^{\upsilon} & 0 & \cdots & -R\_{N}
\end{bmatrix} < 0 \\ \tag{39}
$$

$$
\mathcal{M}\_{\overline{\mu}}^{\upsilon} + R\_{i} \ge 0, \forall i \;. \tag{40}
$$

Moreover, we have the next lemma with respect to the existence of which achieves , (,, ) *T K zw <sup>p</sup>* for given controller variables *Kk* .

Lemma 7 For a given *Kk* achieving the condition (29) there always exists achieving

$$\left\|\overline{T}\_{zw}(\overline{K}, \theta, \beta)\right\|\_{\infty} < \gamma\_p, \forall \, \theta \in \Omega \tag{41}$$

**Proof:** Let us consider the next matrix:

$$A\_{cl}(\overline{\mathbf{K}}\_{k'}\boldsymbol{\theta}) + A\_{cl}(\overline{\mathbf{K}}\_{k'}\boldsymbol{\theta})^T + \begin{bmatrix} B\_{clw} & \mathbf{C}\_{cl}(\overline{\mathbf{K}})^T \\ \mathbf{0} & \boldsymbol{\gamma}\_{cl} \end{bmatrix} \begin{bmatrix} \boldsymbol{\gamma}\_p & \mathbf{0} \\ \mathbf{0} & \boldsymbol{\gamma}\_p \end{bmatrix}^{-1} \begin{bmatrix} B\_{clw}^T \\ \mathbf{C}\_{cl}(\overline{\mathbf{K}}) \end{bmatrix} \tag{42}$$

Then, from (29), we can choose which is larger than the maximum eigenvalue of the next symmetric matrix:

$$\frac{1}{2}\left|A\_{cl}(\overline{\mathbf{K}}\_{k'}\boldsymbol{\theta}) + A\_{cl}(\overline{\mathbf{K}}\_{k'}\boldsymbol{\theta})^T + \begin{bmatrix} B\_{clw} & \mathbf{C}\_{cl}(\overline{\mathbf{K}})^T \\ \mathbf{0} & \boldsymbol{\gamma}\_{cl} \end{bmatrix} \begin{bmatrix} \boldsymbol{\gamma}\_{p} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{\gamma}\_{p} \end{bmatrix}^{-1} \begin{bmatrix} \boldsymbol{B}\_{clw}^{T} \\ \mathbf{C}\_{cl}(\overline{\mathbf{K}}) \end{bmatrix}\right|,\tag{43}$$

which implies that there exists which achieves

$$\mathcal{D}\beta I > A\_{cl}\{\overline{\mathbf{K}}\_{k}, \theta\} + A\_{cl}\{\overline{\mathbf{K}}\_{k}, \theta\}^{T} + \begin{bmatrix} \mathcal{B}\_{clw} & \mathcal{C}\_{cl}\{\overline{\mathbf{K}}\}^{T} \\ \mathcal{C}\_{cl}\{\overline{\mathbf{K}}\}^{T} & \begin{bmatrix} \gamma\_{p} & \mathbf{0} \\ \mathbf{0} & \gamma\_{p} \end{bmatrix}^{-1} \begin{bmatrix} \mathcal{B}\_{clw}^{T} \\ \mathcal{C}\_{cl}\{\overline{\mathbf{K}}\} \end{bmatrix} \theta \in \operatorname{vert } \Omega. \tag{44}$$

This inequality can be transformed into the next inequality:

$$\begin{bmatrix} A\_{cl}(\overline{\mathcal{K}}, \theta) - \beta I + \left( A\_{cl}(\overline{\mathcal{K}}, \theta) - \beta I \right)^{T} & \* & \* \\ B\_{clw}^{T} & -\gamma\_{p}I & \* \\ C\_{cl}(\overline{\mathcal{K}}) & 0 & -\gamma\_{p}I \end{bmatrix} < 0, \theta \in \text{vert } \Omega \tag{45}$$

An Iterative Approach to the Fixed-Order

**Proof** 1. and 2. From (46) and (50),

3. From Step 4 of the algorithm and the fact that

which implies *Kk* achieves ( ,, ) , *T K zw k k p*

( ,, ) , *T K zw k k p* 

 

sufficient condition for ( ,) , *T K zw k*

Fig. 4. Concept of our exterior-point approach.

**Remark 1:** The key idea of our approach is to decrease

Robust *H∞* Control Problem Using a Sequence of Infeasible Controllers 97

 *kk k* 

<sup>1</sup> 0 ( ( ), , , , ) *MP K k k kp* 

<sup>1</sup> ( ( ), , , , ) *MP K k k kp* 

<sup>1</sup> ( ( ), , , , ) *MP K k k kp* 

> 

the shifted imaginary axis approach the original position while the *H* norm constraint

is updated from a non robust *H* controller for the original system to a robust *H* one as *k*

**Remark 2:** Unfortunately, our approach can not always produce a robust *H* controller, in

feasible robust controller is always obtained. Hence a condition for detecting an infeasibility

for obtaining a robust feasible *H* controller may be needed. Moreover, 0

 

 

is achieved(See Fig.4). This fact implies that the controller *Kk*

*k k* 

holds. Moreover, from (47) we have

holds, we have

, (53)

. Hence Theorem 1 holds.

<sup>1</sup> . (52)

*k* so as to approach

1 , 1 and 

*<sup>p</sup>* . Hence we may also need a efficient

*<sup>k</sup>* to 0, i.e.,

so that a

*<sup>k</sup>* is a

*k k* 

increases. In this sense, this approach can be an exterior-point approach.

other words, there does not exist the efficient ways of choices of *K*<sup>1</sup> ,

criterion for *Kk* to be a feasible solution of the problem (11).

and this inequality can be obtained by substituting the common constant Lyapunov function *P I* ( ) into (33). Hence , (,, ) *T K zw <sup>p</sup>* holds.

Using Lemmas 6 and 7, we propose the following iterative approach to obtain a feasible solution of the problem (11):

#### **Algorithm**

**Step 1**: Find any *K*1 and let 1 and 1 be scalars which achieve

$$\beta\_1 > \max\_{\lambda \in \sigma(A\_{cl}(\overline{K}\_1, \theta))} \text{Re}[\lambda]\_\prime \left\| \overline{T}\_{zw}(\overline{K}\_1, \theta, \mu\_1) \right\|\_\infty < \gamma\_p, \ \beta\_1 \le \mu\_1 \tag{46}$$

respectively, for vert . For example, let 1 1 where 1 can be chosen as the solution of the LMI's (45). Let : 1 *k* and choose from 0 to 1. **Step 2:** If 0 *<sup>k</sup>* then let \* : *K K <sup>k</sup>* and exit. Otherwise let

*kk p*

$$
\mu\_{k+1} \coloneqq \alpha \beta\_k + (1 - \alpha)\mu\_k \tag{47}
$$

and go to the next step. **Step 3**: Find ( : 0, , ) *Pi N <sup>i</sup>* which satisfy

$$\overline{M}^{\circ} \langle P^{\circ}(\theta), K\_{k'} \mu\_{k+1'} \gamma\_{p'} \theta \rangle < 0, \forall \theta \in \text{vert } \Omega \tag{48}$$

and let them be *Pik* and define

$$P\_k^{\circ \circ} \{ \theta \} \coloneqq P\_{0k}^{\circ \circ} + \sum\_{i=1}^{N} \theta\_i P\_{ik}^{\circ \circ} \tag{49}$$

**Step 4:** Find *K* and *<sup>t</sup>* which are the solutions of

$$\min\_{\mathcal{K}, \mathcal{J}\_t} \mathcal{J}\_t \text{ s.t. } \mathcal{J}\_t < \mu\_{k+1}. \tag{50}$$

$$
\overline{M}^\*(P\_k^\Rightarrow(\theta), K, \beta\_t, \gamma\_{p'}, \theta) < 0, \forall \theta \in \text{vert } \Omega \tag{51}
$$

and let 1 1 :, : *K K k kt* and : 1 *k k* and go to Step 2. Theorem 1 The next statements hold for our algorithm.


96 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

*T*

and this inequality can be obtained by substituting the common constant Lyapunov function

Using Lemmas 6 and 7, we propose the following iterative approach to obtain a feasible

1 1 

<sup>1</sup> : (1 )

<sup>1</sup> ( ( ), , , , ) 0, vert *MP K*

0

*PP P*

*k k i ik i*

<sup>1</sup> , min s.t. ,

 

*t tk <sup>K</sup>*

( ( ), , , , ) 0, vert *MP K k tp*

1

 

> 

(51)

*N*

*kk p*

*kk k* 

from 0 to 1.

1 be scalars which achieve

 

> 

where

 

 *<sup>p</sup>* ,

1 1 

(47)

(48)

(49)

(50)

\* 0, vert

(45)

(46)

1 can be chosen as the solution

 

(,) ( (,) ) \* \*

 

( ) 0

 *<sup>p</sup>* holds.

*clw p cl p*

*B I C K I*

This inequality can be transformed into the next inequality:

1 and 

1 <sup>1</sup> ( ( , ))

. For example, let

which satisfy

and define

 *A K cl*

*<sup>k</sup>* then let \* : *K K <sup>k</sup>* and exit. Otherwise let

*<sup>t</sup>* which are the solutions of

max Re[ ],

 

( ):

*t*

 

and : 1 *k k* and go to Step 2.

1 1 ( ,, ) *T K zw*

*T*

*AK I AK I*

*cl cl*

into (33). Hence , (,, ) *T K zw*

solution of the problem (11):

**Step 1**: Find any *K*1 and let

respectively, for vert

and go to the next step. **Step 3**: Find ( : 0, , ) *Pi N <sup>i</sup>*

and let them be *Pik*

**Step 4:** Find *K* and

and let 1 1 :, : *K K k kt*

 3. , ( ,, ) *T K zw k k p qm g q* ¥ < " ÎW holds for all *<sup>k</sup>* .

Theorem 1 The next statements hold for our algorithm. 1. *mk* is an upper bound of *b<sup>k</sup>* , i.e., *m b k k* ³ holds. 2. *mk* is monotonically decreasing, i.e., *m m k k* > <sup>+</sup>1 holds.

**Step 2:** If 0 

of the LMI's (45). Let : 1 *k* and choose

*P I* ( ) 

**Algorithm** 

**Proof** 1. and 2. From (46) and (50), *k k* holds. Moreover, from (47) we have

$$
\beta\_k < \mu\_{k+1} < \mu\_k \,. \tag{52}
$$

3. From Step 4 of the algorithm and the fact that *k k* holds, we have

$$0 > \overline{M}^{\circ} \left( P\_{k-1}^{\circ} (\theta), K\_k, \beta\_k, \gamma\_p, \theta \right)$$

$$\geq \overline{M}^{\circ} \left( P\_{k-1}^{\circ} (\theta), K\_k, \mu\_k, \gamma\_p, \theta \right)$$

$$\geq M^{\circ} \left( P\_{k-1}^{\circ} (\theta), K\_k, \mu\_k, \gamma\_p, \theta \right), \tag{53}$$

which implies *Kk* achieves ( ,, ) , *T K zw k k p* . Hence Theorem 1 holds.

**Remark 1:** The key idea of our approach is to decrease *k* so as to approach *<sup>k</sup>* to 0, i.e., the shifted imaginary axis approach the original position while the *H* norm constraint ( ,, ) , *T K zw k k p* is achieved(See Fig.4). This fact implies that the controller *Kk* is updated from a non robust *H* controller for the original system to a robust *H* one as *k* increases. In this sense, this approach can be an exterior-point approach.

**Remark 2:** Unfortunately, our approach can not always produce a robust *H* controller, in other words, there does not exist the efficient ways of choices of *K*<sup>1</sup> , 1 , 1 and so that a feasible robust controller is always obtained. Hence a condition for detecting an infeasibility for obtaining a robust feasible *H* controller may be needed. Moreover, 0 *<sup>k</sup>* is a sufficient condition for ( ,) , *T K zw k <sup>p</sup>* . Hence we may also need a efficient criterion for *Kk* to be a feasible solution of the problem (11).

Fig. 4. Concept of our exterior-point approach.

### **5. Numerical example**

To demonstrate the efficiency of our approach let us consider the following matrices:

$$A\_0 = \begin{bmatrix} -9 & 1 & 2 \\ 6 & -8 & -11 \\ -1 & 4 & -7 \end{bmatrix}, A\_1 = \begin{bmatrix} 0 & 0 & 1 \\ -5 & 1 & 4 \\ 0 & -1 & 1 \end{bmatrix}, A\_2 = \begin{bmatrix} -2 & -3 & -2 \\ 0 & -1 & 5 \\ -3 & 2 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 \\ 0 \\ -3 \end{bmatrix}, B\_w = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \tag{54}$$

$$\mathbf{C} = \begin{bmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} \end{bmatrix}, D = \mathbf{0}.\mathbf{1}, E = \begin{vmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} \end{vmatrix}, \gamma\_p = \mathbf{0}.\mathbf{1} \tag{55}$$

An Iterative Approach to the Fixed-Order

Fig. 6. Placement of the closed-poles via \* *K* .

Fig. 7. The contour plot of \* ( ,) *T K zw*

 on .

\* ( ,) *T K zw* 

problem (11).

Fig. 6 and 7 show locations of eigenvalues of \* *A BK E* ( )

Robust *H∞* Control Problem Using a Sequence of Infeasible Controllers 99

the uncertain closed-loop system via controller variables \* *K* and the contour plot of

on , respectively. From these figures, \* *<sup>K</sup>* is a feasible solution of the

, i.e., the perturbations of poles of

Note that the one-shot type methods (Pipeleers et al. 2009), (Shaked 2001), (Xie 2008) can not use for designing the robust *H* controller because of *E I* .

For this numerical example, we set the initial condition for carrying out our approach as follows:

$$K\_1 = \begin{bmatrix} 0 & 0 & 1 & 0 \\ \hline 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}, \rho = 0.3, \beta\_1 = \mu\_1 = 8.1027\tag{56}$$

where 1 1 ( ) is given as the solution of the LMI's (45).

Fig. 5 shows locations of eigenvalues of 1 *A BK E* ( ) , i.e., the perturbations of poles of the uncertain closed-loop system via initial controller variables *K*<sup>1</sup> . This figure shows that *K*1 is not a robust stabilizing controller.

After 10 iterations the next controller variables are given from our approach:

$$
\overline{K}^\* = \begin{bmatrix}
\hline
35.6459 & -43.3047 & -270.7538 & 85.3833 \\
\end{bmatrix}.
\tag{57}
$$

Fig. 5. Placement of the closed-poles via *K*<sup>1</sup> .

98 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

91 2 0 01 2 3 2 0 0 6 8 11 , 5 1 4 , 0 1 5 , 0 , 1 1 4 7 0 11 3 2 0 3 0

(54)

(55)

(57)

(56)

, i.e., the perturbations of poles of the

1 0 0 , 0.1, , 0.1 <sup>010</sup>

Note that the one-shot type methods (Pipeleers et al. 2009), (Shaked 2001), (Xie 2008) can not

For this numerical example, we set the initial condition for carrying out our approach as

uncertain closed-loop system via initial controller variables *K*<sup>1</sup> . This figure shows that *K*1 is

1.5195 3.6942 8.3794 2.6309 35.6459 43.3047 270.7538 85.3833 . 40.8834 1.7382 131.5127 91.5248

, 1 1

 0.3, 8.1027 

ê ú ë û

To demonstrate the efficiency of our approach let us consider the following matrices:

*A A A BBw* 

[ ] <sup>100</sup>

*C DE <sup>p</sup> g* <sup>é</sup> <sup>ù</sup> = == = ê ú

0 12

use for designing the robust *H* controller because of *E I* .

0010 1000 0100

is given as the solution of the LMI's (45).

After 10 iterations the next controller variables are given from our approach:

1

Fig. 5 shows locations of eigenvalues of 1 *A BK E* ( )

\*

*K*

Fig. 5. Placement of the closed-poles via *K*<sup>1</sup> .

not a robust stabilizing controller.

*K*

**5. Numerical example** 

follows:

where 1 1 ( ) 

Fig. 6 and 7 show locations of eigenvalues of \* *A BK E* ( ) , i.e., the perturbations of poles of the uncertain closed-loop system via controller variables \* *K* and the contour plot of \* ( ,) *T K zw* on , respectively. From these figures, \* *<sup>K</sup>* is a feasible solution of the problem (11).

Fig. 6. Placement of the closed-poles via \* *K* .

Fig. 7. The contour plot of \* ( ,) *T K zw* on .

An Iterative Approach to the Fixed-Order

No. 3, pp. 436-442, ISSN 0018-9286

Oregon, USA, June, 2005

795, Bahama, (2004)

504, ISSN 0018-9286

pp. 1257-1272, ISSN 0020-7179

2209, Shanghai, China, December, 2009

*letters,* Vol. 58, No. 7, pp. 510-518, ISSN 0167-6911

*Contr.*, Vol. 46, No. 4, pp. 652-656, ISSN 0018-9286

No. 1, pp.21-30, ISSN 0167-6911

*Contr.*, Vol. 51, No. 12, pp. 1984-1989, ISSN 0018-9286

*'03*, 4 pages, Cambridge, UK, September, 2003

9286

Robust *H∞* Control Problem Using a Sequence of Infeasible Controllers 101

Chesi, G.; Garulli, A.; Tesi, A. & Vicino, A. (2005) Polynomially parameter-dependent

Gahinet, P.; Apkarian, P. & Chilali, M. (1996) Affine Parameter-Dependent Lyapunov

Geromel, J. C & Korogui R. H. (2006) Analysis and Synthesis of Robust Control Systems

Ichihara, H.; Nobuyama, E. & Ishii, T. (2003) Stability analysis and control synthesis with

Ichihara, H. & Nobuyama, E. (2005) Minimax Polynomial Optimization by using Sum of

Iwasaki, T. & Skelton, R. E (1995) The XY-centering algorithm for the dual LMI problem: a

Kami, Y & E. Nobuyama (2004) A Mixed *H* /D-stability Controller Design Using an

Kami, Y.; Tanaka, H. & Nobuyama, E. An Exterior-point Approach to the Robust D-stability

Leite, V. J. S. & Peres, P. L. D. (2003) An improved LMI condition for robust D-stability of

Oliveira, R. C. L. F. & Peres, P. L. D. (2005) Stability of polytopes of matrices via affine

Peaucelle, D.; Arzelier, D.; Bachelier, O. & Bernussou, J. (2000) A new robust D-stability

Shaked, U. (2001) Improved LMI representations for the analysis and the design of

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Squares Relaxation and its Application to Robust Stability Analysis of Parameterdependent Systems, *Proceedings of American Control Conference*, pp. 3433-3434,

new approach to fixed order control design, *International Journal of Control*, Vol. 62,

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characterizations for stability and performance of linear systems, *System & Control* 

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continuous-time systems with polytopic type uncertainty *IEEE Trans. Automatic* 

Fig. 8 shows behaviours of *k* and *<sup>k</sup>* as a function of iteration number *k* . This figure shows that *k* is an upper bound of *<sup>k</sup>* and monotonically decreasing, which implies that the controller variables *Kk* is updated from a non robust stabilizing controller to a robust *H* controller.

Fig. 8. Behaviours of *k* and *<sup>k</sup>* .

### **6. Conclusions**

In this paper, we have considered the robust *H* control problem against time invariant uncertainties. Firstly, we show the relaxation method for obtaining a feasible solution of the PDMI problem with multi-convex functions. Secondly, we introduce the axis-shifted system and show that this system can be constructed so as to achieve the *H* norm constraint. Next, we propose an iterative approach using the axis-shifted system and multi-convex relaxation method for obtaining the robust *H* controllers. The property of our approach is to construct a controller sequence on which the shifted imaginary axis approaches the original position with the *H* norm constraint achieved and to be able to choose any controller variables as an initial point. Finally we have given a numerical example which shows the efficiency of our approach.

### **7. References**

Boyd, S.; Ghaoui, L. E.; Feron, E. & Balakrishnan, V. (1994). *Linear matrix inequalities in system and control theory*, Society for Industrial and Applied Mathematics, ISBN 978- 0898714852, Philadelphia

100 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

the controller variables *Kk* is updated from a non robust stabilizing controller to a robust

In this paper, we have considered the robust *H* control problem against time invariant uncertainties. Firstly, we show the relaxation method for obtaining a feasible solution of the PDMI problem with multi-convex functions. Secondly, we introduce the axis-shifted system and show that this system can be constructed so as to achieve the *H* norm constraint. Next, we propose an iterative approach using the axis-shifted system and multi-convex relaxation method for obtaining the robust *H* controllers. The property of our approach is to construct a controller sequence on which the shifted imaginary axis approaches the original position with the *H* norm constraint achieved and to be able to choose any controller variables as an initial point. Finally we have given a numerical example which

Boyd, S.; Ghaoui, L. E.; Feron, E. & Balakrishnan, V. (1994). *Linear matrix inequalities in system* 

*and control theory*, Society for Industrial and Applied Mathematics, ISBN 978-

*<sup>k</sup>* as a function of iteration number *k* . This figure

*<sup>k</sup>* and monotonically decreasing, which implies that

Fig. 8 shows behaviours of

shows that

*H* controller.

Fig. 8. Behaviours of

**6. Conclusions** 

**7. References** 

*k* and *<sup>k</sup>* .

shows the efficiency of our approach.

0898714852, Philadelphia

*k* and

*k* is an upper bound of


Vanbierviet, J.; Michiels, W. & S. Vandewalle, (2009) Smooth stabilization and optimal *H*<sup>2</sup> design, *Proceedings of IFAC Workshop on Control Applications of Optimisation,* 6pages, ISBN 978-3-902661-42-5

**0**

**6**

*USA*

**Optimizing the Tracking Performance**

<sup>1</sup>*Department of Electrical & Computer Engineering, Purdue University at Fort Wayne*

A typical control engineering problem deals with the design of a control system subject to closed-loop stability and certain performance requirements. The requirements may include the figures of merit such as gain/phase margin, bandwidth, and tracking error to a reference command. The control system is required to achieve the design objectives against unknown or unmeasurable disturbances. The difficulty arises since the plant is often poorly modeled and the set of performance requirements is typically stringent. The robust control theory attempts to address the question of stability and performance of multivariable systems in the face of

In robust control theory, the question concerning the achievable performance limits is generally posed as an optimization problem in an appropriate mathematical setting. A major benefit of this approach is that it provides a means to optimize the system performance by trading off various stringent, and often conflicting, specifications against each other. In the last three decades, *H*∞ control theory has evolved as the primary multivariable optimization and synthesis tool that can effectively deal with the modeling errors and unknown disturbances

In a tracking problem, the reference command is usually specified as a step or ramp signal. Accordingly, the tracking error is also specified in terms of such signals. This class of signals, however, does not model all command signals of interest. For example, a servo control system may be required to track a periodic signal of a fixed period. For this class of applications, the tracking performance must instead be specified in terms of a periodic command signal. Since every periodic signal can be represented by its Fourier series for all time, the steady state tracking performance of a linear feedback system with a periodic command signal can be studied in terms of the steady state tracking performance of each of its sinusoidal components. Design of the control systems that can track periodic reference signals falls in the category of repetitive control (Hara et al., 1998; Lee & Smith, 1998; Sugimoto & Washida, 1997). This has been an active area of research in the last three decades where many successful applications have been reported in the literature. However, applications of the results to certain high performance positioning systems have proved to be more challenging. For example, in (Broberg & Molyet, 1994) a robust repetitive control system is designed to improve the turn-around sinusoidal tracking performance of the imaging mirror system of

modeling errors and unknown disturbances (Zhou et al., 1996).

(Skogesttad & Postlethwaite, 2007).

**1. Introduction**

<sup>2</sup>*Department of Electrical & Computer Engineering, Northeastern University*

**in Robust Control Systems**

Hossein Oloomi1 and Bahram Shafai<sup>2</sup>

Xie, W. (2008) An equivalent LMI representation of bounded real lemma for continuoustime systems *Journal of Inequalities and Applications*, art. No. 672905, ISSN 1025- 5834

## **Optimizing the Tracking Performance in Robust Control Systems**

Hossein Oloomi1 and Bahram Shafai<sup>2</sup>

<sup>1</sup>*Department of Electrical & Computer Engineering, Purdue University at Fort Wayne* <sup>2</sup>*Department of Electrical & Computer Engineering, Northeastern University USA*

### **1. Introduction**

102 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Vanbierviet, J.; Michiels, W. & S. Vandewalle, (2009) Smooth stabilization and optimal *H*<sup>2</sup>

Xie, W. (2008) An equivalent LMI representation of bounded real lemma for continuous-

ISBN 978-3-902661-42-5

5834

design, *Proceedings of IFAC Workshop on Control Applications of Optimisation,* 6pages,

time systems *Journal of Inequalities and Applications*, art. No. 672905, ISSN 1025-

A typical control engineering problem deals with the design of a control system subject to closed-loop stability and certain performance requirements. The requirements may include the figures of merit such as gain/phase margin, bandwidth, and tracking error to a reference command. The control system is required to achieve the design objectives against unknown or unmeasurable disturbances. The difficulty arises since the plant is often poorly modeled and the set of performance requirements is typically stringent. The robust control theory attempts to address the question of stability and performance of multivariable systems in the face of modeling errors and unknown disturbances (Zhou et al., 1996).

In robust control theory, the question concerning the achievable performance limits is generally posed as an optimization problem in an appropriate mathematical setting. A major benefit of this approach is that it provides a means to optimize the system performance by trading off various stringent, and often conflicting, specifications against each other. In the last three decades, *H*∞ control theory has evolved as the primary multivariable optimization and synthesis tool that can effectively deal with the modeling errors and unknown disturbances (Skogesttad & Postlethwaite, 2007).

In a tracking problem, the reference command is usually specified as a step or ramp signal. Accordingly, the tracking error is also specified in terms of such signals. This class of signals, however, does not model all command signals of interest. For example, a servo control system may be required to track a periodic signal of a fixed period. For this class of applications, the tracking performance must instead be specified in terms of a periodic command signal. Since every periodic signal can be represented by its Fourier series for all time, the steady state tracking performance of a linear feedback system with a periodic command signal can be studied in terms of the steady state tracking performance of each of its sinusoidal components. Design of the control systems that can track periodic reference signals falls in the category of repetitive control (Hara et al., 1998; Lee & Smith, 1998; Sugimoto & Washida, 1997). This has been an active area of research in the last three decades where many successful applications have been reported in the literature. However, applications of the results to certain high performance positioning systems have proved to be more challenging. For example, in (Broberg & Molyet, 1994) a robust repetitive control system is designed to improve the turn-around sinusoidal tracking performance of the imaging mirror system of a weather satellite in face of stringent tracking error specifications. A similar situation has been investigated recently by (Aphale et al., 2008; Salapaka et. al, 2002) who considered a robust control design for a high bandwidth nano-positioning system.

usually done in the classical control theory. The results obtained are simple to interpret and provide insights into the interplay among various design parameters including the peaks of the sensitivity and complementary sensitivity functions and the system bandwidth. In Section 4, we briefly demonstrate how these results can be used to obtain the weights in a robust control mixed sensitivity design of a high bandwidth nano-positioning system. We conclude

Optimizing the Tracking Performance in Robust Control Systems 105

We initiate the discussion by considering the feedback system shown in Figure 1. Let *S*(*s*) = 1 + *G*(*s*)*K*(*s*) and *T*(*s*) = 1 − *S*(*s*) be the sensitivity and complementary sensitivity transfer functions, respectively. In the S/T mixed sensitivity design, the objective is to minimize the

> 

*WPS WTT*

where *WP*(*s*) and *WT*(*s*) are the performance and the stability weights, respectively (Skogesttad and Postlethwaite, 2000; Zhou *et al.*, 1996). These weights are often taken to be

✐ ✐ K(s) G(s)

Typically *MS* and *MT* are chosen to be in the interval 1.5 to 2 so that sufficient gain margin,

*MS*−<sup>1</sup> *PM* <sup>≥</sup> 2 arcsin <sup>1</sup>

However, larger values of *MS* and *MT* are unavoidable for nonminimum phase systems. Ideally, *AS* = *AT* = 0 so that 1/|*WP*| and 1/|*WT*| have the desirable Butterworth highpass and Butterworth lowpass characteristics. This ensures that the frequency responses of 1/|*WP*| and 1/|*WT*| are maximally flat in the high and low frequency ranges respectively, where they take the general shapes of the sensitivity and the complementary sensitivity functions. Although, due to the numerical difficulties (Balas *et al.*, 1998), one is often forced to set the parameters

*MT PM* <sup>≥</sup> 2 arcsin <sup>1</sup>

*GM*, and sufficient phase margin, *PM*, are attained according to the inequalities

✲ ✲ ✲ ✲ ✲

 ∞

, *WT*(*s*) =

*s*/*ω*

✐

❄

d

✻

n

2*MS* 

2*MT* .

✛

<sup>√</sup>*<sup>n</sup> ATs*/*<sup>ω</sup>*

*BT* <sup>+</sup> 1/√*<sup>n</sup> MT*

y

*BT* + 1

*<sup>n</sup>* .

**2. Weights selection in general mixed sensitivity design**

*<sup>s</sup>*/ <sup>√</sup>*<sup>m</sup> MS* <sup>+</sup> *<sup>ω</sup>*

✻

*GM* <sup>≥</sup> *MS*

*GM* <sup>≥</sup> <sup>1</sup> <sup>+</sup> <sup>1</sup>

−

Fig. 1. One degree of freedom feedback control system.

r

*s* + *ω B* <sup>√</sup>*<sup>m</sup> AS*

*B*

*<sup>m</sup>*

The amplitude responses of these weights and their inverses are shown in Figure 2.

the chapter in Section 4.

*WP*(*s*) =

infinity norm

An important step in studying the tracking performance of a control system to a sinusoidal reference signal is to investigate the inherent limitations of a feedback system. These limitations provide a deeper understanding of the problem and help a designer to evaluate his/her design against the best attainable tracking error obtained over all possible controller design. The topic been investigated thoroughly in (Su et al., 2003; 2005). The results show that the best achievable performance can be characterized in terms of the inherent properties, mainly the nonminimum phase zeros of the plant and the frequency of the reference signal.

After gaining the necessary insight into the fundamental limitations on the best achievable tracking performance, the next step is to pose the problem as an *H*∞ robust performance problem. Among the various approaches reported in the literature, the mixed-sensitivity *H*∞ control (Kwakernaak H., 2002), signal-based *H*∞ control (Skogesttad & Postlethwaite, 2007), and *H*∞ loop-shaping design (Balas et al., 998) have perhaps gained more popularity with designers. The mixed-sensitivity *H*∞ design is particularly attractive as it gives the designer the ability to directly shape the sensitivity and complementary sensitivity functions. This, in turn, greatly facilitates the trade-off study among several competing performance objectives. The mixed sensitivity design is a conceptually attractive method, but how easily does it lend itself in practical applications? To apply the design, the designer starts by selecting certain weights such that the *H*∞ optimal controller can provide a good trade-off between conflicting objectives in various frequency ranges. After several iterations, the designer is in a position to assess the design to see if all objectives have been met by the controller. If not, the next logical step is to go back and change the weights and repeat the process until a satisfactory result is obtained. Evidently, this is a tedious and often a long process, especially when the system dimension is high. To shorten the design cycle, it is of great interest to have a set of guidelines that can help the designer in selecting the appropriate weights in the optimization process.

The selection of optimal weights for the *H*∞ control has received attention only very recently (Chiang & Hadaegh, 1994; Lanzon, 2001). In (Lanzon, 2000), the problem is formulated in such a way that the controller and the weights are obtained simultaneously and in an iterative manner. However, the question of the suitability of the weights and the complexity of the algorithm employed are yet to be judged. As an alternative, a new set of simple guidelines have been developed recently that can greatly facilitate the selection of appropriate weights (Oloomi & Shafai, 2003). These guidelines are derived using elementary arguments based on phasors and straight-line approximation of the magnitude response, and in the same spirit as what is usually done in the classical control theory. These results are simple to interpret and provide insights into the interplay among various design parameters including the peaks of the sensitivity and complementary sensitivity functions and the system bandwidth.

The chapter is outlines as follows. In Section 2, we briefly discuss the general guidelines used for the selection of the weighting functions in the mixed S/T sensitivity design. In Section 3, we study the problem of the weights selection for tracking sinusoidal reference signals and obtain certain expressions which relate the parameters of the weights to the steady state tracking error specifications. We then outline a procedure for the selection of the parameters of the weighting functions using the derived expressions. The approximate formulae obtained in this chapter are derived using elementary arguments from phasors and straight-line approximation of the magnitude response, in the same spirit as what is 2 Will-be-set-by-IN-TECH

a weather satellite in face of stringent tracking error specifications. A similar situation has been investigated recently by (Aphale et al., 2008; Salapaka et. al, 2002) who considered a

An important step in studying the tracking performance of a control system to a sinusoidal reference signal is to investigate the inherent limitations of a feedback system. These limitations provide a deeper understanding of the problem and help a designer to evaluate his/her design against the best attainable tracking error obtained over all possible controller design. The topic been investigated thoroughly in (Su et al., 2003; 2005). The results show that the best achievable performance can be characterized in terms of the inherent properties, mainly the nonminimum phase zeros of the plant and the frequency of the reference signal. After gaining the necessary insight into the fundamental limitations on the best achievable tracking performance, the next step is to pose the problem as an *H*∞ robust performance problem. Among the various approaches reported in the literature, the mixed-sensitivity *H*∞ control (Kwakernaak H., 2002), signal-based *H*∞ control (Skogesttad & Postlethwaite, 2007), and *H*∞ loop-shaping design (Balas et al., 998) have perhaps gained more popularity with designers. The mixed-sensitivity *H*∞ design is particularly attractive as it gives the designer the ability to directly shape the sensitivity and complementary sensitivity functions. This, in turn, greatly facilitates the trade-off study among several competing performance objectives. The mixed sensitivity design is a conceptually attractive method, but how easily does it lend itself in practical applications? To apply the design, the designer starts by selecting certain weights such that the *H*∞ optimal controller can provide a good trade-off between conflicting objectives in various frequency ranges. After several iterations, the designer is in a position to assess the design to see if all objectives have been met by the controller. If not, the next logical step is to go back and change the weights and repeat the process until a satisfactory result is obtained. Evidently, this is a tedious and often a long process, especially when the system dimension is high. To shorten the design cycle, it is of great interest to have a set of guidelines that can help the designer in selecting the appropriate weights in the optimization process. The selection of optimal weights for the *H*∞ control has received attention only very recently (Chiang & Hadaegh, 1994; Lanzon, 2001). In (Lanzon, 2000), the problem is formulated in such a way that the controller and the weights are obtained simultaneously and in an iterative manner. However, the question of the suitability of the weights and the complexity of the algorithm employed are yet to be judged. As an alternative, a new set of simple guidelines have been developed recently that can greatly facilitate the selection of appropriate weights (Oloomi & Shafai, 2003). These guidelines are derived using elementary arguments based on phasors and straight-line approximation of the magnitude response, and in the same spirit as what is usually done in the classical control theory. These results are simple to interpret and provide insights into the interplay among various design parameters including the peaks of

the sensitivity and complementary sensitivity functions and the system bandwidth.

The chapter is outlines as follows. In Section 2, we briefly discuss the general guidelines used for the selection of the weighting functions in the mixed S/T sensitivity design. In Section 3, we study the problem of the weights selection for tracking sinusoidal reference signals and obtain certain expressions which relate the parameters of the weights to the steady state tracking error specifications. We then outline a procedure for the selection of the parameters of the weighting functions using the derived expressions. The approximate formulae obtained in this chapter are derived using elementary arguments from phasors and straight-line approximation of the magnitude response, in the same spirit as what is

robust control design for a high bandwidth nano-positioning system.

usually done in the classical control theory. The results obtained are simple to interpret and provide insights into the interplay among various design parameters including the peaks of the sensitivity and complementary sensitivity functions and the system bandwidth. In Section 4, we briefly demonstrate how these results can be used to obtain the weights in a robust control mixed sensitivity design of a high bandwidth nano-positioning system. We conclude the chapter in Section 4.

### **2. Weights selection in general mixed sensitivity design**

We initiate the discussion by considering the feedback system shown in Figure 1. Let *S*(*s*) = 1 + *G*(*s*)*K*(*s*) and *T*(*s*) = 1 − *S*(*s*) be the sensitivity and complementary sensitivity transfer functions, respectively. In the S/T mixed sensitivity design, the objective is to minimize the infinity norm

$$\left\| \begin{array}{c} \mathcal{W}\_{PS} \\ \mathcal{W}\_{T}T \end{array} \right\|\_{\mathcal{R}}$$

where *WP*(*s*) and *WT*(*s*) are the performance and the stability weights, respectively (Skogesttad and Postlethwaite, 2000; Zhou *et al.*, 1996). These weights are often taken to be

$$\mathcal{W}\_{\mathcal{P}}(\mathbf{s}) = \left(\frac{\mathbf{s} / \sqrt[m]{M\_S} + \omega\_{\mathcal{B}}^\star}{\mathbf{s} + \omega\_{\mathcal{B}}^\star \sqrt[m]{A\_S}}\right)^m, \qquad \mathcal{W}\_{\mathcal{T}}(\mathbf{s}) = \left(\frac{\mathbf{s} / \omega\_{\mathcal{B}\mathcal{T}}^\star + 1 / \sqrt[m]{M\_{\mathcal{T}}}}{\sqrt[m]{A\_{\mathcal{T}}} \mathbf{s} / \omega\_{\mathcal{B}\mathcal{T}}^\star + 1}\right)^n.$$

The amplitude responses of these weights and their inverses are shown in Figure 2.

Fig. 1. One degree of freedom feedback control system.

Typically *MS* and *MT* are chosen to be in the interval 1.5 to 2 so that sufficient gain margin, *GM*, and sufficient phase margin, *PM*, are attained according to the inequalities

$$\begin{aligned} GM &\geq \frac{M\_{\mathrm{S}}}{M\_{\mathrm{S}} - 1} & PM &\geq 2 \arcsin\left(\frac{1}{2M\_{\mathrm{S}}}\right) \\ GM &\geq 1 + \frac{1}{M\_{\mathrm{T}}} & PM &\geq 2 \arcsin\left(\frac{1}{2M\_{\mathrm{T}}}\right) .\end{aligned}$$

However, larger values of *MS* and *MT* are unavoidable for nonminimum phase systems. Ideally, *AS* = *AT* = 0 so that 1/|*WP*| and 1/|*WT*| have the desirable Butterworth highpass and Butterworth lowpass characteristics. This ensures that the frequency responses of 1/|*WP*| and 1/|*WT*| are maximally flat in the high and low frequency ranges respectively, where they take the general shapes of the sensitivity and the complementary sensitivity functions. Although, due to the numerical difficulties (Balas *et al.*, 1998), one is often forced to set the parameters

**3. Weights selection for sinusoidal tracking performance**

consider the sinusoidal reference command

Then the sinusoidal steady state output is

where the tracking delay is given by

can be written in the compact form

where

In this section, we study the tracking performance of the feedback system in Figure 1 to a sinusoidal command signal. Tracking of other periodic command waveforms can be reduced to this case since every periodic signal can be represented by its Fourier series and *ω<sup>r</sup>* can be chosen to represent the highest frequency component of *r*(*t*) beyond which all other components are negligible. For example, when tracking a triangular waveform, *ω<sup>r</sup>* can represent the frequency of the third harmonic of *r*(*t*) since higher frequency harmonics have negligible amplitudes for this signal. Thus, let us assume that *d* = *n* = 0 in Figure 1 and

Optimizing the Tracking Performance in Robust Control Systems 107

*<sup>r</sup>*(*t*) = *Ar* cos *<sup>ω</sup>rt*, *<sup>ω</sup><sup>r</sup>* � *<sup>ω</sup>*

*yss*(*t*) = *Ar*|*T*(*jωr*)| cos(*ωrt* + ∠*T*(*jωr*))

= *Ar*|*T*(*jωr*)| cos [*ωr*(*t* − *τe*)] ,

*<sup>τ</sup><sup>e</sup>* <sup>=</sup> <sup>−</sup>∠*T*(*jωr*) *ωr*

In tracking applications, the complementary sensitivity function is shaped so that at least up to the tracking frequency the system behaves as an all-pass filter with negligible phase shift, that is |*T*(*jωr*)| ≈ 1 and ∠*T*(*jωr*) ≈ 0. This ensures that the peak steady state error and delay are small so that *yss*(*t*) ≈ *r*(*t*). However, as was mentioned earlier, for high performance applications even small deviation of *yss*(*t*) from the reference signal *r*(*t*) may exceed the performance requirements. Thus, our objective in this chapter is to address this issue by

Using basic results from trigonometry, it is readily seen that the steady state error signal

*ess*(*t*) = *Ar* cos *ωrt* − *Ar*|*T*(*jωr*)| cos [*ωr*(*t* − *τe*)]

*ess*(*t*) = *Re* cos(*ωrt* + *φe*)

1 − |*T*(*jωr*)| cos *ωrτ<sup>e</sup>*

*ωr t* +

.

<sup>=</sup> *Ar*|*T*(*jωr*)<sup>|</sup> cos

This delay is an increasing function of the tracking frequency.

outlining a procedure for selecting the parameters *m*, *ω*

define what we mean by the steady state tracking errors.

*Re* = *Ar*

*<sup>φ</sup><sup>e</sup>* <sup>=</sup> arctan <sup>|</sup>*T*(*jωr*)<sup>|</sup> sin *<sup>ω</sup>rτ<sup>e</sup>*

*B*.

∠*T*(*jωr*) *ωr*

*<sup>B</sup>*, *<sup>n</sup>*, and *<sup>ω</sup>*

<sup>1</sup> + |*T*(*jωr*)|<sup>2</sup> − <sup>2</sup>|*T*(*jωr*)| cos *<sup>ω</sup>rτe*, (1)

*BT*. To this end, we first

. (2)

*AS* and *AT* to some small non-zero values, the forgoing observations still hold true in the frequency ranges of interest. Keeping this into consideration, *AS* and *AT* are chosen to be sufficiently small so that poles of 1/*WP*(*s*) are at least two decades above the zeros of 1/*WP*(*s*), and zeros of 1/*WT*(*s*) are at least two decades above the poles of 1/*WT*(*s*). In general, it is required to have *AS* � *MS* and *AT* � *MT*. Assuming that *MS*, *AS*, *MT*, and *AT* are chosen based on these observations, we now concentrate on selecting the remaining parameters of the weighting functions, namely, *m*, *ω*� *BT*, *<sup>n</sup>*, and *<sup>ω</sup>*� *BT*. General guidelines for selecting these parameters are given below (Skogesttad and Postlethwaite, 2000).

Fig. 2. Stability and performance weighting functions and their inverses.

### **2.1 General guidelines**


### **3. Weights selection for sinusoidal tracking performance**

In this section, we study the tracking performance of the feedback system in Figure 1 to a sinusoidal command signal. Tracking of other periodic command waveforms can be reduced to this case since every periodic signal can be represented by its Fourier series and *ω<sup>r</sup>* can be chosen to represent the highest frequency component of *r*(*t*) beyond which all other components are negligible. For example, when tracking a triangular waveform, *ω<sup>r</sup>* can represent the frequency of the third harmonic of *r*(*t*) since higher frequency harmonics have negligible amplitudes for this signal. Thus, let us assume that *d* = *n* = 0 in Figure 1 and consider the sinusoidal reference command

$$r(t) = A\_r \cos \omega\_r t, \quad \omega\_r \ll \omega\_B^\star.$$

Then the sinusoidal steady state output is

$$\begin{split} y\_{\rm ss}(t) &= A\_{\rm r} |T(j\omega\_{\rm r})| \cos(\omega\_{\rm r}t + \angle T(j\omega\_{\rm r})) \\ &= A\_{\rm r} |T(j\omega\_{\rm r})| \cos\left[\omega\_{\rm r}\left(t + \frac{\angle T(j\omega\_{\rm r})}{\omega\_{\rm r}}\right)\right] \\ &= A\_{\rm r} |T(j\omega\_{\rm r})| \cos\left[\omega\_{\rm r}(t - \tau\_{\rm c})\right] \end{split}$$

where the tracking delay is given by

$$\tau\_{\mathfrak{e}} = -\frac{\angle T(j\omega\_r)}{\omega\_r}.$$

This delay is an increasing function of the tracking frequency.

In tracking applications, the complementary sensitivity function is shaped so that at least up to the tracking frequency the system behaves as an all-pass filter with negligible phase shift, that is |*T*(*jωr*)| ≈ 1 and ∠*T*(*jωr*) ≈ 0. This ensures that the peak steady state error and delay are small so that *yss*(*t*) ≈ *r*(*t*). However, as was mentioned earlier, for high performance applications even small deviation of *yss*(*t*) from the reference signal *r*(*t*) may exceed the performance requirements. Thus, our objective in this chapter is to address this issue by outlining a procedure for selecting the parameters *m*, *ω <sup>B</sup>*, *<sup>n</sup>*, and *<sup>ω</sup> BT*. To this end, we first define what we mean by the steady state tracking errors.

Using basic results from trigonometry, it is readily seen that the steady state error signal

$$e\_{\rm ss}(t) = A\_r \cos \omega\_r t - A\_r |T(j\omega\_r)| \cos \left[\omega\_r (t - \tau\_\varepsilon)\right],$$

can be written in the compact form

$$e\_{\rm ss}(t) = R\_{\varepsilon} \cos(\omega\_r t + \phi\_{\varepsilon})$$

where

4 Will-be-set-by-IN-TECH

*AS* and *AT* to some small non-zero values, the forgoing observations still hold true in the frequency ranges of interest. Keeping this into consideration, *AS* and *AT* are chosen to be sufficiently small so that poles of 1/*WP*(*s*) are at least two decades above the zeros of 1/*WP*(*s*), and zeros of 1/*WT*(*s*) are at least two decades above the poles of 1/*WT*(*s*). In general, it is required to have *AS* � *MS* and *AT* � *MT*. Assuming that *MS*, *AS*, *MT*, and *AT* are chosen based on these observations, we now concentrate on selecting the remaining parameters of

*BT*, *<sup>n</sup>*, and *<sup>ω</sup>*�

1/MS

Fig. 2. Stability and performance weighting functions and their inverses.

1/MT

1. For systems with *PM* ≤ 90◦, it is well known that *ω<sup>B</sup>* ≤ *ω<sup>c</sup>* ≤ *ωBT* where *ωB*, *ωBT*, and *ω<sup>c</sup>* are the closed loop bandwidth measured on the basis of *S*, the closed loop bandwidth measured on the basis of *T*, and the gain crossover frequency, respectively. Therefore, it

zeros places restriction on the achievable bandwidth. Moreover, for high performance tracking applications with noticeable measurement noise it often becomes necessary to

a peak in the sensitivity curve. This implies that the system will have less stability margins

causes a reduction in the system bandwidth and this manifests itself in a poor tracking

3. When the control objective is to reduce the effect of the measurement noise, the general rule

4. Increasing *m* and *n* can improve the disturbance rejection and measurement noise attenuation, respectively. However, *m* and *n* should be kept as low as possible since large values of these parameters adversely affect the stability margins, and the controller order becomes unnecessarily high. (Controller order is *N* + *n* + *m* where *N* is the order of the

*BT* as much as possible. However, decreasing *<sup>ω</sup>*�

2. When disturbance attenuation is the control objective, the general rule is to increase *ω*�

which manifests itself in an increased overshoot in the step response.

*BT* <sup>&</sup>lt; *<sup>ω</sup>*� *B*.

MT

MS

parameters are given below (Skogesttad and Postlethwaite, 2000).

1/|WP|≈|S|

........ ......... ......... ......... .........


*BT*. General guidelines for selecting these

<sup>ω</sup><sup>r</sup> <sup>ω</sup>

*<sup>B</sup>* more than necessary causes the appearance of

....................................................................................................................................


*BT*. It should be noted that the presence of nonminimum phase

1/|WT|≈|T|

......... ......... ......... ......... ......... ......... ......... ........

BT M1/<sup>n</sup> T

..

........ ........ ........ ........ ........ ........ ......... ......

> ω BT A1/<sup>n</sup> T

*BT* more than necessary

............................................................................................................

AT

*<sup>B</sup>* as

1/AT

the weighting functions, namely, *m*, *ω*�

<sup>ω</sup> <sup>ω</sup><sup>r</sup> BA1/<sup>m</sup> <sup>S</sup> <sup>ω</sup>

......... ......... ........ ........ ........ ........ ........ ........ ........

**2.1 General guidelines**

is required that *ω*�

is to decrease *ω*�

performance.

plant.)

AS

1/AS

BM1/<sup>m</sup> S

*<sup>B</sup>* <sup>≤</sup> *<sup>ω</sup>*�

make a compromise and instead choose *ω*�

much as possible. However, increasing *ω*�

.........................................................................................................................................................................................................................

$$R\_{\varepsilon} = A\_{\prime} \sqrt{1 + |T(j\omega\_{r})|^{2} - 2|T(j\omega\_{r})|\cos\omega\_{r}\tau\_{\ell\nu}} \tag{1}$$

$$\phi\_{\varepsilon} = \arctan\left(\frac{|T(j\omega\_{r})|\sin\omega\_{r}\tau\_{\varepsilon}}{1 - |T(j\omega\_{r})|\cos\omega\_{r}\tau\_{\varepsilon}}\right). \tag{2}$$

phase zeros are located near the origin for which large peaks appear in the sensitivity and

Optimizing the Tracking Performance in Robust Control Systems 109

, *<sup>β</sup>* :<sup>=</sup> *<sup>ω</sup><sup>r</sup>*

*ω*� *BT*

, ∠*T*(*jωr*) ≈ −∠*WT*(*jωr*).

(3)

We first derive an expression for the tracking delay. To simplify notation, let

*<sup>α</sup>* :<sup>=</sup> *<sup>ω</sup><sup>r</sup> ω*� *B*

and note that 0 < *α*, *β* � 1. Using the approximation *WTT* ≈ 1, we have


<sup>|</sup>*T*(*jωr*)| ≈ <sup>1</sup>

= *MT* 

Next, we derive an expression for the peak steady state error. Since *WPS* ≈ 1, we have


1 + *M*<sup>2</sup>

1 + *j <sup>α</sup> m* <sup>√</sup>*AS* 1 + *j <sup>α</sup> <sup>m</sup>* <sup>√</sup>*MS*  

<sup>≈</sup> *<sup>α</sup>m*, for *<sup>m</sup> AS* � *<sup>α</sup>* � *<sup>m</sup> MS*.

*m*


= *AS* 

On the other hand, *E*(*s*) = *S*(*s*)*R*(*s*) so that at the steady state we also have

<sup>|</sup>*S*(*jωr*)| ≈ <sup>1</sup>


1 + *jβ*

1 + *jβ*

≈ *MT*, for *β* �

<sup>√</sup>*<sup>n</sup> AT*

  *n*

1 <sup>√</sup>*<sup>n</sup> MT*

> *β <sup>n</sup> AT*

*Re* <sup>≈</sup> *Arαm*. (6)

*<sup>T</sup>* − 2*MT* cos *ωrτe*. (7)

, (4)

. (5)

<sup>√</sup>*<sup>n</sup> MT*

<sup>−</sup> arctan

, ∠*S*(*jωr*) ≈ −∠*WP*(*jωr*).

<sup>|</sup>*T*(*jωr*)| ≈ <sup>1</sup>

*<sup>τ</sup><sup>e</sup>* <sup>≈</sup> <sup>∠</sup>*WT*(*jωr*) *ωr*

> <sup>≈</sup> *<sup>n</sup> ωr* arctan *β <sup>n</sup> MT*

<sup>|</sup>*S*(*jωr*)| ≈ <sup>1</sup>

Therefore, by equating (1) and (6) and using (4) we obtain

*<sup>α</sup><sup>m</sup>* <sup>≈</sup> 

Therefore, using the straight line approximation

and

Therefore,

complementary sensitivity response curves.

The result is depicted in Figure 3 where the sinusoidal components of the steady state error signal are represented as phasors in the quadrature plane with the reference axis taken as cos *ωrt*. It is seen that the steady state error phasor is rotated by an angle of *φ<sup>e</sup>* in the counter-clockwise direction due to the presence of the tracking delay *τe*, and that the peak amplitude of the steady state tracking error, namely *Re*, is influenced by this rotation as well as the gain of the closed loop system at the tracking frequency *ωr*. It should be noted that when |*T*(*jωr*)| cos *ωrτ<sup>e</sup>* ≤ 1, this phasor resides in the first quadrant so that tan *φ<sup>e</sup>* > 0. However, when |*T*(*jωr*)| cos *ωrτ<sup>e</sup>* > 1, the steady state error phasor moves to the second quadrant for which tan *φ<sup>e</sup>* < 0. Therefore, in obtaining *φ<sup>e</sup>* from tan *φ<sup>e</sup>* in the latter case, we must interpret *φ<sup>e</sup>* as being in the second quadrant ant not in the fourth. Typical sinusoidal tracking waveforms with small peak steady state error and small delay are also shown in Figure 4 where the lead property of the steady state error signal is cleanly seen.

Fig. 3. Phasor diagram for the steady state sinusoidal tracking error.

Fig. 4. Steady state sinusoidal tracking error signal.

We now derive expressions for the parameters of the weighting functions in terms of the tracking error parameters *Re* and *τe*. To this end, recall from (Skogesttad and Postlethwaite, 2000) that in the mixed sensitivity design the weighting functions *WP* and *WT* are used to scale the closed loop transfer functions *S* and *T*, respectively in order to satisfy the performance and stability requirements, and that the inverse of these weighting functions are upper bounds, up to constant scaling factors, on the transfer functions they are used to scale. These constant factors can be absorbed in the weighting functions themselves so that the approximations *WPS* ≈ 1 and *WTT* ≈ 1 are reasonable for appropriate weights. However, the discrepancies can become noticeable if the controller is not designed properly or when the nonminimum phase zeros are located near the origin for which large peaks appear in the sensitivity and complementary sensitivity response curves.

We first derive an expression for the tracking delay. To simplify notation, let

$$\alpha := \frac{\omega\_r}{\omega\_B^{\star}}, \qquad \beta := \frac{\omega\_r}{\omega\_{BT}^{\star}} \tag{3}$$

and note that 0 < *α*, *β* � 1. Using the approximation *WTT* ≈ 1, we have

$$|T(j\omega\_r)| \approx \frac{1}{|\mathcal{W}\_T(j\omega\_r)' } \qquad \angle T(j\omega\_r) \approx -\angle \mathcal{W}\_T(j\omega\_r).$$

Therefore, using the straight line approximation

$$\begin{split} |T(j\omega\_r)| &\approx \frac{1}{|\mathcal{W}\_T(j\omega\_r)|} \\ &= M\_T \left| \frac{1 + j\beta\sqrt[n]{A\_T}}{1 + j\beta\sqrt[n]{M\_T}} \right|^n \\ &\approx M\_{T'} \qquad \text{for } \beta \ll \frac{1}{\sqrt[n]{M\_T}}. \end{split} \tag{4}$$

and

6 Will-be-set-by-IN-TECH

The result is depicted in Figure 3 where the sinusoidal components of the steady state error signal are represented as phasors in the quadrature plane with the reference axis taken as cos *ωrt*. It is seen that the steady state error phasor is rotated by an angle of *φ<sup>e</sup>* in the counter-clockwise direction due to the presence of the tracking delay *τe*, and that the peak amplitude of the steady state tracking error, namely *Re*, is influenced by this rotation as well as the gain of the closed loop system at the tracking frequency *ωr*. It should be noted that when |*T*(*jωr*)| cos *ωrτ<sup>e</sup>* ≤ 1, this phasor resides in the first quadrant so that tan *φ<sup>e</sup>* > 0. However, when |*T*(*jωr*)| cos *ωrτ<sup>e</sup>* > 1, the steady state error phasor moves to the second quadrant for which tan *φ<sup>e</sup>* < 0. Therefore, in obtaining *φ<sup>e</sup>* from tan *φ<sup>e</sup>* in the latter case, we must interpret *φ<sup>e</sup>* as being in the second quadrant ant not in the fourth. Typical sinusoidal tracking waveforms with small peak steady state error and small delay are also shown in Figure 4 where the lead

> 0 Ar ........................................................................................................ ..

> > ............. .. .

... .... ... .... ... ... ... ... ... ...

−ωrτ cos ωrt

Re

......... ... ... ... .. ... .... .... ... ... ... ... ... ... ... ... .... .. ... .... .... ......... ........ ............................................... .. .. .... ...... ...

sin ωrt

Fig. 3. Phasor diagram for the steady state sinusoidal tracking error.

..................................... .. .........

........ ........ ........ .........

ess(t)

. .... ...... ...... .. .... ....... .. ..

φe

....... ....... ....... ...... .......

0 t

We now derive expressions for the parameters of the weighting functions in terms of the tracking error parameters *Re* and *τe*. To this end, recall from (Skogesttad and Postlethwaite, 2000) that in the mixed sensitivity design the weighting functions *WP* and *WT* are used to scale the closed loop transfer functions *S* and *T*, respectively in order to satisfy the performance and stability requirements, and that the inverse of these weighting functions are upper bounds, up to constant scaling factors, on the transfer functions they are used to scale. These constant factors can be absorbed in the weighting functions themselves so that the approximations *WPS* ≈ 1 and *WTT* ≈ 1 are reasonable for appropriate weights. However, the discrepancies can become noticeable if the controller is not designed properly or when the nonminimum

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................................................................................................

..................................... .. .........

........ ....... ........ ....... ........ ....... ........ ........ ....... ........ ........ ........ ....... ....... ........ ....... ........ ........ ....... ........

......................... . ............... ....

r(t)

property of the steady state error signal is cleanly seen.

Ar|T(jω<sup>r</sup> )|

........ ......... ........ ....... ....... ........ ........ ......... .......

yss(t)

Fig. 4. Steady state sinusoidal tracking error signal.

$$\begin{split} \pi\_{\varepsilon} &\approx \frac{\angle \mathcal{W}\_{\mathsf{T}}(j\omega\_{r})}{\omega\_{r}} \\ &\approx \frac{n}{\omega\_{\mathsf{T}}} \left[ \arctan \left( \beta \sqrt[n]{M\_{\mathsf{T}}} \right) - \arctan \left( \beta \sqrt[n]{A\_{\mathsf{T}}} \right) \right]. \end{split} \tag{5}$$

Next, we derive an expression for the peak steady state error. Since *WPS* ≈ 1, we have

$$|S(j\omega\_r)| \approx \frac{1}{|\mathcal{W}\_P(j\omega\_r)' } \qquad \angle S(j\omega\_r) \approx -\angle \mathcal{W}\_P(j\omega\_r).$$

Therefore,

$$\begin{split} |S(j\omega\_{r})| &\approx \frac{1}{|W\_{P}(j\omega\_{r})|} \\ &= A\_{S} \left| \frac{1 + j\frac{\alpha}{\sqrt[m]{A\_{S}}}}{1 + j\frac{\alpha}{\sqrt[m]{M\_{S}}}} \right|^{m} \\ &\approx \alpha^{m}, \qquad \text{for } \sqrt[m]{A\_{S}} \ll \alpha \ll \sqrt[m]{M\_{S}}. \end{split}$$

On the other hand, *E*(*s*) = *S*(*s*)*R*(*s*) so that at the steady state we also have

$$R\_{\mathfrak{e}} \approx A\_{r} \mathfrak{a}^{m}.\tag{6}$$

Therefore, by equating (1) and (6) and using (4) we obtain

$$
\alpha^m \approx \sqrt{1 + M\_T^2 - 2M\_T \cos \omega\_r \tau\_\theta}. \tag{7}
$$

An expression relating (5) to (7) can now be derived noting that

$$
\angle \mathcal{W}\_{\mathcal{P}}(j\omega\_r) \approx m \left[ \arctan \left( \frac{a}{\sqrt[m]{M\_{\mathcal{S}}}} \right) - \arctan \left( \frac{a}{\sqrt[m]{A\_{\mathcal{S}}}} \right) \right]. \tag{8}
$$

Since

$$
\begin{split}
\angle \mathcal{E}(\text{j}\omega\_r) &= \angle \mathcal{S}(\text{j}\omega\_r) + \angle \mathcal{R}(\text{j}\omega\_r) \\ &= \angle \mathcal{S}(\text{j}\omega\_r) \\ &= -\angle \mathcal{W}\_{\text{P}}(\text{j}\omega\_r),
\end{split}
\tag{9}
$$

**3.1 Guidelines for sinusoidal tracking performance**

2. Calculate *MT* from (12) using the values of *m* and *ω*

only specifications to be dealt with. 1. Initially, let *m* = 1 and calculate *ω*

*B*.

satisfactory result is obtained.

and *MT* calculated in Steps 1 and 2.

3. Let *n* = 1 and calculate *ω*

Steps 1 and 2. If *ω*

re-calculate *ω*

**4. Application**

Assume that *Ar*, *ωr*, and the upper bounds on the tracking errors *Re* and *τ<sup>e</sup>* are specified. Further, assume that the parameters *AS* and *AT* are chosen to be some small positive numbers on the basis of our earlier guidelines. A procedure for selecting the remaining parameters of the weighting functions *WP*(*s*) and *WT*(*s*) are given below assuming that *Re* and *τ<sup>e</sup>* are the

Optimizing the Tracking Performance in Robust Control Systems 111

*BT* from (13) with the values of *<sup>m</sup>*, *<sup>ω</sup>*

*BT* is not large enough, increase *<sup>n</sup>* and recalculate *<sup>ω</sup>*

4. Finally, calculate *MS* from (14) and (15), or from (16) if *m* = 1, using the values of *ω*

The importance of nanotechnology has been brought to full attention by the scanning probe microscopy and is the result of new techniques used to explore the properties of near atomic-scale structure (Aphale et al., 2008; Barrett & Quate, 1991; Teoh et al., 2008). However, most schemes of nanotechnology impose severe specifications on positioning systems, making the control system design more challenging. For example, micro/nano positioning systems are essential in auto focus systems, fast mirror scanners, image steering devices in optics; disk spin stands and vibration cancelation in disk drives; wafer and mask positioning in microelectronics; micropumps, needle valve actuation, linear drives, and piezo hammers in precision mechanics; and cell penetration and microdispensing devices in medicine and

In (Salapaka et. al, 2002), a mixed sensitivity robust control has been successfully applied to a noano-poistioning device, suited to biological samples as part of an atomic force microscope, where it is shown that substantial improvement in the positioning and precision is attainable over the conventional PI control. The improvement reported in this chapter is judged on the basis of the system ability to track a "high frequency" triangular reference waveform with a small peak error (in order of micro-meter) and a small delay (in order of milli-seconds). However, it is notable that the success of the design reported in (Salapaka et. al, 2002), as well as other mixed sensitivity designs, depends largely on the appropriate selection of the weights used in the optimization process. While for typical applications appropriate weights are often easily chosen after several trials and errors, the stringent performance requirements imposed for the ultra-high performance applications makes the selection of

In the last section, we derived certain approximate expressions in terms of the tracking performance specifications and provided a guideline for the selection of the weights in the mixed sensitivity design. These expressions should prove valuable to the designer as they expedite the weights selection process in the simulation/design cycle. In order to demonstrate the usefulness of the guideline, consider the mixed sensitivity robust control design for a high

biology (Daniele et al., 1999; Salapaka et. al, 2002; Tamer & Dahleh, 1994).

appropriate weights difficult, or at least time-consuming.

*<sup>B</sup>* from (11). If this value is too large, increase *m* and

*<sup>B</sup>* obtained in Step 1.

*<sup>B</sup>*, and *MT* calculated in

*BT* from (13) till a

*<sup>B</sup>*, *m*,

from (2), (4), (8), and (9) we obtain

$$\arctan\left(\frac{M\_T \sin \omega\_r \tau\_\varepsilon}{1 - M\_T \cos \omega\_r \tau\_\varepsilon}\right) \approx m \left[\arctan\left(\frac{a}{\sqrt[n]{A\_S}}\right) - \arctan\left(\frac{a}{\sqrt[n]{M\_S}}\right)\right].\tag{10}$$

Expressions (5), (6), (7), and (10) are the basic expressions to be used in the selection of the weighting functions. In order to gain insight into the relationships among various parameters involved in these equations, we make further simplifications by noting that *AS*, *AT*, *α* and *β* are small positive numbers. Thus, by neglecting appropriate terms, these equations reduce to

$$
\omega\_B^\star \approx \omega\_r \left(\frac{A\_r}{R\_\ell}\right)^{\frac{1}{m}},
\tag{11}
$$

$$M\_T \approx \cos \omega\_r \tau\_\varepsilon + \sqrt{\left(\frac{\omega\_B^\star}{\omega\_r}\right)^{2m} - \sin^2 \omega\_r \tau\_{\varepsilon \prime}}\tag{12}$$

$$
\omega\_{BT}^{\star} \approx \frac{\omega\_r \sqrt[n]{M\_T}}{\tan \left( \frac{\omega\_r \tau\_r}{n} \right)},
\tag{13}
$$

$$M\_S \approx \left[\frac{\omega\_r}{\omega\_B^\star \left| \tan \left(\frac{m\pi}{2} - \gamma \right) \right|}\right]^m, \qquad (\gamma \neq m\pi/2) \tag{14}$$

where

$$\gamma = \arctan\left(\frac{M\_T \sin \omega\_r \tau\_\ell}{1 - M\_T \cos \omega\_r \tau\_\ell}\right). \tag{15}$$

Note that *ω <sup>B</sup>* � *ω<sup>r</sup>* so that (12) is well defined. For (13), we have used the trigonometric identity tan(*x* − *y*)=(tan *x* − tan *y*)/(1 + tan *x* tan *y*) to obtain the quadratic equation

$$\sqrt[n]{M\_T A\_T} \tan\left(\frac{\omega\_r \tau\_\varepsilon}{n}\right) \beta^2 + \left(\sqrt[n]{A\_T} - \sqrt[n]{M\_T}\right) \beta + \tan\left(\frac{\omega\_r \tau\_\varepsilon}{n}\right) = 0,$$

and then have set *AT* ≈ 0. Derivation of the remaining equations is straightforward. When *m* = 1, (14) and (15) can be combined using the trigonometric identity tan(*x* − *y*)=(tan *x* − tan *y*)/(1 + tan *x* tan *y*) resulting in

$$M\_{\rm S} \approx \frac{\omega\_{\rm r} M\_{\rm T} \sin \omega\_{\rm r} \tau\_{\rm \varepsilon}}{\omega\_{\rm B}^{\star} |1 - M\_{\rm T} \cos \omega\_{\rm r} \tau\_{\rm \varepsilon}|}, \qquad (M\_{\rm T} \cos \omega\_{\rm r} \tau\_{\rm \varepsilon} \neq 1). \tag{16}$$

### **3.1 Guidelines for sinusoidal tracking performance**

Assume that *Ar*, *ωr*, and the upper bounds on the tracking errors *Re* and *τ<sup>e</sup>* are specified. Further, assume that the parameters *AS* and *AT* are chosen to be some small positive numbers on the basis of our earlier guidelines. A procedure for selecting the remaining parameters of the weighting functions *WP*(*s*) and *WT*(*s*) are given below assuming that *Re* and *τ<sup>e</sup>* are the only specifications to be dealt with.


### **4. Application**

8 Will-be-set-by-IN-TECH

 *α* <sup>√</sup>*<sup>m</sup> MS*

∠*E*(*jωr*) = ∠*S*(*jωr*) + ∠*R*(*jωr*) = ∠*S*(*jωr*)

Expressions (5), (6), (7), and (10) are the basic expressions to be used in the selection of the weighting functions. In order to gain insight into the relationships among various parameters involved in these equations, we make further simplifications by noting that *AS*, *AT*, *α* and *β* are small positive numbers. Thus, by neglecting appropriate terms, these equations reduce to

 *α* <sup>√</sup>*<sup>m</sup> AS* − arctan

 *α* <sup>√</sup>*<sup>m</sup> AS*

= −∠*WP*(*jωr*), (9)

− arctan

, (11)

, (13)

*β* + tan

 *ωrτe n* = 0,

, (*MT* cos *ωrτ<sup>e</sup>* �= 1). (16)

<sup>−</sup> sin<sup>2</sup> *<sup>ω</sup>rτe*, (12)

, (*γ* �= *mπ*/2) (14)

. (15)

 *α* <sup>√</sup>*<sup>m</sup> MS* . (10)

. (8)

An expression relating (5) to (7) can now be derived noting that

 arctan

 ≈ *m* arctan

 *Ar Re*

<sup>√</sup>*<sup>n</sup> MT* tan *<sup>ω</sup>rτ<sup>e</sup> n*

*ωr*

*MT* ≈ cos *ωrτ<sup>e</sup>* +

*BT* <sup>≈</sup> *<sup>ω</sup><sup>r</sup>*

 *ωrτe n β*<sup>2</sup> +

*MS* <sup>≈</sup> *<sup>ω</sup>rMT* sin *<sup>ω</sup>rτ<sup>e</sup> ω*

*<sup>B</sup>*|1 − *MT* cos *ωrτe*|

*ω B* tan *<sup>m</sup><sup>π</sup>*

*γ* = arctan

 1 *m*

> *<sup>ω</sup> B ωr*

> > <sup>2</sup> − *γ*

identity tan(*x* − *y*)=(tan *x* − tan *y*)/(1 + tan *x* tan *y*) to obtain the quadratic equation

<sup>2</sup>*<sup>m</sup>*

*<sup>m</sup>*

 *MT* sin *ωrτ<sup>e</sup>* 1 − *MT* cos *ωrτ<sup>e</sup>*

*<sup>n</sup> AT* <sup>−</sup> *<sup>n</sup> MT*

and then have set *AT* ≈ 0. Derivation of the remaining equations is straightforward. When *m* = 1, (14) and (15) can be combined using the trigonometric identity tan(*x* − *y*)=(tan *x* −

*<sup>B</sup>* � *ω<sup>r</sup>* so that (12) is well defined. For (13), we have used the trigonometric

∠*WP*(*jωr*) ≈ *m*

 *MT* sin *ωrτ<sup>e</sup>* 1 − *MT* cos *ωrτ<sup>e</sup>*

> *ω <sup>B</sup>* ≈ *ω<sup>r</sup>*

*ω*

*<sup>n</sup> MT AT* tan

tan *y*)/(1 + tan *x* tan *y*) resulting in

*MS* ≈

from (2), (4), (8), and (9) we obtain

arctan

Since

where

Note that *ω*

The importance of nanotechnology has been brought to full attention by the scanning probe microscopy and is the result of new techniques used to explore the properties of near atomic-scale structure (Aphale et al., 2008; Barrett & Quate, 1991; Teoh et al., 2008). However, most schemes of nanotechnology impose severe specifications on positioning systems, making the control system design more challenging. For example, micro/nano positioning systems are essential in auto focus systems, fast mirror scanners, image steering devices in optics; disk spin stands and vibration cancelation in disk drives; wafer and mask positioning in microelectronics; micropumps, needle valve actuation, linear drives, and piezo hammers in precision mechanics; and cell penetration and microdispensing devices in medicine and biology (Daniele et al., 1999; Salapaka et. al, 2002; Tamer & Dahleh, 1994).

In (Salapaka et. al, 2002), a mixed sensitivity robust control has been successfully applied to a noano-poistioning device, suited to biological samples as part of an atomic force microscope, where it is shown that substantial improvement in the positioning and precision is attainable over the conventional PI control. The improvement reported in this chapter is judged on the basis of the system ability to track a "high frequency" triangular reference waveform with a small peak error (in order of micro-meter) and a small delay (in order of milli-seconds). However, it is notable that the success of the design reported in (Salapaka et. al, 2002), as well as other mixed sensitivity designs, depends largely on the appropriate selection of the weights used in the optimization process. While for typical applications appropriate weights are often easily chosen after several trials and errors, the stringent performance requirements imposed for the ultra-high performance applications makes the selection of appropriate weights difficult, or at least time-consuming.

In the last section, we derived certain approximate expressions in terms of the tracking performance specifications and provided a guideline for the selection of the weights in the mixed sensitivity design. These expressions should prove valuable to the designer as they expedite the weights selection process in the simulation/design cycle. In order to demonstrate the usefulness of the guideline, consider the mixed sensitivity robust control design for a high bandwidth nano-positioning system as discussed in (Salapaka et. al, 2002). A model of the device obtained experimentally is a fourth order nonminimum phase transfer function

expressions. Application of the results in minimizing the tracking errors of a nano-positioning

Optimizing the Tracking Performance in Robust Control Systems 113

Aphale, S. S.; Devasia, S. & Moheimani, S. O. R. (2008). Achieving High-Bandwidth

Balas, G. J.; Doyle, J. C., Glover, K., Packard A. & and Smith, R. (1998). *μ-Analysis and Synthesis*

Barrett, R. C. & Quate, C. F. (1991). Optical Scan Correction System Applied to Atomic Force

Broberg, H. L. & Molyet, R. G. (1994). A New Approach to Phase Cancellation in Repetitive

Chiang R. Y. & Hadaegh F. Y. (1994). Theory and Weighting Strategies of Mixed Sensitivity

Daniele, A.; Salapaka, S., Salapaka, M. V. & Dahleh, M. (1999). Piezoelectric Scanners for

Hara, S.; Yamamoto, Y., Omata T. & and Nakano M. (1998). Repetitive Control System: A New

Kwakernaak H. (2002). Mixed Sensitivity Design, *15th IFAC Triennial World Congress*,

Lee, R. C. H. & Smith, M. C. (1998). Robustness and Trade-offs in Repetitive Control.

Lanzon, A. (2000). Weight Selection in Robust Control: An Optimization Approach. *PhD*

Lanzon, A. (2001). Simultaneous Synthesis of Weights and Controllers in *H*∞ Loop-Shaping,

Lanzon, A. (2005). Weight Optimization in *H*∞ Loop-Shaping. *Automatica*, Vol. 41, No. 7, pp.

Oloomi, H. & Shafai, B. (2003). Weight Selection in Mixed Sensitivity Robust Control for

Salapaka, S.; Sebastian, A., Cleveland, J. P. & and Salapaka, M. V. (2002). High Bandwidth

Skogesttad, S. & Postlethwaite, I. (2007). *Multivariable Feedback Control*, John Wiley & Sons,

Su W.; Qiu L. & Chen J. (2003). Fundamental Performance Limitations in Tracking Sinusoidal

Su W.; Qiu L. & Chen J. (2005). On Performance Limitation in Tracking a Sinusoid. *IEEE Trans.*

Signals. *IEEE Trans. on Automat. Contr.*, Vol. 48, No. 8, pp. 1371-1380.

Improving the Sinusoidal Tracking Performance, *Proc. of the IEEE Conf. on Decision*

Nano-Positioner: A Robust Control Approach. *Rev. Sci. Instruments*, Vol. 73, No. 9,

*on Advanced Intelligent Mechatronics*, pp. 943-948, Xian, China.

Microscopy. *Rev. Sci. Instruments*, Vol. 62, No. 6, pp. 1393-1399.

*Proc. of the American Contr. Conf.*, pp. 253-257, San Diego, CA.

*Proc. of Conf. on Decision and Contr.*, pp. 670-675, Orlando, FL.

*Toolbox*, Mathworks Inc, Natick, MA.

*Society*, Vol. 3, pp. 1766-1770.

Vol. 33, No. 7, pp. 659-668.

*Automatica*, Vol. 34, pp. 889-896.

*Thesis*, Cambridge University.

*and Control*, pp. 300-305, Maui, HI.

ISBN 978-0-470-01168-3, New York.

*on Automat. Contr.*, Vol. 51, No. 8, pp. 1320-1325.

Barcelona, Spain.

1201-1208.

pp. 3232-3241.

*Aerospace*, pp. 12-16, Palo Alto, CA.

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Control, *Conf. Record of the 1994 IEEE Annual Meeting of the Industry Applications*

*H*<sup>∞</sup> Synthesis on a Class of Aerospace Applications, *JFAC Symp. on Automat. Contr. in*

Atomic Force Microscopes: Design of Lateral Sensors, Identification and Control,

Type Servo System for Periodic Exogenous Signals. *IEEE Trans. on Automat. Contr.*,

system is demonstrated.

**6. References**

$$G(s) = \frac{9.7 \times 10^4 (s - (7.2 \pm 7.4j) \times 10^3)}{(s + (1.9 \pm 4.5j) \times 10^3)(s + (1.2 \pm 15.2j) \times 10^2)}.$$

The design considered is a mixed S/T/KS design where the weight on the controller transfer function *KS* is chosen to be *Wu* = 0.1 in order to restrict the magnitude of the input signal within the saturation limit. The other weights chosen are

$$W\_P(s) = \frac{0.1667s + 2827}{s + 2.827}, \qquad W\_T(s) = \frac{s + 235.6}{0.01s + 1414}.$$

A simulation result presented in this chapter which shows a sinusoidal tracking response with *Re* ≈ 1(*μm*) and *τ<sup>e</sup>* ≈ 2 [msec] when system is subjected to a 100 [Hz] command signal with peak value of 5 [*μm*]. From the selected weights, it is seen that

 $m = 1$ ,  $\omega\_B^\star = 2827$ ,  $M\_S \approx 36$ ,  $A\_S = 10^{-6}$ ,  $m = 1$ ,  $\omega\_{BT}^\star = 1414$ ,  $M\_T \approx 36$ ,  $A\_T = 10^{-4}$ .

We like to demonstrate how the initial weights can be obtained using the expressions derived earlier. Starting with *<sup>m</sup>* <sup>=</sup> 1, *Ar*/*Re* <sup>=</sup> 5, and *<sup>ω</sup><sup>r</sup>* <sup>≈</sup> 628 [rad/sec], we obtain *<sup>ω</sup> <sup>B</sup>* ≈ 3142 [rad/sec] which is not too far from the given value of 2827 [rad/sec]. Since *ωrτ<sup>e</sup>* ≈ 70.4 and *ω <sup>B</sup>*/*ω<sup>r</sup>* ≈ 5, equation (12) gives *MT* ≈ 5.245 which is better than the one chosen in (Salapaka et. al, 2002). With the calculated values and from (13) we next obtain *ω BT* ≈ 1176 [rad/sec] which is again not too far from the given value in (Salapaka et. al, 2002). Finally, from (16) we obtain *MS* ≈ 1.5 which is lower than what is considered in that chapter. Therefore, we see that while *ω <sup>B</sup>* and *<sup>ω</sup> BT* are fairly close in the first try, the values of *MS* and *MT* are considerably lower. This is however expected since large values of *MS* and *MT* are unavoidable here due to the presence of a complex pair of RHP zeros (Su et al., 2003; 2005).

In conclusion, we see that using the expressions derived in this chapter, a designer can start off with a fairly reasonable set of parameters and further adjust these parameters for the desired performance. Additionally, if larger values of *MT* and *MS* are to be allowed, the derived expressions can be used to see how these changes affect the remaining parameters like *ω B* and *ω BT*. For example, it is seen from (12) that a larger *MT* is obtained at the expense of a larger value for *ω <sup>B</sup>*. From (13), this in turn implies a larger value for *<sup>ω</sup> BT* as well, and the same can be said for *MS* form (14) and (15). In summary, the values obtain from the derived expressions in this chapter can form the basis of the first try in the simulation and as such should prove valuable to the designers.

#### **5. Conclusion**

In this chapter, the mixed sensitivity robust tracking problem of a feedback system with sinusoidal command waveforms is studied. Approximate expressions relating the tracking errors specifications to various parameters of the weighting functions used in the mixed S/T sensitivity design are derived. The derivation presented in this chapter uses simple arguments using phasors and straight line approximation of magnitude response. We have outlined guidelines for the selection of the weighting functions parameters using the derived expressions. Application of the results in minimizing the tracking errors of a nano-positioning system is demonstrated.

### **6. References**

10 Will-be-set-by-IN-TECH

bandwidth nano-positioning system as discussed in (Salapaka et. al, 2002). A model of the device obtained experimentally is a fourth order nonminimum phase transfer function

The design considered is a mixed S/T/KS design where the weight on the controller transfer function *KS* is chosen to be *Wu* = 0.1 in order to restrict the magnitude of the input signal

A simulation result presented in this chapter which shows a sinusoidal tracking response with *Re* ≈ 1(*μm*) and *τ<sup>e</sup>* ≈ 2 [msec] when system is subjected to a 100 [Hz] command signal with

We like to demonstrate how the initial weights can be obtained using the expressions derived

[rad/sec] which is not too far from the given value of 2827 [rad/sec]. Since *ωrτ<sup>e</sup>* ≈ 70.4 and

which is again not too far from the given value in (Salapaka et. al, 2002). Finally, from (16) we obtain *MS* ≈ 1.5 which is lower than what is considered in that chapter. Therefore, we see that

lower. This is however expected since large values of *MS* and *MT* are unavoidable here due

In conclusion, we see that using the expressions derived in this chapter, a designer can start off with a fairly reasonable set of parameters and further adjust these parameters for the desired performance. Additionally, if larger values of *MT* and *MS* are to be allowed, the derived expressions can be used to see how these changes affect the remaining parameters like *ω*

*<sup>B</sup>*. From (13), this in turn implies a larger value for *<sup>ω</sup>*

same can be said for *MS* form (14) and (15). In summary, the values obtain from the derived expressions in this chapter can form the basis of the first try in the simulation and as such

In this chapter, the mixed sensitivity robust tracking problem of a feedback system with sinusoidal command waveforms is studied. Approximate expressions relating the tracking errors specifications to various parameters of the weighting functions used in the mixed S/T sensitivity design are derived. The derivation presented in this chapter uses simple arguments using phasors and straight line approximation of magnitude response. We have outlined guidelines for the selection of the weighting functions parameters using the derived

*BT*. For example, it is seen from (12) that a larger *MT* is obtained at the expense of a

*BT* are fairly close in the first try, the values of *MS* and *MT* are considerably

*<sup>B</sup>*/*ω<sup>r</sup>* ≈ 5, equation (12) gives *MT* ≈ 5.245 which is better than the one chosen in (Salapaka

earlier. Starting with *<sup>m</sup>* <sup>=</sup> 1, *Ar*/*Re* <sup>=</sup> 5, and *<sup>ω</sup><sup>r</sup>* <sup>≈</sup> 628 [rad/sec], we obtain *<sup>ω</sup>*

et. al, 2002). With the calculated values and from (13) we next obtain *ω*

to the presence of a complex pair of RHP zeros (Su et al., 2003; 2005).

(*<sup>s</sup>* + (1.9 ± 4.5*j*) × 103)(*<sup>s</sup>* + (1.2 ± 15.2*j*) × 102)

*<sup>s</sup>* <sup>+</sup> 2.827 , *WT*(*s*) = *<sup>s</sup>* <sup>+</sup> 235.6

*<sup>B</sup>* <sup>=</sup> 2827, *MS* <sup>≈</sup> 36, *AS* <sup>=</sup> <sup>10</sup>−6,

*BT* <sup>=</sup> 1414, *MT* <sup>≈</sup> 36, *AT* <sup>=</sup> <sup>10</sup>−4.

.

*<sup>B</sup>* ≈ 3142

*B*

*BT* ≈ 1176 [rad/sec]

*BT* as well, and the

0.01*<sup>s</sup>* <sup>+</sup> <sup>1414</sup> .

*<sup>G</sup>*(*s*) = 9.7 <sup>×</sup> <sup>104</sup>(*<sup>s</sup>* <sup>−</sup> (7.2 <sup>±</sup> 7.4*j*) <sup>×</sup> <sup>103</sup>)

within the saturation limit. The other weights chosen are

*WP*(*s*) = 0.1667*<sup>s</sup>* <sup>+</sup> <sup>2827</sup>

peak value of 5 [*μm*]. From the selected weights, it is seen that

*m* = 1, *ω*

*n* = 1, *ω*

*ω*

while *ω*

and *ω*

larger value for *ω*

**5. Conclusion**

should prove valuable to the designers.

*<sup>B</sup>* and *<sup>ω</sup>*


Sugimoto, H. & Washida, K. (1997). A Proposition of Design Method for Modified Repetitive Control System with Corrected Dead Time Using Sensitivity Function Shaping and its Application to Motor Control System, *Proc. of the Power Conversion Conf.*, Vol. 2, pp. 619-624, Nagaoka.

**Part 2** 

**Robust Control of Robotic Systems** 


## **Part 2**

**Robust Control of Robotic Systems** 

12 Will-be-set-by-IN-TECH

114 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Sugimoto, H. & Washida, K. (1997). A Proposition of Design Method for Modified Repetitive

Tamer, N. & Dahleh, M. (1994). Feedback Control of Piezoelectric Tube Scanners, *Proc. of the*

Teoh J. N.; Du, C. & Xie, L. (2008). Combined *H*<sup>2</sup> and KYP Lemma Based Control for

Zhou, K.; Doyle, J. C. & and Glover, K. (1996). *Robust and Optimal Control*, Prentice Hall, ISBN

*American Contr. Conf.*, pp. 1826-1831, Lake Buena Vista, FL.

pp. 619-624, Nagaoka.

0-13-456567-3, New Jersey.

pp. 828-833.

Control System with Corrected Dead Time Using Sensitivity Function Shaping and its Application to Motor Control System, *Proc. of the Power Conversion Conf.*, Vol. 2,

Positioning Error Minimization and Specific Narrowband Disturbance Rejections, *IEEE Int. Conf. on Control Applications (IEEE Multi-Conference on System and Control)*,

**0**

**7**

*Japan*

**Robust Adaptive Position/Force Control**

A mobile manipulator is a class of mobile robot on which the multi-link manipulator is mounted. This system is expected to play an important role both in the production process of factory and in the medical care system of welfare business. To come up to this expectation, a mobile manipulator is required to simultaneously track to both the desired position trajectory and force trajectory. However, these tracking performances are subject to nonholonomic and holonomic constraints. Furthermore, mobile manipulators possess complex and strongly coupled dynamics of mobile bases and manipulators. Then, there are very few studies on

In (Chang & Chen, 2002; Oya et al., 2003; Su et al., 1999), position and force control methods for mobile robot without manipulators have been addressed. Since in these studies holonomic constraints representing the interaction between end-effector of the manipulator and environment have not been considered, those approaches could not be applied to the position/force control problems of the mobile manipulators. In (Dong, 2002; Li et al., 2007; 2008), adaptive and robust control approaches have been applied to the position/force control problems of the mobile manipulators. In these approaches, since the chained form transforms are required, synthesis methods of the control torques and adaptation laws of these approaches are too complicated to apply. On the other hand, we have derived the stabilizing controllers for a class of mobile manipulators(Narikiyo et al., 2008). In (Narikiyo et al., 2008) we have proposed robust adaptive control scheme for the system with dynamic uncertainties and external disturbances directly from the reduced order dynamics subject to both the holonomic and nonholonomic constraints. Furthermore, in (Narikiyo et al., 2009) we have developed this control scheme to control the system with both kinematic and dynamic uncertainties. In these studies usefulness of these control schemes have been demonstrated by numerical examples. However, proof of the closed loop stability has not been completed

In this study we complete the proof and relax the assumptions of (Narikiyo et al., 2009). Then we implement these control schemes (Narikiyo et al., 2008; 2009) experimentally and apply to the prototype shown in Fig.1 to demonstrate the effectiveness of these proposed control schemes. It is also guaranteed theoretically that the tracking position and force errors to the desired trajectories are asymptotically converged to zero by the proposed control schemes.

the problems of stabilization position/force control for mobile manipulators.

under an inadequate assumption(Narikiyo et al., 2009).

**1. Introduction**

**of Mobile Manipulators**

*Toyota Technological Institute*

Tatsuo Narikiyo and Michihiro Kawanishi

## **Robust Adaptive Position/Force Control of Mobile Manipulators**

Tatsuo Narikiyo and Michihiro Kawanishi *Toyota Technological Institute Japan*

### **1. Introduction**

A mobile manipulator is a class of mobile robot on which the multi-link manipulator is mounted. This system is expected to play an important role both in the production process of factory and in the medical care system of welfare business. To come up to this expectation, a mobile manipulator is required to simultaneously track to both the desired position trajectory and force trajectory. However, these tracking performances are subject to nonholonomic and holonomic constraints. Furthermore, mobile manipulators possess complex and strongly coupled dynamics of mobile bases and manipulators. Then, there are very few studies on the problems of stabilization position/force control for mobile manipulators.

In (Chang & Chen, 2002; Oya et al., 2003; Su et al., 1999), position and force control methods for mobile robot without manipulators have been addressed. Since in these studies holonomic constraints representing the interaction between end-effector of the manipulator and environment have not been considered, those approaches could not be applied to the position/force control problems of the mobile manipulators. In (Dong, 2002; Li et al., 2007; 2008), adaptive and robust control approaches have been applied to the position/force control problems of the mobile manipulators. In these approaches, since the chained form transforms are required, synthesis methods of the control torques and adaptation laws of these approaches are too complicated to apply. On the other hand, we have derived the stabilizing controllers for a class of mobile manipulators(Narikiyo et al., 2008). In (Narikiyo et al., 2008) we have proposed robust adaptive control scheme for the system with dynamic uncertainties and external disturbances directly from the reduced order dynamics subject to both the holonomic and nonholonomic constraints. Furthermore, in (Narikiyo et al., 2009) we have developed this control scheme to control the system with both kinematic and dynamic uncertainties. In these studies usefulness of these control schemes have been demonstrated by numerical examples. However, proof of the closed loop stability has not been completed under an inadequate assumption(Narikiyo et al., 2009).

In this study we complete the proof and relax the assumptions of (Narikiyo et al., 2009). Then we implement these control schemes (Narikiyo et al., 2008; 2009) experimentally and apply to the prototype shown in Fig.1 to demonstrate the effectiveness of these proposed control schemes. It is also guaranteed theoretically that the tracking position and force errors to the desired trajectories are asymptotically converged to zero by the proposed control schemes.

#### **2. Modeling of mobile manipulator**

#### Fig. 1. Mobile manipulator

Fig.1 shows the prototype of mobile manipulator employed in experiments. Let *qB* <sup>∈</sup> *<sup>R</sup>n*, *qM* <sup>∈</sup> *<sup>R</sup><sup>m</sup>* and *<sup>q</sup>* = [*q<sup>T</sup> <sup>B</sup> <sup>q</sup><sup>T</sup> M*] *<sup>T</sup>* <sup>∈</sup> *<sup>R</sup>n*+*<sup>m</sup>* be the generalized coordinates of the mobile base, manipulator and whole system, respectively. Then the equations of nonholonomic constraints imposed on the mobile base are written as

$$J\_B(q\_B)\dot{q}\_B = 0,\tag{1}$$

where *qB* = [*q<sup>T</sup> <sup>B</sup>*<sup>1</sup> *<sup>q</sup><sup>T</sup> B*2] *<sup>T</sup>* and *JB*(*qB*)=[*JB*<sup>1</sup> *JB*2] <sup>∈</sup> *<sup>R</sup>*(*n*−*k*)×*n*, *detJB*<sup>1</sup> �<sup>=</sup> 0. The equations of holonomic constraints imposed on the manipulator are given by

$$\Phi(q) = 0,\tag{2}$$

where <sup>Φ</sup>(*q*) <sup>∈</sup> *<sup>R</sup>m*−*h*. Let *JM*(*q*) = *<sup>∂</sup>*Φ/*∂<sup>q</sup>* <sup>∈</sup> *<sup>R</sup>*(*m*−*h*)×(*n*+*m*) ,*rankJM* = *m* − *h*. Then (2) can be rewritten as

$$J\_M(q)\dot{q} = 0.\tag{3}$$

Furthermore, let

$$J\_M(q) = \left[\frac{\partial \Phi}{\partial q\_B} \begin{array}{c} \frac{\partial \Phi}{\partial q\_{M1}} \end{array} \frac{\partial \Phi}{\partial q\_{M2}}\right] = \left[J\_{M0} \begin{array}{c} J\_{M1} \ J\_{M2} \end{array}\right] \mu$$

*qM* = [*q<sup>T</sup> <sup>M</sup>*<sup>1</sup> *<sup>q</sup><sup>T</sup> <sup>M</sup>*2] *<sup>T</sup>*, *qM*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>h*, *qM*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>m*−*<sup>h</sup>* and *detJM*<sup>2</sup> �<sup>=</sup> 0. Then the equations of motion of the mobile manipulator is written as

$$M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) + d(q,t) = f^T(q)\lambda + B(q)\tau\_\prime \tag{4}$$

$$J(q)\dot{q} = 0,\tag{5}$$

of Mobile Manipulators 3

Robust Adaptive Position/Force Control of Mobile Manipulators 119

Indices {*i*, *j* = 1, 2} correspond to decompositions of *qB*, *qM*. *d*(*t*) denotes uncertain

from the floor and *<sup>λ</sup><sup>M</sup>* <sup>∈</sup> *<sup>R</sup>m*−*<sup>h</sup>* denote reaction forces acted on the end-effector from the

*ξ* + *C*(*q*, *q*˙)*ξ* + *G*(*q*) = *Y*(*q*, *q*˙, *ξ*, ˙

where *<sup>p</sup>*<sup>0</sup> <sup>∈</sup> *<sup>R</sup>s*<sup>0</sup> denotes unknown parameter vector and *<sup>Y</sup>* <sup>∈</sup> *<sup>R</sup>*(*n*+*m*)×*s*<sup>0</sup> is called regressor

Let *fB*(*qB*)=[ *f*1(*qB*), ..., *fk*(*qB*)] be the bases of null space of *JB*(*qB*), then there exists *η* =

Using the suitable selection of *fB*(*qB*), *η* can be specified to be equal to forward linear velocity

loss of generality. Since *η* corresponds to angular velocity of wheels *νB*, there exists *ϕ* such

 −*J* −1 *<sup>B</sup>*<sup>1</sup> *JB*<sup>2</sup> *Ik*

*<sup>S</sup>*(*qB*) = *Blockdiag* {*SB*(*qB*), *Im*} <sup>∈</sup> *<sup>R</sup>*(*n*+*m*)×(*k*+*m*)

*M*(*q*)*S*˙(*qB*) + *C*(*q*, *q*˙)*S*(*qB*)

*JM*<sup>0</sup> *fBϕ*−1*ν<sup>B</sup>* + *JM*1*q*˙*M*<sup>1</sup>

Differentiating (8), substituting it into (4) and multiplying both sides by *ST*(*qB*) from the left,

*G*1(*q*) = *ST*(*qB*)*G*(*q*), *d*1(*q*,*t*) = *ST*(*qB*)*d*(*q*, *t*),

It is well known that Property 1 and 2 are invariant under changes of

*M*1(*q*)*ν*˙ + *C*1(*q*, *q*˙)*ν* + *G*1(*q*) + *d*1(*q*, *t*) = *B*1(*q*)*τ* + *J*

*M*1(*q*) = *ST*(*qB*)*M*(*q*)*S*(*qB*),

*<sup>B</sup>*1(*q*) = *<sup>S</sup>T*(*qB*)*B*(*q*), *JM* =

coordinates(Murray et al., 1993). Then (9) has following properties similarly to (4).

*C*1(*q*, *q*˙) = *ST*(*qB*)

environment. The equation (4) has following properties(Slotine & Li, 1991).

*u* and angular velocity *ω* of the mobile base, that is, *k* = 2 and *η* = [*η*<sup>1</sup> *η*2]

*SB*(*qB*) = *fB*(*qB*)*ϕ*−<sup>1</sup> =

*<sup>T</sup>*, *<sup>λ</sup><sup>B</sup>* <sup>∈</sup> *<sup>R</sup>n*−*<sup>k</sup>* denote reaction forces acted on the wheels

*ξ*)*p*0,

*q*˙*<sup>B</sup>* = *fB*(*qB*)*η*. (6)

*q*˙*<sup>B</sup>* = *SB*(*qB*)*νB*, (7)

 .

*<sup>T</sup><sup>T</sup>*

*q*˙ = *S*(*qB*)*ν*. (8)

*T*

 ,

> .

*JM*<sup>0</sup> *fBϕ*−<sup>1</sup> *JM*<sup>1</sup> *JM*<sup>2</sup>

*<sup>T</sup>* = [*u ω*]

,

*<sup>M</sup>*(*q*)*λM*, (9)

<sup>∈</sup> *<sup>R</sup>k*+*m*,

*<sup>T</sup>*, without

disturbance. For *λ* = [*λ<sup>T</sup>*

**Property 2**:For any *ξ*

[*η*1, ..., *ηk*]

where

Furthermore, let

then we have

where

**Property 1**:*M*˙ <sup>−</sup> <sup>2</sup>*<sup>C</sup>* is skew symmetric.

*<sup>B</sup> <sup>λ</sup><sup>T</sup> M*]

*M*(*q*) ˙

matrix whose elements consist of known functions.

*<sup>T</sup>* such that (1) is equivalent to

that *ν<sup>B</sup>* = *ϕη*. Therefore (6) is rewritten as

*ν* = *νT <sup>B</sup>* , *q*˙ *T <sup>M</sup>*1, − *J* −1 *M*2 

we have(Yamamoto & Yun, 1996)

where

$$\begin{split} M(q) &= \begin{bmatrix} M\_{11}(q) \ M\_{12}(q) \\ M\_{21}(q) \ M\_{22}(q) \end{bmatrix}, G(q) = \begin{bmatrix} \mathcal{G}\_{11}(q) \\ \mathcal{G}\_{21}(q) \end{bmatrix}, \mathcal{C}(q, \dot{q}) = \begin{bmatrix} \mathcal{C}\_{11}(q, \dot{q}) \\ \mathcal{C}\_{21}(q, \dot{q}) \end{bmatrix}, \\ B(q) &= \begin{bmatrix} B\_{11}(q\_{B}) & 0 \\ 0 & I\_{m} \end{bmatrix}, d(q, t) = \begin{bmatrix} d\_{11}(q, t) \\ d\_{21}(q, t) \end{bmatrix}, \\ J(q) &= \begin{bmatrix} J\_{B} & 0 & 0 \\ J\_{M0} & J\_{M1} & J\_{M2} \end{bmatrix}, \tau = \begin{bmatrix} \tau\_{B} \\ \tau\_{M} \end{bmatrix}, \lambda = \begin{bmatrix} \lambda\_{B} \\ \lambda\_{M} \end{bmatrix}. \end{split}$$

Indices {*i*, *j* = 1, 2} correspond to decompositions of *qB*, *qM*. *d*(*t*) denotes uncertain disturbance. For *λ* = [*λ<sup>T</sup> <sup>B</sup> <sup>λ</sup><sup>T</sup> M*] *<sup>T</sup>*, *<sup>λ</sup><sup>B</sup>* <sup>∈</sup> *<sup>R</sup>n*−*<sup>k</sup>* denote reaction forces acted on the wheels from the floor and *<sup>λ</sup><sup>M</sup>* <sup>∈</sup> *<sup>R</sup>m*−*<sup>h</sup>* denote reaction forces acted on the end-effector from the environment. The equation (4) has following properties(Slotine & Li, 1991).

**Property 1**:*M*˙ <sup>−</sup> <sup>2</sup>*<sup>C</sup>* is skew symmetric.

**Property 2**:For any *ξ*

2 Will-be-set-by-IN-TECH

Fig.1 shows the prototype of mobile manipulator employed in experiments. Let *qB* <sup>∈</sup> *<sup>R</sup>n*,

manipulator and whole system, respectively. Then the equations of nonholonomic constraints

*<sup>T</sup>* <sup>∈</sup> *<sup>R</sup>n*+*<sup>m</sup>* be the generalized coordinates of the mobile base,

*<sup>T</sup>* and *JB*(*qB*)=[*JB*<sup>1</sup> *JB*2] <sup>∈</sup> *<sup>R</sup>*(*n*−*k*)×*n*, *detJB*<sup>1</sup> �<sup>=</sup> 0. The equations of

= [*JM*<sup>0</sup> *JM*<sup>1</sup> *JM*2] ,

*<sup>T</sup>*, *qM*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>h*, *qM*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>m*−*<sup>h</sup>* and *detJM*<sup>2</sup> �<sup>=</sup> 0. Then the equations of motion of

 ,

 *λ<sup>B</sup> λM* .

, *C*(*q*, *q*˙) =

*JB*(*qB*)*q*˙*<sup>B</sup>* = 0, (1)

Φ(*q*) = 0, (2)

*JM*(*q*)*q*˙ = 0. (3)

,*rankJM* = *m* − *h*. Then (2) can be

*<sup>T</sup>*(*q*)*λ* + *B*(*q*)*τ*, (4) *J*(*q*)*q*˙ = 0, (5)

> *C*11(*q*, *q*˙) *C*21(*q*, *q*˙)

 ,

**2. Modeling of mobile manipulator**

Fig. 1. Mobile manipulator

*<sup>B</sup> <sup>q</sup><sup>T</sup> M*]

holonomic constraints imposed on the manipulator are given by

 *∂*Φ *∂qB*

*∂*Φ *∂qM*<sup>1</sup>

*M*(*q*)*q*¨ + *C*(*q*, *q*˙)*q*˙ + *G*(*q*) + *d*(*q*, *t*) = *J*

, *d*(*q*, *t*) =

, *G*(*q*) =

 *τ<sup>B</sup> τM* , *λ* =

*∂*Φ *∂qM*<sup>2</sup>  *G*11(*q*) *G*21(*q*)

*d*11(*q*,*t*) *d*21(*q*,*t*)

where <sup>Φ</sup>(*q*) <sup>∈</sup> *<sup>R</sup>m*−*h*. Let *JM*(*q*) = *<sup>∂</sup>*Φ/*∂<sup>q</sup>* <sup>∈</sup> *<sup>R</sup>*(*m*−*h*)×(*n*+*m*)

*M*11(*q*) *M*12(*q*) *M*21(*q*) *M*22(*q*)

 , *τ* =

*B*11(*qB*) 0 0 *Im*

 *JB* 0 0 *JM*<sup>0</sup> *JM*<sup>1</sup> *JM*<sup>2</sup>

*JM*(*q*) =

imposed on the mobile base are written as

*<sup>B</sup>*<sup>1</sup> *<sup>q</sup><sup>T</sup> B*2]

the mobile manipulator is written as

*M*(*q*) =

*B*(*q*) =

*J*(*q*) =

*qM* <sup>∈</sup> *<sup>R</sup><sup>m</sup>* and *<sup>q</sup>* = [*q<sup>T</sup>*

where *qB* = [*q<sup>T</sup>*

rewritten as

*qM* = [*q<sup>T</sup>*

where

Furthermore, let

*<sup>M</sup>*<sup>1</sup> *<sup>q</sup><sup>T</sup> <sup>M</sup>*2]

$$M(q)\dot{\xi} + \mathcal{C}(q, \dot{q})\xi + G(q) = Y(q, \dot{q}, \xi, \dot{\xi})p\_{0\prime}$$

where *<sup>p</sup>*<sup>0</sup> <sup>∈</sup> *<sup>R</sup>s*<sup>0</sup> denotes unknown parameter vector and *<sup>Y</sup>* <sup>∈</sup> *<sup>R</sup>*(*n*+*m*)×*s*<sup>0</sup> is called regressor matrix whose elements consist of known functions.

Let *fB*(*qB*)=[ *f*1(*qB*), ..., *fk*(*qB*)] be the bases of null space of *JB*(*qB*), then there exists *η* = [*η*1, ..., *ηk*] *<sup>T</sup>* such that (1) is equivalent to

$$d\_B = f\_B(q\_B)\eta. \tag{6}$$

Using the suitable selection of *fB*(*qB*), *η* can be specified to be equal to forward linear velocity *u* and angular velocity *ω* of the mobile base, that is, *k* = 2 and *η* = [*η*<sup>1</sup> *η*2] *<sup>T</sup>* = [*u ω*] *<sup>T</sup>*, without loss of generality. Since *η* corresponds to angular velocity of wheels *νB*, there exists *ϕ* such that *ν<sup>B</sup>* = *ϕη*. Therefore (6) is rewritten as

$$
\dot{q}\_B = S\_B(q\_B) \upsilon\_{B\_{\prime}} \tag{7}
$$

where

$$S\_B(q\_B) = f\_B(q\_B) \, q^{-1} = \begin{bmatrix} -I\_{B1}^{-1} I\_{B2} \\ I\_k \end{bmatrix}.$$

Furthermore, let

$$S(q\_B) = \text{Blockdiag}\left\{ \mathcal{S}\_B(q\_B), I\_m \right\} \in \mathcal{R}^{(n+m)\times(k+m)},$$

$$\nu = \left[ \boldsymbol{\nu}\_{B'}^T \dot{\boldsymbol{\eta}}\_{M1'}^T - \left\{ \boldsymbol{J}\_{M2}^{-1} \left( \boldsymbol{J}\_{M0} f\_B \boldsymbol{\varphi}^{-1} \boldsymbol{\nu}\_B + \boldsymbol{J}\_{M1} \dot{\boldsymbol{\eta}}\_{M1} \right) \right\}^T \right]^T \in \mathbb{R}^{k+m},$$

then we have

$$
\dot{q} = \mathcal{S}(q\_B)\nu.\tag{8}
$$

Differentiating (8), substituting it into (4) and multiplying both sides by *ST*(*qB*) from the left, we have(Yamamoto & Yun, 1996)

$$M\_1(q)\dot{\upsilon} + \mathbb{C}\_1(q, \dot{q})\upsilon + \mathbb{G}\_1(q) + d\_1(q, t) = B\_1(q)\tau + \overline{f}\_M^T(q)\lambda\_{M\nu} \tag{9}$$

where

$$\begin{aligned} M\_1(q) &= S^T(q\_B)M(q)S(q\_B), \\ C\_1(q, \dot{q}) &= S^T(q\_B) \left\{ M(q)\dot{S}(q\_B) + \mathcal{C}(q, \dot{q})S(q\_B) \right\}, \\ G\_1(q) &= S^T(q\_B)G(q), \\ B\_1(q) &= S^T(q\_B)B(q), \overline{J}\_M = \left[ J\_{M0}f\_B\dot{\sigma}^{-1}J\_{M1}f\_{M2} \right]. \end{aligned}$$

It is well known that Property 1 and 2 are invariant under changes of coordinates(Murray et al., 1993). Then (9) has following properties similarly to (4).

**Property 3**:*M*˙ <sup>1</sup> <sup>−</sup> <sup>2</sup>*C*<sup>1</sup> is skew-symmetric. **Property 4**:For any ¯ *ξ*

$$M\_1(q)\dot{\overline{\xi}} + \mathcal{C}\_1(q, \dot{q})\overline{\xi} + G\_1(q) = Y\_1(q, \dot{q}, \overline{\xi}, \dot{\overline{\xi}})p\_{1/2}$$

where *<sup>p</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>s*<sup>1</sup> denotes unknown parameter vector and *<sup>Y</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>*(*k*+*m*)×*s*<sup>1</sup> is called the regressor matrix whose elements consist of known functions. Furthermore, kinematic uncertainties of the system give the following properties(Cheah et al., 2003; Fukao et al., 2000). **Property 5**:*SB*(*qB*)*ν<sup>B</sup>* in (7) can be written as

$$S\_B(q\_B)\nu\_B = \sum\_{i=1}^k \left(\sigma\_{i0}(q\_B) + \sum\_{j=1}^{h\_i} \theta\_{ij}\sigma\_{ij}(q\_B)\right)\nu\_{Bi}.$$

**Property 6**:*J T <sup>M</sup>*(*q*)*λ<sup>M</sup>* in (9) can be written as

$$\mathcal{T}\_M^\mathsf{T}(q)\lambda\_M = Z\_1(q,\lambda\_M)\psi\_\lambda$$

where *θij* is unknown parameter which consists of unknown parameters of mobile base, and *<sup>σ</sup>ij* is known functions which consists of the coordinate *qB*, (*<sup>i</sup>* <sup>=</sup> 1, ..., *<sup>k</sup>*, *<sup>j</sup>* <sup>=</sup> 1, ..., *hi*). *<sup>ψ</sup>* <sup>∈</sup> *<sup>R</sup><sup>c</sup>* is unknown parameter vector of the whole system and *<sup>Z</sup>*1(*q*, *<sup>λ</sup>M*) <sup>∈</sup> *<sup>R</sup>*(*k*+*m*)×*<sup>c</sup>* is known matrix function of the position/force coordinate *q* and *λ<sup>M</sup>* , respectively.

Following assumptions are required to synthesize the control scheme.

**Assump.1**:There are no unknown parameters in *B*1(*q*) and *detB*1(*q*) � 0 for all *q*. *d*<sup>1</sup> and its derivative are bounded and �*d*1� ≤ *D*. Where *D* is unknown.

**Assump.2**:*JB*, *JM*, *J* −1 *<sup>B</sup>*<sup>1</sup> , *J* −1 *<sup>M</sup>*<sup>2</sup> ∈ *L*<sup>∞</sup> and these matrices are all continuously differentiable with respect to *q* and kinematic parameters, and these derivatives are bounded.

#### **3. Hybrid position/force control scheme**

Let *q*∗ be the desired position trajectory, then there exist desired velocity input *ν*∗ = [*ν*∗ <sup>1</sup> , ..., *ν*<sup>∗</sup> *<sup>k</sup>* , *<sup>ν</sup>*∗*<sup>T</sup> <sup>M</sup>* ] *<sup>T</sup>* such that

$$
\dot{q}^\* = \mathcal{S}(q\_B^\*) \nu^\*.\tag{10}
$$

of Mobile Manipulators 5

Robust Adaptive Position/Force Control of Mobile Manipulators 121

(, ) *<sup>x</sup> <sup>y</sup>*

real system

Fig. 2. Reference robot and tracking errors

*eB* =

*eM*<sup>1</sup> = *q*<sup>∗</sup>

trajectory tracking are written as the following.

⎡ ⎣ *e*1 *e*2 *e*3 ⎤ ⎦ = ⎡ ⎣

Using the results in (Fukao et al., 2000), desired velocity inputs *ν<sup>c</sup>* = [*ν<sup>T</sup>*

*νBc* = *ϕuBc*, *uBc* =

∗

For the system (7) the following Lemma is shown in (Fukao et al., 2000).

−1 *M*2 �

=

*νM*1*<sup>c</sup>* = *q*˙

*νM*2*<sup>c</sup>* = −*J*

Where *Ki* > 0, *i* = 1, 2, 3 and *KM*<sup>1</sup> are arbitrarily assigned.

*given by the following equations.*

*Then,* lim*t*→<sup>∞</sup> *eB* = 0*.*

*<sup>M</sup>*<sup>1</sup> <sup>−</sup> *qM*1, *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>=</sup> *<sup>λ</sup><sup>M</sup>* <sup>−</sup> *<sup>λ</sup>*<sup>∗</sup>

by

I

� *uc ωc* �

*JM*<sup>0</sup> *fB ϕ*−1*νBc* + *JM*1*νM*1*<sup>c</sup>*

*e*˙3 = −*u*∗*K*2*e*<sup>2</sup> − *K*<sup>3</sup> sin *e*<sup>3</sup> (13)

� *u*<sup>∗</sup> cos *e*<sup>3</sup> + *K*1*e*<sup>1</sup> *ω*∗ + *u*∗*K*2*e*<sup>2</sup> + *K*<sup>3</sup> sin *e*<sup>3</sup>

**Lemma 1.** *If ν<sup>B</sup>* = *νBc is applied to (7), then the first derivatives of trajectory error coordinates are*

*e*˙1 = −*K*1*e*<sup>1</sup> + (*ω*<sup>∗</sup> + *u*∗*K*2*e*<sup>2</sup> + *K*<sup>3</sup> sin *e*3)*e*<sup>2</sup> *e*˙2 = −(*ω*<sup>∗</sup> + *u*∗*K*2*e*<sup>2</sup> + *K*<sup>3</sup> sin *e*3)*e*<sup>1</sup> + *u*<sup>∗</sup> sin *e*<sup>3</sup>

*<sup>M</sup>*<sup>1</sup> + *KM*1*eM*<sup>1</sup>

*y y*

(*x*<sup>∗</sup> − *x*) cos *φ* + (*y*<sup>∗</sup> − *y*) sin *φ* −(*x*<sup>∗</sup> − *x*) sin *φ* + (*y*<sup>∗</sup> − *y*) cos *φ φ*<sup>∗</sup> − *φ*

�

�

(, ) *x y*

reference system

**2** *e*

**3** *e*

I

> ⎤ ⎦

*<sup>M</sup>* (11)

*Bc <sup>ν</sup><sup>T</sup>*

*<sup>M</sup>*1*<sup>c</sup> <sup>ν</sup><sup>T</sup> <sup>M</sup>*2*c*] *<sup>T</sup>* for

(12)

**1** *e*

*x x*

Since [*ν*∗ <sup>1</sup> , ..., *ν*<sup>∗</sup> *k* ] *<sup>T</sup>* are desired velocities of the mobile base, we can set [*ν*<sup>∗</sup> <sup>1</sup> *ν*<sup>∗</sup> 2 ] *<sup>T</sup>* = *ϕ*[*u*<sup>∗</sup> *ω*∗] *T* and *k* = 2 without loss of generality. Where desired forward linear velocity *u*∗ and desired angular velocity *ω*∗ of the mobile base. Using the relations such as *ν*∗ *<sup>M</sup>*<sup>1</sup> = *q*˙ ∗ *<sup>M</sup>*<sup>1</sup> and *ν*<sup>∗</sup> *<sup>M</sup>*<sup>2</sup> = *q*˙ ∗ *<sup>M</sup>*2, *ν*∗ *<sup>M</sup>*2(= *q*˙ ∗ *<sup>M</sup>*2) can be determined by *ν*<sup>∗</sup> *<sup>M</sup>*<sup>1</sup> and *u*∗, *ω*∗. For these values following assumptions are required.

**Assump.3**:*q*∗, *u*∗, *ω*∗, *q*˙ <sup>∗</sup>, *u*˙ <sup>∗</sup>, *ω*˙ <sup>∗</sup>, *q*¨∗, *u*¨∗, *ω*¨ <sup>∗</sup> and ˙*q*¨ <sup>∗</sup> are bounded globally. And *u*<sup>∗</sup> � 0.

#### **3.1 Reference robot**

To specify error dynamics of trajectory tracking system we introduce the reference robot shown in Fig.1. Trajectory error *eB* for base coordinates *qB* = [*x y φ*] *<sup>T</sup>*, trajectory error *eM*<sup>1</sup> for manipulator coordinates *qM*<sup>1</sup> and trajectory error *λ*˜ *<sup>M</sup>* for constrained forces are given 4 Will-be-set-by-IN-TECH

*<sup>ξ</sup>* <sup>+</sup> *<sup>C</sup>*1(*q*, *<sup>q</sup>*˙)*<sup>ξ</sup>* <sup>+</sup> *<sup>G</sup>*1(*q*) = *<sup>Y</sup>*1(*q*, *<sup>q</sup>*˙, *<sup>ξ</sup>*, ˙

where *<sup>p</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>s*<sup>1</sup> denotes unknown parameter vector and *<sup>Y</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>*(*k*+*m*)×*s*<sup>1</sup> is called the regressor matrix whose elements consist of known functions. Furthermore, kinematic uncertainties of

*<sup>M</sup>*(*q*)*λ<sup>M</sup>* = *Z*1(*q*, *λM*)*ψ*,

where *θij* is unknown parameter which consists of unknown parameters of mobile base, and *<sup>σ</sup>ij* is known functions which consists of the coordinate *qB*, (*<sup>i</sup>* <sup>=</sup> 1, ..., *<sup>k</sup>*, *<sup>j</sup>* <sup>=</sup> 1, ..., *hi*). *<sup>ψ</sup>* <sup>∈</sup> *<sup>R</sup><sup>c</sup>* is unknown parameter vector of the whole system and *<sup>Z</sup>*1(*q*, *<sup>λ</sup>M*) <sup>∈</sup> *<sup>R</sup>*(*k*+*m*)×*<sup>c</sup>* is known matrix

**Assump.1**:There are no unknown parameters in *B*1(*q*) and *detB*1(*q*) � 0 for all *q*. *d*<sup>1</sup> and its

Let *q*∗ be the desired position trajectory, then there exist desired velocity input *ν*∗ =

and *k* = 2 without loss of generality. Where desired forward linear velocity *u*∗ and desired

*q*¨

To specify error dynamics of trajectory tracking system we introduce the reference robot

*eM*<sup>1</sup> for manipulator coordinates *qM*<sup>1</sup> and trajectory error *λ*˜ *<sup>M</sup>* for constrained forces are given

*q*˙<sup>∗</sup> = *S*(*q*<sup>∗</sup>

*<sup>T</sup>* are desired velocities of the mobile base, we can set [*ν*<sup>∗</sup>

*hi* ∑ *j*=1

*θijσij*(*qB*)

*<sup>M</sup>*<sup>2</sup> ∈ *L*<sup>∞</sup> and these matrices are all continuously differentiable with

*<sup>B</sup>*)*ν*∗. (10)

*<sup>M</sup>*<sup>1</sup> = *q*˙ ∗ *<sup>M</sup>*<sup>1</sup> and *ν*<sup>∗</sup>

*<sup>M</sup>*<sup>1</sup> and *u*∗, *ω*∗. For these values following assumptions

<sup>∗</sup> are bounded globally. And *u*<sup>∗</sup> � 0.

<sup>1</sup> *ν*<sup>∗</sup> 2 ]

*<sup>T</sup>* = *ϕ*[*u*<sup>∗</sup> *ω*∗]

*<sup>T</sup>*, trajectory error

*<sup>M</sup>*<sup>2</sup> = *q*˙ ∗ *<sup>M</sup>*2,

*T*

⎞ ⎠ *νBi*.

the system give the following properties(Cheah et al., 2003; Fukao et al., 2000).

⎛

⎝*σi*0(*qB*) +

*k* ∑ *i*=1

*J T*

function of the position/force coordinate *q* and *λ<sup>M</sup>* , respectively. Following assumptions are required to synthesize the control scheme.

respect to *q* and kinematic parameters, and these derivatives are bounded.

angular velocity *ω*∗ of the mobile base. Using the relations such as *ν*∗

<sup>∗</sup>, *u*˙ <sup>∗</sup>, *ω*˙ <sup>∗</sup>, *q*¨∗, *u*¨∗, *ω*¨ <sup>∗</sup> and ˙

shown in Fig.1. Trajectory error *eB* for base coordinates *qB* = [*x y φ*]

derivative are bounded and �*d*1� ≤ *D*. Where *D* is unknown.

*ξ*)*p*1,

**Property 3**:*M*˙ <sup>1</sup> <sup>−</sup> <sup>2</sup>*C*<sup>1</sup> is skew-symmetric.

*ξ*

**Property 5**:*SB*(*qB*)*ν<sup>B</sup>* in (7) can be written as

*<sup>M</sup>*1(*q*)˙

*SB*(*qB*)*ν<sup>B</sup>* =

*<sup>M</sup>*(*q*)*λ<sup>M</sup>* in (9) can be written as

**Property 4**:For any ¯

**Property 6**:*J*

**Assump.2**:*JB*, *JM*, *J*

*<sup>k</sup>* , *<sup>ν</sup>*∗*<sup>T</sup> <sup>M</sup>* ]

<sup>1</sup> , ..., *ν*<sup>∗</sup> *k* ]

**Assump.3**:*q*∗, *u*∗, *ω*∗, *q*˙

**3.1 Reference robot**

[*ν*∗ <sup>1</sup> , ..., *ν*<sup>∗</sup>

*ν*∗ *<sup>M</sup>*2(= *q*˙ ∗

Since [*ν*∗

are required.

−1 *<sup>B</sup>*<sup>1</sup> , *J* −1

*<sup>T</sup>* such that

**3. Hybrid position/force control scheme**

*<sup>M</sup>*2) can be determined by *ν*<sup>∗</sup>

*T*

Fig. 2. Reference robot and tracking errors by

$$e\_B = \begin{bmatrix} e\_1 \\ e\_2 \\ e\_3 \end{bmatrix} = \begin{bmatrix} (\mathbf{x}^\* - \mathbf{x})\cos\phi + (\mathbf{y}^\* - \mathbf{y})\sin\phi \\ -(\mathbf{x}^\* - \mathbf{x})\sin\phi + (\mathbf{y}^\* - \mathbf{y})\cos\phi \\ \phi^\* - \phi \end{bmatrix}$$

$$e\_{M1} = q\_{M1}^\* - q\_{M1}, \ \tilde{\lambda}\_M = \lambda\_M - \lambda\_M^\* \tag{11}$$

Using the results in (Fukao et al., 2000), desired velocity inputs *ν<sup>c</sup>* = [*ν<sup>T</sup> Bc <sup>ν</sup><sup>T</sup> <sup>M</sup>*1*<sup>c</sup> <sup>ν</sup><sup>T</sup> <sup>M</sup>*2*c*] *<sup>T</sup>* for trajectory tracking are written as the following.

$$\begin{aligned} \nu\_{Bc} &= q \mu\_{Bc} \quad \mu\_{Bc} = \begin{bmatrix} \mu\_c \\ \omega\_c \end{bmatrix} \\ &= \begin{bmatrix} \mu^\* \cos e\_3 + K\_1 e\_1 \\ \omega^\* + \mu^\* K\_2 e\_2 + K\_3 \sin e\_3 \end{bmatrix} \\ \nu\_{M1c} &= \dot{\eta}\_{M1}^\* + K\_{M1} e\_{M1} \\ \nu\_{M2c} &= -J\_{M2}^{-1} \left( J\_{M0} f\_B \boldsymbol{\rho}^{-1} \nu\_{Bc} + J\_{M1} \nu\_{M1c} \right) \end{aligned} \tag{12}$$

Where *Ki* > 0, *i* = 1, 2, 3 and *KM*<sup>1</sup> are arbitrarily assigned. For the system (7) the following Lemma is shown in (Fukao et al., 2000).

**Lemma 1.** *If ν<sup>B</sup>* = *νBc is applied to (7), then the first derivatives of trajectory error coordinates are given by the following equations.*

$$\begin{aligned} \dot{e}\_1 &= -K\_1 e\_1 + (\omega^\* + u^\* K\_2 e\_2 + K\_3 \sin e\_3) e\_2 \\ \dot{e}\_2 &= -(\omega^\* + u^\* K\_2 e\_2 + K\_3 \sin e\_3) e\_1 + u^\* \sin e\_3 \\ \dot{e}\_3 &= -u^\* K\_2 e\_2 - K\_3 \sin e\_3 \end{aligned} \tag{13}$$

*Then,* lim*t*→<sup>∞</sup> *eB* = 0*.*

However, since Lemma1 has not considered kinematic parameter uncertainties, *νBc* cannot be applied to our problem. Therefore, we give the following assumption similar manner to (Fukao et al., 2000).

**Assump.4**:There exist velocity inputs and adaptive laws:

$$\nu\_B = \nu\_{Bc}(q\_{B'} q\_{B'}^\* \, ^\* \mathbf{a})$$

$$\dot{\mathbf{a}}\_i = T\_i(q\_{B'} q\_{B'}^\* \, ^\* \mathbf{a}) \tag{14}$$

of Mobile Manipulators 7

Robust Adaptive Position/Force Control of Mobile Manipulators 123

**Theorem 1.** *Applying the following control law and adaptive laws to the mobile manipulator (4) and*

 *∂V*<sup>1</sup> *∂q*

<sup>1</sup> (*q*, *q*˙, *χ*, *χ*˙)*δ* (17)

*S*ˆ(*qB*, ˆ *θ*) *<sup>T</sup>*

*<sup>t</sup>*→<sup>∞</sup> *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>=</sup> 0, (18)

], 1 ≤ *i* ≤ *k, and Kd*, Γ1, Γ2, Λ*<sup>i</sup> are positive definite matrix*

*S*ˆ(*qB*, ˆ *θ*) *<sup>T</sup>*

*<sup>θ</sup>*, *<sup>ψ</sup>*ˆ, *<sup>a</sup>*ˆ, *<sup>D</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>L</sup>*∞*.*

*D*˜ 2. (20)

*D*ˆ *γ* .

−*Kdδ* − *F* + *Y*1(*q*, *q*˙, *χ*, *χ*˙)*p*ˆ1 −

*<sup>M</sup>*(*q*, *<sup>ψ</sup>*ˆ*λ*˜ *<sup>M</sup>*) <sup>−</sup> *<sup>Z</sup>*1(*q*, *<sup>λ</sup>M*)*ψ*<sup>ˆ</sup>

*(5),*

*where* ˆ

(Proof) We set *V*<sup>2</sup> as

written as follows.

*τ* = *B*−<sup>1</sup> 1

˙

˙ *ψ*ˆ = Γ2*Z<sup>T</sup>*

˙

˙ ˆ *θ<sup>i</sup>* = Λ*<sup>i</sup>*

˙ *<sup>D</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>γ</sup>*�*δ*�

*then all internal signals are bounded and*

*M*<sup>1</sup> ˙

Differentiating *V*<sup>2</sup> along (19), we have

<sup>−</sup> *<sup>∂</sup>V*<sup>1</sup> *∂q*

*V*˙ <sup>2</sup> <sup>=</sup> *<sup>∂</sup>V*<sup>1</sup> *∂q*

*θ<sup>i</sup> is estimate of θi, σ<sup>i</sup>* = [*σi*1, ··· , *σihi*

*with appropriate dimensions, α*<sup>2</sup> *is arbitrary constant and*

+*α*<sup>2</sup> ˆ *J T*

*V*<sup>2</sup> = *V*<sup>1</sup> +

+ 1 2

Proof of this theorem is shown by the following Lemmas.

+*α*<sup>2</sup> ˆ *J T*

*<sup>p</sup>*ˆ1 <sup>=</sup> <sup>−</sup>Γ1*Y<sup>T</sup>*

*a*ˆ*<sup>i</sup>* = *Ti*(*qB*, *q*<sup>∗</sup>

<sup>1</sup> (*q*, *λM*)*δ*

 *∂V*<sup>1</sup> *∂q σi <sup>T</sup> ν*˜*i*

*<sup>B</sup>*, *a*ˆ)

lim

*δ* = −(*C*<sup>1</sup> + *Kd*)*δ* + *Y*1(*q*, *q*˙, *χ*, *χ*˙)*p*˜1 −

**Lemma 2.** *For the closed loop system, <sup>δ</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2*, and <sup>ν</sup>*˜, *<sup>p</sup>*ˆ,*eM*1,*e*˙*M*1, *<sup>ν</sup>c*, *<sup>χ</sup>*, *<sup>q</sup>*, *<sup>q</sup>*˙, <sup>ˆ</sup>

*δTM*1*δ* +

1 2 *α*2*α*−<sup>1</sup>

*<sup>∂</sup>q*<sup>∗</sup> *<sup>S</sup>*(*q*<sup>∗</sup>

*k* ∑ *i*=1 ˜ *θT <sup>i</sup>* <sup>Λ</sup>−<sup>1</sup> *i* ˙ ˜

<sup>1</sup> *<sup>β</sup><sup>T</sup> <sup>β</sup>* <sup>+</sup>

*g* ∑ *i*=1

*<sup>B</sup>*)*ν*<sup>∗</sup> +

1 2

*k* ∑ *i*=1 ˜ *θT <sup>i</sup>* <sup>Λ</sup>−<sup>1</sup> *<sup>i</sup>* ˜ *θ<sup>i</sup>* + 1 2 *ψ*˜*T*Γ−<sup>1</sup> <sup>2</sup> *<sup>ψ</sup>*˜ +

*<sup>S</sup>*(*qB*)(*ν<sup>c</sup>* <sup>+</sup> *<sup>ν</sup>*˜) + *<sup>∂</sup>V*<sup>1</sup>

*<sup>S</sup>*ˆ(*qB*)*ν*˜ <sup>−</sup> *<sup>α</sup>*2*βT<sup>β</sup>* <sup>+</sup>

*<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>* <sup>=</sup> 0, lim

*<sup>F</sup>*(*t*) = *<sup>δ</sup>D*<sup>ˆ</sup> <sup>2</sup>

Letting parameter estimation errors be *<sup>p</sup>*˜ <sup>=</sup> *<sup>p</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>p</sup>* and *<sup>D</sup>*˜ <sup>=</sup> *<sup>D</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>D</sup>*, closed loop system can be

�*δ*�*D*<sup>ˆ</sup> <sup>+</sup> *<sup>ρ</sup>*(*t*)

.

 *∂V*<sup>1</sup> *∂q*

> 1 2 *p*˜ *T* <sup>1</sup> <sup>Γ</sup>−<sup>1</sup> *<sup>p</sup>*˜1

1 2*γ*

*∂V*<sup>1</sup> *∂a*ˆ*<sup>i</sup>*

*Ti* <sup>−</sup> *<sup>δ</sup>TKd<sup>δ</sup>*

*<sup>θ</sup><sup>i</sup>* <sup>−</sup> *<sup>δ</sup>T*(*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1) + *<sup>D</sup>*˜ ˙

*<sup>M</sup>λ*˜ *<sup>M</sup>* <sup>−</sup> *<sup>Z</sup>*1*ψ*˜ <sup>−</sup> (*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1) (19)

such that the closed loop system of (7) is stable at *q*∗ *<sup>B</sup>*. Furthermore, there exists Lyapunov function *V*1(*qB*, *q*<sup>∗</sup> *<sup>B</sup>*, *a*˜) such that the time derivative of *V*<sup>1</sup> along the closed loop system of (7) with (14) is negative semi-definite. Where *a*ˆ is the estimate of an unknown parameter vector *a* = [*a*1, ..., *ak*] *<sup>T</sup>*, which is composed of *<sup>θ</sup>ij*, and *<sup>a</sup>*˜ <sup>=</sup> *<sup>a</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>a</sup>* is the estimated error.

#### **3.2 Control laws for the system with both kinematic and dynamic uncertainties**

In this section we propose the robust and adaptive position/force control scheme of the mobile manipulators with both the kinematic and dynamic uncertainties. To begin with, we introduce filter coordinates in a similar way to (Yuan, 1997) as follows. For any constant *α*<sup>1</sup> we set *<sup>β</sup><sup>M</sup>* <sup>∈</sup> *<sup>R</sup><sup>m</sup>* as

$$
\dot{\beta}\_M = -\alpha\_1 \beta\_M - \alpha\_1 \left[ \hat{\vec{f}}\_{M1} \, \hat{\vec{f}}\_{M2} \right]^T \hat{\lambda}\_{M\nu} \tag{15}
$$

where <sup>ˆ</sup> *JM*(*q*, *ψ*ˆ) denotes the Jacobian matrix which is replaced *ψ* with estimate *ψ*ˆ and

$$
\hat{\vec{f}}\_M(q, \hat{\psi}) = \left[ \hat{f}\_{M0}(q, \hat{\psi}) f\_B(q\_B) \hat{\phi}^{-1} \hat{f}\_{M1}(q, \hat{\psi}) \, \hat{f}\_{M2}(q, \hat{\psi}) \right],
$$

$$
= \left[ \hat{\vec{f}}\_{M0}(q, \hat{\psi}) \, \hat{\vec{f}}\_{M1}(q, \hat{\psi}) \, \hat{\vec{f}}\_{M2}(q, \hat{\psi}) \right].
$$

Secondly, we set

$$
\vec{\nu} = \boldsymbol{\nu} - \boldsymbol{\nu}\_c, \boldsymbol{\beta} = \begin{bmatrix} \boldsymbol{0}\_k^T \ \boldsymbol{\beta}\_M^T \end{bmatrix}^T, \vec{\nu} = R\vec{s}, \boldsymbol{\delta} = \vec{\nu} + \boldsymbol{\beta},
$$

$$
\boldsymbol{\chi} = \boldsymbol{\nu}\_c - \boldsymbol{\beta}, \vec{s} = \begin{bmatrix} \vec{\nu}\_B^T \ \vec{\nu}\_{M1}^T \end{bmatrix}^T, \boldsymbol{e} = \begin{bmatrix} \boldsymbol{e}\_B^T \ \boldsymbol{e}\_M^T \end{bmatrix}^T. \tag{16}
$$

where

$$R = \begin{bmatrix} I\_k & 0 \\ 0 & I\_l \\ -\hat{J}\_{M2}^{-1}\hat{J}\_{M0} & -\hat{J}\_{M2}^{-1}\hat{J}\_{M1} \end{bmatrix}.$$

Finally we introduce variable *ρ*(*t*) which satisfies following conditions (Li et al., 2008).

$$\begin{aligned} &(1) \quad \rho(t) > 0, \; \forall t \in [0, \infty) \\ &(2) \quad \lim\_{t \to \infty} \rho(t) = 0 \\ &(3) \quad \lim\_{t \to \infty} \int\_0^t \rho(\tau) d\tau = \rho\_0 < \infty \end{aligned}$$

Under assumptions from Assump.1 to Assump.4, following theorem is derived.

**Theorem 1.** *Applying the following control law and adaptive laws to the mobile manipulator (4) and (5),*

$$\begin{aligned} \tau &= B\_1^{-1} \left[ -K\_d \delta - F + Y\_1(q, \dot{q}, \chi, \dot{\chi}) \dot{\rho}\_1 - \left( \frac{\partial V\_1}{\partial q} \dot{S}(q\_B, \dot{\theta}) \right)^T \right] \\ &+ a\_2 \dot{\tilde{I}}\_M^T(q, \dot{\psi} \tilde{\lambda}\_M) - Z\_1(q, \lambda\_M) \dot{\psi} \right] \\ \dot{\rho}\_1 &= -\Gamma\_1 Y\_1^T(q, \dot{q}, \chi, \dot{\chi}) \delta \\ \dot{\psi} &= \Gamma\_2 Z\_1^T(q, \lambda\_M) \delta \\ \dot{\theta}\_i &= T\_i(q\_B, \psi\_B^\*, \dot{\theta}) \\ \dot{\theta}\_i &= \Lambda\_i \left( \frac{\partial V\_1}{\partial q} \sigma\_i \right)^T \tilde{v}\_i \\ \dot{D} &= \gamma \| \dot{\delta} \| \end{aligned} \tag{17}$$

*then all internal signals are bounded and*

$$\lim\_{t \to \infty} e = 0, \lim\_{t \to \infty} \tilde{\lambda}\_M = 0,\tag{18}$$

*where* ˆ *θ<sup>i</sup> is estimate of θi, σ<sup>i</sup>* = [*σi*1, ··· , *σihi* ], 1 ≤ *i* ≤ *k, and Kd*, Γ1, Γ2, Λ*<sup>i</sup> are positive definite matrix with appropriate dimensions, α*<sup>2</sup> *is arbitrary constant and*

$$F(t) = \frac{\delta \hat{D}^2}{||\delta||\hat{D} + \rho(t)}.$$

Letting parameter estimation errors be *<sup>p</sup>*˜ <sup>=</sup> *<sup>p</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>p</sup>* and *<sup>D</sup>*˜ <sup>=</sup> *<sup>D</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>D</sup>*, closed loop system can be written as follows.

$$M\_1 \dot{\delta} = -(\mathbf{C}\_1 + \mathbf{K}\_d)\delta + Y\_1(q, \dot{q}, \chi, \dot{\chi})\tilde{p}\_1 - \left(\frac{\partial V\_1}{\partial q}\hat{\mathbf{S}}(q\_B, \dot{\theta})\right)^T$$

$$+ a\_2 \hat{\tilde{f}}\_M^T \tilde{\lambda}\_M - Z\_1 \tilde{\psi} - (F + d\_1) \tag{19}$$

Proof of this theorem is shown by the following Lemmas.

**Lemma 2.** *For the closed loop system, <sup>δ</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2*, and <sup>ν</sup>*˜, *<sup>p</sup>*ˆ,*eM*1,*e*˙*M*1, *<sup>ν</sup>c*, *<sup>χ</sup>*, *<sup>q</sup>*, *<sup>q</sup>*˙, <sup>ˆ</sup> *<sup>θ</sup>*, *<sup>ψ</sup>*ˆ, *<sup>a</sup>*ˆ, *<sup>D</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>L</sup>*∞*.* (Proof)

We set *V*<sup>2</sup> as

6 Will-be-set-by-IN-TECH

However, since Lemma1 has not considered kinematic parameter uncertainties, *νBc* cannot be applied to our problem. Therefore, we give the following assumption similar manner to

*ν<sup>B</sup>* = *νBc*(*qB*, *q*∗

*a*ˆ*<sup>i</sup>* = *Ti*(*qB*, *q*<sup>∗</sup>

with (14) is negative semi-definite. Where *a*ˆ is the estimate of an unknown parameter vector

In this section we propose the robust and adaptive position/force control scheme of the mobile manipulators with both the kinematic and dynamic uncertainties. To begin with, we introduce filter coordinates in a similar way to (Yuan, 1997) as follows. For any constant *α*<sup>1</sup> we set

> � ˆ *JM*<sup>1</sup> <sup>ˆ</sup> *JM*<sup>2</sup> �*T*

ˆ*JM*0(*q*, *ψ*ˆ)*fB*(*qB*)*ϕ*ˆ−<sup>1</sup> ˆ*JM*1(*q*, *ψ*ˆ) ˆ*JM*2(*q*, *ψ*ˆ)

*<sup>T</sup>*,*e* = [*e<sup>T</sup>*

*ρ*(*τ*)*dτ* = *ρ*<sup>0</sup> < ∞

*Ik* 0 0 *Il*

*JM*2(*q*, *ψ*ˆ)

*<sup>T</sup>*, *ν*˜ = *Rs*˜, *δ* = *ν*˜ + *β*,

*<sup>B</sup> <sup>e</sup><sup>T</sup> M*]

> ⎤ ⎥ ⎦ .

� .

*JM*(*q*, *ψ*ˆ) denotes the Jacobian matrix which is replaced *ψ* with estimate *ψ*ˆ and

*JM*1(*q*, *<sup>ψ</sup>*ˆ) <sup>ˆ</sup>

*<sup>k</sup> <sup>β</sup><sup>T</sup> M*]

*T <sup>B</sup> ν*˜ *T <sup>M</sup>*1]

−ˆ *J* −1 *M*2 ˆ *JM*<sup>0</sup> <sup>−</sup><sup>ˆ</sup> *J* −1 *M*2 ˆ *JM*<sup>1</sup>

Finally we introduce variable *ρ*(*t*) which satisfies following conditions (Li et al., 2008).

*<sup>t</sup>*→<sup>∞</sup> *<sup>ρ</sup>*(*t*) = <sup>0</sup>

� *t* 0

Under assumptions from Assump.1 to Assump.4, following theorem is derived.

(1) *ρ*(*t*) > 0, ∀*t* ∈ [0, ∞)

*<sup>T</sup>*, which is composed of *<sup>θ</sup>ij*, and *<sup>a</sup>*˜ <sup>=</sup> *<sup>a</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>a</sup>* is the estimated error.

˙

**3.2 Control laws for the system with both kinematic and dynamic uncertainties**

*<sup>M</sup>* = −*α*1*β<sup>M</sup>* − *α*<sup>1</sup>

*JM*0(*q*, *<sup>ψ</sup>*ˆ) <sup>ˆ</sup>

*<sup>B</sup>*, *a*ˆ)

*<sup>B</sup>*, *a*˜) such that the time derivative of *V*<sup>1</sup> along the closed loop system of (7)

*<sup>B</sup>*, *a*ˆ) (14)

*<sup>B</sup>*. Furthermore, there exists Lyapunov

*λ*˜ *<sup>M</sup>*, (15)

*<sup>T</sup>*, (16)

�

(Fukao et al., 2000).

function *V*1(*qB*, *q*<sup>∗</sup>

*a* = [*a*1, ..., *ak*]

*<sup>β</sup><sup>M</sup>* <sup>∈</sup> *<sup>R</sup><sup>m</sup>* as

Secondly, we set

where <sup>ˆ</sup>

where

**Assump.4**:There exist velocity inputs and adaptive laws:

such that the closed loop system of (7) is stable at *q*∗

*β*˙

�

*<sup>ν</sup>*˜ <sup>=</sup> *<sup>ν</sup>* <sup>−</sup> *<sup>ν</sup>c*, *<sup>β</sup>* = [0*<sup>T</sup>*

*χ* = *ν<sup>c</sup>* − *β*,*s*˜ = [*ν*˜

*R* =

(2) lim

(3) lim *t*→∞

⎡ ⎢ ⎣

= � ˆ

ˆ

*JM*(*q*, *ψ*ˆ) =

$$\begin{split} V\_{2} &= V\_{1} + \frac{1}{2} \boldsymbol{\delta}^{T} M\_{1} \boldsymbol{\delta} + \frac{1}{2} a\_{2} a\_{1}^{-1} \boldsymbol{\beta}^{T} \boldsymbol{\beta} + \frac{1}{2} \boldsymbol{\tilde{\rho}\_{1}^{T}} \boldsymbol{\Gamma}^{-1} \boldsymbol{\tilde{\rho}\_{1}} \\ &+ \frac{1}{2} \sum\_{i=1}^{k} \boldsymbol{\tilde{\theta}\_{i}^{T}} \boldsymbol{\Lambda}\_{i}^{-1} \boldsymbol{\tilde{\theta}\_{i}} + \frac{1}{2} \boldsymbol{\tilde{\psi}}^{T} \boldsymbol{\Gamma}\_{2}^{-1} \boldsymbol{\tilde{\psi}} + \frac{1}{2\gamma} \boldsymbol{\tilde{\rho}^{2}}. \end{split} \tag{20}$$

Differentiating *V*<sup>2</sup> along (19), we have

$$\dot{V}\_{2} = \frac{\partial V\_{1}}{\partial q} \mathcal{S}(q\_{B})(\nu\_{c} + \tilde{\nu}) + \frac{\partial V\_{1}}{\partial q^{\*}} \mathcal{S}(q\_{B}^{\*})\nu^{\*} + \sum\_{i=1}^{\mathcal{S}} \frac{\partial V\_{1}}{\partial \dot{a}\_{i}} T\_{i} - \boldsymbol{\delta}^{T} \mathcal{K}\_{d} \boldsymbol{\delta}$$

$$-\frac{\partial V\_{1}}{\partial q} \mathcal{S}(q\_{B})\tilde{\nu} - \alpha\_{2} \boldsymbol{\beta}^{T} \boldsymbol{\beta} + \sum\_{i=1}^{k} \tilde{\theta}\_{i}^{T} \Lambda\_{i}^{-1} \dot{\tilde{\theta}}\_{i} - \boldsymbol{\delta}^{T} (\boldsymbol{F} + d\_{1}) + \frac{\tilde{D}\dot{D}}{\gamma}.$$

of Mobile Manipulators 9

Robust Adaptive Position/Force Control of Mobile Manipulators 125

of variable *ν*˜, we have *ν*˜ ∈ *L*∞. Since unknown parameters are constant and bounded,

*<sup>θ</sup>*, *<sup>ψ</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>L</sup>*∞. From (12) and definitions of *<sup>ν</sup>* and *<sup>ν</sup>c*, *<sup>ν</sup>*˜*M*<sup>1</sup> <sup>=</sup> <sup>−</sup>*e*˙*M*<sup>1</sup> <sup>−</sup> *KM*1*eM*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*∞. Then *eM*1,*e*˙*M*<sup>1</sup> ∈ *L*∞. From (12), Assump.3 and Assump.4, we have *νc*, *χ* ∈ *L*∞. From Assump.2 and Assump.3, we have *qM*, *q*˙*<sup>M</sup>* ∈ *L*∞. Similarly from *ν*˜, *ν<sup>c</sup>* ∈ *L*<sup>∞</sup> and Assump.2, we have

*M. If there exist α*<sup>1</sup> *and α*<sup>2</sup> *such that*

*<sup>s</sup>*˜ <sup>+</sup> *<sup>R</sup>*˙*s*˜)+(*C*<sup>1</sup> <sup>+</sup> *Kd*)*<sup>δ</sup>* <sup>−</sup> *<sup>Y</sup>*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙)*p*˜1 <sup>−</sup> *<sup>α</sup>*1*M*<sup>ˆ</sup> <sup>1</sup>*<sup>β</sup>*

*<sup>M</sup>λ*˜ *<sup>M</sup>* <sup>−</sup> <sup>Δ</sup>(*ψ*, *<sup>ψ</sup>*ˆ)<sup>ˆ</sup>

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*) + *α*<sup>2</sup> *Ik*<sup>+</sup>*<sup>m</sup>* + Δ(*ψ*, *ψ*ˆ)

<sup>1</sup> {(*C*<sup>1</sup> + *Kd*)*δ*

*J T Mλ*<sup>∗</sup>

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*) + *α*<sup>2</sup> *Ik*<sup>+</sup>*<sup>m</sup>* + Δ(*ψ*, *ψ*ˆ)

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*) + *α*<sup>2</sup> *Ik*<sup>+</sup>*<sup>m</sup>* + Δ(*ψ*, *ψ*ˆ)

+ <sup>Δ</sup>(*ψ*, *<sup>ψ</sup>*ˆ)<sup>ˆ</sup>

*<sup>Y</sup>*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙)*p*˜1 <sup>=</sup> *<sup>Y</sup>*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙)*p*˜1 <sup>−</sup>

<sup>=</sup> <sup>−</sup>Δ(*ψ*, *<sup>ψ</sup>*ˆ)<sup>ˆ</sup>

<sup>1</sup> from left, we have

<sup>−</sup>*Y*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙)*p*˜1 <sup>+</sup> *<sup>α</sup>*1*M*<sup>ˆ</sup> <sup>1</sup>*<sup>β</sup>*

*JM* 

*S*ˆ(*qB*, ˆ *θ*) *<sup>T</sup>* *J T*

*R*˙*s*˜ + *M*−<sup>1</sup>

+ <sup>Δ</sup>(*ψ*, *<sup>ψ</sup>*ˆ)<sup>ˆ</sup>

 ˆ *J T <sup>M</sup>* − *J T M λ*˜ *<sup>M</sup>* + ˆ *J T <sup>M</sup>* − *J T M λ*∗ *M*

*J T Mλ*<sup>∗</sup>

*V*2(*t*) < *V*2(0) + *ρ*<sup>0</sup> < ∞. (24)

− *Ik*<sup>+</sup>*m*,

*<sup>M</sup>* + (*F* + *d*1). (25)

 *β*˙

. (26)

 ˆ *J T <sup>M</sup>λ*˜ *<sup>M</sup>*

*<sup>M</sup>*<sup>ˆ</sup> <sup>1</sup>(*q*) <sup>−</sup> *<sup>M</sup>*1(*q*)

*J T Mλ*<sup>∗</sup> *M*

*<sup>M</sup>* + (*F* + *d*1)

*<sup>θ</sup>*, *<sup>ψ</sup>*˜ <sup>∈</sup> *<sup>L</sup>*∞, and *<sup>δ</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2. From definition

This shows

*p*ˆ, *D*ˆ , *a*ˆ, ˆ

*where* ˆ *J T M* †

(Proof)

Therefore, *V*2(*t*) is bounded, that is, *qB*, *a*˜, *δ*, *β*, *p*˜1, *D*˜ , ˜

Δ(*ψ*, *ψ*ˆ) = *J*

*is left annihilator of* ˆ

Substituting (15) and (16) into (19), we have

+ *∂V*<sup>1</sup> *∂q*

Multiplying (25) by ˆ

= *M*1(*R*˙

In this calculation, following relations are used.

*JMM*<sup>−</sup><sup>1</sup>

ˆ *JMM*<sup>−</sup><sup>1</sup> 1 

×ˆ *J T <sup>M</sup>λ*˜ *<sup>M</sup>* <sup>=</sup> <sup>ˆ</sup>

+ *∂V*<sup>1</sup> *∂q*

*S*ˆ(*qB*, ˆ *θ*) *<sup>T</sup>*

*Z*1*ψ*˜ =

*T M* ˆ *JM*<sup>ˆ</sup> *J T M* −<sup>1</sup> ˆ *JM* + ˆ *J T M* †

*J T*

*q*˙*B*,*e*˙*<sup>B</sup>* ∈ *L*∞. Therefore *q*, *q*˙ ∈ *L*∞. **Lemma 3.** *Let M*ˆ <sup>1</sup> = *M*1(*p*ˆ1) *and*

*is nonsingular, then <sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞*.*

In this computation, we used the relations *<sup>∂</sup>V*<sup>1</sup> *<sup>∂</sup><sup>q</sup> <sup>S</sup>*ˆ*<sup>β</sup>* <sup>=</sup> 0 and:

$$
\begin{split}
\hat{\boldsymbol{\hat{J}}}\_{M}\boldsymbol{\mathcal{R}} &= \begin{bmatrix}
\hat{\boldsymbol{\hat{J}}}\_{M0}\hat{\boldsymbol{\hat{J}}}\_{M1}\hat{\boldsymbol{\hat{J}}}\_{M2} \end{bmatrix} \begin{bmatrix}
\boldsymbol{I}\_{k} & \boldsymbol{0} \\
\boldsymbol{0} & \boldsymbol{I}\_{l} \\
\boldsymbol{0}\_{(m-l)\times k} \ \boldsymbol{0}\_{(m-l)\times l}
\end{bmatrix}.
\end{split}
$$

Furthermore, by using the relation

$$\begin{split} \frac{\partial V\_{1}}{\partial q} \tilde{S}(q\_{B}) \tilde{\nu} &= \frac{\partial V\_{1}}{\partial q} (\hat{S} - S) \tilde{\nu} = \frac{\partial V\_{1}}{\partial q} \left[ \sum\_{i=1}^{k} \left( \sigma\_{i0}(q\_{B}) + \sum\_{j=1}^{l\_{i}} \hat{\theta}\_{ij} \sigma\_{ij}(q\_{B}) \right) \right] \\ &- \sum\_{i=1}^{k} \left( \sigma\_{i0}(q\_{B}) + \sum\_{j=1}^{l\_{i}} \theta\_{ij} \sigma\_{ij}(q\_{B}) \right) \Bigg] \nu\_{Bi} = \frac{\partial V\_{1}}{\partial q} \sum\_{i=1}^{k} \sigma\_{i} \tilde{\eta}\_{i} \tilde{\nu}\_{i} \end{split}$$

we have

$$\dot{V}\_2 = \dot{V}\_1^{\nu\_{\text{B}} = \nu\_{\text{BC}}} - \delta^T \mathcal{K}\_d \delta - a\_2 \beta^T \beta - \delta^T (F + d\_1) + \frac{\tilde{D} \dot{\tilde{D}}}{\gamma} \tag{21}$$

where

$$\dot{V}\_1^{\nu\_B = \nu\_{BC}} = \frac{\partial V\_1}{\partial q} S(q\_B) \nu\_c + \frac{\partial V\_1}{\partial q^\*} S(q\_B^\*) \nu^\* + \sum\_{i=1}^{\mathcal{S}} \frac{\partial V\_1}{\partial \mathfrak{A}\_i} T\_i \le 0.$$

Last inequality sign ≤ is given by Assump.4. From the definition of *F*(*t*) and adaptive law of *D*ˆ following inequality is derived.

$$\begin{split}-\delta^{T}(\boldsymbol{F}+\boldsymbol{d}\_{1}) + \frac{\boldsymbol{\tilde{D}\dot{\boldsymbol{D}}}}{\gamma} &= -\delta^{T}\frac{\delta\dot{\boldsymbol{D}}^{2}}{\|\delta\|\boldsymbol{\tilde{D}}+\rho(t)} - \delta^{T}\boldsymbol{d}\_{1} + \frac{\boldsymbol{\tilde{D}\dot{\boldsymbol{D}}}}{\gamma} - \frac{D\dot{\boldsymbol{D}}}{\gamma} \\ &< -\frac{\|\boldsymbol{\delta}\|^{2}\dot{\boldsymbol{D}}^{2}}{\|\delta\|\boldsymbol{\tilde{D}}+\rho(t)} + \|\delta\|\boldsymbol{D} + \frac{\dot{\boldsymbol{D}}\dot{\boldsymbol{D}}}{\gamma} - \frac{D\dot{\boldsymbol{D}}}{\gamma} \\ &= -\frac{\|\|\boldsymbol{\delta}\|^{2}\dot{\boldsymbol{D}}^{2}}{\|\|\boldsymbol{\delta}\|\boldsymbol{\tilde{D}}+\rho(t)} + \frac{\dot{\boldsymbol{D}}(\gamma\|\|\boldsymbol{\delta}\|)}{\gamma} + \frac{D}{\gamma}\left(\gamma\|\boldsymbol{\delta}\| - \dot{\boldsymbol{D}}\right) \\ &= \frac{\rho(t)\|\|\boldsymbol{\delta}\|\|\boldsymbol{\tilde{D}}}{\|\boldsymbol{\delta}\|\|\boldsymbol{\tilde{D}}+\rho(t)} < \rho(t) \end{split}$$

These inequalities lead the right hand of (32) to

$$
\dot{V}\_2 < -\delta^T K\_d \delta - a\_2 \beta^T \beta + \rho(t). \tag{22}
$$

Integrating both sides of this inequality and using definition of *ρ*(*t*), we have

$$V\_2(t) - V\_2(0) < -\int\_0^t \delta^T K\_d \delta d\tau - \mathfrak{a}\_2 \int\_0^t \mathfrak{F}^T \mathfrak{F} d\tau + \rho\_0 < \infty. \tag{23}$$

This shows

8 Will-be-set-by-IN-TECH

−ˆ *J* −1 *M*2 ˆ *JM*<sup>0</sup> <sup>−</sup><sup>ˆ</sup> *J* −1 *M*2 ˆ *JM*<sup>1</sup>

� .

⎡ ⎣ *k* ∑ *i*=1 ⎛

*θijσij*(*qB*)

*<sup>B</sup>*)*ν*<sup>∗</sup> +

⎝*σi*0(*qB*) +

⎞ ⎠ ⎤

*g* ∑ *i*=1

<sup>−</sup> *<sup>δ</sup>Td*<sup>1</sup> <sup>+</sup>

*<sup>D</sup>*<sup>ˆ</sup> (*γ*�*δ*�) *γ*

+ �*δ*�*D* +

� *t* 0

+

*∂V*<sup>1</sup> *∂a*ˆ*<sup>i</sup>*

> *D*ˆ ˙ *D*ˆ *<sup>γ</sup>* <sup>−</sup> *<sup>D</sup>* ˙ *D*ˆ *γ*

*D*ˆ ˙ *D*ˆ *<sup>γ</sup>* <sup>−</sup> *<sup>D</sup>* ˙ *D*ˆ *γ*

> + *D γ* �

<sup>2</sup> <sup>&</sup>lt; <sup>−</sup>*<sup>δ</sup>TKd<sup>δ</sup>* <sup>−</sup> *<sup>α</sup>*2*βT<sup>β</sup>* <sup>+</sup> *<sup>ρ</sup>*(*t*). (22)

*<sup>γ</sup>*�*δ*� − ˙ *D*ˆ �

*β<sup>T</sup> βdτ* + *ρ*<sup>0</sup> < ∞. (23)

*∂q*

*li* ∑ *j*=1

<sup>1</sup> <sup>−</sup> *<sup>δ</sup>TKd<sup>δ</sup>* <sup>−</sup> *<sup>α</sup>*2*βT<sup>β</sup>* <sup>−</sup> *<sup>δ</sup>T*(*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1) + *<sup>D</sup>*˜ ˙

*∂V*<sup>1</sup> *<sup>∂</sup>q*<sup>∗</sup> *<sup>S</sup>*(*q*<sup>∗</sup>

Last inequality sign ≤ is given by Assump.4. From the definition of *F*(*t*) and adaptive law of

�*δ*�*D*<sup>ˆ</sup> <sup>+</sup> *<sup>ρ</sup>*(*t*)

�*δ*�*D*<sup>ˆ</sup> <sup>+</sup> *<sup>ρ</sup>*(*t*) <sup>&</sup>lt; *<sup>ρ</sup>*(*t*)

*<sup>δ</sup>TKdδd<sup>τ</sup>* <sup>−</sup> *<sup>α</sup>*<sup>2</sup>

<sup>0</sup>(*m*−*l*)×*<sup>k</sup>* <sup>0</sup>(*m*−*l*)×*<sup>l</sup>*

*<sup>∂</sup><sup>q</sup>* (*S*<sup>ˆ</sup> <sup>−</sup> *<sup>S</sup>*)*ν*˜ <sup>=</sup> *<sup>∂</sup>V*<sup>1</sup>

⎝*σi*0(*qB*) +

*S*(*qB*)*ν<sup>c</sup>* +

*<sup>γ</sup>* <sup>=</sup> <sup>−</sup>*δ<sup>T</sup> <sup>δ</sup>D*<sup>ˆ</sup> <sup>2</sup>

<sup>&</sup>lt; <sup>−</sup> �*δ*�2*D*<sup>ˆ</sup> <sup>2</sup> �*δ*�*D*<sup>ˆ</sup> <sup>+</sup> *<sup>ρ</sup>*(*t*)

<sup>=</sup> <sup>−</sup> �*δ*�2*D*<sup>ˆ</sup> <sup>2</sup> �*δ*�*D*<sup>ˆ</sup> <sup>+</sup> *<sup>ρ</sup>*(*t*)

<sup>=</sup> *<sup>ρ</sup>*(*t*)�*δ*�*D*<sup>ˆ</sup>

Integrating both sides of this inequality and using definition of *ρ*(*t*), we have

� *t* 0

⎛

*<sup>∂</sup><sup>q</sup> <sup>S</sup>*ˆ*<sup>β</sup>* <sup>=</sup> 0 and:

*Ik* 0 0 *Il* ⎤ ⎥ ⎦

*li* ∑ *j*=1 ˆ *θijσij*(*qB*)

<sup>⎦</sup> *<sup>ν</sup>Bi* <sup>=</sup> *<sup>∂</sup>V*<sup>1</sup> *∂q*

*D*ˆ

*Ti* ≤ 0.

⎞ ⎠

*<sup>γ</sup>* , (21)

*k* ∑ *i*=1 *σi* ˜ *θiν*˜*<sup>i</sup>*

In this computation, we used the relations *<sup>∂</sup>V*<sup>1</sup>

ˆ *JMR* =

Furthermore, by using the relation

*V*˙

*V*˙ *<sup>ν</sup>B*=*νBC* <sup>1</sup> <sup>=</sup> *<sup>∂</sup>V*<sup>1</sup>

<sup>−</sup> *<sup>δ</sup>T*(*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1) + *<sup>D</sup>*˜ ˙

These inequalities lead the right hand of (32) to

*V*2(*t*) − *V*2(0) < −

*V*˙

*D*ˆ following inequality is derived.

*∂V*<sup>1</sup> *∂q*

we have

where

� ˆ *JM*<sup>0</sup> <sup>ˆ</sup> *JM*<sup>1</sup> <sup>ˆ</sup> *JM*<sup>2</sup> � ⎡ ⎢ ⎣

= �

> − *k* ∑ *i*=1

> > *∂q*

*D*ˆ

*<sup>S</sup>*˜(*qB*)*ν*˜ <sup>=</sup> *<sup>∂</sup>V*<sup>1</sup>

<sup>2</sup> = *<sup>V</sup>*˙ *<sup>ν</sup>B*=*νBC*

$$V\_2(t) < V\_2(0) + \rho\_0 < \infty. \tag{24}$$

Therefore, *V*2(*t*) is bounded, that is, *qB*, *a*˜, *δ*, *β*, *p*˜1, *D*˜ , ˜ *<sup>θ</sup>*, *<sup>ψ</sup>*˜ <sup>∈</sup> *<sup>L</sup>*∞, and *<sup>δ</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2. From definition of variable *ν*˜, we have *ν*˜ ∈ *L*∞. Since unknown parameters are constant and bounded, *p*ˆ, *D*ˆ , *a*ˆ, ˆ *<sup>θ</sup>*, *<sup>ψ</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>L</sup>*∞. From (12) and definitions of *<sup>ν</sup>* and *<sup>ν</sup>c*, *<sup>ν</sup>*˜*M*<sup>1</sup> <sup>=</sup> <sup>−</sup>*e*˙*M*<sup>1</sup> <sup>−</sup> *KM*1*eM*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*∞. Then *eM*1,*e*˙*M*<sup>1</sup> ∈ *L*∞. From (12), Assump.3 and Assump.4, we have *νc*, *χ* ∈ *L*∞. From Assump.2 and Assump.3, we have *qM*, *q*˙*<sup>M</sup>* ∈ *L*∞. Similarly from *ν*˜, *ν<sup>c</sup>* ∈ *L*<sup>∞</sup> and Assump.2, we have *q*˙*B*,*e*˙*<sup>B</sup>* ∈ *L*∞. Therefore *q*, *q*˙ ∈ *L*∞.

**Lemma 3.** *Let M*ˆ <sup>1</sup> = *M*1(*p*ˆ1) *and*

$$\Delta(\psi, \hat{\psi}) = \overline{f}\_M^T \left(\hat{\overline{f}}\_M \hat{\overline{f}}\_M^T\right)^{-1} \hat{\overline{f}}\_M + \left(\hat{\overline{f}}\_M^T\right)^\dagger - I\_{k+m\prime}$$

*where* ˆ *J T M* † *is left annihilator of* ˆ *J T M. If there exist α*<sup>1</sup> *and α*<sup>2</sup> *such that*

$$\left\{\mathfrak{a}\_1 \hat{\mathsf{M}}\_1 \boldsymbol{B} \boldsymbol{l} \boldsymbol{k} \boldsymbol{l} \boldsymbol{\deg}(\boldsymbol{0}\_{k\prime} \boldsymbol{I}\_m) + \mathfrak{a}\_2 \boldsymbol{I}\_{k+m} + \Delta(\boldsymbol{\psi}, \hat{\boldsymbol{\psi}}) \right\}$$

*is nonsingular, then <sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞*.*

(Proof)

Substituting (15) and (16) into (19), we have

$$\begin{aligned} & \left\{ \alpha\_1 \hat{M}\_1 \text{Blockdiag}(0\_k, I\_m) + \alpha\_2 I\_{k+m} + \Delta(\psi, \hat{\mathfrak{p}}) \right\} \hat{\mathfrak{f}}\_M^T \vec{\lambda}\_M \\ &= M\_1 (R\ddot{\mathfrak{s}} + \dot{R}\ddot{\mathfrak{s}}) + (\mathbf{C}\_1 + \mathbf{K}\_d)\delta - Y\_1(q, \dot{q}, \chi\_\lambda \chi\_\lambda \dot{\mathfrak{x}} + \dot{\mathfrak{f}}) \vec{p}\_1 - \alpha\_1 \hat{M}\_1 \beta \\ &+ \left( \frac{\partial V\_1}{\partial q} \hat{S}(q\_B, \dot{\theta}) \right)^T + \Delta(\psi, \hat{\psi}) \hat{\tilde{f}}\_M^T \lambda\_M^\* + (F + d\_1). \end{aligned} \tag{25}$$

In this calculation, following relations are used.

$$\begin{aligned} Y\_1(q, \dot{q}, \chi, \dot{\chi}) \dot{p}\_1 &= Y\_1(q, \dot{q}, \chi, \dot{\chi} + \dot{\beta}) \dot{p}\_1 - \left\{ \hat{M}\_1(q) - M\_1(q) \right\} \dot{\beta}\_1 \\ Z\_1 \tilde{\boldsymbol{\psi}} &= \left( \tilde{\boldsymbol{f}}\_M^T - \tilde{\boldsymbol{f}}\_M^T \right) \tilde{\boldsymbol{\lambda}}\_M + \left( \tilde{\boldsymbol{f}}\_M^T - \tilde{\boldsymbol{f}}\_M^T \right) \boldsymbol{\lambda}\_M^\* \\ &= -\Delta(\boldsymbol{\psi}, \dot{\boldsymbol{\psi}}) \tilde{\boldsymbol{\tilde{f}}}\_M^T \tilde{\boldsymbol{\lambda}}\_M - \Delta(\boldsymbol{\psi}, \dot{\boldsymbol{\psi}}) \dot{\tilde{\boldsymbol{f}}}\_M^T \boldsymbol{\lambda}\_M^\* \end{aligned}$$

Multiplying (25) by ˆ *JMM*<sup>−</sup><sup>1</sup> <sup>1</sup> from left, we have

$$\begin{aligned} &\hat{\boldsymbol{M}}\_{M}^{-1}\boldsymbol{M}\_{1}^{-1}\left\{\boldsymbol{\alpha}\_{1}\hat{\boldsymbol{M}}\_{1}\text{Blockdiag}\left(\boldsymbol{0}\_{k},\boldsymbol{I}\_{m}\right)+\boldsymbol{\alpha}\_{2}\boldsymbol{I}\_{k+m}+\Delta(\boldsymbol{\psi},\hat{\boldsymbol{\psi}})\right\} \\ &\times \hat{\boldsymbol{f}}\_{M}^{T}\hat{\boldsymbol{\lambda}}\_{M} = \hat{\boldsymbol{f}}\_{M}\left[\dot{\boldsymbol{R}}\boldsymbol{\tilde{s}}+\boldsymbol{M}\_{1}^{-1}\left\{(\boldsymbol{C}\_{1}+\boldsymbol{K}\_{d})\boldsymbol{\delta}\right.\right.} \\ &-\boldsymbol{Y}\_{1}(\boldsymbol{q},\dot{\boldsymbol{q}},\boldsymbol{\chi},\dot{\boldsymbol{\chi}}+\dot{\boldsymbol{\beta}})\boldsymbol{\bar{p}}\_{1}+\boldsymbol{\alpha}\_{1}\boldsymbol{\hat{M}}\_{1}\boldsymbol{\beta} \\ &+\left(\frac{\partial\boldsymbol{V}\_{1}}{\partial\boldsymbol{q}}\boldsymbol{\hat{S}}(\boldsymbol{q}\_{B},\boldsymbol{\dot{\theta}})\right)^{T}+\Delta(\boldsymbol{\psi},\boldsymbol{\dot{\psi}})\hat{\boldsymbol{I}}\_{M}^{T}\boldsymbol{\lambda}\_{M}^{\*}+(\boldsymbol{F}+\boldsymbol{d}\_{1})\right\}\Big].\tag{26} \end{aligned}$$

of Mobile Manipulators 11

Robust Adaptive Position/Force Control of Mobile Manipulators 127

In this section we propose the robust and adaptive position/force control scheme of the mobile manipulators with only the dynamic uncertainties. Since kinetic parameters are known, the

*<sup>M</sup>* = −*α*1*β<sup>M</sup>* − *α*<sup>1</sup> [*JM*<sup>1</sup> *JM*2]

*<sup>M</sup>*<sup>2</sup> *JM*<sup>0</sup> *fB <sup>ϕ</sup>*−<sup>1</sup> <sup>−</sup>*<sup>J</sup>*

Then the robust adaptive control scheme proposed in (Narikiyo et al., 2008) can be applied to the system with dynamic uncertainties. This control scheme is shown in the following

**Theorem 2.** *Let the kinematic parameters be known. Applying the following control law and adaptive*

<sup>1</sup> [−*Kdδ* − *F* + *Y*1(*q*, *q*˙, *χ*, *χ*˙)*p*ˆ

�*<sup>T</sup>* + *J T M*(−*λ*<sup>∗</sup>

*S*(*qB*)

*<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>* <sup>=</sup> 0, lim

*<sup>F</sup>*(*t*) = *<sup>δ</sup>D*<sup>ˆ</sup> <sup>2</sup>

�*δ*�*D*<sup>ˆ</sup> <sup>+</sup> *<sup>ρ</sup>*(*t*)

*<sup>T</sup>λ*˜ *<sup>M</sup>*

*<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>*<sup>3</sup> <sup>=</sup> 0.

Proof of the theorem 2 is completed by the following Lemmas as similar to the proof of

.

� *∂V*<sup>1</sup> *∂q*

lim

Substituting (28) into (4), we can obtain the following closed loop system.

+(1 + *α*2)¯*J*

*p*ˆ = −Γ*Y*1(*q*, *q*˙, *χ*, *χ*˙)*δ*

*<sup>M</sup>* = −*α*1*β<sup>M</sup>* − *α*1[*JM*<sup>1</sup> *JM*2]

lim

*δ* = −(*C*<sup>1</sup> + *Kd*)*δ* + *Y*1(*q*, *q*˙, *χ*, *χ*˙)*p*˜ −

**Lemma 7.** *For the closed loop system, δ*, *β* ∈ *L*2*,ν*˜, *Rs*˜, *p*ˆ,*eM*1,*e*˙*M*1, *νc*, *χ*, *q*, *q*˙ ∈ *L*<sup>∞</sup> *and*

*<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>*<sup>1</sup> <sup>=</sup> 0, lim

*T*

*JM*(*q*, *ψ*ˆ) has no uncertainties. Therefore (15) and *R*ˆ are replaced with

*Ik* 0 0 *Il*

−1 *<sup>M</sup>*<sup>2</sup> *JM*<sup>1</sup> ⎤ ⎦ .

*<sup>M</sup>* <sup>+</sup> *<sup>α</sup>*2*λ*˜ *<sup>M</sup>*)]

<sup>1</sup> (*q*, *q*˙, *χ*, *χ*˙)*δ* (28)

*<sup>t</sup>*→<sup>∞</sup> *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>=</sup> 0, (29)

*S*(*qB*)

*<sup>M</sup>λ*˜ *<sup>M</sup>* <sup>−</sup> (*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1) (30)

�*<sup>T</sup>*

*<sup>T</sup> <sup>λ</sup>*˜ *<sup>M</sup>*, (27)

**3.3 Control laws for the system only with the dynamic uncertainties**

⎡ ⎣

−*J* −1

*β*˙

*R* =

*τ* = *B*−<sup>1</sup>

˙ *<sup>p</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup>Γ*Y<sup>T</sup>*

˙ *<sup>D</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>γ</sup>*�*δ*�

− � *∂V*<sup>1</sup> *∂q*

*laws to the mobile manipulator (4) and (5),*

*then internal signals are bounded and*

*M*<sup>1</sup> ˙

˙

˙ *<sup>D</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>γ</sup>*�*δ*�

*β*˙

Jacobian matrix ˆ

theorem.

*where*

Theorem 1.

By definition and Assump.3 *<sup>ν</sup>*˙*<sup>c</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup> and *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙ <sup>=</sup> *<sup>ν</sup>*˙*c*, we have

$$\mathcal{Y}\_1(q, \dot{q}, \chi, \dot{\chi} + \dot{\beta}) \in L\_{\infty}.$$

Therefore, if *α*<sup>1</sup> and *α*<sup>2</sup> are selected such that

$$\left\{ \mathfrak{a}\_1 \hat{\mathcal{M}}\_1 \boldsymbol{B} \boldsymbol{l} \boldsymbol{c} \boldsymbol{k} \boldsymbol{d} \boldsymbol{i} \boldsymbol{a} \boldsymbol{g} \left( \mathbf{0}\_{k\prime} \boldsymbol{I}\_m \right) + \mathfrak{a}\_2 \boldsymbol{I}\_{k+m} + \Delta \left( \boldsymbol{\psi}\_{\prime} \boldsymbol{\psi} \right) \right\}$$

is nonsingular, then *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞.

$$\mathbf{Lemma}\\
\mathbf{4.}\quad Z\_1, \dot{\upsilon}\_{\mathcal{C}}, \dot{\beta}, \dot{\chi}\_{\mathcal{I}}\\
\text{Y}\_1(\boldsymbol{q}, \dot{\boldsymbol{q}}, \boldsymbol{\chi}, \dot{\chi}), F(t) \in L\_{\infty}.$$

#### (Proof)

*Z*<sup>1</sup> ∈ *L*<sup>∞</sup> is derived from Lemma 2 and 3. *ν*˙*<sup>c</sup>* ∈ *L*<sup>∞</sup> is derived from definition of *νc*, Assump.2 and adaptive laws of *<sup>a</sup>*ˆ, *<sup>ψ</sup>*ˆ. *<sup>β</sup>*˙ <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup> is derived directly from (15). Then *<sup>χ</sup>*˙ <sup>=</sup> *<sup>ν</sup>*˙*<sup>c</sup>* <sup>−</sup> *<sup>β</sup>*˙ <sup>∈</sup> *<sup>L</sup>*∞. These relations imply *Y*1(*q*, *q*˙, *χ*, *χ*˙) ∈ *L*∞. And finally,

$$\|\|F(t)\|\| = \|\frac{\delta \hat{D}^2}{\|\delta\|\|\hat{D} + \rho(t)}\|\| < \|\frac{\delta \hat{D}^2}{\|\delta\|\|\hat{D}}\|\| = \hat{D} \in L\_{\infty}.$$

**Lemma 5.**

$$\lim\_{t \to \infty} e = 0$$

(Proof)

From definitions and Lemma 4, we have ˙ *δ* ∈ *L*∞. Since *δ*, *β* ∈ *L*2, by Barbalatat's Lemma(Kristic et al., 1995; Slotine & Li, 1991) we have lim*t*→<sup>∞</sup> *δ* = 0 and lim*t*→<sup>∞</sup> *β* = 0. This means lim*t*→<sup>∞</sup> *ν*˜ = 0. By Lemma1 lim*t*→<sup>∞</sup> *eB* = 0. Since lim*t*→<sup>∞</sup> *ν*˜ = 0 and

$$
\tilde{\nu}\_{M1} = -\dot{e}\_{M1} - K\_{M1}e\_{M1\nu}
$$

lim*t*→<sup>∞</sup> *eM*<sup>1</sup> = 0. Then lim*t*→<sup>∞</sup> *e* = 0. From above 5 Lemmas and next Lemma the proof of Theorem 1 is completed.

#### **Lemma 6.**

$$\lim\_{t \to \infty} \lambda\_M = \lambda\_M^\*$$

(Proof)

From (19), Lemma 3 and Lemma 4, ˙ *<sup>δ</sup>* <sup>∈</sup> *<sup>L</sup>*∞. And from definitions, *<sup>β</sup>*˙ <sup>∈</sup> *<sup>L</sup>*∞, ˙ *ν*˜ ∈ *L*∞. Then *<sup>q</sup>*¨ <sup>∈</sup> *<sup>L</sup>*∞, *<sup>R</sup>*¨ and *<sup>ν</sup>*¨*<sup>c</sup>* <sup>∈</sup> *<sup>L</sup>*∞. These lead us to *<sup>Y</sup>*˙ <sup>1</sup>(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙) <sup>∈</sup> *<sup>L</sup>*∞.

Differentiating both sides of (26), we have ˙ *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞, that is *<sup>λ</sup>*˙ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞.Then, by above Lemmas and definition of *<sup>β</sup>M*, *<sup>β</sup>*¨*<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞. This shows that *<sup>β</sup>*˙ *<sup>M</sup>* is uniformly continuous. And *β<sup>M</sup>* → 0 is shown in previous Lemma. Therefore from Barbalat's Lemma(Kristic et al., 1995; Slotine & Li, 1991) we have lim*t*→<sup>∞</sup> *<sup>β</sup>*˙ *<sup>M</sup>* = 0. Since [ ˆ *JM*<sup>1</sup> <sup>ˆ</sup> *JM*2] *<sup>T</sup>* in (15) is full column rank, we have

$$\lim\_{t \to \infty} \bar{\lambda}\_M = 0.$$

#### **3.3 Control laws for the system only with the dynamic uncertainties**

In this section we propose the robust and adaptive position/force control scheme of the mobile manipulators with only the dynamic uncertainties. Since kinetic parameters are known, the Jacobian matrix ˆ *JM*(*q*, *ψ*ˆ) has no uncertainties. Therefore (15) and *R*ˆ are replaced with

$$
\dot{\mathcal{P}}\_M = -\boldsymbol{\alpha}\_1 \boldsymbol{\beta}\_M - \boldsymbol{\alpha}\_1 \left[ \boldsymbol{I}\_{M1} \, \boldsymbol{I}\_{M2} \right]^\top \tilde{\boldsymbol{\Lambda}}\_{M\prime} \tag{27}
$$

$$
\mathcal{R} = \begin{bmatrix}
I\_k & \mathbf{0} \\
\mathbf{0} & I\_l \\
$$

Then the robust adaptive control scheme proposed in (Narikiyo et al., 2008) can be applied to the system with dynamic uncertainties. This control scheme is shown in the following theorem.

**Theorem 2.** *Let the kinematic parameters be known. Applying the following control law and adaptive laws to the mobile manipulator (4) and (5),*

$$\begin{aligned} \tau &= B\_1^{-1}[-K\_d \delta - F + Y\_1(q, \dot{q}, \chi, \dot{\chi}) \dot{\mathcal{P}} \\ &- \left(\frac{\partial V\_1}{\partial q} S(q\_B)\right)^T + \overline{f}\_M^T (-\lambda\_M^\* + a\_2 \tilde{\lambda}\_M)] \\ \dot{\mathcal{P}} &= -\Gamma Y\_1^T (q, \dot{q}, \chi, \dot{\chi}) \delta \\ \dot{\tilde{D}} &= \gamma \|\delta\| \end{aligned} \tag{28}$$

*then internal signals are bounded and*

$$\lim\_{t \to \infty} e = 0, \lim\_{t \to \infty} \tilde{\lambda}\_M = 0,\tag{29}$$

*where*

10 Will-be-set-by-IN-TECH

*<sup>Y</sup>*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙) <sup>∈</sup> *<sup>L</sup>*∞.

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*) + *α*<sup>2</sup> *Ik*<sup>+</sup>*<sup>m</sup>* + Δ(*ψ*, *ψ*ˆ)

*Z*<sup>1</sup> ∈ *L*<sup>∞</sup> is derived from Lemma 2 and 3. *ν*˙*<sup>c</sup>* ∈ *L*<sup>∞</sup> is derived from definition of *νc*, Assump.2 and adaptive laws of *<sup>a</sup>*ˆ, *<sup>ψ</sup>*ˆ. *<sup>β</sup>*˙ <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup> is derived directly from (15). Then *<sup>χ</sup>*˙ <sup>=</sup> *<sup>ν</sup>*˙*<sup>c</sup>* <sup>−</sup> *<sup>β</sup>*˙ <sup>∈</sup> *<sup>L</sup>*∞. These

� <sup>&</sup>lt; � *<sup>δ</sup>D*<sup>ˆ</sup> <sup>2</sup>

�*δ*�*D*<sup>ˆ</sup> � <sup>=</sup> *<sup>D</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>L</sup>*∞.

*δ* ∈ *L*∞. Since *δ*, *β* ∈ *L*2, by Barbalatat's

*<sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞, that is *<sup>λ</sup>*˙ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞.Then, by above Lemmas

*<sup>T</sup>* in (15) is full column rank, we have

*<sup>M</sup>* is uniformly continuous. And *β<sup>M</sup>* → 0 is

*ν*˜ ∈ *L*∞. Then

By definition and Assump.3 *<sup>ν</sup>*˙*<sup>c</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup> and *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙ <sup>=</sup> *<sup>ν</sup>*˙*c*, we have

Therefore, if *α*<sup>1</sup> and *α*<sup>2</sup> are selected such that

is nonsingular, then *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞.

(Proof)

**Lemma 5.**

(Proof)

**Lemma 6.**

(Proof)

**Lemma 4.** *<sup>Z</sup>*1, *<sup>ν</sup>*˙*c*, *<sup>β</sup>*˙, *<sup>χ</sup>*˙,*Y*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙), *<sup>F</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*∞*.*

relations imply *Y*1(*q*, *q*˙, *χ*, *χ*˙) ∈ *L*∞. And finally,

From definitions and Lemma 4, we have ˙

lim*t*→<sup>∞</sup> *eM*<sup>1</sup> = 0. Then lim*t*→<sup>∞</sup> *e* = 0.

From (19), Lemma 3 and Lemma 4, ˙

1991) we have lim*t*→<sup>∞</sup> *<sup>β</sup>*˙

*<sup>q</sup>*¨ <sup>∈</sup> *<sup>L</sup>*∞, *<sup>R</sup>*¨ and *<sup>ν</sup>*¨*<sup>c</sup>* <sup>∈</sup> *<sup>L</sup>*∞. These lead us to *<sup>Y</sup>*˙

Differentiating both sides of (26), we have ˙

and definition of *<sup>β</sup>M*, *<sup>β</sup>*¨*<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞. This shows that *<sup>β</sup>*˙

*<sup>M</sup>* = 0. Since [

�*F*(*t*)� <sup>=</sup> � *<sup>δ</sup>D*<sup>ˆ</sup> <sup>2</sup>

�*δ*�*D*<sup>ˆ</sup> <sup>+</sup> *<sup>ρ</sup>*(*t*)

means lim*t*→<sup>∞</sup> *ν*˜ = 0. By Lemma1 lim*t*→<sup>∞</sup> *eB* = 0. Since lim*t*→<sup>∞</sup> *ν*˜ = 0 and

From above 5 Lemmas and next Lemma the proof of Theorem 1 is completed.

ˆ *JM*<sup>1</sup> <sup>ˆ</sup> *JM*2]

> lim *<sup>t</sup>*→<sup>∞</sup> *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>=</sup> 0.

lim *<sup>t</sup>*→<sup>∞</sup> *<sup>λ</sup><sup>M</sup>* <sup>=</sup> *<sup>λ</sup>*<sup>∗</sup>

lim *<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>* <sup>=</sup> <sup>0</sup>

Lemma(Kristic et al., 1995; Slotine & Li, 1991) we have lim*t*→<sup>∞</sup> *δ* = 0 and lim*t*→<sup>∞</sup> *β* = 0. This

*ν*˜*M*<sup>1</sup> = −*e*˙*M*<sup>1</sup> − *KM*1*eM*1,

shown in previous Lemma. Therefore from Barbalat's Lemma(Kristic et al., 1995; Slotine & Li,

*M*

<sup>1</sup>(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙) <sup>∈</sup> *<sup>L</sup>*∞.

*<sup>δ</sup>* <sup>∈</sup> *<sup>L</sup>*∞. And from definitions, *<sup>β</sup>*˙ <sup>∈</sup> *<sup>L</sup>*∞, ˙

$$F(t) = \frac{\delta \hat{D}^2}{||\delta||\hat{D} + \rho(t)}.$$

Substituting (28) into (4), we can obtain the following closed loop system.

$$\begin{aligned} M\_1 \dot{\delta} &= -(\mathbf{C}\_1 + \mathbf{K}\_d)\delta + Y\_1(q, \dot{q}, \chi, \dot{\chi})\dot{\mathcal{p}} - \left(\frac{\partial V\_1}{\partial q} S(q\_B)\right)^T \\ &+ (1 + \mathbf{a}\_2) \bar{I}\_M^T \tilde{\lambda}\_M - (F + d\_1) \\ \dot{\mathcal{p}} &= -\Gamma Y\_1(q, \dot{q}, \chi, \dot{\chi})\delta \\ \dot{\mathcal{D}} &= \gamma ||\delta|| \\ \dot{\mathcal{P}}\_M &= -\mathbf{a}\_1 \boldsymbol{\beta}\_M - \mathbf{a}\_1 [f\_{M1} \ f\_{M2}]^T \tilde{\lambda}\_M \end{aligned} \tag{30}$$

Proof of the theorem 2 is completed by the following Lemmas as similar to the proof of Theorem 1.

**Lemma 7.** *For the closed loop system, δ*, *β* ∈ *L*2*,ν*˜, *Rs*˜, *p*ˆ,*eM*1,*e*˙*M*1, *νc*, *χ*, *q*, *q*˙ ∈ *L*<sup>∞</sup> *and*

$$\lim\_{t \to \infty} e\_1 = 0, \lim\_{t \to \infty} e\_3 = 0.$$

(Proof) We set *V*<sup>2</sup> as

$$V\_2 = V\_1 + \frac{1}{2}\delta^T M\_1 \delta + \frac{1+a\_2}{2a\_1}\beta^T \beta + \frac{1}{2}\vec{p}^T \Gamma^{-1} \vec{p}$$

$$+ \frac{1}{2\gamma} \vec{D}^2. \tag{31}$$

of Mobile Manipulators 13

Robust Adaptive Position/Force Control of Mobile Manipulators 129

*ν*˜*M*<sup>1</sup> = −*e*˙*M*<sup>1</sup> − *KM*1*eM*<sup>1</sup> ∈ *L*∞. Then *eM*1,*e*˙*M*<sup>1</sup> ∈ *L*∞. Furthermore, from (12) and Assump.3, we have *νc*, *χ* ∈ *L*<sup>∞</sup> and *eM*1,*e*˙*M*<sup>1</sup> ∈ *L*∞. From Assump.2, we have *qM*, *q*˙*<sup>M</sup>* ∈ *L*∞. Similarly from *ν*˜, *ν<sup>c</sup>* ∈ *L*<sup>∞</sup> and Assump.2, we have *q*˙*B*,*e*˙*<sup>B</sup>* ∈ *L*∞. Therefore *q*, *q*˙ ∈ *L*∞. Finally, from

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*)+(1 + *α*2)*Ik*<sup>+</sup>*<sup>m</sup>*

*Y*1(*q*, *q*˙, *χ*, *χ*˙)*p*˜ = *Y*1(*q*, *q*˙, *χ*, *χ*˙ + *β*˙)*p*˜ −

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*)+(1 + *α*2)*Ik*<sup>+</sup>*<sup>m</sup>*

<sup>1</sup> from left, we have

*<sup>M</sup>λ*˜ *<sup>M</sup>* = ¯*JM*

Furthermore, since (12) and Assump.3 lead to *<sup>ν</sup>*˙*<sup>c</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup> and *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙ <sup>=</sup> *<sup>ν</sup>*˙*c*, we have

Since from these Lemmas and Barbalat's Lemma(Slotine & Li, 1991) we have

*<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>* <sup>=</sup> 0, lim

lim

× ¯*J T* = *M*1(*R*˙

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*)+(1 + *α*2)*Ik*<sup>+</sup>*<sup>m</sup>*

*R*˙*s*˜ + *M*−<sup>1</sup>

*<sup>t</sup>*→<sup>∞</sup> *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>=</sup> 0,

<sup>−</sup>*Y*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙)*p*˜ <sup>−</sup> *<sup>α</sup>*1*M*<sup>ˆ</sup> <sup>1</sup>*<sup>β</sup>* + (*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1)

*<sup>Y</sup>*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙) <sup>∈</sup> *<sup>L</sup>*∞.

*α*1*M*ˆ <sup>1</sup>*Blockdiag*(0*k*, *Im*)+(1 + *α*2)*Ik*<sup>+</sup>*<sup>m</sup>*

*<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>*<sup>3</sup> <sup>=</sup> 0.

−*Y*1(*q*, *q*˙, *χ*, *χ*˙)*p*˜ + (*F* + *d*1).

*<sup>M</sup>*<sup>ˆ</sup> <sup>1</sup>(*q*) <sup>−</sup> *<sup>M</sup>*1(*q*)

*s*˜ + *R*˙*s*˜) + *M*1*β*˙ + (*C*<sup>1</sup> + *Kd*)*δ*

*β*˙,

 ¯*J T <sup>M</sup>λ*˜ *<sup>M</sup>*

<sup>1</sup> {(*C*<sup>1</sup> + *Kd*)*δ*

. (37)

*s*˜ + *R*˙*s*˜)+(*C*<sup>1</sup> + *Kd*)*δ* <sup>−</sup>*Y*1(*q*, *<sup>q</sup>*˙, *<sup>χ</sup>*, *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙)*p*˜ <sup>−</sup> *<sup>α</sup>*1*M*<sup>ˆ</sup> <sup>1</sup>*<sup>β</sup>* + (*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1). (36)

*e*1, sin *e*<sup>3</sup> ∈ *L*<sup>2</sup> and Barbalat's Lemma(Slotine & Li, 1991), we have

**Lemma 8.** *Let M*ˆ <sup>1</sup> = *M*1(*p*ˆ)*. If there exist α*1, *α*<sup>2</sup> *such that* 

(1 + *α*2)¯*J*

¯*JMM*−<sup>1</sup> 1 

Therefore, if *α*1*andα*<sup>2</sup> are selected such that

is nonsingular, then *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞.

proof of the theorem is completed.

*T*

*is nonsingular, then <sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞*.*

(Proof)

From (30) we have

By using the relation

above equation is converted into

Multiplying (36) by ¯*JMM*−<sup>1</sup>

lim

*<sup>M</sup>λ*˜ *<sup>M</sup>* <sup>=</sup> *<sup>M</sup>*1(*R*˙

*<sup>t</sup>*→<sup>∞</sup> *<sup>e</sup>*<sup>1</sup> <sup>=</sup> 0, lim

Differentiating *V*<sup>2</sup> along (30), we have

$$
\dot{V}\_2 = \frac{\partial V\_1}{\partial q} S(q\_B)(\upsilon\_c + \tilde{\upsilon}) + \frac{\partial V\_1}{\partial q^\*} S(q\_B^\*) \upsilon^\*
$$

$$
$$

$$
+ (1 + \alpha\_2) \tilde{\lambda}\_M \tilde{\mathbb{J}}\_M R \tilde{s} - \delta^T (F + d\_1).
$$

Furthermore, by using definitions of *V*<sup>1</sup> = *V*1(*qB*, *q*<sup>∗</sup> *<sup>B</sup>*) and *δ*,

$$\frac{\partial V\_1}{\partial q} S(q\_B) \tilde{\nu} = \frac{\partial V\_1}{\partial q} S(q\_B) \delta$$

is derived and by using the relation

$$\begin{aligned} \bar{J}\_M \mathcal{R} &= \begin{bmatrix} J\_{M0} f\_B \varphi^{-1} \ J\_{M1} \ J\_{M2} \end{bmatrix} \begin{bmatrix} I\_k & 0 \\ 0 & I\_l \\ -J\_{M2}^{-1} f\_{M0} f\_B \varphi^{-1} & -J\_{M2}^{-1} f\_{M1} \end{bmatrix} \\ &= \begin{bmatrix} \mathbf{0}\_{(m-l)\times k} \ \mathbf{0}\_{(m-l)\times l} \end{bmatrix} .\end{aligned}$$

we have

$$\dot{V}\_2 = -K\_1 e\_1^2 - \frac{K\_3}{K\_2} \sin^2 e\_3 - \delta^T K\_d \delta - (1 + a\_2) \beta^T \beta - \delta^T (F + d\_1). \tag{32}$$

*F*(*t*) and adaptive laws lead the right hand of (32) to

$$
\dot{V}\_2 < -K\_1 e\_1^2 - \frac{K\_3}{K\_2} \sin^2 e\_3 - \delta^T K\_d \delta - (1 + \alpha\_2) \beta^T \beta + \rho(t). \tag{33}
$$

Integrating both side of this inequality and using definition of *ρ*(*t*), we have

$$\begin{split} V\_2(t) - V\_2(0) &< -K\_1 \int\_0^t e\_1^2 d\tau - \frac{K\_3}{K\_2} \int\_0^t \sin^2 e\_3 d\tau \\ &- \int\_0^t \delta^T K\_d \delta d\tau - (1 + a\_2) \int\_0^t \beta^T \beta d\tau + a < \infty. \end{split} \tag{34}$$

This shows

$$V\_2(t) < V\_2(0) + a < \infty. \tag{35}$$

Therefore, *<sup>V</sup>*2(*t*) is bounded, that is, *<sup>e</sup>*1,*e*2, *<sup>δ</sup>*, *<sup>β</sup>*, *<sup>p</sup>*˜, *<sup>D</sup>*˜ <sup>∈</sup> *<sup>L</sup>*∞, and *<sup>e</sup>*1, sin *<sup>e</sup>*3, *<sup>δ</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2. From definitions of variables *<sup>ν</sup>*˜, we have *Rs*˜, *<sup>p</sup>*ˆ, *<sup>D</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>L</sup>*∞. From (12) and definitions of *<sup>ν</sup>* and *<sup>ν</sup>c*,

*ν*˜*M*<sup>1</sup> = −*e*˙*M*<sup>1</sup> − *KM*1*eM*<sup>1</sup> ∈ *L*∞. Then *eM*1,*e*˙*M*<sup>1</sup> ∈ *L*∞. Furthermore, from (12) and Assump.3, we have *νc*, *χ* ∈ *L*<sup>∞</sup> and *eM*1,*e*˙*M*<sup>1</sup> ∈ *L*∞. From Assump.2, we have *qM*, *q*˙*<sup>M</sup>* ∈ *L*∞. Similarly from *ν*˜, *ν<sup>c</sup>* ∈ *L*<sup>∞</sup> and Assump.2, we have *q*˙*B*,*e*˙*<sup>B</sup>* ∈ *L*∞. Therefore *q*, *q*˙ ∈ *L*∞. Finally, from *e*1, sin *e*<sup>3</sup> ∈ *L*<sup>2</sup> and Barbalat's Lemma(Slotine & Li, 1991), we have

$$\lim\_{t \to \infty} e\_1 = 0, \lim\_{t \to \infty} e\_3 = 0.$$

**Lemma 8.** *Let M*ˆ <sup>1</sup> = *M*1(*p*ˆ)*. If there exist α*1, *α*<sup>2</sup> *such that*

$$\left\{\alpha\_1 \hat{M}\_1 B \text{lock} \text{diag} \left(0\_{k\prime} I\_m\right) + \left(1 + \alpha\_2\right) I\_{k+m} \right\}$$

*is nonsingular, then <sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞*.*

(Proof)

12 Will-be-set-by-IN-TECH

1 + *α*<sup>2</sup> 2*α*<sup>1</sup>

*<sup>∂</sup>q*<sup>∗</sup> *<sup>S</sup>*(*q*<sup>∗</sup>

*β<sup>T</sup> β* + 1 <sup>2</sup> *<sup>p</sup>*˜*T*Γ−<sup>1</sup> *<sup>p</sup>*˜

*B*)*ν*<sup>∗</sup>

*<sup>B</sup>*) and *δ*,

*S*(*qB*)*δ*

*D*˜ 2. (31)

� *∂V*<sup>1</sup> *∂q*

*Ik* 0 0 *Il*

sin2 *<sup>e</sup>*<sup>3</sup> <sup>−</sup> *<sup>δ</sup>TKd<sup>δ</sup>* <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>α</sup>*2)*β<sup>T</sup> <sup>β</sup>* <sup>−</sup> *<sup>δ</sup>T*(*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1). (32)

sin2 *<sup>e</sup>*<sup>3</sup> <sup>−</sup> *<sup>δ</sup>TKd<sup>δ</sup>* <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>α</sup>*2)*βT<sup>β</sup>* <sup>+</sup> *<sup>ρ</sup>*(*t*). (33)

sin2 *e*3*dτ*

*V*2(*t*) < *V*2(0) + *a* < ∞. (35)

� *t* 0

*βTβdτ* + *a* < ∞.

−1 *<sup>M</sup>*<sup>2</sup> *JM*<sup>1</sup> ⎤ ⎦

(34)

*<sup>M</sup>*<sup>2</sup> *JM*<sup>0</sup> *fB <sup>ϕ</sup>*−<sup>1</sup> <sup>−</sup>*<sup>J</sup>*

�*<sup>T</sup>*

*V*<sup>2</sup> = *V*<sup>1</sup> +

Differentiating *V*<sup>2</sup> along (30), we have

is derived and by using the relation

*V*˙

¯*JMR* =

<sup>2</sup> = −*K*1*e*

*V*˙

*V*˙ <sup>2</sup> <sup>=</sup> *<sup>∂</sup>V*<sup>1</sup> *∂q*

Furthermore, by using definitions of *V*<sup>1</sup> = *V*1(*qB*, *q*<sup>∗</sup>

�

= �

> 2 <sup>1</sup> <sup>−</sup> *<sup>K</sup>*<sup>3</sup> *K*2

*F*(*t*) and adaptive laws lead the right hand of (32) to

− � *t* 0

2 <sup>1</sup> <sup>−</sup> *<sup>K</sup>*<sup>3</sup> *K*2

<sup>2</sup> < −*K*1*e*

+ 1 2*γ* 1 2

*∂V*<sup>1</sup> *∂q*

*JM*<sup>0</sup> *fB ϕ*−<sup>1</sup> *JM*<sup>1</sup> *JM*<sup>2</sup>

<sup>0</sup>(*m*−*l*)×*<sup>k</sup>* <sup>0</sup>(*m*−*l*)×*<sup>l</sup>*

Integrating both side of this inequality and using definition of *ρ*(*t*), we have

*<sup>δ</sup>TKdδd<sup>τ</sup>* <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>α</sup>*2)

*V*2(*t*) − *V*2(0) < −*K*<sup>1</sup>

*δTM*1*δ* +

*<sup>S</sup>*(*qB*)(*ν<sup>c</sup>* <sup>+</sup> *<sup>ν</sup>*˜) + *<sup>∂</sup>V*<sup>1</sup>

<sup>−</sup>*δTKd<sup>δ</sup>* <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>α</sup>*2)*β<sup>T</sup> <sup>β</sup>* <sup>−</sup> *<sup>δ</sup>TST*(*qB*)

*<sup>S</sup>*(*qB*)*ν*˜ <sup>=</sup> *<sup>∂</sup>V*<sup>1</sup>

� ⎡ ⎣

� ,

� *t* 0 *e*2 <sup>1</sup>*d<sup>τ</sup>* <sup>−</sup> *<sup>K</sup>*<sup>3</sup> *K*2

Therefore, *<sup>V</sup>*2(*t*) is bounded, that is, *<sup>e</sup>*1,*e*2, *<sup>δ</sup>*, *<sup>β</sup>*, *<sup>p</sup>*˜, *<sup>D</sup>*˜ <sup>∈</sup> *<sup>L</sup>*∞, and *<sup>e</sup>*1, sin *<sup>e</sup>*3, *<sup>δ</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2. From definitions of variables *<sup>ν</sup>*˜, we have *Rs*˜, *<sup>p</sup>*ˆ, *<sup>D</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>L</sup>*∞. From (12) and definitions of *<sup>ν</sup>* and *<sup>ν</sup>c*,

� *t* 0

*∂q*

−*J* −1

+(<sup>1</sup> <sup>+</sup> *<sup>α</sup>*2)*λ*˜ *<sup>M</sup>* ¯*JMRs*˜ <sup>−</sup> *<sup>δ</sup>T*(*<sup>F</sup>* <sup>+</sup> *<sup>d</sup>*1).

(Proof) We set *V*<sup>2</sup> as

we have

This shows

From (30) we have

$$\begin{aligned} (1+\varkappa\_2)\tilde{f}\_M^T \tilde{\lambda}\_M &= M\_1(R\dot{\tilde{s}} + \dot{R}\tilde{s}) + M\_1 \dot{\tilde{\rho}} + (\mathcal{C}\_1 + \mathcal{K}\_d)\delta \\ &- Y\_1(q, \dot{q}, \chi, \dot{\chi})\tilde{\rho} + (F + d\_1). \end{aligned}$$

By using the relation

$$\begin{aligned} Y\_1(q, \dot{q}, \chi\_{\prime} \dot{\chi}) \tilde{p} &= Y\_1(q, \dot{q}, \chi\_{\prime} \dot{\chi} + \dot{\beta}) \tilde{p} \\ &- \left\{ \hat{M}\_1(q) - M\_1(q) \right\} \dot{\beta} \tilde{\rho} \end{aligned}$$

above equation is converted into

$$\begin{aligned} \left\{ \mathfrak{a}\_1 \hat{M}\_1 \text{Blockdiag}(\mathbf{0}\_k, I\_m) + (1 + \mathfrak{a}\_2) I\_{k+m} \right\} \tilde{f}\_M^T \bar{\lambda}\_M \\ = M\_1 (R \dot{\tilde{s}} + \dot{R} \tilde{s}) + (\mathbf{C}\_1 + \mathbf{K}\_d) \delta \\ - Y\_1 (q\_\prime \dot{q}\_\prime \chi\_\prime \dot{\chi} + \dot{\tilde{\beta}}) \tilde{p} - \mathfrak{a}\_1 \hat{M}\_1 \beta + (F + d\_1) . \end{aligned} \tag{36}$$

Multiplying (36) by ¯*JMM*−<sup>1</sup> <sup>1</sup> from left, we have

$$
\begin{split}
\bar{f}\_M M\_1^{-1} \left\{ \mathfrak{a}\_1 \hat{\mathsf{M}}\_1 \text{Blockdiag} (\mathbf{0}\_{\mathsf{k}} \, I\_m) + (1 + \mathfrak{a}\_2) I\_{k+m} \right\} \\
\times \bar{f}\_M^T \tilde{\lambda}\_M = \bar{f}\_M \left[ \dot{\mathsf{R}} \tilde{\mathsf{s}} + M\_1^{-1} \left\{ (\mathsf{C}\_1 + \mathsf{K}\_d) \delta \right. \\
$$

Furthermore, since (12) and Assump.3 lead to *<sup>ν</sup>*˙*<sup>c</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup> and *<sup>χ</sup>*˙ <sup>+</sup> *<sup>β</sup>*˙ <sup>=</sup> *<sup>ν</sup>*˙*c*, we have

$$\mathcal{Y}\_1(q, \dot{q}, \chi, \dot{\chi} + \dot{\beta}) \in L\_{\infty}.$$

Therefore, if *α*1*andα*<sup>2</sup> are selected such that

$$\{\alpha\_1 \hat{M}\_1 B \text{lock} \& \text{diag}(0\_{k\nu} I\_m) + (1 + \alpha\_2) I\_{k+m}\}$$

is nonsingular, then *<sup>λ</sup>*˜ *<sup>M</sup>* <sup>∈</sup> *<sup>L</sup>*∞. Since from these Lemmas and Barbalat's Lemma(Slotine & Li, 1991) we have

$$\lim\_{t \to \infty} e = 0, \lim\_{t \to \infty} \vec{\lambda}\_M = 0,$$

proof of the theorem is completed.

of Mobile Manipulators 15

Robust Adaptive Position/Force Control of Mobile Manipulators 131

<sup>2</sup>), *<sup>p</sup>*<sup>1</sup>

<sup>2</sup>), *<sup>p</sup>*<sup>1</sup>

<sup>9</sup> <sup>=</sup> *rd* 2*b*

<sup>11</sup> <sup>=</sup> *<sup>r</sup>*2*<sup>d</sup>*

*m*3*rl*3*L*2, *p*<sup>1</sup>

<sup>4</sup>*b*<sup>2</sup> *<sup>m</sup>*3*r*2*L*2*l*3,

<sup>25</sup> <sup>=</sup> *<sup>I</sup>*2, *<sup>p</sup>*<sup>1</sup>

� � *u ω* � .

> ⎤ ⎥ ⎥ ⎦

⎡ ⎣ *νr νl νM*<sup>1</sup> ⎤

⎦ . (38)

<sup>13</sup> <sup>=</sup> *<sup>m</sup>*3*l*3*g*, *<sup>p</sup>*<sup>1</sup>

(*m*2*l*<sup>2</sup> + *m*3*L*2),

<sup>4</sup>*b*<sup>2</sup> (*m*2*l*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3*L*2),

<sup>20</sup> <sup>=</sup> <sup>1</sup> 4*b m*3*rl*<sup>2</sup> 3,

<sup>26</sup> = *I*3.

*θr*

*θr*

<sup>14</sup> <sup>=</sup> <sup>1</sup> 2 *m*3*rl*3,

(*m*2*l*<sup>2</sup> + *m*3*L*2), *p*<sup>1</sup>

<sup>16</sup> <sup>=</sup> <sup>1</sup> 2 *m*3*l* 2 <sup>3</sup>, *<sup>p</sup>*<sup>1</sup> <sup>17</sup> <sup>=</sup> <sup>1</sup> 4*b m*3*r* <sup>2</sup>*l*3,

<sup>19</sup> <sup>=</sup> <sup>1</sup> 2*b*

<sup>22</sup> <sup>=</sup> <sup>1</sup>

<sup>24</sup> <sup>=</sup> *<sup>I</sup>*1, *<sup>p</sup>*<sup>1</sup>

Generalized coordinates are also shown in this Figure. Base coordinates (*x*, *y*, *φ*) can be detected by 3D camera system and others can be detected by encoders. Nonholonomic

> *x*˙ sin *φ* − *y*˙ cos *φ* = 0 *x*˙ cos *φ* + *y*˙ sin *φ* + *bφ*˙ = *r* ˙

> *<sup>x</sup>*˙ cos *<sup>φ</sup>* <sup>+</sup> *<sup>y</sup>*˙ sin *<sup>φ</sup>* <sup>−</sup> *<sup>b</sup>φ*˙ <sup>=</sup> *<sup>r</sup>* ˙

Then (*θr*, *θl*) can be eliminated from the equations of motion by using (*u*, *ω*) as generalized

<sup>2</sup> sin *<sup>φ</sup>* <sup>0</sup> *<sup>r</sup>* <sup>2</sup>*<sup>b</sup>* <sup>−</sup> *<sup>r</sup>*

0 01

Mobile manipulator used for simulation is shown in Fig.3. In this simulation both the

In this simulation, the mobile base is controlled to track to the desired position/orientation trajectory on the floor and the end-effector of the manipulator is controlled to be constrained on the ceiling with desired reaction force. Then, nonholonomic constraints imposed on this system are same as shown before. On the other hand, holonomic constarint imposed on this

Φ(*q*) = *L*<sup>1</sup> + *L*<sup>2</sup> sin *θ*<sup>2</sup> + *L*<sup>3</sup> sin(*θ*<sup>2</sup> + *θ*3) − *L* = 0.

<sup>2</sup>*<sup>b</sup>* 0

Coordinates (*θr*, *θl*) are related with forward velocity *u* and angular velocity *ω* as

� ˙ *θr* ˙ *θl* � = � 1 *<sup>r</sup> <sup>b</sup> <sup>r</sup>* <sup>1</sup> *<sup>r</sup>* <sup>−</sup> *<sup>b</sup> r*

⎡ ⎢ ⎢ ⎣

*r* <sup>2</sup> cos *<sup>φ</sup> <sup>r</sup>* <sup>2</sup> cos *<sup>φ</sup>* <sup>0</sup> *<sup>r</sup>* <sup>2</sup> sin *<sup>φ</sup> <sup>r</sup>*

**4.2 Simulation for the system with both the kinetic and dynamic uncertainties**

*p*1 <sup>8</sup> <sup>=</sup> *<sup>r</sup>*<sup>2</sup> 4*b*

*p*1 <sup>10</sup> <sup>=</sup> *<sup>r</sup>* 4*b* (*m*2*l* 2 <sup>2</sup> <sup>+</sup> *<sup>m</sup>*3*L*<sup>2</sup>

*p*1 <sup>12</sup> <sup>=</sup> *<sup>r</sup>*<sup>2</sup>

*p*1

*p*1 <sup>18</sup> <sup>=</sup> <sup>1</sup> 2

*p*1 <sup>21</sup> <sup>=</sup> <sup>1</sup>

*p*1 <sup>23</sup> <sup>=</sup> <sup>1</sup>

<sup>8</sup>*b*<sup>2</sup> (*m*2*<sup>l</sup>*

*m*3*rdl*3, *p*<sup>1</sup>

<sup>4</sup>*b*<sup>2</sup> *<sup>m</sup>*3*r*2*dl*3, *<sup>p</sup>*<sup>1</sup>

2 <sup>3</sup>, *<sup>p</sup>*<sup>1</sup>

<sup>8</sup>*b*<sup>2</sup> *<sup>m</sup>*3*r*2*<sup>l</sup>*

constraints imposed on this system are written as follows.

velocities. Therefore, kinematic equations (8) are given by

*x*˙ *y*˙ *φ*˙ ˙ *θ*1 ⎤ ⎥ ⎥ ⎦ =

⎡ ⎢ ⎢ ⎣

kinematic and dynamic parameters are all unknown.

system is

<sup>15</sup> <sup>=</sup> *<sup>m</sup>*3*L*2*l*3, *<sup>p</sup>*<sup>1</sup>

2 <sup>2</sup> <sup>+</sup> *<sup>m</sup>*3*L*<sup>2</sup>

Fig. 3. Mobile manipulator used for simulations and experiments

#### **4. Simulations and experiments**

#### **4.1 Mobile manipulator**

Schematic model of mobile manipulator used for simulations and experiments is shown in Fig.3. This mobile manipulator consists of 3 wheel mobile base and 3 link manipulator. *mB*, *mW* and *m*1, *m*2, *m*<sup>3</sup> denote masses of base, wheels and manipulator links, respectively. *IB*,*IW*,*Im* and *I*1, *I*2, *I*<sup>3</sup> denote moment of inertia of base, wheel axis, wheel and manipulator links, respectively. These dynamic parameters are unknown. Kinematic parameters are denoted in Fig.3. Numerical values of kinematic parameters are estimated as follows. 2*b* = 0.316,*r* = 0.098, *d* = 0.11, *L*<sup>1</sup> = 0.143, *L*<sup>2</sup> = 0.19, *L*<sup>3</sup> = 0.342, *l*<sup>1</sup> = 0.0715, *l*<sup>2</sup> = 0.095, *l*<sup>3</sup> = 0.171. On the other hand, dynamic parameters are hardly identified. However, in simulations we use following estimates; *mB* = 5.0, *mW* = 1.25, *IB* = 0.137, *IW* = 0.00313, *Im* = 0.00582, 2*b* = 0.316, *m*<sup>1</sup> = 1.25, *m*<sup>2</sup> = 0.5, *m*<sup>3</sup> = 0.75, *I*<sup>1</sup> = 0.00259, *I*<sup>2</sup> = 0.00173, *I*<sup>3</sup> = 0.00201. Unknown parameter vector *<sup>p</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>*<sup>26</sup> is consisting of these parameters and is given by

$$p\_1 = [p\_{1'}^1 p\_{2'}^1 \cdots \cdot p\_{26}^1]^T \ .$$

where

$$\begin{aligned} p\_1^1 &= \frac{r^2}{4} (m\_B + m\_1 + m\_2 + m\_3), \\ p\_2^1 &= \frac{r^2 d}{2b} (m\_B + m\_1 + m\_2 + m\_3), \\ p\_3^1 &= \frac{r^2}{4b^2} \left\{ (m\_B + m\_1 + m\_2 + m\_3)d^2 + I\_B + 2I\_W \right\}, \\ p\_4^1 &= m\_W r^2 + I\_{m\_1} p\_5^1 = (m\_2 l\_2 + m\_3 L\_2) g\_{\prime\prime} \\ p\_6^1 &= \frac{r}{2} (m\_2 l\_2 + m\_3 L\_2)\_{\prime\prime} p\_7^1 = \frac{1}{2} (m\_2 l\_2^2 + m\_3 L\_2^2)\_{\prime\prime} \end{aligned}$$

14 Will-be-set-by-IN-TECH

*y*

*x*

2*b d*

2*r*

parameter vector *<sup>p</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>*<sup>26</sup> is consisting of these parameters and is given by

*p*<sup>1</sup> = [*p*<sup>1</sup>

<sup>4</sup> (*mB* <sup>+</sup> *<sup>m</sup>*<sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3),

<sup>2</sup>*<sup>b</sup>* (*mB* <sup>+</sup> *<sup>m</sup>*<sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3),

(*m*2*l*<sup>2</sup> + *m*3*L*2), *p*<sup>1</sup>

<sup>1</sup>, *<sup>p</sup>*<sup>1</sup>

<sup>2</sup>, ··· , *<sup>p</sup>*<sup>1</sup> 26] *T*,

(*mB* + *m*<sup>1</sup> + *m*<sup>2</sup> + *m*3)*d*<sup>2</sup> + *IB* + 2*IW*

<sup>5</sup> = (*m*2*l*<sup>2</sup> + *m*3*L*2)*g*,

<sup>7</sup> <sup>=</sup> <sup>1</sup> 2 (*m*2*l* 2 <sup>2</sup> <sup>+</sup> *<sup>m</sup>*3*L*<sup>2</sup>  ,

2),

Fig. 3. Mobile manipulator used for simulations and experiments

*z*

*O*

**4. Simulations and experiments**

*p*1 <sup>1</sup> <sup>=</sup> *<sup>r</sup>*<sup>2</sup>

*p*1 <sup>2</sup> <sup>=</sup> *<sup>r</sup>*2*<sup>d</sup>*

*p*1 <sup>3</sup> <sup>=</sup> *<sup>r</sup>*<sup>2</sup> 4*b*<sup>2</sup> 

*p*1

*p*1 <sup>6</sup> <sup>=</sup> *<sup>r</sup>* 2

<sup>4</sup> <sup>=</sup> *mWr*<sup>2</sup> <sup>+</sup> *Im*, *<sup>p</sup>*<sup>1</sup>

**4.1 Mobile manipulator**

where

O*M*

O

*L*

3 3

T3 

<sup>2</sup>*<sup>l</sup> <sup>m</sup>*<sup>2</sup> *<sup>m</sup>*<sup>3</sup>

*L*1

*L*2

T 2 

1 1

*l*

passive wheel driving wheel

T

 <sup>0</sup> *Pxy* , , I

Schematic model of mobile manipulator used for simulations and experiments is shown in Fig.3. This mobile manipulator consists of 3 wheel mobile base and 3 link manipulator. *mB*, *mW* and *m*1, *m*2, *m*<sup>3</sup> denote masses of base, wheels and manipulator links, respectively. *IB*,*IW*,*Im* and *I*1, *I*2, *I*<sup>3</sup> denote moment of inertia of base, wheel axis, wheel and manipulator links, respectively. These dynamic parameters are unknown. Kinematic parameters are denoted in Fig.3. Numerical values of kinematic parameters are estimated as follows. 2*b* = 0.316,*r* = 0.098, *d* = 0.11, *L*<sup>1</sup> = 0.143, *L*<sup>2</sup> = 0.19, *L*<sup>3</sup> = 0.342, *l*<sup>1</sup> = 0.0715, *l*<sup>2</sup> = 0.095, *l*<sup>3</sup> = 0.171. On the other hand, dynamic parameters are hardly identified. However, in simulations we use following estimates; *mB* = 5.0, *mW* = 1.25, *IB* = 0.137, *IW* = 0.00313, *Im* = 0.00582, 2*b* = 0.316, *m*<sup>1</sup> = 1.25, *m*<sup>2</sup> = 0.5, *m*<sup>3</sup> = 0.75, *I*<sup>1</sup> = 0.00259, *I*<sup>2</sup> = 0.00173, *I*<sup>3</sup> = 0.00201. Unknown

*m*1

*l*

$$\begin{split} p\_{8}^{1} &= \frac{r^{2}}{4b} (m\_{2}l\_{2} + m\_{3}l\_{2}), p\_{9}^{1} = \frac{rd}{2b} (m\_{2}l\_{2} + m\_{3}l\_{2}),\\ p\_{10}^{1} &= \frac{r}{4b} (m\_{2}l\_{2}^{2} + m\_{3}l\_{2}^{2}), p\_{11}^{1} = \frac{r^{2}d}{4b^{2}} (m\_{2}l\_{2} + m\_{3}l\_{2}),\\ p\_{12}^{1} &= \frac{r^{2}}{8b^{2}} (m\_{2}l\_{2}^{2} + m\_{3}l\_{2}^{2}), p\_{13}^{1} = m\_{3}l\_{3}g, p\_{14}^{1} = \frac{1}{2}m\_{3}rl\_{3},\\ p\_{15}^{1} &= m\_{3}l\_{2}l\_{3}, p\_{16}^{1} = \frac{1}{2}m\_{3}l\_{3}^{2}, p\_{17}^{1} = \frac{1}{4b}m\_{3}r^{2}l\_{3},\\ p\_{18}^{1} &= \frac{1}{2}m\_{3}rl\_{3}, p\_{19}^{1} = \frac{1}{2b}m\_{3}rl\_{3}L\_{2}, p\_{20}^{1} = \frac{1}{4b}m\_{3}rl\_{3}^{2},\\ p\_{21}^{1} &= \frac{1}{4b^{2}}m\_{3}r^{2}dl\_{3}, p\_{22}^{1} = \frac{1}{4b^{2}}m\_{3}r^{2}L\_{2}l\_{3},\\ p\_{23}^{1} &= \frac{1}{8b^{2}}m\_{3}r^{2}l\_{3}^{2}, p\_{24}^{1} = l\_{1}, p\_{25}^{1} = l\_{2}, p\_{26}^{1} = l\_{3}.\end{split}$$

Generalized coordinates are also shown in this Figure. Base coordinates (*x*, *y*, *φ*) can be detected by 3D camera system and others can be detected by encoders. Nonholonomic constraints imposed on this system are written as follows.

$$\begin{aligned} \dot{x}\sin\phi - \dot{y}\cos\phi &= 0\\ \dot{x}\cos\phi + \dot{y}\sin\phi + b\dot{\phi} &= r\dot{\theta}r\\ \dot{x}\cos\phi + \dot{y}\sin\phi - b\dot{\phi} &= r\dot{\theta}r \end{aligned}$$

Coordinates (*θr*, *θl*) are related with forward velocity *u* and angular velocity *ω* as

$$
\begin{bmatrix}
\dot{\theta}\_r \\
\dot{\theta}\_l
\end{bmatrix} = \begin{bmatrix}
\frac{1}{r} & \frac{b}{r} \\
\frac{1}{r} & -\frac{b}{r}
\end{bmatrix} \begin{bmatrix} u \\ \omega \end{bmatrix}
$$

Then (*θr*, *θl*) can be eliminated from the equations of motion by using (*u*, *ω*) as generalized velocities. Therefore, kinematic equations (8) are given by

$$
\begin{bmatrix}
\dot{\boldsymbol{x}} \\
\dot{\boldsymbol{y}} \\
\dot{\boldsymbol{\phi}} \\
\dot{\boldsymbol{\theta}}\_{1}
\end{bmatrix} = \begin{bmatrix}
\frac{r}{2}\cos\phi & \frac{r}{2}\cos\phi & 0 \\
\frac{r}{2}\sin\phi & \frac{r}{2}\sin\phi & 0 \\
\frac{r}{2b} & -\frac{r}{2b} & 0 \\
0 & 0 & 1
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\nu}\_{r} \\
\boldsymbol{\nu}\_{l} \\
\boldsymbol{\nu}\_{M1}
\end{bmatrix}.\tag{38}
$$

.

#### **4.2 Simulation for the system with both the kinetic and dynamic uncertainties**

Mobile manipulator used for simulation is shown in Fig.3. In this simulation both the kinematic and dynamic parameters are all unknown.

In this simulation, the mobile base is controlled to track to the desired position/orientation trajectory on the floor and the end-effector of the manipulator is controlled to be constrained on the ceiling with desired reaction force. Then, nonholonomic constraints imposed on this system are same as shown before. On the other hand, holonomic constarint imposed on this system is

$$\Phi(\eta) = L\_1 + L\_2 \sin \theta\_2 + L\_3 \sin(\theta\_2 + \theta\_3) - L = 0.1$$

Fig. 4. Trajectory on (*x*, *y*) plane

This holonomic constraint represents that the end-effector of manipulator is constrained on the ceiling.

Desired trajectory of the manipulator is given by

$$
\theta\_1^\* = 1.2(1 - \cos 0.25t), \; \theta\_2^\* = \frac{\pi}{4}
$$

and desired force trajectory *λ*∗ *<sup>M</sup>* is 10. Desired position trajectories of the base *q*<sup>∗</sup> *<sup>B</sup>* = [*x*<sup>∗</sup> *y*<sup>∗</sup> *φ*∗] *T* are generated by the reference robot given by

$$
\dot{\boldsymbol{q}}^\* = \boldsymbol{f}\_B(\boldsymbol{q}\_B^\*) \boldsymbol{\eta}^\*.\tag{39}
$$

of Mobile Manipulators 17

Robust Adaptive Position/Force Control of Mobile Manipulators 133

Fig.4 shows a desired trajectory and a tracking trajectory. Broken line shows a desired trajectory generated by the reference robot and solid line shows a tracking trajectory of the mobile manipulator. In spite of quite a large initial tracking error, tracking error is converged sufficiently small. Fig.5 and Fig.6 show the trajectory tracking errors of each coordinate and

Fig.7 and 8 show a snapshot of experiments and the end-effector of the mobile manipulator respectively. In order to reduce the adverse effects of friction from the wall rolling ball is mounted on the top of the end-effector and to detect the reaction force from the wall force sensor is equipped under the rolling ball. In experiments mobile manipulator is controlled to move on the straight line parallel to the wall with assigned speed and simultaneously end-effector is controlled to press against the wall with assigned force. Since the end-effector

Fig. 5. Position tracking errors

Fig. 6. Force tracking error

force tracking error, respectively.

**4.3 Experiments**

To obtain the mixed straight and curved line, desired velocities *η*∗ = [*u*∗ *ω*∗] *<sup>T</sup>* are defined as follows.

$$\begin{cases} u^\* = & 0.1 \left(1 - \cos \frac{\pi}{25} t \right) \\ \omega^\* = 0 & (0 \le t < 2.5) \end{cases} \quad \begin{cases} u^\* = & 0.2 \\ \omega^\* = 0 & (2.5 \le t < 5) \end{cases}$$

$$\begin{cases} u^\* = & 0.1 \left(1 + \cos \frac{\pi}{25} t \right) \\ \omega^\* = 0 & (5 \le t < 7.5) \end{cases} \quad \begin{cases} u^\* = & 0.1 \pi \left(1 - \cos \frac{2\pi}{25} t \right) \\ \omega^\* = -u^\* \end{cases}$$

$$\begin{cases} u^\* = & 0.1 \pi \left(1 - \cos \frac{2\pi}{25} t \right) \\ \omega^\* = u^\* \end{cases} \quad \begin{cases} u^\* = & 0.1 \left(1 + \cos \frac{\pi}{25} t \right) \\ \omega^\* = 0 & (12.5 \le t < 15) \end{cases}$$

$$\begin{cases} u^\* = & 0.2 \\ \omega^\* = 0 & (15 \le t) \end{cases}$$

Initial conditions are *q*˙(0) = 0, *q*(0)=[0 0 − *π*/20 0; *π*/4, *π*/3] *<sup>T</sup>* and *p*ˆ1(0) = 0, *a*ˆ*i*(0) = 0, ˆ *θi*(0) = 0, *D*ˆ (0) = 0, *ψ*ˆ(0)=[0.1 0.1] *<sup>T</sup>*. Disturbance vector is *d*1(*q*)=[1 1 0.5 0.5 0.5] *<sup>T</sup>*. For this system design parameters are asigned as *Kd* = 10 × *I*4, Γ<sup>1</sup> = Γ<sup>2</sup> = *γ* = Λ<sup>1</sup> = Λ<sup>2</sup> = 1, *α*<sup>1</sup> = 5 × *I*3, *α*<sup>2</sup> = 25, *γ*<sup>1</sup> = *γ*<sup>2</sup> = 100, *K*<sup>1</sup> = *K*<sup>2</sup> = *K*<sup>3</sup> = *KM*<sup>11</sup> = *KM*<sup>12</sup> = 10 and

$$\rho(t) = \frac{1}{(1 + \frac{t}{10})^2}.$$

Fig. 5. Position tracking errors

16 Will-be-set-by-IN-TECH

This holonomic constraint represents that the end-effector of manipulator is constrained on

<sup>2</sup> <sup>=</sup> *<sup>π</sup>* 4

*<sup>B</sup>*)*η*∗. (39)

*<sup>B</sup>* = [*x*<sup>∗</sup> *y*<sup>∗</sup> *φ*∗]

*<sup>T</sup>* are defined as

*<sup>T</sup>* and *p*ˆ1(0) = 0, *a*ˆ*i*(0) =

*<sup>T</sup>*. For

*T*

*<sup>M</sup>* is 10. Desired position trajectories of the base *q*<sup>∗</sup>

*u*<sup>∗</sup> = 0.2

*u*∗ = 0.1*π*

*u*<sup>∗</sup> = 0.1

*ω*<sup>∗</sup> = 0 (2.5 ≤ *t* < 5)

*ω*<sup>∗</sup> = −*u*<sup>∗</sup> (7.5 ≤ *t* < 10)

*ω*<sup>∗</sup> = 0 (12.5 ≤ *t* < 15)

*<sup>T</sup>*. Disturbance vector is *d*1(*q*)=[1 1 0.5 0.5 0.5]

<sup>1</sup> <sup>−</sup> cos <sup>2</sup>*<sup>π</sup>* 2.5 *t* 

1 + cos *<sup>π</sup>* 2.5 *t* 

<sup>1</sup> = 1.2(1 − cos 0.25*t*), *θ*<sup>∗</sup>

<sup>∗</sup> = *fB*(*q*<sup>∗</sup>

this system design parameters are asigned as *Kd* = 10 × *I*4, Γ<sup>1</sup> = Γ<sup>2</sup> = *γ* = Λ<sup>1</sup> = Λ<sup>2</sup> = 1, *α*<sup>1</sup> =

(1 + *<sup>t</sup>* <sup>10</sup> )<sup>2</sup> .

*<sup>ρ</sup>*(*t*) = <sup>1</sup>

*q*˙

To obtain the mixed straight and curved line, desired velocities *η*∗ = [*u*∗ *ω*∗]

<sup>1</sup> <sup>−</sup> cos *<sup>π</sup>* 2.5 *t* 

1 + cos *<sup>π</sup>* 2.5 *t* 

> <sup>1</sup> <sup>−</sup> cos <sup>2</sup>*<sup>π</sup>* 2.5 *t*

*ω*<sup>∗</sup> = 0 (0 ≤ *t* < 2.5)

*ω*<sup>∗</sup> = 0 (5 ≤ *t* < 7.5)

*ω*<sup>∗</sup> = *u*<sup>∗</sup> (10 ≤ *t* < 12.5)

Initial conditions are *q*˙(0) = 0, *q*(0)=[0 0 − *π*/20 0; *π*/4, *π*/3]

5 × *I*3, *α*<sup>2</sup> = 25, *γ*<sup>1</sup> = *γ*<sup>2</sup> = 100, *K*<sup>1</sup> = *K*<sup>2</sup> = *K*<sup>3</sup> = *KM*<sup>11</sup> = *KM*<sup>12</sup> = 10 and

Fig. 4. Trajectory on (*x*, *y*) plane

and desired force trajectory *λ*∗

Desired trajectory of the manipulator is given by

are generated by the reference robot given by

*u*<sup>∗</sup> = 0.1

*u*<sup>∗</sup> = 0.1

*u*∗ = 0.1*π*

*u*<sup>∗</sup> = 0.2

*θi*(0) = 0, *D*ˆ (0) = 0, *ψ*ˆ(0)=[0.1 0.1]

*ω*<sup>∗</sup> = 0 (15 ≤ *t*)

*θ*∗

the ceiling.

follows.

0, ˆ

Fig. 6. Force tracking error

Fig.4 shows a desired trajectory and a tracking trajectory. Broken line shows a desired trajectory generated by the reference robot and solid line shows a tracking trajectory of the mobile manipulator. In spite of quite a large initial tracking error, tracking error is converged sufficiently small. Fig.5 and Fig.6 show the trajectory tracking errors of each coordinate and force tracking error, respectively.

#### **4.3 Experiments**

Fig.7 and 8 show a snapshot of experiments and the end-effector of the mobile manipulator respectively. In order to reduce the adverse effects of friction from the wall rolling ball is mounted on the top of the end-effector and to detect the reaction force from the wall force sensor is equipped under the rolling ball. In experiments mobile manipulator is controlled to move on the straight line parallel to the wall with assigned speed and simultaneously end-effector is controlled to press against the wall with assigned force. Since the end-effector

of Mobile Manipulators 19

Robust Adaptive Position/Force Control of Mobile Manipulators 135

0 2 4 6 8 10 12 14

time [s]

0 2 4 6 8 10 12 14

times [s]

with constant force 2*N*. Fig.9 shows position error trajectories of the mobile base. Even though the mobile base is declined about 20 degrees parallel to the wall initially, position errors are settled to the neighbourhood of origin. Fig.10 shows force error. Also the force error is settled

In this experiment we assume that not only dynamic parameters but also kinematic parameters are unknown. Then we apply control laws given in Theorem 1. In these control

> , *<sup>a</sup>*<sup>2</sup> <sup>=</sup> *<sup>b</sup> r* ,

, *θp*<sup>2</sup> =

 *r* 2 *r* 2*b*  .

**4.3.2 Application to the system with both the kinetic and dynamic uncertainties**

*<sup>a</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> *r*

> *r* 2 *r* 2*b*

*θp*<sup>1</sup> =

0

–0.2

–0.4

–0.5

force error

x–x\* [ m ]

y–y\* [ m ] –0.3 φ– φ\* [rad]

[rad]

–0.1

0.1

0

0.005

–0.01

–0.015

–0.02

–0.025

20

15

5

fforce [N]

0

–5

–15

–10

10

Fig. 9. Position error trajectories

Fig. 10. Force error trajectory

similarly to the position error trajectories.

laws, unknown parameters are defined as follows.

.005

–0

[m]

### Fig. 7. Snapshot of experiments

is constrained on the wall, following holonomic constraint Φ(*q*) is imposed.

$$\begin{aligned} \Phi(\boldsymbol{\eta}) &= \boldsymbol{y} + d \sin \phi + \{L\_2 \cos \theta\_2 + L\_3 \cos(\theta\_2 + \theta\_3)\}, \\ &\times \sin(\phi + \theta\_1) - L = 0, \end{aligned}$$

where *L* denotes distance between the wall and *P*<sup>0</sup> which is center of wheel axis. Then *θ*<sup>2</sup> is also eliminated by using the holonomic constraint.

Fig. 8. End-effector

#### **4.3.1 Application to the system only with the dynamic uncertainties**

In case when kinetic parameters are known, we can apply control laws given in Theorem 2 and we can use estimates of kinematic parameters shown in Subsection 4.1. Control parameters are give as follows. *Kd* = 5, *K*<sup>1</sup> = 5, *K*<sup>2</sup> = 50, *K*<sup>3</sup> = 10, *KM*<sup>1</sup> = [10 10], *α*<sup>1</sup> = 5, *α*<sup>2</sup> = 5, Γ = 0.2, *γ* = 1. *ρ*(*t*) is given as

$$\rho(t) = \frac{1}{(t/K\_{\rho} + 1)^2},$$

where *K<sup>ρ</sup>* is constant and is assigned 500 in this experiments.

Experimental situation is shown in Fig.7. Mobile manipulator is controlled to move on the straight line with constant speed 5*cm*/*sec*. End-effector is controlled to press against the wall

Fig. 9. Position error trajectories

18 Will-be-set-by-IN-TECH

is constrained on the wall, following holonomic constraint Φ(*q*) is imposed.

× sin(*φ* + *θ*1) − *L* = 0,

**4.3.1 Application to the system only with the dynamic uncertainties**

where *K<sup>ρ</sup>* is constant and is assigned 500 in this experiments.

Φ(*q*) = *y* + *d* sin *φ* + {*L*<sup>2</sup> cos *θ*<sup>2</sup> + *L*<sup>3</sup> cos(*θ*<sup>2</sup> + *θ*3)}

where *L* denotes distance between the wall and *P*<sup>0</sup> which is center of wheel axis. Then *θ*<sup>2</sup> is

1.5cm

In case when kinetic parameters are known, we can apply control laws given in Theorem 2 and we can use estimates of kinematic parameters shown in Subsection 4.1. Control parameters are give as follows. *Kd* = 5, *K*<sup>1</sup> = 5, *K*<sup>2</sup> = 50, *K*<sup>3</sup> = 10, *KM*<sup>1</sup> = [10 10], *α*<sup>1</sup> = 5, *α*<sup>2</sup> = 5, Γ =

Experimental situation is shown in Fig.7. Mobile manipulator is controlled to move on the straight line with constant speed 5*cm*/*sec*. End-effector is controlled to press against the wall

(*t*/*K<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)<sup>2</sup> ,

*<sup>ρ</sup>*(*t*) = <sup>1</sup>

Force

sensor 2.8cm

Fig. 7. Snapshot of experiments

Fig. 8. End-effector

0.2, *γ* = 1. *ρ*(*t*) is given as

also eliminated by using the holonomic constraint.

Fig. 10. Force error trajectory

with constant force 2*N*. Fig.9 shows position error trajectories of the mobile base. Even though the mobile base is declined about 20 degrees parallel to the wall initially, position errors are settled to the neighbourhood of origin. Fig.10 shows force error. Also the force error is settled similarly to the position error trajectories.

### **4.3.2 Application to the system with both the kinetic and dynamic uncertainties**

In this experiment we assume that not only dynamic parameters but also kinematic parameters are unknown. Then we apply control laws given in Theorem 1. In these control laws, unknown parameters are defined as follows.

$$\begin{aligned} a\_1 &= \frac{1}{r}, a\_2 = \frac{b}{r}, \\ \theta\_{p1} &= \begin{bmatrix} \frac{r}{2} \\ \frac{r}{2b} \end{bmatrix}, \theta\_{p2} = \begin{bmatrix} \frac{r}{2} \\ \frac{r}{2b} \end{bmatrix}. \end{aligned}$$

of Mobile Manipulators 21

Robust Adaptive Position/Force Control of Mobile Manipulators 137

–2 0 5 10 15 20

force error

time [s]

As similar in case when kinetic parameters are known, mobile manipulator is controlled to move on the straight line with constant speed 5*cm*/*sec*. End-effector is controlled to presss against the wall with constant force 2*N*. Fig.11 shows position error trajectories of the mobile base. Similarly to the previous experiment, even though the mobile base is declined about 20 degrees to the wall initially, position errors are settled to the neighbourhood of origin. Fig.12 shows force error. Also the force error is settled similarly to the position error trajectories.

In this study robust adaptive hybrid position/force control problems have been investigated. Proposed control schemes can be applied to the system which has not only dynamic uncertainties but also both the kinematic and dynamic uncertainties. Furthermore unknown disturbances have been considered. It is guaranteed theoretically that the tracking position errors and force errors are asymptotically converged to zero and all internal signals are bounded. This means that all estimated parameters are also bounded and still remained to be small. However, some estimated parameters are monotonically increasing, especially *D*ˆ .

˙ *<sup>D</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>γ</sup><sup>δ</sup>*, *<sup>D</sup>*<sup>ˆ</sup> is increased in so far as *<sup>δ</sup>* <sup>=</sup> 0. Furthermore in this experiments sensor noise and unmodeled nonlinearities hinder the proposed control schemes from achievement of the perfect regulation *δ* = 0. These lead us to the fact that the estimate *D*ˆ becomes large with the passage of time. Therefore, in the practical situation to avoid the numerical difficulties of *D*ˆ resulted from the long-term control, *D*ˆ should be set constant value when the estimate *D*ˆ exceeds the designated threshold or *δ* should be set 0 in the computation of the adaptive

Usefulness of the proposed control schemes has been demonstrated by experiments. Especially, since environmental uncertainties can be considered as the kinematic uncertainties, the proposed control scheme given by Theorem 1 can be applied to the case when

8 10

–8 –6 –4

–10

laws if *δ* < *�*, where *�* is specified small number.

environmental uncertainties arise.

Fig. 12. Force error trajectory

**5. Concluding remark**

Since *D*ˆ is updated by

[N]

Furthermore, from Property 6

$$\begin{aligned} \tilde{I}\_M^I(q)\lambda\_M &= Z\_1(q,\lambda\_M)\psi\\ &= \begin{bmatrix} \lambda\_M \sin\phi & \lambda\_M \cos\phi & \lambda\_M \cos(\phi+\theta\_1)\cos\theta\_2\\ \lambda\_M \sin\phi & -\lambda\_M \cos\phi & -\lambda\_M \cos(\phi+\theta\_1)\cos\theta\_2\\ 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \\ \lambda\_M \cos(\phi+\theta\_1)\cos(\theta\_2+\theta\_3) &= 0\\ -\lambda\_M \cos(\phi+\theta\_1)\cos(\theta\_2+\theta\_3) &= 0\\ 0 & \lambda\_M \cos(\phi+\theta\_1)\cos\theta\_2\\ 0 & -\lambda\_M \sin(\phi+\theta\_1)\sin\theta\_2\\ 0 & 0 & 0\\ 0 & 0 & \lambda\_M\\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \psi\_1\\ \psi\_2\\ \psi\_3\\ \psi\_4\\ \psi\_5\\ \psi\_6 \end{bmatrix} \\ \text{and} \\ -\lambda\_M \sin(\phi+\theta\_1)\sin(\theta\_2+\theta\_3) &= 0 \end{aligned}$$

then we have

$$\begin{aligned} \psi\_1 &= \frac{1}{2}r\_\prime \psi\_2 = \frac{1}{2}\frac{rd}{b\prime}\psi\_3 = \frac{1}{2}\frac{L\_2d}{b\prime},\\ \psi\_4 &= \frac{1}{2}\frac{L\_3d}{b\prime}\psi\_5 = L\_2\prime \psi\_6 = L\_3. \end{aligned}$$

Therefore control scheme given in Theorem 1 is overparameterized scheme. Control

Fig. 11. Position error trajectories

parameters are as follows.*Kd* = 5, *K*<sup>1</sup> = 5, *K*<sup>2</sup> = 50, *K*<sup>3</sup> = 10, *KM*<sup>1</sup> = [10 10], *α*<sup>1</sup> = 5, *α*<sup>2</sup> = 5, Γ<sup>1</sup> = 0.01, Γ<sup>2</sup> = 0.01, *γ* = 0.01, *γ*<sup>1</sup> = 10, *γ*<sup>2</sup> = 10, Λ<sup>1</sup> = 1, Λ<sup>2</sup> = 1. *ρ*(*t*) is same as that given in case when kinetic parameters are known.

Fig. 12. Force error trajectory

20 Will-be-set-by-IN-TECH

*λ<sup>M</sup>* sin *φ λ<sup>M</sup>* cos *φ λ<sup>M</sup>* cos(*φ* + *θ*1) cos *θ*<sup>2</sup> *λ<sup>M</sup>* sin *φ* −*λ<sup>M</sup>* cos *φ* −*λ<sup>M</sup>* cos(*φ* + *θ*1) cos *θ*<sup>2</sup> 00 0 00 0 00 0

0 *λ<sup>M</sup>* cos(*φ* + *θ*1) cos *θ*<sup>2</sup> 0 −*λ<sup>M</sup>* sin(*φ* + *θ*1) sin *θ*<sup>2</sup>

0 0 0 −*λ<sup>M</sup>* sin(*φ* + *θ*1) sin(*θ*<sup>2</sup> + *θ*3) −*λ<sup>M</sup>* sin(*φ* + *θ*1) sin(*θ*<sup>2</sup> + *θ*3)

*<sup>b</sup>* , *<sup>ψ</sup>*<sup>5</sup> <sup>=</sup> *<sup>L</sup>*2, *<sup>ψ</sup>*<sup>6</sup> <sup>=</sup> *<sup>L</sup>*3.

Therefore control scheme given in Theorem 1 is overparameterized scheme. Control

0.05 0.02

0 5 10 15 20

–0.14 –0.4 time [s]

parameters are as follows.*Kd* = 5, *K*<sup>1</sup> = 5, *K*<sup>2</sup> = 50, *K*<sup>3</sup> = 10, *KM*<sup>1</sup> = [10 10], *α*<sup>1</sup> = 5, *α*<sup>2</sup> = 5, Γ<sup>1</sup> = 0.01, Γ<sup>2</sup> = 0.01, *γ* = 0.01, *γ*<sup>1</sup> = 10, *γ*<sup>2</sup> = 10, Λ<sup>1</sup> = 1, Λ<sup>2</sup> = 1. *ρ*(*t*) is same as that given

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*ψ*1 *ψ*2 *ψ*3 *ψ*4 *ψ*5 *ψ*6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

–0.15 –0.1 –0. 05 0

x – x\* [m] y – y\* [m] φ – φ\* [m] –0.35 –0.3 –0.25 –0.2

rad]

[

0 0

Furthermore, from Property 6

then we have

*J T*

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

*<sup>M</sup>*(*q*)*λ<sup>M</sup>* = *Z*1(*q*, *λM*)*ψ*

*<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> 2

–0.04 –0.02 0

–0.06

–0.12 –0.1 –0.08

[m]

Fig. 11. Position error trajectories

in case when kinetic parameters are known.

*<sup>ψ</sup>*<sup>4</sup> <sup>=</sup> <sup>1</sup> 2 *L*3*d*

*<sup>r</sup>*, *<sup>ψ</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> 2 *rd <sup>b</sup>* , *<sup>ψ</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> 2 *L*2*d b* ,

*λ<sup>M</sup>* cos(*φ* + *θ*1) cos(*θ*<sup>2</sup> + *θ*3) 0 −*λ<sup>M</sup>* cos(*φ* + *θ*1) cos(*θ*<sup>2</sup> + *θ*3) 0

> As similar in case when kinetic parameters are known, mobile manipulator is controlled to move on the straight line with constant speed 5*cm*/*sec*. End-effector is controlled to presss against the wall with constant force 2*N*. Fig.11 shows position error trajectories of the mobile base. Similarly to the previous experiment, even though the mobile base is declined about 20 degrees to the wall initially, position errors are settled to the neighbourhood of origin. Fig.12 shows force error. Also the force error is settled similarly to the position error trajectories.

### **5. Concluding remark**

In this study robust adaptive hybrid position/force control problems have been investigated. Proposed control schemes can be applied to the system which has not only dynamic uncertainties but also both the kinematic and dynamic uncertainties. Furthermore unknown disturbances have been considered. It is guaranteed theoretically that the tracking position errors and force errors are asymptotically converged to zero and all internal signals are bounded. This means that all estimated parameters are also bounded and still remained to be small. However, some estimated parameters are monotonically increasing, especially *D*ˆ . Since *D*ˆ is updated by

$$
\dot{D} = \gamma ||\delta||\_{\prime}
$$

*<sup>D</sup>*<sup>ˆ</sup> is increased in so far as *<sup>δ</sup>* <sup>=</sup> 0. Furthermore in this experiments sensor noise and unmodeled nonlinearities hinder the proposed control schemes from achievement of the perfect regulation *δ* = 0. These lead us to the fact that the estimate *D*ˆ becomes large with the passage of time. Therefore, in the practical situation to avoid the numerical difficulties of *D*ˆ resulted from the long-term control, *D*ˆ should be set constant value when the estimate *D*ˆ exceeds the designated threshold or *δ* should be set 0 in the computation of the adaptive laws if *δ* < *�*, where *�* is specified small number.

Usefulness of the proposed control schemes has been demonstrated by experiments. Especially, since environmental uncertainties can be considered as the kinematic uncertainties, the proposed control scheme given by Theorem 1 can be applied to the case when environmental uncertainties arise.

**0**

**8**

*Japan*

*Okayama University*

**Positioning Control of One Link Arm with**

Takayuki Kuwashima, Jun Imai and Masami Konishi

**Parametric Uncertainty Using the QFT Method**

Many manufacturing robots are currently operated in various factories, with the aim of saving labor and cost. In particular, automatic sorting robots for products in assortment lines are used to improve productivity. In case of having the robots operate in such ways, the manipulators are demanded to have positioning performance of high precision to the target position. But in condition that products of various weights are handled, control performance such as positioning precision and settling time would be deteriorated if the controller is designed based on a fixed mathematical model; and they include parameters such as weights of grabbing product with their robot hands. So the controller needs to have robust control performance against parameter uncertainty in plant dynamics. It means that a control system should maintain its performance even if there exists uncertainty. In decades, control problems under plant uncertainty have been much studied because conventional control theory is based on the assumption that the dynamics of plant is fully well-known and certain, and that the mathematical model accurately reflects behavior of controlled plant in the real world. Nowadays, robust control theory is developed in order to cope with such problems. *H*∞ control theory, among robust control theories, has been applied to many control systems and can handle unstructured uncertainty of plant, but it can not directly deal with structured

In this chapter, a robust controller design using Quantitative Feedback Theory(QFT) (Houpis et al., 2006) is presented for one link arm with parametric uncertainty. QFT is a robust control theory developed based mainly on classical control. QFT can cope with parametric uncertainty in a plant(Khodabakhshian & Golbon, 2005; Barve & Nataraj, 1998; Zolotas & Halikias, 1999; Ryoo et al., 2002). In design procedure, the region where the plant can exists in accordance with parameter variation is illustrated on Nichols chart for each frequency, which is called template. By using templates, the controller is designed in order to

Let us consider the one link arm as shown in Fig.1. It simulates the sorting robot in manufacturing facility. It consists of a Direct-Drive (DD) motor as an actuator, a rigid arm and a payload mass at free end of the arm, simulating a carried product. The arm is driven by the DD motor and rotates in the horizontal plane. The DD motor is actuated by input torque *τ*[Nm]. The specification of the motor is as shown in Table. 1 and the arm is made of stainless

**1. Introduction**

uncertainty such as parameter variation.

**2. Model of one link arm**

satisfy performance specification for all possible plants.

### **6. Acknowledgements**

This study was supported in part by both the scientific Research Fund (B)19360110 and Hitech Research Center, Project for Private University from the Ministry of Education, Culture, Sports, Science and Technology.

### **7. References**


## **Positioning Control of One Link Arm with Parametric Uncertainty Using the QFT Method**

Takayuki Kuwashima, Jun Imai and Masami Konishi *Okayama University Japan*

### **1. Introduction**

22 Will-be-set-by-IN-TECH

138 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

This study was supported in part by both the scientific Research Fund (B)19360110 and Hitech Research Center, Project for Private University from the Ministry of Education, Culture,

Chang, Y. C. & Chen, B. S. (2002). Adaptive tracking control for nonholonomic Caplygin systems. *IEEE Trans. Control Systems Technology*, Vol. 10, No. 1, pp. 96-104 Cheah, C. C.;Kawamura, S. & Arimoto, S. (2003). Stability of Hybrid Position and Force

Dong, W. (2002). On trajectory and force tracking control of constrained mobile manipulators

Fukao, T.;Hasegawa, H. & Adachi, H. (2000). Adaptive Tracking Control of a Nonholonomic Mobile Robot. *IEEE Trans. Robotics and Automation*, Vol. 16, No. 5, pp. 609-615 Kristic, M.;Kanellakopoulos, I. & Kokotovic, P. (1995). *Nonlinear and Adaptive Control Design*,

Li, Z.;Yang, J.;Luo, J.;Wang, Z. & Ming, A. (2007). Robust motion/force control of

Li, Z.;Adams, M. & Wijesoma, W. S. (2008). Robust Adaptive control of uncertain force/motion constrained nonholonomic mobile manipulators. *Automatica*, Vol. 44, pp. 776-784 Murray, R. M.; Li, Z. & Sastry, S. S. (1993). *An Introduction to Robotic Manipulation*, CRC Press Narikiyo, T.; Kawanishi, M. & Nakagawa, M. (2008). Robust Adaptive Position/Force Control

Narikiyo, T.; Kawanishi, M. & Mizuno, T. (2009). Robust Adaptive Position/Force Control

Su, C. Y.;Stepanenko, Y. & Goldenberg, A. A. (1999). Reduced Order Model and

Yamamoto, Y. & Yun, X. (1996). Effect of the Dynamic Interaction on Coordinated Control of

Yuan, J. (1997). Adaptive Control of a Constrained Robot-Ensuring Zero Tracking and Zero Force Errors. *IEEE Trans. Automatic Control*, Vol. 42, No. 12, pp. 1709-1714

*ICROS-SICE International Joint Conference*, pp. 4704–4709, Fukuoka, Japan Oya, M.;Su, C. Y.& Katoh, R. (2003). Robust Adaptive Motion/Force Tracking Control of

nonholonomic mobile manipulators using hybrid joints. *Advanced Robotics*, Vol. 21,

of Mobile Manipulators with Dynamic Uncertainties, *Proceedings of IASTED CA2009*,

of Mobile Manipulators with Kinematic and Dynamic Uncertainties, *Proceedings of*

Uncertain Nonholonomic Mechanical Systems. *IEEE Trans. Robotics and Automation*,

Robust Control Architecture for Mechanical Systems with Nonholonomic Pfaffian Constraints. *IEEE Trans. Systems, Man, and Cybernetics-Part A*, Vol. 29, No. 3, pp.

Mobile Manipulators. *IEEE Trans. Robotics and Automation*, Vol. 12, No. 5, pp.816-824

with parameter uncertainty. *Automatica*, Vol. 32, pp. 1475-1484

Control for Robotic Manipulator with Kinamatic and Dynamics Uncertainties.

**6. Acknowledgements**

**7. References**

Sports, Science and Technology.

*Automatica*, Vol.39, pp.847-855

John Wiley and Sons

No. 11, pp. 1231-1252

pp. 264-269, ISBN, Cambridge, UK

Slotine, J-J. E. & Li, W. (1991). *Applied Nonlinear Control*, Prentice Hall

Vol. 19, No. 1, pp. 175-181

307-313

Many manufacturing robots are currently operated in various factories, with the aim of saving labor and cost. In particular, automatic sorting robots for products in assortment lines are used to improve productivity. In case of having the robots operate in such ways, the manipulators are demanded to have positioning performance of high precision to the target position. But in condition that products of various weights are handled, control performance such as positioning precision and settling time would be deteriorated if the controller is designed based on a fixed mathematical model; and they include parameters such as weights of grabbing product with their robot hands. So the controller needs to have robust control performance against parameter uncertainty in plant dynamics. It means that a control system should maintain its performance even if there exists uncertainty. In decades, control problems under plant uncertainty have been much studied because conventional control theory is based on the assumption that the dynamics of plant is fully well-known and certain, and that the mathematical model accurately reflects behavior of controlled plant in the real world. Nowadays, robust control theory is developed in order to cope with such problems. *H*∞ control theory, among robust control theories, has been applied to many control systems and can handle unstructured uncertainty of plant, but it can not directly deal with structured uncertainty such as parameter variation.

In this chapter, a robust controller design using Quantitative Feedback Theory(QFT) (Houpis et al., 2006) is presented for one link arm with parametric uncertainty. QFT is a robust control theory developed based mainly on classical control. QFT can cope with parametric uncertainty in a plant(Khodabakhshian & Golbon, 2005; Barve & Nataraj, 1998; Zolotas & Halikias, 1999; Ryoo et al., 2002). In design procedure, the region where the plant can exists in accordance with parameter variation is illustrated on Nichols chart for each frequency, which is called template. By using templates, the controller is designed in order to satisfy performance specification for all possible plants.

### **2. Model of one link arm**

Let us consider the one link arm as shown in Fig.1. It simulates the sorting robot in manufacturing facility. It consists of a Direct-Drive (DD) motor as an actuator, a rigid arm and a payload mass at free end of the arm, simulating a carried product. The arm is driven by the DD motor and rotates in the horizontal plane. The DD motor is actuated by input torque *τ*[Nm]. The specification of the motor is as shown in Table. 1 and the arm is made of stainless

Fig. 1. One link arm


Table 1. Specification of the DD motor

steel with length of L=0.3m, line density *ρ*=0.79 Kg/m and cross-section 1cm2. Now, let *θ*(*t*) be the angle of rotation from the initial position, an equation of motion describing this system is

$$J\ddot{\theta} + D\dot{\theta} = \pi. \tag{1}$$

**3. Effect of parameter variation**

We employ average values

Designed controller is

in parameters of *P*¯(*s*).

as boundary.

Fig. 2. Feedback Control System

**4. Control system design using QFT**

QFT design procedure is as follows:

**STEP5.** Design the filter *F*(*s*).

Target angle

In this section, the effect of plat parameter variation on control performance of the one link arm system is studied. As described in Section 2, values of *J* and *D* vary in a certain range.

Positioning Control of One Link Arm with Parametric Uncertainty Using the QFT Method 141

for each parameter as the nominal parameter. Let *P*¯(*s*) donates the nominal plant. A PD controller has been designed for *P*¯(*s*) in order to satisfy following performance specifications:

Numerical experiment has been done on the control system shown in Fig. 2. The target rotary angle *θr*(*t*) is aimed at 1 rad and computational interval is 0.005s. Simulation result to control *P*¯(*s*) is shown in Fig. 3. On the other hand, *J* and *D* can take various values. Control results for P are shown in Fig. 4, and it is found that the responses of output *θ*(*t*) are scattered and some responses violate performance specifications. From these results, it can be seen that the controller designed for *P*¯(*s*) does not always ensure the desired performance with variation

C(s)

θ<sup>r</sup> θ

In control systems, it is desirable for controlled output of plant *θ*(*t*) to satisfy performance specification in spite of parameter uncertainty in a plant. In QFT design procedure, designer can determine the performance specifications that response of *θ*(*t*) should lie in. It has certain width as tolerance in time and frequency domains. Controller is designed on Nichols Chart (NC) so that all *θ*(*t*) lie between these specifications. These specifications are described on NC

Consider feedback control system shown in Fig.5. Control objective is to control angular position of arm. We design *C*(*s*) and *F*(*s*) in order to satisfy performance specifications.

**STEP1.** Determine performance specification in time and frequency domains. **STEP2.** Gain the area where varying plant can exist on NC, called template.

**STEP4.** Form the open loop transfer function *C*(*s*)*P*(*s*) to satisfy bounds.

**STEP3.** Construct performance specification on NC as bounds.

τ Input torque

0.5 1 + 0.001*s*

Controller Plant Output

P

*C*(*s*) = 3.0 +

¯*<sup>J</sup>* <sup>=</sup> 0.233kg · <sup>m</sup>2, *<sup>D</sup>*¯ <sup>=</sup> 0.644N · <sup>m</sup> · <sup>s</sup> (6)

Overshoot : *Os* < 10% (7) Peak time : *tp* < 1.0sec. (8)

. (9)

The Laplace transform of Eq.(1) gives the transfer function from *τ* to *θ* as

$$P(s) = \frac{1}{Js^2 + Ds}.\tag{2}$$

Here, *J* donates rotating inertia of the whole arm system, and is represented as

$$J = J\_0 + \frac{\rho L^3}{3} + ML^3. \tag{3}$$

*D* is the friction coefficient around the motor rotary axis. From Eq.(3), it is found that value of *J* varies depending on weight of product. This time, we considered 6 samples of products, and their weights are from 0.5 through 3.0 at 0.5kg intervals. On the other hand, *D* varies with time. So observation experiment of *D* using step response method has been performed for 200 times. The variation ranges for *J* and *D* are as

$$J \in \left( 0.098, \, 0.368 \right), \, D \in \left( 0.475, \, 0.814 \right). \tag{4}$$

Then, we consider 7 samples of *J* and 10 samples of *D* as parameters of *P*(*s*), and let P donates the set of the plant transfer functions, that is

$$\mathcal{P} = \{P\_l\} \quad \text{where } l = 1, 2, \ldots, 70. \tag{5}$$

### **3. Effect of parameter variation**

2 Will-be-set-by-IN-TECH

Rigid arm

Rated voltage[V] 200 Rated torque[N·m] 25 Rated current[A] 2.7 Rated velocity[min−1] 150 Inertia moment *<sup>J</sup>*0[kg·m2] 0.0909

steel with length of L=0.3m, line density *ρ*=0.79 Kg/m and cross-section 1cm2. Now, let *θ*(*t*) be the angle of rotation from the initial position, an equation of motion describing this system

> *J* ¨ *θ* + *D* ˙

Here, *J* donates rotating inertia of the whole arm system, and is represented as

*<sup>J</sup>* <sup>=</sup> *<sup>J</sup>*<sup>0</sup> <sup>+</sup> *<sup>ρ</sup>L*<sup>3</sup>

*D* is the friction coefficient around the motor rotary axis. From Eq.(3), it is found that value of *J* varies depending on weight of product. This time, we considered 6 samples of products, and their weights are from 0.5 through 3.0 at 0.5kg intervals. On the other hand, *D* varies with time. So observation experiment of *D* using step response method has been performed for 200

Then, we consider 7 samples of *J* and 10 samples of *D* as parameters of *P*(*s*), and let P donates

*<sup>P</sup>*(*s*) = <sup>1</sup>

The Laplace transform of Eq.(1) gives the transfer function from *τ* to *θ* as

θ Load

*θ* = *τ*. (1)

*Js*<sup>2</sup> <sup>+</sup> *Ds*. (2)

<sup>3</sup> <sup>+</sup> *ML*3. (3)

*J* ∈ (0.098, 0.368), *D* ∈ (0.475, 0.814). (4)

P = {*Pl*} where *l* = 1, 2, ..., 70. (5)

DD motor

τ

Fig. 1. One link arm

is

Table 1. Specification of the DD motor

times. The variation ranges for *J* and *D* are as

the set of the plant transfer functions, that is

In this section, the effect of plat parameter variation on control performance of the one link arm system is studied. As described in Section 2, values of *J* and *D* vary in a certain range. We employ average values

$$
\vec{f} = 0.233 \text{kg} \cdot \text{m}^2, \quad \vec{D} = 0.644 \text{N} \cdot \text{m} \cdot \text{s} \tag{6}
$$

for each parameter as the nominal parameter. Let *P*¯(*s*) donates the nominal plant. A PD controller has been designed for *P*¯(*s*) in order to satisfy following performance specifications:

Overshoot : *Os* < 10% (7)

$$\text{Peak time}: \ t\_p < 1.0 \text{sec.}\tag{8}$$

Designed controller is

$$C(s) = 3.0 + \frac{0.5}{1 + 0.001s}.\tag{9}$$

Numerical experiment has been done on the control system shown in Fig. 2. The target rotary angle *θr*(*t*) is aimed at 1 rad and computational interval is 0.005s. Simulation result to control *P*¯(*s*) is shown in Fig. 3. On the other hand, *J* and *D* can take various values. Control results for P are shown in Fig. 4, and it is found that the responses of output *θ*(*t*) are scattered and some responses violate performance specifications. From these results, it can be seen that the controller designed for *P*¯(*s*) does not always ensure the desired performance with variation in parameters of *P*¯(*s*).

Fig. 2. Feedback Control System

### **4. Control system design using QFT**

In control systems, it is desirable for controlled output of plant *θ*(*t*) to satisfy performance specification in spite of parameter uncertainty in a plant. In QFT design procedure, designer can determine the performance specifications that response of *θ*(*t*) should lie in. It has certain width as tolerance in time and frequency domains. Controller is designed on Nichols Chart (NC) so that all *θ*(*t*) lie between these specifications. These specifications are described on NC as boundary.

Consider feedback control system shown in Fig.5. Control objective is to control angular position of arm. We design *C*(*s*) and *F*(*s*) in order to satisfy performance specifications. QFT design procedure is as follows:

**STEP1.** Determine performance specification in time and frequency domains.

**STEP2.** Gain the area where varying plant can exist on NC, called template.

**STEP3.** Construct performance specification on NC as bounds.

**STEP4.** Form the open loop transfer function *C*(*s*)*P*(*s*) to satisfy bounds.

**STEP5.** Design the filter *F*(*s*).

F(s) C(s)

θr Target angle

Fig. 5. QFT Control System

[rad]

0

Fig. 6. Time domain performance specification

0.2

0.4

Rotary angle of arm

**4.2 Template**

each frequency.

0.6

0.8

1

1.2

Filter Controller Plant Output

Positioning Control of One Link Arm with Parametric Uncertainty Using the QFT Method 143

<sup>±</sup> 2% 10%

0 1 2 3 4 5 6

< 2.0[sec] <sup>s</sup> t

*<sup>s</sup>*<sup>2</sup> <sup>+</sup> 4.6148*<sup>s</sup>* <sup>+</sup> 15.2 , (12)

(*<sup>s</sup>* <sup>+</sup> 3.3)(*<sup>s</sup>* <sup>+</sup> <sup>4</sup>)<sup>2</sup> . (13)

Time [sec]

In QFT, more concrete specification is constructed so as to have certain width by determining upper and lower bounds according to Eqs.(10) and (11). Transfer function *TRU* and *TRL* that

*TRU* (*s*) = 0.5(*<sup>s</sup>* <sup>+</sup> <sup>30</sup>)

One of the features of QFT is to describe the plant as a contour. On NC, a plant is expressed as a point at certain frequency. If parameters of the plant model vary, the plant is expressed on NC as contour for each frequency. It is referred to as a *template*. Therefore the template represents the region where the plant can exist. So the designer can recognize uncertainty in plant visually from templates. Fig.8 shows templates of *P*(*s*) with variations of *J* and *D* at

*TRL* (*s*) = 52.8

The time and frequency responses of these functions are shown in Figs.6 and 7.

represents upper and lower bounds respectively, is designed as follows:

τ Input torque P

θ

Fig. 3. Simulation result for *P*¯(*s*)

Fig. 4. Simulation result for P(*s*)

### **4.1 Performance specification**

Here, performance specifications in time domain is determined as

$$\text{Overshoot}: \ O\_s = 0 \sim 10\% \,\tag{10}$$

$$\text{Setting time}: \ t\_s < 2.0 \text{sec.} \tag{11}$$

Fig. 5. QFT Control System

4 Will-be-set-by-IN-TECH

Performance Specification

Performance Specification

 0 0.5 1 1.5 2 2.5 3 3.5 4 time[s]

 0 0.5 1 1.5 2 2.5 3 3.5 4 time[s]

> Overshoot : *Os* = 0 ∼ 10%, (10) Settling time : *ts* < 2.0sec. (11)

<1.0[sec] <sup>P</sup>t

Here, performance specifications in time domain is determined as

<1.0[sec] <sup>P</sup>t

0

Fig. 3. Simulation result for *P*¯(*s*)

0

Fig. 4. Simulation result for P(*s*) **4.1 Performance specification**

0.2

0.4

0.6

Rotary angle of arm [rad]

0.8

1

1.2

1.4

0.2

0.4

0.6

Rotary angle of arm [rad]

0.8

1

1.2

O<sup>S</sup> <10%

O<sup>S</sup> <10%

1.4

Fig. 6. Time domain performance specification

In QFT, more concrete specification is constructed so as to have certain width by determining upper and lower bounds according to Eqs.(10) and (11). Transfer function *TRU* and *TRL* that represents upper and lower bounds respectively, is designed as follows:

$$T\_{R\_{\rm II}}(s) = \frac{0.5(s+30)}{s^2 + 4.6148s + 15.2} \,\text{/}\tag{12}$$

$$T\_{\mathbb{R}\_L}(s) = \frac{52.8}{(s+3.3)(s+4)^2}.\tag{13}$$

The time and frequency responses of these functions are shown in Figs.6 and 7.

#### **4.2 Template**

One of the features of QFT is to describe the plant as a contour. On NC, a plant is expressed as a point at certain frequency. If parameters of the plant model vary, the plant is expressed on NC as contour for each frequency. It is referred to as a *template*. Therefore the template represents the region where the plant can exist. So the designer can recognize uncertainty in plant visually from templates. Fig.8 shows templates of *P*(*s*) with variations of *J* and *D* at each frequency.


variation of templates, *V*(dB) take the values

Then, U-contour has form as shown in Fig.9.

Fig. 9. U-contour on NC

**4.3 Bound on NC**

**4.4 Loop shaping**


Positioning Control of One Link Arm with Parametric Uncertainty Using the QFT Method 145

Let us determine performance specification on NC. As one of the features of NC, we can obtain a peak gain of a closed loop system by plotting gain-phase curve of open loop transfer function on NC. This peak gain is given by constant M curve to which gain-phase curve is tangent, called M-contour. Therefore, if peak gain *Mp*(dB) of performance specification in frequency domain is given, M-contour representing *Mp* can be obtained on NC, which indicate performance specification about the peak gain. So this specification can be satisfied unless gain-phase curve of open loop transfer function *C*(*s*)*P*(*s*) does not enter this M-contour. However, gain-phase curve of *C*(*s*)*P*(*s*) is represented as certain region if plant model *P*(*s*) has parametric uncertainty, as stated in former section. So consider that we take down lower half of M-contour by *V*(dB), which is maximum variation of templates in bandwidth frequency, to make gain margin. The region constructed by M-contour and gain margin is called *U-contour*. If gain-phase curve of *C*(*s*)*P*(*s*) does not enter U-contour, it means that

performance specification about the peak gain is satisfied for all varying plant P.

From upper specification in Eq.(12), the main object of controller design is written as

 

 

*CP* 1 + *CP*

This time, peak gain of frequency domain performance specification, *Mp*, and maximum

*Mp* = 1.05dB, *V* = 11.49dB. (14)

< *Mp* = 1.05dB. (15)

Phase angle [deg]

M-Contour ( <sup>=</sup>1.05[dB]) <sup>M</sup> <sup>p</sup>

V =11.49[dB]

U-contour




0

Magnitude [dB]

10

20

30

40

Fig. 7. Frequency domain performance specification

Fig. 8. Plant's template on NC

Fig. 9. U-contour on NC

#### **4.3 Bound on NC**

6 Will-be-set-by-IN-TECH

 0.1 1 10 100 Frequancy[rad/s]

ω =1[rad]


ω =10[rad]

RU T

ω = 0.1[rad]

RL T



Fig. 8. Plant's template on NC


ω =1000[rad]

ω =100[rad]




Magnitude [dB]


0

20

40

Fig. 7. Frequency domain performance specification




Magnitude[dB]


0

20

Let us determine performance specification on NC. As one of the features of NC, we can obtain a peak gain of a closed loop system by plotting gain-phase curve of open loop transfer function on NC. This peak gain is given by constant M curve to which gain-phase curve is tangent, called M-contour. Therefore, if peak gain *Mp*(dB) of performance specification in frequency domain is given, M-contour representing *Mp* can be obtained on NC, which indicate performance specification about the peak gain. So this specification can be satisfied unless gain-phase curve of open loop transfer function *C*(*s*)*P*(*s*) does not enter this M-contour. However, gain-phase curve of *C*(*s*)*P*(*s*) is represented as certain region if plant model *P*(*s*) has parametric uncertainty, as stated in former section. So consider that we take down lower half of M-contour by *V*(dB), which is maximum variation of templates in bandwidth frequency, to make gain margin. The region constructed by M-contour and gain margin is called *U-contour*. If gain-phase curve of *C*(*s*)*P*(*s*) does not enter U-contour, it means that performance specification about the peak gain is satisfied for all varying plant P.

This time, peak gain of frequency domain performance specification, *Mp*, and maximum variation of templates, *V*(dB) take the values

$$M\_p = 1.05 \text{dB}, V = 11.49 \text{dB}. \tag{14}$$

Then, U-contour has form as shown in Fig.9.

#### **4.4 Loop shaping**

From upper specification in Eq.(12), the main object of controller design is written as

$$\left|\frac{CP}{1+CP}\right| < M\_p = 1.05 \text{dB}.\tag{15}$$

**5. Simulation**

the following PID controller

0

Fig. 11. Simulation result with QFT

Table 2. Standard deviation of *ts* and *Os*

0.2

0 0.2 0.4 0.6

0.4

Rotary angle of arm

[rad]

0.6

0.8

1

1.2

In this section, based on numerical simulation, control performance of QFT control system designed in former section are compared with conventional PID control system. We consider 70 samples of set of *J* and *D* in plant dynamics. As a conventional controller to be compared,

Positioning Control of One Link Arm with Parametric Uncertainty Using the QFT Method 147

0.001 *s* +

is adopted. The target rotary angle *θr*(*t*) is 1 rad and computational time step is 0.005s. The simulation results for two control system are shown in Figs.11 and 12. We found that scattering of the output *θ*(*t*) due to the parametric uncertainty are suppressed, and all *θ*(*t*) lies between upper and lower performance specifications, while the PID controller can not cope with parametric uncertainty. Now let us consider settling time *ts*[s] and overshoot *Os*[rad] as performance indices and use standard deviation to assess the scattering of these indices. The results are shown in Table 2. From this table, it is found that standard deviation of *ts*[s] is

0 2 4 6 8 10

time[s]

PID QFT *ts*[s] 64.0 <sup>×</sup> <sup>10</sup>−<sup>2</sup> 0.875 <sup>×</sup> <sup>10</sup>−<sup>2</sup> *Os*[rad] 1.95 <sup>×</sup> <sup>10</sup>−<sup>3</sup> 0.278 <sup>×</sup> <sup>10</sup>−<sup>3</sup>

0.8 1 1.2 0 2 4 6 8 10

Response of θ

0.8

Performance Specification

<sup>1</sup> <sup>+</sup> 0.001*<sup>s</sup>* (18)

*CP*(*s*) = 1.3 +

reduced to 1/80 and that of *Os*[s] is decreased to 1/7, compared with PID.

Fig. 10. M-contour and templates on NC

*Mp* is a peak magnitude of closed loop transfer function. This specification is described on NC as bound, called as U-contour (See Fig.9). As stated in Sec.4.3, this bound means the region that the gain-phase curve of open loop transfer function *C*(*s*)*P*(*s*) must not enter to satisfy Eq.(15) for all possible plant. *C*(*s*) is designed by trial and error in order not to penetrate U-contour. This time, *C*(*s*) is consist of integrator and phase-delay compensator. Then, *C*(*s*) has been decided as

$$\mathcal{C}(s) = 5.2 \cdot 10^3 \frac{(s+1)(s+3.2)}{s(s+285)}.\tag{16}$$

Fig.10 shows templates of *C*(*s*)P(*s*) and M-contour representing *Mp*. From this figure, it is found that all templates do not enter the M-contour. It means that the system including P(*s*) satisfies Eq.(15).

#### **4.5 Design of** *F*(*s*)

In the final step, filter *F*(*s*) is designed so that transfer function from input to output

$$\frac{F(s)\mathcal{C}(s)\mathcal{P}(s)}{1+\mathcal{C}(s)\mathcal{P}(s)}$$

lies between performance specifications *TRU* and *TRL* . Then, *F*(*s*) was decided as

$$F(\mathbf{s}) = \frac{(\mathbf{s}/2\mathbf{5} + 1)^2}{(\mathbf{s}/\mathbf{3} + 1)(\mathbf{s}/\mathbf{4} + 1)}.\tag{17}$$

#### **5. Simulation**

8 Will-be-set-by-IN-TECH


Phase angle [deg]

*Mp* is a peak magnitude of closed loop transfer function. This specification is described on NC as bound, called as U-contour (See Fig.9). As stated in Sec.4.3, this bound means the region that the gain-phase curve of open loop transfer function *C*(*s*)*P*(*s*) must not enter to satisfy Eq.(15) for all possible plant. *C*(*s*) is designed by trial and error in order not to penetrate U-contour. This time, *C*(*s*) is consist of integrator and phase-delay compensator. Then, *C*(*s*)

*<sup>C</sup>*(*s*) = 5.2 · <sup>103</sup> (*<sup>s</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>s</sup>* <sup>+</sup> 3.2)

Fig.10 shows templates of *C*(*s*)P(*s*) and M-contour representing *Mp*. From this figure, it is found that all templates do not enter the M-contour. It means that the system including P(*s*)

> *F*(*s*)*C*(*s*)P(*s*) 1 + *C*(*s*)P(*s*)

In the final step, filter *F*(*s*) is designed so that transfer function from input to output

lies between performance specifications *TRU* and *TRL* . Then, *F*(*s*) was decided as

*<sup>F</sup>*(*s*) = (*s*/25 <sup>+</sup> <sup>1</sup>)<sup>2</sup>

(*s*/3 + 1)(*s*/4 + 1)

*<sup>s</sup>*(*<sup>s</sup>* <sup>+</sup> <sup>285</sup>) . (16)

. (17)



has been decided as

satisfies Eq.(15).

**4.5 Design of** *F*(*s*)


Fig. 10. M-contour and templates on NC



0

20

Magnitude [dB]

40

60

80

ω = 0.1[rad]

ω =1000[rad]

M-Contour ( <sup>=</sup>1.05[dB]) <sup>M</sup> <sup>p</sup>

100

In this section, based on numerical simulation, control performance of QFT control system designed in former section are compared with conventional PID control system. We consider 70 samples of set of *J* and *D* in plant dynamics. As a conventional controller to be compared, the following PID controller

$$\mathcal{C}\_P(s) = 1.3 + \frac{0.001}{s} + \frac{0.8}{1 + 0.001s} \tag{18}$$

is adopted. The target rotary angle *θr*(*t*) is 1 rad and computational time step is 0.005s. The simulation results for two control system are shown in Figs.11 and 12. We found that scattering of the output *θ*(*t*) due to the parametric uncertainty are suppressed, and all *θ*(*t*) lies between upper and lower performance specifications, while the PID controller can not cope with parametric uncertainty. Now let us consider settling time *ts*[s] and overshoot *Os*[rad] as performance indices and use standard deviation to assess the scattering of these indices. The results are shown in Table 2. From this table, it is found that standard deviation of *ts*[s] is reduced to 1/80 and that of *Os*[s] is decreased to 1/7, compared with PID.

Fig. 11. Simulation result with QFT


Table 2. Standard deviation of *ts* and *Os*

**0**

**9**

*Mexico*

**Robust Visual Servoing of Robot Manipulators Based on Passivity**

Servo applications (regulation and tracking) are an important class of tasks for robots. Robustness in a servo controller must be guaranteed when the robot manipulator operates in an uncertain environment to ensure the stability and performance in the presence of uncertain robot manipulator dynamics, objects to be manipulated by the robot, and obstacles to be avoid. Also, robots must have sensing capability to adapt to the new tasks without reprogramming. Servoing based on visual measurements, also referred to as *visual servoing* (Hutchinson et al. (1996)), provided an alternative solution to these applications. Commonly used schemes include position-based and image-based visual servoing, being the main difference how to use the visual measurements: if the visual measurements are used to infer the end-effector pose to implement a cartesian control, we get a position-based visual servoing (PBVS) (see, *e.g.*, Fujita et al. (2007)); if the visual measurements are used directly to calculate the control torque to the manipulator, we get an image-based visual servoing (IBVS) (see, *e.g.*, Espiau et al. (1992) and Kelly (1996)). It is generally recognized that IBVS has the advantages of having less on-line computation burden, being more accurate, while the PBVS is more flexible

*Dynamic visual servoing* was proposed by Weiss et al. (1987). A key point in this approach is to view the visual measurements as the output of a dynamic system. By adopting this point of view, Dickmanns & Graefe (1988) set up a dynamic model of curvature evolution of the road in a driving application. However, stability and robustness issues were not addressed . To address these important questions and to investigate further the applications of IBVS for general scenarios, Hashimoto et al. (1996) and Ma et al. (1999) studied the image dynamics, *i.e.*, how the image feature of an object moving in a 3-D space evolutes in the 2-D image plane. The visual system in a robotics application is linearized in Hashimoto et al. (1996) at the desired point yielding a linear-time-invariant (LTI) multi-input-multi-output (MIMO) model. Ma et al. (1999) proposed a curve dynamic model in vision guided navigation application and based on it designed a linearizing control law that controls the curvature dynamics in the image plane using only perspective projection thanks to a state-observer. By combining passivity of the visual feedback system and the manipulator dynamics, Fujita et al. (2007) addressed the PBVS to track a 3-D object in a camera-in-hand configuration. However, the resulting control law was more complicated than that obtained with the transposed jacobian approach

**1. Introduction**

for its implementation.

A. Luis Rodríguez and Yu Tang

*National University of Mexico*

Fig. 12. Simulation result with PID

### **6. Conclusion**

We presented a controller design for a one link arm with parametric uncertainty. In condition that there is uncertainty in plant's dynamics, the desirable control performance may not be obtained because the controller is designed based on only nominal model of plant. In this chapter, robust controller based on QFT is designed for plant with parametric uncertainty. By numerical experiments, systems designed by using QFT and conventional method are compared. It is found that the control system designed by QFT shows robust performance and can suppress the undesirable output due to parametric uncertainty.

### **7. References**


## **Robust Visual Servoing of Robot Manipulators Based on Passivity**

A. Luis Rodríguez and Yu Tang *National University of Mexico Mexico*

### **1. Introduction**

10 Will-be-set-by-IN-TECH

148 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Performance Specification

0 2 4 6 8 10<sup>0</sup>

We presented a controller design for a one link arm with parametric uncertainty. In condition that there is uncertainty in plant's dynamics, the desirable control performance may not be obtained because the controller is designed based on only nominal model of plant. In this chapter, robust controller based on QFT is designed for plant with parametric uncertainty. By numerical experiments, systems designed by using QFT and conventional method are compared. It is found that the control system designed by QFT shows robust performance

Houpis, C. H.; Rasmussen, S. J. & Garcia-Sanz, M. (2006). *Quantitative Feedback Theory:*

Khodabakhshian, A. & Golbon, N. (2005). Design of a New Load Frequency PID Controller

Barve, J. J. & Nataraj, P. S. V. (1998). Synthesis of a robust pH controller using QFT, *Proceedings*

*Energy, Computer, Communication and Control*, pp. 214-217, vol.1, New Delhi. Zolotas, A. C. & Halikias, G. D. (1999). Optimal design of PID controllers using the QFT

Ryoo, J. R.; Doh, T. & Chung, Y. M. J. (2002). A QFT Design of Disturbance Observer for the

*on Advanced Motion Control. 2002. Proceedings*, pp. 209 - 214, Slovenia.

using QFT, *Proceedings of the 13th Mediterranean Conference on Control and Automation*,

*of IEEE TENCON '98. IEEE Region 10 International Conference on Global Connectivity in*

method, *IEE Proceedings-Control Theory and Applications*., Vol. 146, Issue 6, pp. 585-589.

Track-following Control System of an Optical Disk Drive, *7th International Workshop*

and can suppress the undesirable output due to parametric uncertainty.

*Fundamentals and Applications*, pp.22-99,Taylor & Francis.

pp. 970-975, Limassol, Cyprus.

time[s]

1.2 0 2 4 6 8 10

Response of θ

0

Fig. 12. Simulation result with PID

**6. Conclusion**

**7. References**

0.2

0.2 0.4

0.4

Rotary angle of arm

[rad]

0.6

0.8

0.6 0.8 1

1

1.2

Servo applications (regulation and tracking) are an important class of tasks for robots. Robustness in a servo controller must be guaranteed when the robot manipulator operates in an uncertain environment to ensure the stability and performance in the presence of uncertain robot manipulator dynamics, objects to be manipulated by the robot, and obstacles to be avoid. Also, robots must have sensing capability to adapt to the new tasks without reprogramming. Servoing based on visual measurements, also referred to as *visual servoing* (Hutchinson et al. (1996)), provided an alternative solution to these applications. Commonly used schemes include position-based and image-based visual servoing, being the main difference how to use the visual measurements: if the visual measurements are used to infer the end-effector pose to implement a cartesian control, we get a position-based visual servoing (PBVS) (see, *e.g.*, Fujita et al. (2007)); if the visual measurements are used directly to calculate the control torque to the manipulator, we get an image-based visual servoing (IBVS) (see, *e.g.*, Espiau et al. (1992) and Kelly (1996)). It is generally recognized that IBVS has the advantages of having less on-line computation burden, being more accurate, while the PBVS is more flexible for its implementation.

*Dynamic visual servoing* was proposed by Weiss et al. (1987). A key point in this approach is to view the visual measurements as the output of a dynamic system. By adopting this point of view, Dickmanns & Graefe (1988) set up a dynamic model of curvature evolution of the road in a driving application. However, stability and robustness issues were not addressed . To address these important questions and to investigate further the applications of IBVS for general scenarios, Hashimoto et al. (1996) and Ma et al. (1999) studied the image dynamics, *i.e.*, how the image feature of an object moving in a 3-D space evolutes in the 2-D image plane. The visual system in a robotics application is linearized in Hashimoto et al. (1996) at the desired point yielding a linear-time-invariant (LTI) multi-input-multi-output (MIMO) model. Ma et al. (1999) proposed a curve dynamic model in vision guided navigation application and based on it designed a linearizing control law that controls the curvature dynamics in the image plane using only perspective projection thanks to a state-observer. By combining passivity of the visual feedback system and the manipulator dynamics, Fujita et al. (2007) addressed the PBVS to track a 3-D object in a camera-in-hand configuration. However, the resulting control law was more complicated than that obtained with the transposed jacobian approach (Kelly et al. (2000)) and had to be restricted to consider a steady object in order to establish the stability of the control system.

*Property 3*: The dynamics of (1) is linearly parameterizable:

� *<sup>∂</sup>g*(*q*)

robot coordinate frame in terms of the joint position *q*

*Property 4*: The gravity torque satisfies

by considering the stored energy *V*(*q*, *q*˙) = <sup>1</sup>

**2.2 Robot forward kinematics**

the joint velocity *q*˙

**2.3 Camera model**

its optical axis by *θ* radians,

*H* is the magnification matrix,

the optical axis at the robot plane.

vector of manipulator parameters, *na* the number of parameters.

*M*(*q*)*q*¨ + *C*(*q*, *q*˙)*q*˙ + *g*(*q*) = *Y*(*q*, *q*˙, *q*¨)*a* = *τ*, (4)

*<sup>∂</sup><sup>q</sup>* �≤ *cg*, for some *cg* <sup>&</sup>gt; 0 and <sup>∀</sup>*<sup>q</sup>* ∈ �3. (5)

*TM*(*q*)*q*˙ + *P*(*q*) and its time derivatives along (1)

*X* = *f*(*q*), (6)

*X*˙ = *J*(*q*)*q*˙. (7)

*x* = *HR*(*θ*)[*X* − *X*0], (8)

⎤ ⎦ ,

� ,

where the regressor *<sup>Y</sup>*(*q*, *<sup>q</sup>*˙, *<sup>q</sup>*¨) ∈ �3×*na* , contains known functions of *<sup>q</sup>*, *<sup>q</sup>*˙, *<sup>q</sup>*¨, and *<sup>a</sup>* ∈ �*na* is the

Robust Visual Servoing of Robot Manipulators Based on Passivity 151

Passivity of the mapping *τ* → *q*˙ in the the manipulator dynamics (1) follows from Property 2

*V*˙ = *τTq*˙ (Ortega & Spong (1989)). Based on the passivity, simple and robust control laws have been proposed (see, *e.g.*, Slotine & Li (1987); Takegaki & Arimoto (1981)) for robot control.

The forward kinematics *<sup>f</sup>* : �<sup>3</sup> → �<sup>3</sup> gives the cartesian position *<sup>X</sup>* of a feature point in the

and the velocity kinematics *<sup>J</sup>*(*q*) = *<sup>∂</sup> <sup>f</sup>*(*q*)/*∂<sup>q</sup>* ∈ �3×<sup>3</sup> relates the feature point velocity *<sup>X</sup>*˙ with

The pin-hole model of a CCD camera (Hutchinson et al. (1996)) is considered. In this model, a point in the robot space seen in the camera frame *X* = [*X*<sup>1</sup> *X*<sup>2</sup> *X*3] (meter) is transformed into

where *R*(*θ*) ∈ *SO*(3) is the rotation matrix generated by clockwise rotating the camera about

cos(*θ*) − sin(*θ*) 0 sin(*θ*) cos(*θ*) 0 0 01

> � *α*<sup>1</sup> 0 0 0 *α*<sup>2</sup> 0

with *λ* the focus length, *X*<sup>3</sup> the depth (the distance from the lens to the image plane), *α*<sup>1</sup> and *α*<sup>2</sup> (pixels/m) the scale factors of length units in the image plane, *X*<sup>0</sup> the intersection point of

the image position *<sup>x</sup>* ∈ �<sup>2</sup> (pixel) in the image plane by perspective projection (Fig. 1)

*R*(*θ*) =

⎡ ⎣

*<sup>H</sup>* <sup>=</sup> *<sup>λ</sup>*

*λ* − *X*<sup>3</sup>

2 *q*˙

Motivated by these works, we consider in this chapter the servo problem in a robot manipulator based only on the visual measurements by following the dynamic visual servoing approach to design an IBVS with a fixed camera configuration. To model the whole visual servoing system, we "lift" the manipulator dynamics up to the image space, reconstructed based on the perspective projection of the robot space in the image plane, and model it with the lagrangian formalism by formulating the kinetic and potential energy in the image space. The resulting motion equation has the same structure as that obtained in the joint space using the lagrangian modeling (Spong et al. (2006)), and therefore inheriting the passivity property. Robust control schemes based on the passivity of the motion equation are then designed for visual servoing. The main features of this robust control law are (1) No image derivative is required as it uses only the image position for feedback, (2) no camera parameters and robot inertia parameters are needed for the implementation, making it robust to the parameter uncertainties in both the camera and robot manipulator, (3) no other measurements as from optical encoders are needed but the visual measurements from a single fixed camera.

The rest of this chapter is organized as follows: Section 2 presents the robot image dynamics obtained by gathering together manipulator dynamics, manipulator kinematics and the camera model into a single dynamic system, and its experimental validation in a laboratory set. Section 3 gives the controller design and the main results. Experiments results are shown in Section 4 to illustrate the performance of the proposed IBVS. Concluding remarks and future works are given in Section 5.

### **2. Robot image dynamics**

#### **2.1 Robot joint dynamics**

Consider a 3-DOF articulated (RRR) manipulator moving in a (robot) space, whose motion equation in the joint space is

$$M(q)\ddot{q} + \mathcal{C}(q,\dot{q})\dot{q} + \mathcal{g}(q) = \tau,\tag{1}$$

where *<sup>τ</sup>*(*t*) ∈ �<sup>3</sup> is the control torque and *<sup>q</sup>*(*t*) ∈ �3, the joint position of the manipulator. *<sup>M</sup>*(*q*) ∈ �3×<sup>3</sup> represents the inertia matrix, and *<sup>C</sup>*(*q*, *<sup>q</sup>*˙)*q*˙ ∈ �<sup>3</sup> and *<sup>g</sup>*(*q*) ∈ �<sup>3</sup> are Coriolis/centrifugal and gravity torques, respectively. This equation can be obtained by modeling the manipulator in the joint space using the lagragian formalism (Spong et al. (2006)) with kinetic energy *K*(*q*, *q*˙) = <sup>1</sup> 2 *q*˙ *TM*(*q*)*q*˙ and potential energy *P*(*q*). The motion equation (1) has the following properties:

*Property 1* : The inertia matrix *M*(*q*) is positive definite, ı.e.,

$$\underline{\mathbf{m}} \le q^T M(q) q \le \overline{\mathbf{m}}, \ \forall q \in \mathbb{R}^3, \ q \ne 0, \ 0 < \underline{\mathbf{m}} \le \overline{\mathbf{m}}.\tag{2}$$

*Property 2*: The matrix *<sup>M</sup>*˙ (*q*) <sup>−</sup> <sup>2</sup>*C*(*q*, *<sup>q</sup>*˙) is skew-symmetric, ı.e.,

$$\|\mathbf{x}^T[\dot{M}(q) - 2\mathbf{C}(q, \dot{q})]\mathbf{x} = 0, \ \forall \mathbf{x} \in \mathfrak{R}^3. \tag{3}$$

*Property 3*: The dynamics of (1) is linearly parameterizable:

$$\mathcal{M}(q)\ddot{q} + \mathcal{C}(q,\dot{q})\dot{q} + \mathcal{g}(q) = \mathcal{Y}(q,\dot{q},\ddot{q})\mathcal{a} = \tau,\tag{4}$$

where the regressor *<sup>Y</sup>*(*q*, *<sup>q</sup>*˙, *<sup>q</sup>*¨) ∈ �3×*na* , contains known functions of *<sup>q</sup>*, *<sup>q</sup>*˙, *<sup>q</sup>*¨, and *<sup>a</sup>* ∈ �*na* is the vector of manipulator parameters, *na* the number of parameters. *Property 4*: The gravity torque satisfies

$$\|\left|\begin{array}{c}\frac{\partial g(q)}{\partial q}\right|\|\leq c\_{\mathcal{S}'}\text{ for some }c\_{\mathcal{S}}>0\text{ and }\,\forall q\in\mathfrak{R}^3.\tag{5}$$

Passivity of the mapping *τ* → *q*˙ in the the manipulator dynamics (1) follows from Property 2 by considering the stored energy *V*(*q*, *q*˙) = <sup>1</sup> 2 *q*˙ *TM*(*q*)*q*˙ + *P*(*q*) and its time derivatives along (1) *V*˙ = *τTq*˙ (Ortega & Spong (1989)). Based on the passivity, simple and robust control laws have been proposed (see, *e.g.*, Slotine & Li (1987); Takegaki & Arimoto (1981)) for robot control.

#### **2.2 Robot forward kinematics**

The forward kinematics *<sup>f</sup>* : �<sup>3</sup> → �<sup>3</sup> gives the cartesian position *<sup>X</sup>* of a feature point in the robot coordinate frame in terms of the joint position *q*

$$X = f(q)\_\prime \tag{6}$$

and the velocity kinematics *<sup>J</sup>*(*q*) = *<sup>∂</sup> <sup>f</sup>*(*q*)/*∂<sup>q</sup>* ∈ �3×<sup>3</sup> relates the feature point velocity *<sup>X</sup>*˙ with the joint velocity *q*˙

$$
\dot{X} = f(q)\dot{q}.\tag{7}
$$

#### **2.3 Camera model**

2 Will-be-set-by-IN-TECH

(Kelly et al. (2000)) and had to be restricted to consider a steady object in order to establish the

Motivated by these works, we consider in this chapter the servo problem in a robot manipulator based only on the visual measurements by following the dynamic visual servoing approach to design an IBVS with a fixed camera configuration. To model the whole visual servoing system, we "lift" the manipulator dynamics up to the image space, reconstructed based on the perspective projection of the robot space in the image plane, and model it with the lagrangian formalism by formulating the kinetic and potential energy in the image space. The resulting motion equation has the same structure as that obtained in the joint space using the lagrangian modeling (Spong et al. (2006)), and therefore inheriting the passivity property. Robust control schemes based on the passivity of the motion equation are then designed for visual servoing. The main features of this robust control law are (1) No image derivative is required as it uses only the image position for feedback, (2) no camera parameters and robot inertia parameters are needed for the implementation, making it robust to the parameter uncertainties in both the camera and robot manipulator, (3) no other measurements as from optical encoders are needed but the visual measurements from a single

The rest of this chapter is organized as follows: Section 2 presents the robot image dynamics obtained by gathering together manipulator dynamics, manipulator kinematics and the camera model into a single dynamic system, and its experimental validation in a laboratory set. Section 3 gives the controller design and the main results. Experiments results are shown in Section 4 to illustrate the performance of the proposed IBVS. Concluding remarks and

Consider a 3-DOF articulated (RRR) manipulator moving in a (robot) space, whose motion

where *<sup>τ</sup>*(*t*) ∈ �<sup>3</sup> is the control torque and *<sup>q</sup>*(*t*) ∈ �3, the joint position of the manipulator. *<sup>M</sup>*(*q*) ∈ �3×<sup>3</sup> represents the inertia matrix, and *<sup>C</sup>*(*q*, *<sup>q</sup>*˙)*q*˙ ∈ �<sup>3</sup> and *<sup>g</sup>*(*q*) ∈ �<sup>3</sup> are Coriolis/centrifugal and gravity torques, respectively. This equation can be obtained by modeling the manipulator in the joint space using the lagragian formalism (Spong et al. (2006))

*M*(*q*)*q*¨ + *C*(*q*, *q*˙)*q*˙ + *g*(*q*) = *τ*, (1)

*TM*(*q*)*q*˙ and potential energy *P*(*q*). The motion equation (1)

<sup>m</sup> <sup>≤</sup> *<sup>q</sup>TM*(*q*)*<sup>q</sup>* <sup>≤</sup> *<sup>m</sup>*, <sup>∀</sup>*<sup>q</sup>* ∈ �3, *<sup>q</sup>* �<sup>=</sup> 0, 0 <sup>&</sup>lt; <sup>m</sup> <sup>≤</sup> *<sup>m</sup>*. (2)

*<sup>x</sup>T*[*M*˙ (*q*) <sup>−</sup> <sup>2</sup>*C*(*q*, *<sup>q</sup>*˙)]*<sup>x</sup>* <sup>=</sup> 0, <sup>∀</sup>*<sup>x</sup>* ∈ �3. (3)

stability of the control system.

fixed camera.

future works are given in Section 5.

**2. Robot image dynamics**

equation in the joint space is

with kinetic energy *K*(*q*, *q*˙) = <sup>1</sup>

has the following properties:

2 *q*˙

*Property 2*: The matrix *<sup>M</sup>*˙ (*q*) <sup>−</sup> <sup>2</sup>*C*(*q*, *<sup>q</sup>*˙) is skew-symmetric, ı.e.,

*Property 1* : The inertia matrix *M*(*q*) is positive definite, ı.e.,

**2.1 Robot joint dynamics**

The pin-hole model of a CCD camera (Hutchinson et al. (1996)) is considered. In this model, a point in the robot space seen in the camera frame *X* = [*X*<sup>1</sup> *X*<sup>2</sup> *X*3] (meter) is transformed into the image position *<sup>x</sup>* ∈ �<sup>2</sup> (pixel) in the image plane by perspective projection (Fig. 1)

$$\mathbf{x} = HR(\theta)[\mathbf{X} - \mathbf{X}\_0].\tag{8}$$

where *R*(*θ*) ∈ *SO*(3) is the rotation matrix generated by clockwise rotating the camera about its optical axis by *θ* radians,

$$R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0\\ \sin(\theta) & \cos(\theta) & 0\\ 0 & 0 & 1 \end{bmatrix} \prime$$

*H* is the magnification matrix,

$$H = \frac{\lambda}{\lambda - X\_3} \begin{bmatrix} \alpha\_1 & 0 & 0\\ 0 & \alpha\_2 & 0 \end{bmatrix} / 2$$

with *λ* the focus length, *X*<sup>3</sup> the depth (the distance from the lens to the image plane), *α*<sup>1</sup> and *α*<sup>2</sup> (pixels/m) the scale factors of length units in the image plane, *X*<sup>0</sup> the intersection point of the optical axis at the robot plane.

Fig. 2. Graphic representation of the robot image dynamics (9).

*Property 3'*: The dynamics of (9) is linearly parameterizable:

� *<sup>∂</sup>gφ*(*φ*)

the control design based on visual measurements.

system in Fig. 2 with that of the model (9).

may be obtained (Spong et al. (2006)). *Property 4'*: The gravity torque satisfies

(2004).

*Property 2'*: The matrix *<sup>M</sup>*˙ *<sup>φ</sup>*(*φ*) <sup>−</sup> <sup>2</sup>*Cφ*(*φ*, *<sup>φ</sup>*˙) is skew-symmetric, ı.e.,

*<sup>x</sup>T*[*M*˙ *<sup>φ</sup>*(*φ*) <sup>−</sup> <sup>2</sup>*Cφ*(*φ*, *<sup>φ</sup>*˙)]*<sup>x</sup>* <sup>=</sup> 0, <sup>∀</sup>*<sup>x</sup>* ∈ �3. (11)

*∂φ* �≤ *cg<sup>φ</sup>* , for some *cg<sup>φ</sup>* <sup>&</sup>gt; 0 and <sup>∀</sup>*<sup>φ</sup>* ∈ �3. (13)

*Mφ*(*φ*)*φ*¨ + *Cφ*(*φ*, *φ*˙)*φ*˙ + *gφ*(*φ*) = *Yφ*(*φ*, *φ*˙, *φ*¨)*a<sup>φ</sup>* = *τ*, (12)

For this 3-DOF articulated robot manipulator, a parametrization with *<sup>a</sup>φ*, *<sup>Y</sup>φ*(*φ*, *<sup>φ</sup>*˙, *<sup>φ</sup>*¨) ∈ �<sup>15</sup>

Robust Visual Servoing of Robot Manipulators Based on Passivity 153

*Remark 1:* As a consequence of this paradigm, the parameters of the manipulator dynamics, forward kinematics and camera transformation are all lumped together into the parameters of the model (9). The passivity property followed from Property 2' will simplify significantly

*Remark 2:* The aforementioned robot plane (Y-Z) and the image-plane (y-z) in Fig.1 are uniquely defined once the camera position and orientation are set, and are parallel from perspective projection (8). In the experiments, *X*<sup>0</sup> will be defined as the anti-image of the principal point in the image plane determined using the method in Grammatikopoulos et al.

Before moving to consider the control design, we validate our point of view through experiments. The validation was carried out by first identifying the robot image dynamics using an off-line least-square algorithm, and then by comparing the output of the robot image

The experiment platform is shown in Fig 3. It consists of a three-link manipulator (made in the laboratory) moving in the (robot) space, and a fixed IEEE 1394 digital camera from Basler (model A601 FC). The camera has focus length *λ* = 0.9091 (cm), and scaling factors *α*<sup>1</sup> = *α*<sup>2</sup> = 4.6 (pixels/cm). The rotation angle of the camera about its optical axis was set to *θ* = 0 (rad). The image plane has a resolution of (horizontal x vertical) 320 × 240 pixels. Before the experiments, the feature point corresponding to the robot base point (origin of the robot coordinate frame) and length of the link 1, link 2 and link 3 are determined in the

Fig. 1. The robot plane (Y-Z) is the plane perpendicular to the camera optical axis and containing the origin of the robot coordinate frame. The image-plane (y-z) is its perspective projection.

### **2.4 Robot image dynamics**

By robot image dynamics we refer to the dynamics of the robot image evolving in the image space (defined below) when the robot moves in the robot space.

Let the robot plane (Y-Z) be the plane perpendicular to the camera optical axis and containing the origin of the robot coordinate frame. The robot manipulator moves in the robot space defined by the robot coordinate frame (X-Y-Z) (Fig. 1). The image-plane (y-z) is the perspective projection of the robot plane (Y-Z), and the image space (x-y-z) is the space reconstructed based on the perspective projection of the robot space in the image plane (y-z). Common methods used for this reconstruction include range identification (Chen & Kano (2002); Dixon et al. (2003); Karagiannis & Astolfi (2005)), utilization of multiple feature points attached to the end effector as well as to the target (Kelly et al. (2006); Yuan (1989)). In this work, we use the feature points attached to the links to reconstruct the joint angles of the robot image in the image space needed for the controller implementation.

The robot image dynamics we consider in this chapter is obtained by lumping together manipulator dynamics, forward kinematics and the camera transformation into a single dynamics (Fig. 2). Consider the kinetic energy *Kφ*(*φ*, *φ*˙) = <sup>1</sup> <sup>2</sup>*φ*˙ *TMφ*(*φ*)*φ*˙ and potential energy *Pφ*(*φ*) in the image space, where, similar to modeling the manipulator in the joint space, *φ* is the joint image position (rad)1, *Mφ*(*φ*) the inertia matrix seen in the image space. The motion equation as seen in the image space is obtained by modeling the visual servo system (manipulator and camera) with lagragian formalism:

$$M\_{\Phi}(\phi)\ddot{\phi} + \mathbb{C}\_{\Phi}(\phi, \phi)\dot{\phi} + \mathbb{g}\_{\Phi}(\phi) = \tau. \tag{9}$$

As in the joint-space model, this motion equation has the following properties: *Property 1'*: The inertia matrix *Mφ*(*φ*) is positive definite, ı.e.,

$$\underline{\mathbf{m}}\_{\Phi} \le \boldsymbol{\phi}^{T} M\_{\Phi}(\boldsymbol{\phi}) \boldsymbol{\phi} \le \overline{m}\_{\Phi'} \; \forall \boldsymbol{\phi} \in \mathbb{R}^{3}, \; \boldsymbol{\phi} \ne \boldsymbol{0}, \; 0 < \underline{m}\_{\Phi} \le \overline{m}\_{\Phi}. \tag{10}$$

<sup>1</sup> Joint image position is meant here the joint position in the image space.

4 Will-be-set-by-IN-TECH

Fig. 1. The robot plane (Y-Z) is the plane perpendicular to the camera optical axis and containing the origin of the robot coordinate frame. The image-plane (y-z) is its perspective

space (defined below) when the robot moves in the robot space.

image space needed for the controller implementation.

(manipulator and camera) with lagragian formalism:

dynamics (Fig. 2). Consider the kinetic energy *Kφ*(*φ*, *φ*˙) = <sup>1</sup>

*Property 1'*: The inertia matrix *Mφ*(*φ*) is positive definite, ı.e.,

<sup>1</sup> Joint image position is meant here the joint position in the image space.

By robot image dynamics we refer to the dynamics of the robot image evolving in the image

Let the robot plane (Y-Z) be the plane perpendicular to the camera optical axis and containing the origin of the robot coordinate frame. The robot manipulator moves in the robot space defined by the robot coordinate frame (X-Y-Z) (Fig. 1). The image-plane (y-z) is the perspective projection of the robot plane (Y-Z), and the image space (x-y-z) is the space reconstructed based on the perspective projection of the robot space in the image plane (y-z). Common methods used for this reconstruction include range identification (Chen & Kano (2002); Dixon et al. (2003); Karagiannis & Astolfi (2005)), utilization of multiple feature points attached to the end effector as well as to the target (Kelly et al. (2006); Yuan (1989)). In this work, we use the feature points attached to the links to reconstruct the joint angles of the robot image in the

The robot image dynamics we consider in this chapter is obtained by lumping together manipulator dynamics, forward kinematics and the camera transformation into a single

*Pφ*(*φ*) in the image space, where, similar to modeling the manipulator in the joint space, *φ* is the joint image position (rad)1, *Mφ*(*φ*) the inertia matrix seen in the image space. The motion equation as seen in the image space is obtained by modeling the visual servo system

As in the joint-space model, this motion equation has the following properties:

*Mφ*(*φ*)*φ*¨ + *Cφ*(*φ*, *φ*˙)*φ*˙ + *gφ*(*φ*) = *τ*. (9)

<sup>m</sup>*<sup>φ</sup>* <sup>≤</sup> *<sup>φ</sup>TMφ*(*φ*)*<sup>φ</sup>* <sup>≤</sup> *<sup>m</sup>φ*, <sup>∀</sup>*<sup>φ</sup>* ∈ �3, *<sup>φ</sup>* �<sup>=</sup> 0, 0 <sup>&</sup>lt; <sup>m</sup>*<sup>φ</sup>* <sup>≤</sup> *<sup>m</sup>φ*. (10)

<sup>2</sup>*φ*˙ *TMφ*(*φ*)*φ*˙ and potential energy

projection.

**2.4 Robot image dynamics**

Fig. 2. Graphic representation of the robot image dynamics (9).

*Property 2'*: The matrix *<sup>M</sup>*˙ *<sup>φ</sup>*(*φ*) <sup>−</sup> <sup>2</sup>*Cφ*(*φ*, *<sup>φ</sup>*˙) is skew-symmetric, ı.e.,

$$\propto \mathbf{x}^T [\dot{M}\_\Phi(\phi) - 2\mathbf{C}\_\Phi(\phi, \dot{\phi})] \mathbf{x} = \mathbf{0}, \ \forall \mathbf{x} \in \mathfrak{R}^3. \tag{11}$$

*Property 3'*: The dynamics of (9) is linearly parameterizable:

$$M\_{\Phi}(\phi)\ddot{\phi} + \mathbb{C}\_{\Phi}(\phi, \dot{\phi})\dot{\phi} + \mathbb{g}\_{\Phi}(\phi) = Y\_{\Phi}(\phi, \dot{\phi}, \ddot{\phi})a\_{\phi} = \tau,\tag{12}$$

For this 3-DOF articulated robot manipulator, a parametrization with *<sup>a</sup>φ*, *<sup>Y</sup>φ*(*φ*, *<sup>φ</sup>*˙, *<sup>φ</sup>*¨) ∈ �<sup>15</sup> may be obtained (Spong et al. (2006)).

*Property 4'*: The gravity torque satisfies

$$\|\|\frac{\partial g\_{\phi}(\phi)}{\partial \phi}\|\| \le c\_{\mathcal{S}\phi'} \text{ for some } c\_{\mathcal{S}\phi} > 0 \text{ and } \forall \phi \in \mathfrak{K}^3. \tag{13}$$

*Remark 1:* As a consequence of this paradigm, the parameters of the manipulator dynamics, forward kinematics and camera transformation are all lumped together into the parameters of the model (9). The passivity property followed from Property 2' will simplify significantly the control design based on visual measurements.

*Remark 2:* The aforementioned robot plane (Y-Z) and the image-plane (y-z) in Fig.1 are uniquely defined once the camera position and orientation are set, and are parallel from perspective projection (8). In the experiments, *X*<sup>0</sup> will be defined as the anti-image of the principal point in the image plane determined using the method in Grammatikopoulos et al. (2004).

Before moving to consider the control design, we validate our point of view through experiments. The validation was carried out by first identifying the robot image dynamics using an off-line least-square algorithm, and then by comparing the output of the robot image system in Fig. 2 with that of the model (9).

The experiment platform is shown in Fig 3. It consists of a three-link manipulator (made in the laboratory) moving in the (robot) space, and a fixed IEEE 1394 digital camera from Basler (model A601 FC). The camera has focus length *λ* = 0.9091 (cm), and scaling factors *α*<sup>1</sup> = *α*<sup>2</sup> = 4.6 (pixels/cm). The rotation angle of the camera about its optical axis was set to *θ* = 0 (rad). The image plane has a resolution of (horizontal x vertical) 320 × 240 pixels.

Before the experiments, the feature point corresponding to the robot base point (origin of the robot coordinate frame) and length of the link 1, link 2 and link 3 are determined in the

Parameter *aφ*<sup>1</sup> *aφ*<sup>2</sup> *aφ*<sup>3</sup> *aφ*<sup>4</sup> *aφ*<sup>5</sup> *aφ*<sup>6</sup> *aφ*<sup>7</sup> *aφ*<sup>8</sup> Value 0.0036 0.0537 0.0389 0.0021 0.0914 0.0596 0.0225 0.0402

Robust Visual Servoing of Robot Manipulators Based on Passivity 155

Fig. 4. Identification of the robot image dynamics: Left column shows the torque inputs, right

In this section, motivated from joint space control in Escobar et al. (1999); Kelly (1999); Tomei (1991) we design an output feedback control law with bounded control action and desired gravity compensation to regulate the joint image position. The results are summarized in the

*gφ*(*φd*)=[0 *aφ*4*g* cos(*φd*2) + *aφ*5*g* cos(*φd*<sup>2</sup> + *φd*3)

is the desired gravity compensation term, *<sup>φ</sup>*˜ <sup>=</sup> *<sup>φ</sup>* <sup>−</sup> *<sup>φ</sup><sup>d</sup>* the image position error, *<sup>φ</sup><sup>d</sup>* ∈ �<sup>3</sup> the

*aφ*5*g* cos(*φd*<sup>2</sup> + *φd*3)],

*<sup>τ</sup>* <sup>=</sup> <sup>−</sup>*ψ*1(*φ*˜) <sup>−</sup> *<sup>ψ</sup>*2(*z*) + *<sup>g</sup>φ*(*φd*), (15)

*<sup>i</sup>* (*x*)=[*ψi*1(*x*1) *<sup>ψ</sup>i*2(*x*2) *<sup>ψ</sup>i*3(*x*3)] : �<sup>3</sup> → �3, *<sup>i</sup>* <sup>=</sup> 1, 2, are

column shows the joint image position for Link 1 (up), 2 (middle) and 3 (down).

*Theorem 1:* Consider the robot image dynamics (9) and the control law

**3. Controller design for IBVS**

desired joint image position, *ψ<sup>T</sup>*

following.

where

Parameter *aφ*<sup>9</sup> *aφ*<sup>10</sup> *aφ*<sup>11</sup> *aφ*<sup>12</sup> *aφ*<sup>13</sup> *aφ*<sup>14</sup> *aφ*<sup>15</sup> Value 0.0758 0.0463 0.0073 0.0052 0.0641 0.0598 0.0047

Table 1. Estimated parameters of the robot image dynamics (12).

Fig. 3. Experiment platform.

image plane. Using this information, the joint image position (*φ*) is obtained by measuring two feature points at the second and the third joint (end-effector).

To attenuate the noise in the image measurements image thresholding was first applied to the images of the feature points, then the centroid of each feature point image was calculated, which was used as the image positions of the feature points to calculate the joint image position.

The applied voltage, which is proportional to the applied torque, is fed through a D/A converter (AI-1608AY card from CONTEC) to the power amplification unit. A PC Pentium D running at 3 GHz was used in the experiments. The sampling time was set to 50 (frames/sec). The joint image position together with the applied voltage is shown in Fig. 4.

First the robot image dynamics (9) is identified using an off-line least square algorithm and the parametrization (12). In order to avoid from using the image acceleration, a first order low-pass filter is applied to both sides of the parametrization (12) to get a filtered version of the parametrization (12):

where

$$Y\_f(\phi, \dot{\phi})a\_\phi = \tau\_{f, \prime} \tag{14}$$

$$\tau\_f = \frac{\beta}{s \pm \beta} \tau\_\prime \ Y\_f = \frac{\beta}{s \pm \beta} Y\_\phi$$

*Yφ*

$$\tau\_f = \frac{\beta}{s+\beta}\tau\_r \ Y\_f = \frac{\beta}{s+\beta}Y\_\theta$$
 are the filtered regression and the filtered input torque, respectively, and  $\beta/(s+\beta)$ , with  $\beta = 0.9$  and  $s$  the Laplace transform variable, the filter transfer function. In order to further avoid from using the derivatives of the joint image position,  $\dot{\phi}$  is substituted by its approximate derivative  $\dot{\phi} = \frac{s}{1+\varepsilon\_f\beta}\phi$ , with  $\varepsilon\_f = 0.001$ . 900 samples were used. The estimated parameters are shown in Table 1.

After the parameter vector of the robot image dynamics (12) is obtained, this dynamical model is validated by comparing the output of the robot image dynamics (Fig. 2) and the output of the identified robot image dynamics (9). The applied torques and the measured joint image positions and the outputs of the identified robot image dynamics are shown in Fig. 5.


Table 1. Estimated parameters of the robot image dynamics (12).

Fig. 4. Identification of the robot image dynamics: Left column shows the torque inputs, right column shows the joint image position for Link 1 (up), 2 (middle) and 3 (down).

#### **3. Controller design for IBVS**

In this section, motivated from joint space control in Escobar et al. (1999); Kelly (1999); Tomei (1991) we design an output feedback control law with bounded control action and desired gravity compensation to regulate the joint image position. The results are summarized in the following.

*Theorem 1:* Consider the robot image dynamics (9) and the control law

$$
\pi = -\psi\_1(\vec{\phi}) - \psi\_2(z) + \mathfrak{g}\_\phi(\phi\_d),
\tag{15}
$$

where

6 Will-be-set-by-IN-TECH

image plane. Using this information, the joint image position (*φ*) is obtained by measuring

To attenuate the noise in the image measurements image thresholding was first applied to the images of the feature points, then the centroid of each feature point image was calculated, which was used as the image positions of the feature points to calculate the joint image

The applied voltage, which is proportional to the applied torque, is fed through a D/A converter (AI-1608AY card from CONTEC) to the power amplification unit. A PC Pentium D running at 3 GHz was used in the experiments. The sampling time was set to 50 (frames/sec).

First the robot image dynamics (9) is identified using an off-line least square algorithm and the parametrization (12). In order to avoid from using the image acceleration, a first order low-pass filter is applied to both sides of the parametrization (12) to get a filtered version of

*<sup>s</sup>* <sup>+</sup> *<sup>β</sup> <sup>τ</sup>*, *Yf* <sup>=</sup> *<sup>β</sup>*

are the filtered regressor and the filtered input torque, respectively, and *β*/(*s* + *β*), with *β* = 0.9 and *s* the Laplace transform variable, the filter transfer function. In order to further avoid from using the derivatives of the joint image position, *φ*˙ is substituted by its approximate derivative *<sup>φ</sup>*˙ <sup>=</sup> *<sup>s</sup>* <sup>1</sup>+*�fs <sup>φ</sup>*, with *�<sup>f</sup>* <sup>=</sup> 0.001. 900 samples were used. The estimated parameters

After the parameter vector of the robot image dynamics (12) is obtained, this dynamical model is validated by comparing the output of the robot image dynamics (Fig. 2) and the output of the identified robot image dynamics (9). The applied torques and the measured joint image

positions and the outputs of the identified robot image dynamics are shown in Fig. 5.

*s* + *β Yφ*

*Yf*(*φ*, *φ*˙)*a<sup>φ</sup>* = *τ<sup>f</sup>* , (14)

The joint image position together with the applied voltage is shown in Fig. 4.

*<sup>τ</sup><sup>f</sup>* <sup>=</sup> *<sup>β</sup>*

two feature points at the second and the third joint (end-effector).

Fig. 3. Experiment platform.

the parametrization (12):

are shown in Table 1.

position.

where

$$g\_{\phi}(\phi\_d) = \begin{bmatrix} 0 & a\_{\phi4}\mathfrak{g}\cos(\phi\_{d2}) + a\_{\phi5}\mathfrak{g}\cos(\phi\_{d2} + \phi\_{d3}) \end{bmatrix}$$

$$a\_{\phi5}\mathfrak{g}\cos(\phi\_{d2} + \phi\_{d3})\mathfrak{l}$$

is the desired gravity compensation term, *<sup>φ</sup>*˜ <sup>=</sup> *<sup>φ</sup>* <sup>−</sup> *<sup>φ</sup><sup>d</sup>* the image position error, *<sup>φ</sup><sup>d</sup>* ∈ �<sup>3</sup> the desired joint image position, *ψ<sup>T</sup> <sup>i</sup>* (*x*)=[*ψi*1(*x*1) *<sup>ψ</sup>i*2(*x*2) *<sup>ψ</sup>i*3(*x*3)] : �<sup>3</sup> → �3, *<sup>i</sup>* <sup>=</sup> 1, 2, are

On the other hand, it follows from Property 4' and (17) that

*P*<sup>1</sup> =

Then *<sup>φ</sup>* = *<sup>φ</sup><sup>d</sup>* is the global minimum of *<sup>P</sup>*1(*φ*) since *<sup>∂</sup>P*<sup>1</sup>

2

Consider the Lyapunov function candidate

*<sup>V</sup>*(*φ*˜, *<sup>φ</sup>*˙, *<sup>z</sup>*) = <sup>1</sup>

Its time derivative along the error dynamics (19) is

(Khalil (2002)) by considering the invariant set

*∂gφ*(*φ*) *∂φ* > 0.

follows.

as in Tomei (1991).

 *φ* 0 

The above two expressions imply that *φ* = *φd*. Rewriting the observer dynamics (18) as

� *<sup>g</sup>φ*(*φd*) <sup>−</sup> *<sup>g</sup>φ*(*φ*) �≤ *cg<sup>φ</sup>* � *<sup>φ</sup><sup>d</sup>* <sup>−</sup> *<sup>φ</sup>* �≤� *<sup>ψ</sup>*1(*φ*˜) � .

Robust Visual Servoing of Robot Manipulators Based on Passivity 157

*<sup>ψ</sup>*1(*φ*˜) <sup>−</sup> *<sup>g</sup>φ*(*φd*) + *<sup>g</sup>φ*(*φ*)

*<sup>φ</sup>*˙ *TMφ*(*φ*)*φ*˙ <sup>+</sup> *<sup>P</sup>*1(*φ*) <sup>−</sup> *<sup>P</sup>*1(*φd*) + *<sup>z</sup>*

<sup>2</sup> (*z*)[−*z*˙ <sup>+</sup> *<sup>φ</sup>*˙]

In order to conclude the asymptotic stability of the equilibrium, we invoke LaSalle theorem

which contains only the equilibrium. Therefore the asymptotic stability of the equilibrium

*Remark 3:* Since the control law is developed based on the robot image dynamics, no other measurements than the joint image positions are need for its implementation. This is important because taking time derivative of image measurements are in general not acceptable given noisy image measurements. Also, notice that the only parameters in the control law are the gravity term of the robot image dynamics, which may be also tuned on-line

Remark 4: Due to the occlusion effect in a single fixed camera configuration, it is impossible to determine uniquely the joint image positions for certain poses of the manipulator by the

*<sup>T</sup>ψ*2(*z*)

<sup>Ω</sup> <sup>=</sup> {(*φ*˜ *<sup>φ</sup>*˙ *<sup>z</sup>*) ∈ �<sup>3</sup> × �<sup>3</sup> × �<sup>3</sup> : *<sup>V</sup>*˙ (*φ*˜, *<sup>φ</sup>*˙, *<sup>z</sup>*) = <sup>0</sup>}, (25)

*<sup>V</sup>*˙ <sup>=</sup> <sup>−</sup>*φ*˙ *<sup>T</sup>ψ*2(*z*) + *<sup>z</sup>*˙

<sup>=</sup> <sup>−</sup>*ψ<sup>T</sup>*

Therefore, (*φ*˜, *φ*˙, *z*)=(0, 0, 0) is the unique equilibrium of the closed-loop system. Consider the following potential energy-like function for the closed-loop system

*<sup>z</sup>*˙ <sup>=</sup> <sup>−</sup>*α<sup>z</sup>* <sup>+</sup> *<sup>φ</sup>*˙. (21)

*∂φ* <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>φ</sup>* <sup>=</sup> *<sup>φ</sup><sup>d</sup>* and *<sup>∂</sup>*2*P*<sup>1</sup>

0

<sup>=</sup> <sup>−</sup>*αzTψ*2(*z*) <sup>≤</sup> 0. (24)

*∂φ*<sup>2</sup> <sup>=</sup> *∂ψ*1(*φ*˜)

*ψ*2(*z*)*dz*. (23)

*∂φ* +

 *dφ*

<sup>=</sup> *<sup>φ</sup>Tψ*1(*φ*˜) <sup>−</sup> *<sup>φ</sup>Tgφ*(*φd*) + *<sup>P</sup>φ*(*φ*). (22)

Fig. 5. Validation of the robot image dynamics: Left column shows the torque inputs, right column shows the joint image position measured from the robot image dynamics (Fig. 2) (solid line) and the joint image position from the model (9) (dashed line) for Link 1 (up), 2 (middle) and 3 (down).

continuous functions satisfying

$$\text{tr}\psi\_{\overline{i}\rangle}(\mathbf{x}) > 0, \forall \mathbf{x} \in \mathfrak{R}, \ \psi\_{\overline{i}\rangle}(0) = 0, \ i = 1, 2, \ j = 1, 2, 3,\tag{16}$$

$$\frac{\partial \psi\_1}{\partial \mathbf{x}} = \text{diag}[\frac{\partial \psi\_{11}}{\partial \mathbf{x}\_1} \, \frac{\partial \psi\_{12}}{\partial \mathbf{x}\_2} \, \frac{\partial \psi\_{13}}{\partial \mathbf{x}\_3}] \ge c\_{\mathcal{g}\_\theta} I\_\prime \tag{17}$$

where *cg<sup>φ</sup>* is given in Property 4' and

$$
\dot{w} = -\kappa (w + \phi), \; \alpha > 0,
$$

$$
z = w + \phi. \tag{18}
$$

Then all the signals in the control loop are bounded and *<sup>φ</sup>*˜ <sup>→</sup> 0 asymptotically.

*Proof.* Substituting the control law (15) into the robot image dynamics (9) gets the closed-loop equation

$$M\_{\Phi}(\phi)\ddot{\phi} + \mathbb{C}\_{\Phi}(\phi, \phi)\dot{\phi} + g\_{\phi}(\phi) = -\psi\_1(\tilde{\phi}) - \psi\_2(z) + g\_{\phi}(\phi\_d). \tag{19}$$

First, we show that (*φ*˜, *φ*˙, *z*) has a unique equilibrium at the origin, and then using the Lyapunov analysis to show this equilibrium is asymptotically stable. In fact, at the equilibria we have

$$
\psi\_1(\tilde{\boldsymbol{\phi}}) = \mathcal{g}\_{\boldsymbol{\phi}}(\phi\_d) - \mathcal{g}\_{\boldsymbol{\phi}}(\boldsymbol{\phi}).\tag{20}
$$

On the other hand, it follows from Property 4' and (17) that

$$\left| \parallel \mathcal{g}\_{\Phi}(\phi\_{d}) - \mathcal{g}\_{\Phi}(\phi) \parallel \leq c\_{\mathcal{G}\phi} \parallel \phi\_{d} - \phi \parallel \leq \parallel \; \psi\_{1}(\bar{\phi}) \parallel \cdot \; \vert$$

The above two expressions imply that *φ* = *φd*. Rewriting the observer dynamics (18) as

8 Will-be-set-by-IN-TECH

Fig. 5. Validation of the robot image dynamics: Left column shows the torque inputs, right column shows the joint image position measured from the robot image dynamics (Fig. 2) (solid line) and the joint image position from the model (9) (dashed line) for Link 1 (up), 2

*xψij*(*x*) > 0, ∀*x* ∈ �, *ψij*(0) = 0, *i* = 1, 2, *j* = 1, 2, 3, (16)

*<sup>M</sup>φ*(*φ*)*φ*¨ <sup>+</sup> *<sup>C</sup>φ*(*φ*, *<sup>φ</sup>*˙)*φ*˙ <sup>+</sup> *<sup>g</sup>φ*(*φ*) = <sup>−</sup>*ψ*1(*φ*˜) <sup>−</sup> *<sup>ψ</sup>*2(*z*) + *<sup>g</sup>φ*(*φd*). (19)

] ≥ *cg<sup>φ</sup> I*, (17)

*z* = *w* + *φ*. (18)

*<sup>ψ</sup>*1(*φ*˜) = *<sup>g</sup>φ*(*φd*) <sup>−</sup> *<sup>g</sup>φ*(*φ*). (20)

(middle) and 3 (down).

equation

continuous functions satisfying

where *cg<sup>φ</sup>* is given in Property 4' and

In fact, at the equilibria we have

*∂ψ*1

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> *diag*[

*∂ψ*11 *∂x*<sup>1</sup>

Then all the signals in the control loop are bounded and *<sup>φ</sup>*˜ <sup>→</sup> 0 asymptotically.

Lyapunov analysis to show this equilibrium is asymptotically stable.

*∂ψ*12 *∂x*<sup>2</sup>

*w*˙ = −*α*(*w* + *φ*), *α* > 0,

*Proof.* Substituting the control law (15) into the robot image dynamics (9) gets the closed-loop

First, we show that (*φ*˜, *φ*˙, *z*) has a unique equilibrium at the origin, and then using the

*∂ψ*13 *∂x*<sup>3</sup>

$$
\dot{z} = -\alpha z + \dot{\phi}.\tag{21}
$$

Therefore, (*φ*˜, *φ*˙, *z*)=(0, 0, 0) is the unique equilibrium of the closed-loop system. Consider the following potential energy-like function for the closed-loop system

$$P\_1 = \int\_0^{\phi} \left[ \psi\_1(\tilde{\phi}) - g\_{\phi}(\phi\_d) + g\_{\phi}(\phi) \right] d\phi$$

$$= \phi^T \psi\_1(\tilde{\phi}) - \phi^T g\_{\phi}(\phi\_d) + P\_{\phi}(\phi). \tag{22}$$

Then *<sup>φ</sup>* = *<sup>φ</sup><sup>d</sup>* is the global minimum of *<sup>P</sup>*1(*φ*) since *<sup>∂</sup>P*<sup>1</sup> *∂φ* <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>φ</sup>* <sup>=</sup> *<sup>φ</sup><sup>d</sup>* and *<sup>∂</sup>*2*P*<sup>1</sup> *∂φ*<sup>2</sup> <sup>=</sup> *∂ψ*1(*φ*˜) *∂φ* + *∂gφ*(*φ*) *∂φ* > 0.

Consider the Lyapunov function candidate

$$V(\tilde{\phi}, \phi, z) = \frac{1}{2} \dot{\phi}^T M\_{\phi}(\phi) \dot{\phi} + P\_1(\phi) - P\_1(\phi\_d) + \int\_0^z \psi\_2(z) dz. \tag{23}$$

Its time derivative along the error dynamics (19) is

$$\dot{V} = -\phi^T \psi\_2(z) + \dot{z}^T \psi\_2(z)$$

$$= -\psi\_2^T(z)[-\dot{z} + \dot{\phi}]$$

$$= -az^T \psi\_2(z) \le 0. \tag{24}$$

In order to conclude the asymptotic stability of the equilibrium, we invoke LaSalle theorem (Khalil (2002)) by considering the invariant set

$$\Omega = \{ (\tilde{\phi}, \dot{\phi} \, z) \in \mathfrak{R}^3 \times \mathfrak{R}^3 \times \mathfrak{R}^3 : \dot{V}(\tilde{\phi}, \dot{\phi} \, z) = 0 \}, \tag{25}$$

which contains only the equilibrium. Therefore the asymptotic stability of the equilibrium follows.

*Remark 3:* Since the control law is developed based on the robot image dynamics, no other measurements than the joint image positions are need for its implementation. This is important because taking time derivative of image measurements are in general not acceptable given noisy image measurements. Also, notice that the only parameters in the control law are the gravity term of the robot image dynamics, which may be also tuned on-line as in Tomei (1991).

Remark 4: Due to the occlusion effect in a single fixed camera configuration, it is impossible to determine uniquely the joint image positions for certain poses of the manipulator by the geometrical method used here. For practical purposes, the given desired image joint position *φd*<sup>1</sup> must be restricted away ±20 (deg) away from the camera optical axis.

Remark 5: Typically, *ψ<sup>T</sup> <sup>i</sup>* (*x*)=[*ψi*1(*x*1) *ψi*2(*x*2) *ψi*3(*x*3)] is taken a sigmoid function, *e.g.*, 2*k*<sup>1</sup> *<sup>π</sup> atan*( *<sup>x</sup> k*2 ) with *k*1, *k*<sup>2</sup> > 0 determining the magnitude and shape of the sigmoid function, to avoid the control signal from saturating (Escobar et al. (1999)).

### **4. Experiments results**

In order to evaluate the robustness of the proposed control in the presence of quantization errors in camera transformation, lens distortion, possible misalignment of the camera rotation angle about its optical axis, laboratory experiments were carried out.

The control law (15) with gravity compensation term *gφ*(*φd*) was applied to the robot the experiment platform in Fig.3. The sigmoid function as in Remark 5 was used with *k*<sup>1</sup> = 7.65, 2.25, 1.75 and *k*<sup>2</sup> = 4.35, 1.98, 1.15 for the joint 1, 2 3, respectively, and *α* = 5 in (18).

Fig. 6 depicts the regulation of the joint image position to its desired position, which corresponds to moving the end-effector from an initial position corresponding to *φ*(0) = [10 -100 -110] (deg) to a desired position *φd*(0)=[−30 − 30 − 30] (deg) at *t* = 0 (sec.) and *φd*(5)=[30 40 40] (deg) at *t* = 5 (sec.). Although not established in theory here, the results of tracking a desired image trajectory, which corresponds to drawing a figure in the image space shown in Fig. 7, with the proposed control law are shown in Fig. 8 and Fig. 9.

Fig. 7. Experiments of the visual servoing: tracking of the joint image trajectory in the image

Robust Visual Servoing of Robot Manipulators Based on Passivity 159

Fig. 8. Experiments of the visual servoing: control torques for joint 1 (up), 2 (middle) and 3

space.

(down).

Fig. 6. Experiments of the visual servoing: regulation of the joint image position for joint 1 (up), 2 (middle) and 3 (down). Left column shows the torque inputs, right column shows the joint image position and the desired position.

10 Will-be-set-by-IN-TECH

geometrical method used here. For practical purposes, the given desired image joint position

In order to evaluate the robustness of the proposed control in the presence of quantization errors in camera transformation, lens distortion, possible misalignment of the camera rotation

The control law (15) with gravity compensation term *gφ*(*φd*) was applied to the robot the experiment platform in Fig.3. The sigmoid function as in Remark 5 was used with *k*<sup>1</sup> = 7.65, 2.25, 1.75 and *k*<sup>2</sup> = 4.35, 1.98, 1.15 for the joint 1, 2 3, respectively, and *α* = 5 in (18). Fig. 6 depicts the regulation of the joint image position to its desired position, which corresponds to moving the end-effector from an initial position corresponding to *φ*(0) = [10 -100 -110] (deg) to a desired position *φd*(0)=[−30 − 30 − 30] (deg) at *t* = 0 (sec.) and *φd*(5)=[30 40 40] (deg) at *t* = 5 (sec.). Although not established in theory here, the results of tracking a desired image trajectory, which corresponds to drawing a figure in the image space shown in Fig. 7, with the proposed control law are shown in Fig. 8 and Fig. 9.

Fig. 6. Experiments of the visual servoing: regulation of the joint image position for joint 1 (up), 2 (middle) and 3 (down). Left column shows the torque inputs, right column shows the

joint image position and the desired position.

) with *k*1, *k*<sup>2</sup> > 0 determining the magnitude and shape of the sigmoid function, to

*<sup>i</sup>* (*x*)=[*ψi*1(*x*1) *ψi*2(*x*2) *ψi*3(*x*3)] is taken a sigmoid function, *e.g.*,

*φd*<sup>1</sup> must be restricted away ±20 (deg) away from the camera optical axis.

avoid the control signal from saturating (Escobar et al. (1999)).

angle about its optical axis, laboratory experiments were carried out.

Remark 5: Typically, *ψ<sup>T</sup>*

**4. Experiments results**

2*k*<sup>1</sup> *<sup>π</sup> atan*( *<sup>x</sup> k*2

Fig. 7. Experiments of the visual servoing: tracking of the joint image trajectory in the image space.

Fig. 8. Experiments of the visual servoing: control torques for joint 1 (up), 2 (middle) and 3 (down).

**5. Conclusions**

**6. Acknowledgement**

**7. References**

This chapter has presented an IBVS based on passivity of the robot image dynamics obtained by lumping together the manipulator dynamics, forward kinematics and perspective projection the camera. Using the passivity, controller design was considerably simplified. Regulation and tracking performances were illustrated through laboratory experiments. Although the general basic idea presented in this chapter is applicable to *n*-DOF scenarios, much research works related to more precise state observation of the robot image dynamics in the image space using its perspective projection, and analysis of the robustness issues remain

Robust Visual Servoing of Robot Manipulators Based on Passivity 161

The authors acknowledge the financial support from CONACyT-Mexico through Project

Chen, X. & Kano, H. (2002). A new state observer for perspective systems, *IEEE Transactions*

Dickmanns, E. & Graefe, V. (1988). Dynamic monocular machine vision, *Machine Vision and*

Dixon, W., Fang, Y., Dawson, D. & Flynn, T. (2003). Range identification for perspective vision

Escobar, G., Ortega, R. & Sira-Ramirez, H. (1999). Output-feedback global stabilization of

Espiau, B., Chaumette, B. & Rives, P. (1992). A New Approach to Visual Servoing in Robotics,

Fujita, M., Kawai, H. & Spong, M. (2007). Passivity-Based Dynamic Visual Feedback Control

Grammatikopoulos, L., Karras, G. & Petsa, E. (2004). Camera calibration combining images

Hashimoto, K., Ebine, T. & Kimura, H. (1996). Visual servoing with hand-eye

Hutchinson, S., Hager, G. D. & Corke, P. I. (1996). A tutorial on visual servo control, *IEEE*

Karagiannis, D. & Astolfi, A. (2005). A new solution to the problem of range identification in

Kelly, R. (1999). Regulation of manipulators in generic task space: an energy

a nonlinear benchmarksystem using a saturated passivity-based controller, *IEEE*

for Three-Dimensional Target Tracking: Stability and *L*2-Gain Performance Analysis,

with two vanishing points, *International Archives of the Photogrammetry, Remote Sensing*

manipulator-optimal control approach, *IEEE Transactions on Robotics and Automation*

perspective vision systems, *IEEE Transactions on Automatic Control* 50(12): 2074–2077.

shapingplus damping injection approach, *IEEE Transactions on Robotics and*

systems, *IEEE Transactions on Automatic Control* 48(12): 2232–2238.

*Transactions on Control Systems Technology* 7(2): 289–293.

*IEEE Trans. Robotics and Automation* 8(3): 313–326.

*IEEE Transactions on Control Tech.* 15(1): 40.

*& Spatial Information Sciences* 35(5): 99–104.

*Trans. Robot. Autom.* 12(9): 651.

*Automation* 15(2): 381–386.

to be done. Further works in these directions are undergoing.

129800 and PAPIIT-UNAM Project IN120009.

*Applications* 1(4): 223–240.

12(5): 766–774.

*on Automatic Control* 47(4): 658–663.

Fig. 9. Experiments of the visual servoing: the joint image trajectory and the desired trajectory for joint 1 (up), 2 (middle) and 3 (down).

### **5. Conclusions**

12 Will-be-set-by-IN-TECH

Fig. 9. Experiments of the visual servoing: the joint image trajectory and the desired

trajectory for joint 1 (up), 2 (middle) and 3 (down).

This chapter has presented an IBVS based on passivity of the robot image dynamics obtained by lumping together the manipulator dynamics, forward kinematics and perspective projection the camera. Using the passivity, controller design was considerably simplified. Regulation and tracking performances were illustrated through laboratory experiments.

Although the general basic idea presented in this chapter is applicable to *n*-DOF scenarios, much research works related to more precise state observation of the robot image dynamics in the image space using its perspective projection, and analysis of the robustness issues remain to be done. Further works in these directions are undergoing.

### **6. Acknowledgement**

The authors acknowledge the financial support from CONACyT-Mexico through Project 129800 and PAPIIT-UNAM Project IN120009.

### **7. References**


Kelly, R., Bugarin, E., Cervantes, I. & Alvarez-Ramirez, J. (2006). Monocular direct visual servoing for regulation of manipulators moving in the 3d cartesian space, *Decision and Control, 2006 45th IEEE Conference on*, pp. 1782–1787.

**10** 

Recep Burkan

*Turkey* 

*Istanbul University, Faculty of Engineering, Mechanical Engineering Department, Istanbul* 

**Modelling of Bound Estimation Laws and** 

**Robust Controllers for Robot Manipulators** 

**Using Functions and Integration Techniques** 

Some robust control methods have been developed in the past in order to increase tracking performance in the presence of parametric uncertainties. In the presence of parametric uncertainty, unmodelled dynamics and other sources of uncertainties, robust control laws are used. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. In early application of Corless-Leitmann [1] approach to robot manipulators [2, 3], it is difficult to compute uncertainty bound precisely. Because, uncertainty bound on parameters depends on the inertia parameters, the reference trajectory and manipulator state vector. Spong [4] proposed a new robust controller for robot manipulators using the Lyapunov theory that guaranties stability of uncertain systems. In this approach, Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. One of the advantage of Spong's approach [4] is that uncertainty on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. Yaz [6] proposed a robust control law based on Spong's study [4] and global exponential stability of uncertain system is guaranteed. However, disturbance and unmodelled dynamics are not considered in algorithm of [4, 6]. Danesh at al [7] develop Spong's approach [4] in such a manner that control scheme is made robust not only to uncertain inertia parameters but also to robust unmodelled dynamics and disturbances. Koo and Kim [8] introduce adaptive scheme of uncertainty bound on parameters for robust control of robot manipulators. In [8], upper uncertainty bound is not known as would be in robust controller [4] and uncertainty bound is estimated with estimation law in order to control the uncertain system. A new robust control approach is proposed by Liu and Goldenerg [9] for robot manipulators based on a decomposition of model uncertainty. Parameterized uncertainty is distinguished from unparameterized uncertainty and a compensator is designed for parameterized and unparameterized uncertainty. A decomposition-based control design framework for mechanical systems with

In order to increases tracking performance of uncertain systems, design of uncertainty bound estimation functions are considered. For this purpose, some uncertainty bound estimation functions are developed [11-15] based on a Lyapunov function, thus, stability of

**1. Introduction** 

model uncertainties is proposed by Liu [10].


## **Modelling of Bound Estimation Laws and Robust Controllers for Robot Manipulators Using Functions and Integration Techniques**

Recep Burkan

*Istanbul University, Faculty of Engineering, Mechanical Engineering Department, Istanbul Turkey* 

### **1. Introduction**

14 Will-be-set-by-IN-TECH

162 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Kelly, R., Bugarin, E., Cervantes, I. & Alvarez-Ramirez, J. (2006). Monocular direct visual

Kelly, R., Carelli, R., Nasisi, O., Kuchen, B. & Reyes, F. (2000). Stable visual servoing

Ma, Y., Kosecka, J. & Sastry, S. (1999). Vision guided navigation for a nonholonomic mobile

Ortega, R. & Spong, M. (1989). Adaptive motion control of rigid robots: A tutorial, *Automatica*

Slotine, J. & Li, W. (1987). On the Adaptive Control of Robot Manipulators, *The International*

Spong, M., Hutchinson, S. & Vidyasagar, M. (2006). *Robot modeling and control*, John Wiley &

Takegaki, M. & Arimoto, S. (1981). A new feedback method for dynamic control of

Tomei, P. (1991). Adaptive PD controller for robot manipulators, *IEEE Transactions on Robotics*

Weiss, L., Sanderson, A. & Neuman, C. (1987). Dynamic sensor-based control of robots with visual feedback, *Robotics and Automation, IEEE Journal of* 3(5): 404–417. Yuan, J. (1989). A general photogrammetric method for determining object positionand orientation, *IEEE Transactions on Robotics and Automation* 5(2): 129–142.

manipulators, *ASME, Transactions, Journal of Dynamic Systems, Measurement and*

robot, *Robotics and Automation, IEEE Transactions on* 15(3): 521–536.

*and Control, 2006 45th IEEE Conference on*, pp. 1782–1787.

Khalil, H. K. (2002). *Nonlinear Systems*, Prentice-Hall, New Jersey.

*Journal of Robotics Research* 6(3): 49.

5(1): 39–48.

25(6): 877–888.

*Control* 103: 119–125.

*and Automation* 7(4): 565–570.

Sons.

servoing for regulation of manipulators moving in the 3d cartesian space, *Decision*

of camera-in-hand robotic systems, *Mechatronics, IEEE/ASME Transactions on*

Some robust control methods have been developed in the past in order to increase tracking performance in the presence of parametric uncertainties. In the presence of parametric uncertainty, unmodelled dynamics and other sources of uncertainties, robust control laws are used. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. In early application of Corless-Leitmann [1] approach to robot manipulators [2, 3], it is difficult to compute uncertainty bound precisely. Because, uncertainty bound on parameters depends on the inertia parameters, the reference trajectory and manipulator state vector. Spong [4] proposed a new robust controller for robot manipulators using the Lyapunov theory that guaranties stability of uncertain systems. In this approach, Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. One of the advantage of Spong's approach [4] is that uncertainty on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. Yaz [6] proposed a robust control law based on Spong's study [4] and global exponential stability of uncertain system is guaranteed. However, disturbance and unmodelled dynamics are not considered in algorithm of [4, 6]. Danesh at al [7] develop Spong's approach [4] in such a manner that control scheme is made robust not only to uncertain inertia parameters but also to robust unmodelled dynamics and disturbances. Koo and Kim [8] introduce adaptive scheme of uncertainty bound on parameters for robust control of robot manipulators. In [8], upper uncertainty bound is not known as would be in robust controller [4] and uncertainty bound is estimated with estimation law in order to control the uncertain system. A new robust control approach is proposed by Liu and Goldenerg [9] for robot manipulators based on a decomposition of model uncertainty. Parameterized uncertainty is distinguished from unparameterized uncertainty and a compensator is designed for parameterized and unparameterized uncertainty. A decomposition-based control design framework for mechanical systems with model uncertainties is proposed by Liu [10].

In order to increases tracking performance of uncertain systems, design of uncertainty bound estimation functions are considered. For this purpose, some uncertainty bound estimation functions are developed [11-15] based on a Lyapunov function, thus, stability of

Modelling of Bound Estimation Laws and

0 0r 0 r0

action. The corrected velocity error is given as

bound on parametric uncertainty such that

candidate is defined as [15, 16].

time derivative of V of system (9) is

Equation (12) is arranged as

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 165

where is a positive definite matrix. Then the following nominal control law is considered:

τ M (q)q C (q,q)q G (q) Kσ

where *π*0Rp represents the fixed parameters in dynamic model and Kσ is the vector of PD

Where u(t) is the additional robust control input. It is assumed that there exists an unknown

Since ρR+p is assumed to be unknown, ρ should be estimated with the estimation law to

 M(q)σ C(q,q)σ Kσ Y(q,q,q ,q )( r r π u(t)) (9) By taking into account above parameters and control algorithm, the Lyapunov function

where BRnxn is a positive diagonal matrix, *(t)* is chosen as a pxp dimensional diagonal

(11)

TT T T T T 2 V -q Kq-q ΛKΛq σ Yu(t) σ Yπ ρ(t) (t) (t)ρ(t) ρ(t) (t) ρ(t) (12)

T T 2

(ρ ρ*ˆ ˆˆˆ* (t)) (t) (t)(ρ ρ(t)) (ρ ρ(t)) (t) ρ(t) 0

 *[ M(q) C(q,q)] <sup>n</sup>*

(13)

(4)

<sup>σ</sup> *q qr* <sup>q</sup> <sup>Λ</sup><sup>q</sup> (5)

π πρ <sup>0</sup> (7)

ρ(t) ρ ρ *ˆ* (t) (8)

*R* [17, 18], and taking

0 rr rr 0 τ τ Y(q,q,q ,q )u(t) Y(q,q,q ,q )( π u(t))-Kσ (6)

(10)

rr 0

Y(q,q,q ,q )π Kσ

The control input τ is defined in terms of the nominal control vector 0 τ as

control the system properly. ρ*ˆ* (t) shows the estimate of ρ and ρ(t) is

1 11 T T T2 V( ,q,ρ(t)) <sup>σ</sup> M(q)<sup>σ</sup> q Bq <sup>ρ</sup>(t) (t) <sup>ρ</sup>(t) 2 22

matrix changes in time. The time derivative of V along the trajectories is

 <sup>T</sup> <sup>1</sup> T TT T 2 <sup>V</sup> <sup>σ</sup> M(q) <sup>σ</sup> M(q)<sup>σ</sup> q Bq <sup>ρ</sup>(t) (t) (t)ρ(t) <sup>ρ</sup>(t) (t) (t) <sup>2</sup> *= + ++ +*

TT T T

V -q Kq-q ΛKΛ q σ Yu(t) σ Yπ

Substituting (6) into (1) and after some algebra yields

Taking *B K* 2 , using the property - 2 0 *<sup>T</sup>*

uncertain system is guaranteed. In early derivation of uncertainty bound estimation laws [11-13], only a single derivation is possible because selection of variable function is difficult for other derivation and first order differential equation is used. Only exponential function and logarithmic functions are used for derivations because it is diffucut to define variable functions for other derivations.

In previous studies, some robust control laws are introduced, however, a method for derivation of adaptive bound estimation law for robust controllers is not proposed. Recently, a new approach for derivation of bound estimation laws for robust control of robot manipulators is proposed [14, 15]. A general equation is developed based on the Lyapunov theory in order to derive adaptive bound estimation laws and stability of uncertain system is guaranteed. In the approach [15], some functions depending on robot kinematics and control parameters and proper integration techniques can be used for derivation of new bound estimation laws. Then, new bound estimation laws are derived and this derivations also show how the general rule can be used for derivation of different bound estimation laws. After that, four new robust controllers are designed based on each bound estimation law. Lyapunov theory based on Corless-Leitmann [1] approach is used and uniform boundedness error convergence is achieved. This study also shows that bound estimation laws for robust control input do not only include these derivations but also allows derivation of other bound estimation laws for robust controllers provided that appropriate function and proper integration techniques are chosen. In this work, based on the study [15], some appropriate functions are developed and proper integration techniques are chosen. As results, new uncertainty bound estimation laws for robust control input are developed and new robust controllers are proposed. In derivations, some functions and integration techniques are used.

### **2. A method for derivation of bound estimation laws**

In the absence of friction or other disturbances, the dynamic model of an n-link manipulator can be written as [16]

$$\mathbf{M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = \tau} \tag{1}$$

where q denotes generalised coordinates, τ is the n-dimensional vector of applied torques (or forces), *M(q)* is a positive definite mass matrix, C(q,q)q is the n-dimensional vector of centripetal and Coriolis terms and G(q) is the n-dimensional vector of gravitational terms. Equation (1) can also be expressed in the following form.

$$
\pi = \text{Y}(\mathbf{q}\_l \dot{\mathbf{q}}\_l \ddot{\mathbf{q}}\_l) \text{n} \tag{2}
$$

where *π* is a p-dimensional vector of robot inertia parameters and Y is an nxp matrix which is a function of joint position, velocity and acceleration. For any specific trajectory, the desired position, velocity and acceleration vectors are qd, *<sup>d</sup> q* and *<sup>d</sup> q* . The measured actual position and velocity errors are *<sup>d</sup> qqq* , and *<sup>d</sup> qqq* . Using the above information, the corrected desired velocity and acceleration vectors for nonlinearities and decoupling effects are proposed as:

$$
\dot{\mathbf{q}}\_r = \dot{\mathbf{q}}\_d - \Lambda \tilde{\mathbf{q}} \; ; \; \ddot{\mathbf{q}}\_r = \ddot{\mathbf{q}}\_d - \Lambda \dot{\tilde{\mathbf{q}}} \tag{3}
$$

164 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

uncertain system is guaranteed. In early derivation of uncertainty bound estimation laws [11-13], only a single derivation is possible because selection of variable function is difficult for other derivation and first order differential equation is used. Only exponential function and logarithmic functions are used for derivations because it is diffucut to define variable

In previous studies, some robust control laws are introduced, however, a method for derivation of adaptive bound estimation law for robust controllers is not proposed. Recently, a new approach for derivation of bound estimation laws for robust control of robot manipulators is proposed [14, 15]. A general equation is developed based on the Lyapunov theory in order to derive adaptive bound estimation laws and stability of uncertain system is guaranteed. In the approach [15], some functions depending on robot kinematics and control parameters and proper integration techniques can be used for derivation of new bound estimation laws. Then, new bound estimation laws are derived and this derivations also show how the general rule can be used for derivation of different bound estimation laws. After that, four new robust controllers are designed based on each bound estimation law. Lyapunov theory based on Corless-Leitmann [1] approach is used and uniform boundedness error convergence is achieved. This study also shows that bound estimation laws for robust control input do not only include these derivations but also allows derivation of other bound estimation laws for robust controllers provided that appropriate function and proper integration techniques are chosen. In this work, based on the study [15], some appropriate functions are developed and proper integration techniques are chosen. As results, new uncertainty bound estimation laws for robust control input are developed and new robust controllers are proposed. In derivations, some functions and

In the absence of friction or other disturbances, the dynamic model of an n-link manipulator

where q denotes generalised coordinates, τ is the n-dimensional vector of applied torques (or forces), *M(q)* is a positive definite mass matrix, C(q,q)q is the n-dimensional vector of centripetal and Coriolis terms and G(q) is the n-dimensional vector of gravitational terms.

where *π* is a p-dimensional vector of robot inertia parameters and Y is an nxp matrix which is a function of joint position, velocity and acceleration. For any specific trajectory, the

position and velocity errors are *<sup>d</sup> qqq* , and *<sup>d</sup> qqq* . Using the above information, the corrected desired velocity and acceleration vectors for nonlinearities and decoupling effects

*r d qq q* ; q q *r d q*

M(q)q C(q,q)q G(q) *τ* (1)

Y(q,q,q ) π (2)

(3)

*q* . The measured actual

functions for other derivations.

integration techniques are used.

can be written as [16]

are proposed as:

**2. A method for derivation of bound estimation laws** 

Equation (1) can also be expressed in the following form.

 

desired position, velocity and acceleration vectors are qd, *<sup>d</sup> q* and *<sup>d</sup>*

where is a positive definite matrix. Then the following nominal control law is considered:

$$\begin{aligned} \mathbf{r}\_0 &= \mathbf{M}\_0(\mathbf{q})\ddot{\mathbf{q}}\_\mathbf{r} + \mathbf{C}\_0(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}}\_\mathbf{r} + \mathbf{G}\_0(\mathbf{q}) - \mathbf{K}\sigma\\ &= \mathbf{Y}(\mathbf{q}, \dot{\mathbf{q}}\_\mathbf{r}\dot{\mathbf{q}}\_\mathbf{r}, \ddot{\mathbf{q}}\_\mathbf{r})\mathbf{r}\_0 - \mathbf{K}\sigma \end{aligned} \tag{4}$$

where *π*0Rp represents the fixed parameters in dynamic model and Kσ is the vector of PD action. The corrected velocity error is given as

$$
\sigma \mathbf{o} = \mathbf{q} - \dot{\mathbf{q}}\_r = \dot{\mathbf{\tilde{q}}} + \Lambda \mathbf{\tilde{q}} \tag{5}
$$

The control input τ is defined in terms of the nominal control vector 0 τ as

$$\mathbf{r} = \mathbf{r}\_0 + \mathbf{Y} (\mathbf{q}\_l \dot{\mathbf{q}}\_l \dot{\mathbf{q}}\_r, \ddot{\mathbf{q}}\_r) \mathbf{u}(\mathbf{t}) = \mathbf{Y} (\mathbf{q}\_l \dot{\mathbf{q}}\_l \dot{\mathbf{q}}\_r, \ddot{\mathbf{q}}\_r) (\mathbf{n}\_0 + \mathbf{u}(\mathbf{t})) \text{-K}\mathbf{o} \tag{6}$$

Where u(t) is the additional robust control input. It is assumed that there exists an unknown bound on parametric uncertainty such that

$$
\tilde{\pi} = \mathbf{n}\_0 - \mathbf{n} \le \mathbf{p} \tag{7}
$$

Since ρR+p is assumed to be unknown, ρ should be estimated with the estimation law to control the system properly. ρ*ˆ* (t) shows the estimate of ρ and ρ(t) is

$$
\tilde{\boldsymbol{\phi}}(\mathbf{t}) = \mathbf{p} - \hat{\boldsymbol{\phi}}(\mathbf{t}) \tag{8}
$$

Substituting (6) into (1) and after some algebra yields

$$\mathbf{M(q)\dot{\sigma} + C(q,\dot{q})\sigma + K\sigma = Y(q,\dot{q},\dot{q}\_r,\ddot{q}\_r)(\tilde{\mathbf{u}} + \mathbf{u}(t))}\tag{9}$$

By taking into account above parameters and control algorithm, the Lyapunov function candidate is defined as [15, 16].

$$\mathbf{V}(\sigma, \mathbf{q}, \tilde{\mathbf{p}}(\mathbf{t})) = \frac{1}{2} \sigma^{\mathrm{T}} \mathbf{M}(\mathbf{q}) \sigma + \frac{1}{2} \tilde{\mathbf{q}}^{\mathrm{T}} \mathbf{B} \tilde{\mathbf{q}} + \frac{1}{2} \tilde{\mathbf{p}}(\mathbf{t})^{\mathrm{T}} \Phi(\mathbf{t})^{2} \tilde{\boldsymbol{\rho}}(\mathbf{t}) \tag{10}$$

where BRnxn is a positive diagonal matrix, *(t)* is chosen as a pxp dimensional diagonal matrix changes in time. The time derivative of V along the trajectories is

$$\dot{\mathbf{V}} = \mathbf{o}^{\mathrm{T}} \mathbf{M}(\mathbf{q}) \dot{\boldsymbol{\sigma}} + \frac{1}{2} \mathbf{o}^{\mathrm{T}} \dot{\mathbf{M}}(\mathbf{q}) \mathbf{o} + \tilde{\mathbf{q}}^{\mathrm{T}} \mathbf{B} \dot{\tilde{\mathbf{q}}} + \tilde{\mathbf{p}}(\mathbf{t})^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t}) \dot{\boldsymbol{\Phi}}(\mathbf{t}) \tilde{\boldsymbol{\rho}}(\mathbf{t}) + \tilde{\boldsymbol{\rho}}(\mathbf{t})^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t})^{2} \dot{\tilde{\boldsymbol{\rho}}}(\mathbf{t}) \tag{11}$$

Taking *B K* 2 , using the property - 2 0 *<sup>T</sup> [ M(q) C(q,q)] <sup>n</sup> R* [17, 18], and taking time derivative of V of system (9) is

$$\dot{\mathbf{V}} = \dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \mathbf{\tilde{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \mathbf{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \mathbf{\tilde{u}} + \tilde{\rho}(\mathbf{t})^{\mathrm{T}} \Phi(\mathbf{t}) \dot{\Phi}(\mathbf{t}) \tilde{\rho}(\mathbf{t}) + \tilde{\rho}(\mathbf{t})^{\mathrm{T}} \Phi(\mathbf{t})^{2} \dot{\tilde{\rho}}(\mathbf{t}) \tag{12}$$

Equation (12) is arranged as

$$\begin{split} \dot{\mathbf{V}} &= -\dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \ \ddot{\mathbf{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \mathbf{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\mathbf{n}} \\ &+ \left( \boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t}) \right)^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t}) \dot{\boldsymbol{\Phi}}(\mathbf{t}) (\boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t})) - \left( \boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t}) \right)^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t})^{2} \dot{\boldsymbol{\hat{\rho}}}(\mathbf{t}) \le 0 \end{split} \tag{13}$$

*(t) (t) ˆ* (since is a constant). Remembering that and if u(t) is taken as the estimated term of uncertainty bound, that is u(t) ρ*ˆ* (t) then Equation (13 is written as

$$\begin{aligned} \dot{\mathbf{V}} &= -\dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \dot{\mathbf{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \mathbf{q} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} (-\hat{\rho}(\mathbf{t})) \; \mathbf{+} \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{\rho} \\ &+ [\boldsymbol{\rho} - \hat{\rho}(\mathbf{t})]^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t}) \dot{\boldsymbol{\Phi}}(\mathbf{t}) [\boldsymbol{\rho} - \hat{\rho}(\mathbf{t})] - (\boldsymbol{\rho} - \hat{\rho}(\mathbf{t}))^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t})^{\mathrm{2}} \; \dot{\boldsymbol{\rho}}(\mathbf{t})] \leq 0 \end{aligned} \tag{14}$$

Modelling of Bound Estimation Laws and

**2.1 First choice of (t)-1** For the first derivation of

Then

After integration, the result is

derived from Lyapunov function. As a result,

1 1 <sup>1</sup>

 

 

 

1

2

e e

p

Substituting Equation (24) into (23) yields

guarantess stability of uncertain system. However, (t)-1 and

search for appropriate function of (t)-1 as a solution of the Equation (23).

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 167

The Equation (23) is a general equation for derivation of the bound estimation law and it is

derived depending on the function (t)-1. For derivation, selection of (t)-1 and integration techniques are very important. There is no certain rule for selection of (t)-1 and integration techniques for this systems. System state parameters and mathematical insight are used to

> T T <sup>1</sup> <sup>Y</sup> <sup>σ</sup>dt)i i <sup>Y</sup> <sup>σ</sup>dt) e e *i i ( ( <sup>i</sup> (t) diag( sin( ))*

T T 11 11 T T 22 22

e e e e

*sin( )(Y ) ˆ(t)*

e e *p p*

*ˆ(t) sin( )(Y ) (t) [ dt] ..... ... ................*

*p ( ( T*

*( / )cos( ) ˆ(t)*

 

 

 

*ˆ(t) ( / )cos( )*

*ˆ(t) sin( )(Y )*

Y σdt) Y σdt) 1 1 1 1 <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>2</sup> <sup>1</sup> <sup>2</sup> 2 2

*( ( T ( ( T*

 

 

T T p p

*p p*

 

<sup>Y</sup> <sup>σ</sup>dt) <sup>2</sup> 1 1 <sup>2</sup> 2 2

*ˆ(t) ( / )cos( ) (t) (t)C ..... ..... ..... ................*

*( (*

e *<sup>p</sup>*

T T p p

 

2 Y σdt) Y σdt) Y σdt)

*(( ( (( (*

Y σdt) Y σdt)

1 1 <sup>1</sup> <sup>1</sup>

*p ( p p p*

<sup>2</sup> <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>2</sup> 2 2

/ )e e e / )e e e

2 Y σdt) Y σdt)

 

 

*ˆ(t) ( sin(* <sup>T</sup> <sup>p</sup>

*sin( )*

*( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( )*

/ )e e

 

*p p*

*sin( ) <sup>C</sup> .....*

T T 11 11 T T 22 22

Y σdt) Y σdt)

Y σdt) Y σdt)

T T p p

 

*sin( )*

Y σdt) Y σdt)

*p p*

*p ( ( p p*

e e

*( ( ( (*

e e

*( (*

*..... ................*

*ˆ(t)* , <sup>1</sup> *(t)* is chosen as a time varying function such that

 

 

 

<sup>1</sup>

<sup>T</sup> 1 1 <sup>T</sup> 2 2

 

Y σdt)

<sup>T</sup> <sup>p</sup>

TT T 11 11 11 TT T 22 22 22

Y σdt)

 

e 1 e 1

> 

 

e

*p*

*)cos( )*

Y σdt)

*( p*

(24)

*ˆ(t)* all derived from Equation (23)

*ˆ(t)* are unknown and

*ˆ(t)* is

1 1

(25)

(26)

(27)

*<sup>p</sup>* 1 *(t) .. .....*

1

 

> 1 2

*.....*

 

Equation (14) can be arranged as

$$\begin{aligned} \dot{\mathbf{V}} &= -\dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \dot{\mathbf{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \, \mathbf{q} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} [\mathbf{p} \cdot \dot{\mathbf{p}}(t)] \\ &+ [\mathbf{b} \cdot \dot{\mathbf{p}}(t)]^{\mathrm{T}} \Phi(t) \dot{\Phi}(t) [\mathbf{b} \cdot \dot{\mathbf{p}}(t)] - (\mathbf{p} \cdot \dot{\mathbf{p}}(t))^{\mathrm{T}} \Phi(t)^{2} \dot{\hat{\rho}}(t)] \le 0 \end{aligned} \tag{15}$$

Consequently, a suitable expression for the time derivative of V is obtained.

$$\dot{\mathbf{V}} = \dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} - \dot{\mathbf{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \mathbf{\tilde{q}} + \left[ (\boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t}))^{\mathrm{T}} \right] \mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma} + \Phi(\mathbf{t}) \dot{\Phi}(\mathbf{t}) (\boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t})) - \Phi(\mathbf{t})^{2} \dot{\boldsymbol{\rho}}(\mathbf{t}) \Big] \leq 0 \tag{16}$$

where T T -q Kq-q ΛKΛq 0 . If the rest of Equation (16) is zero, system will be stable. Remaining terms in Equation (16) are

$$\left[ \left[ (\rho \cdot \dot{\rho}(t)) \right]^T \right] \mathbf{Y}^T \mathbf{o} + \Phi(t) \dot{\Phi}(t) (\rho - \dot{\rho}(t)) - \Phi(t)^2 \dot{\rho}(t) \right] = 0 \tag{17}$$

[(ρ-ρ*ˆ* (t)]is considered as a common multiplier then

$$\mathbf{Y}^T \mathbf{o} + \Phi(\mathbf{t})\dot{\Phi}(\mathbf{t})(\mathbf{o} - \hat{\mathbf{p}}(\mathbf{t})) - \Phi(\mathbf{t})^2 \dot{\hat{\mathbf{p}}}(\mathbf{t}) = \mathbf{0} \tag{18}$$

Hence, we look for the conditions for which the equation

T 2 Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t) 0

is satisfied. Equation (18) can be written as

$$
\Phi(\mathbf{t})\dot{\Phi}(\mathbf{t})(\mathbf{b}-\dot{\mathbf{p}}(\mathbf{t}))-\Phi(\mathbf{t})\Phi(\mathbf{t})\dot{\hat{\mathbf{p}}}(\mathbf{t})=-\mathbf{Y}^{\mathrm{T}}\mathbf{a}\tag{19}
$$

Then

$$
\Phi(\mathbf{t})\dot{\hat{\boldsymbol{\phi}}}(\mathbf{t}) + \dot{\Phi}(\mathbf{t})\dot{\hat{\boldsymbol{\phi}}}(\mathbf{t}) = \Phi(\mathbf{t})^{-1}\mathbf{Y}^{\mathrm{T}}\mathbf{o} + \dot{\Phi}(\mathbf{t})\mathbf{b} \tag{20}
$$

Equation (20) is arranged as

$$\frac{d}{dt}(\Phi(\mathbf{t})\hat{\boldsymbol{\phi}}(\mathbf{t})) = \Phi(\mathbf{t})^{-1}\mathbf{Y}^{\mathrm{T}}\boldsymbol{\alpha} + \dot{\Phi}(\mathbf{t})\boldsymbol{\phi} \tag{21}$$

Integration both side of Equation (21) yields

$$\Phi(\mathbf{t})\hat{\phi}(\mathbf{t}) = \int \Phi(\mathbf{t})^{-1} \mathbf{Y}^{\mathrm{T}} \mathbf{c} dt + \int \dot{\Phi}(\mathbf{t}) \mathbf{p} d\mathbf{t} + \mathbf{C} \tag{22}$$

Then, a general equation for derivation derivation of bound estimation law is developed as [14, 15]

$$\hat{\phi}(\mathbf{t}) = \Phi(\mathbf{t})^{-1} \{ \left[ \Phi(\mathbf{t})^{\cdot 1} \mathbf{Y}^{\Gamma} \mathbf{c} dt \right] + \rho + \Phi(\mathbf{t})^{\cdot 1} \mathbf{C} \tag{23}$$

The Equation (23) is a general equation for derivation of the bound estimation law and it is derived from Lyapunov function. As a result, *ˆ(t)* all derived from Equation (23) guarantess stability of uncertain system. However, (t)-1 and *ˆ(t)* are unknown and *ˆ(t)* is derived depending on the function (t)-1. For derivation, selection of (t)-1 and integration techniques are very important. There is no certain rule for selection of (t)-1 and integration techniques for this systems. System state parameters and mathematical insight are used to search for appropriate function of (t)-1 as a solution of the Equation (23).

### **2.1 First choice of (t)-1**

166 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

T T 2

T T 2

*ˆ ˆ ˆˆ ˆ*

T T T T <sup>2</sup> V -q Kq-q ΛKΛq [( ρ ρ*ˆ*(t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 (16)

T T <sup>2</sup> [(ρ-ρ*ˆ* (t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 (17)

T 2 Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t) 0 (18)

<sup>T</sup> (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) (t)ρ(t) Y σ (19)

*ˆ ˆ*ρ(t) (t) Y σ (t)ρ (20)


*ˆ* (21)

(t) C (23)

*ˆ*

[ρ ρ(t)] (t) (t)[ρ ρ(t)] (ρ ρ(t)) (t) ρ(t)] 0

[ρ-ρ(t)] (t) (t)[ρ-ρ(t)] (ρ -ρ(t)) (t) (t)] 0

where T T -q Kq-q ΛKΛq 0 . If the rest of Equation (16) is zero, system will be stable.

T 2 Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t) 0


<sup>d</sup> -1 T (t) (t)) (t) Y <sup>σ</sup> (t)<sup>ρ</sup> dt*(*

Then, a general equation for derivation derivation of bound estimation law is developed as


*ˆ ˆˆ ˆ*

estimated term of uncertainty bound, that is u(t) ρ*ˆ* (t) then Equation (13 is written as

TT T T

V -q Kq-q ΛKΛq σ Y( ρ(t)) σ Yρ

TT T

V -q Kq-q ΛKΛq σ Y[ρ-ρ(t)]

Consequently, a suitable expression for the time derivative of V is obtained.

 

(15)

and if u(t) is taken as the

(14)

*ˆ* (since is a constant). Remembering that

Then

[14, 15]

*(t) (t)* 

Equation (14) can be arranged as

Remaining terms in Equation (16) are

[(ρ-ρ*ˆ* (t)]is considered as a common multiplier then

is satisfied. Equation (18) can be written as

Integration both side of Equation (21) yields

Equation (20) is arranged as

Hence, we look for the conditions for which the equation

For the first derivation of *ˆ(t)* , <sup>1</sup> *(t)* is chosen as a time varying function such that

$$\Phi(\mathbf{t})^{-1} = \text{diag}(\mathcal{J}\_{\mathbf{i}} \mathbf{e}^{(a\_i \int \mathbf{Y}^{\mathbb{T}} \text{odt} \mathbf{t})\_i} \sin(\mathbf{e}^{(a\_i \int \mathbf{Y}^{\mathbb{T}} \text{odt} \mathbf{t})\_i})) \tag{24}$$

Substituting Equation (24) into (23) yields

$$
\begin{bmatrix}
\hat{\rho}(t)\_{1} \\
\hat{\rho}(t)\_{2} \\
\dots \\
\hat{\rho}(t)\_{r}
\end{bmatrix} = \Phi(t)^{-1} \left| \begin{bmatrix}
\rho\_{1}\mathbf{e}^{(a\_{1}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{1}} \sin(\mathbf{e}^{(a\_{1}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{1}}) \mathbf{Y}^{T}\sigma \\
\rho\_{2}\mathbf{e}^{(a\_{2}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{2}} \sin(\mathbf{e}^{(a\_{2}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{2}}) \mathbf{Y}^{T}\sigma \\
\dots \\
\dots \\
\end{bmatrix} \begin{bmatrix}
\rho\_{1} \\
\rho\_{2} \\
\dots \\
\dots \\
\rho\_{p}
\end{bmatrix} + \Phi(t)^{-1} \begin{bmatrix} 1 \\ 1 \\ \dots \\
1 \end{bmatrix} \right| \tag{25}
$$

After integration, the result is

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
\end{bmatrix} + \begin{bmatrix}
\rho\_1\\ \rho\_2\\ \dots\\ \rho\_p
\end{bmatrix} + \Phi(t) \mathbf{C}^{-1} \begin{bmatrix} 1\\ 1\\ 1\\ \dots\\ 1 \end{bmatrix} \tag{26}
$$

Then

TT T 11 11 11 TT T 22 22 22 T T p p 2 Y σdt) Y σdt) Y σdt) 1 1 <sup>1</sup> <sup>2</sup> <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>2</sup> 2 2 2 Y σdt) Y σdt) / )e e e / )e e e / )e e *p p (( ( (( ( p ( ( p p ( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( ) ..... ................ ˆ(t) ( sin(* <sup>T</sup> <sup>p</sup> T T 11 11 T T 22 22 T T p p 1 2 Y σdt) Y σdt) Y σdt) 1 Y σdt) Y σdt) 2 Y σdt) Y σdt) p e e e e e e e *p p p ( p ( ( ( ( ( ( ..... )cos( ) sin( ) sin( ) <sup>C</sup> ..... sin( )* (27)

If *ˆ( )* 0 is taken as initial condition, constant C is equivalent to Cos(1). So, the estimation law for the uncertainty bound is derived as.

TT T 11 11 11 TT T 22 22 22 T T p p 2 Y σdt) Y σdt) Y σdt) 1 1 <sup>1</sup> <sup>2</sup> <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>2</sup> 2 2 2 Y σdt) Y σdt) / )e e e / )e e e / )e e *p p (( ( (( ( p ( ( p p ( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( ) ..... ................ ˆ(t) ( sin(* <sup>T</sup> <sup>p</sup> T T 11 11 T T 22 22 T T p p 1 2 Y σdt) Y σdt) Y σdt) 1 Y σdt) Y σdt) 2 Y σdt) Y σdt) p e e e e e 1 e e *p p p ( p ( ( ( ( ( ( ..... )cos( ) sin( ) sin( ) Cos( ) ..... sin( )* (28)

### **2.2 Second choice of (t)-1**

For the second derivation of *ˆ(t)* , <sup>1</sup> *(t)* is defined as

$$\Phi(t)^{-1} = \operatorname{diag}(\beta\_i \frac{\mathbf{e}^{\left[\left(a\_i \mathbf{Y}^\top \text{cdt}\right)\_i\right]}}{1 + \mathbf{e}^{\left(\left[\frac{2a\_i \mathbf{Y}^\top \text{cdt}}{\mathbf{c}}\right]\_i\right)}}\tag{29}$$

Modelling of Bound Estimation Laws and

 

 

2

 

*p*

*ˆ(t)*

If 0 *ˆ( )* 

*.....*

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 169

e

( Y σdt) <sup>1</sup> 2 Y <sup>σ</sup>dt)

1 ( Y σdt) <sup>2</sup> <sup>2</sup> 2 Y <sup>σ</sup>dt)

1 ( Y σdt) <sup>2</sup> <sup>2</sup> 2 Y <sup>σ</sup>dt)

<sup>1</sup> 1 e

e 1 e e 1 e

*(*

*(*

*.....*

e 1 e *<sup>p</sup>*

*<sup>p</sup> (*

<sup>T</sup> <sup>p</sup>

(32)

2 Y σdt)

( Y σdt) <sup>1</sup> 2 Y <sup>σ</sup>dt)

e 1 e e

*(*

*(*

*.....*

(35)

e 1 e *<sup>p</sup>*

*<sup>p</sup> (*

T

2 Y σdt)

<sup>T</sup> <sup>p</sup>

(33)

<sup>T</sup> p p <sup>T</sup> p p

( Y <sup>σ</sup>dt) ( Y <sup>σ</sup>dt)

p p <sup>T</sup> p p

*T T*

 

> 

> >

*p*

( Y <sup>σ</sup>dt) ( Y <sup>σ</sup>dt)

*arctan( ) .....*

*p*

 i i ii i Sin ( Y <sup>σ</sup>dt) Cos( Y <sup>σ</sup>dt) *)* (34)

 *p*

*)*

*<sup>C</sup> .....*

1 1

is taken as initial condition, constant C is equivalent to -arctan(1). So, the

*)*

e

e

<sup>T</sup> 1 1 <sup>T</sup> 1 1

<sup>T</sup> 2 2 <sup>T</sup> 2 2

<sup>T</sup> 1 1

*( /) arctan( )*

( Y σdt) 2 ( Y σdt)

<sup>2</sup> ( Y <sup>σ</sup>dt) <sup>2</sup> 2 2 2 Y <sup>σ</sup>dt)

*ˆ(t) (/) arctan( )*

1 1 2 Y <sup>σ</sup>dt)

*(*

*(*

1 e e

1 e

e

1 e *<sup>p</sup>*

estimation law for the uncertainty bound is derived as.

1 1 2 Y <sup>σ</sup>dt)

*(*

*(*

1 e e

1 e

e

1 e *<sup>p</sup>*

e

1 ( Y σdt)

*......... ˆ(t)*

*p p (*

2

 

**2.3 Third choice of (t)-1** For the third derivation of

After integration, the result is

 

 

*p*

*ˆ(t)*

*.....*

e

1 ( Y σdt)

*......... ˆ(t)*

*p p (*

<sup>T</sup> 2 2

<sup>T</sup> <sup>p</sup>

<sup>T</sup> 1 1 <sup>T</sup>

<sup>T</sup> 2 2 <sup>T</sup> 2 2

e

e

e

*ˆ(t)* , <sup>1</sup> *(t)* is defined as

*(Y ) ˆ(t)*

1

 

1

*ˆ(t) (Y )*

1 2T <sup>T</sup> *(t) diag(*

2 T T 1 1 11 11 <sup>1</sup> <sup>1</sup> 2 T T 2 1 2 2 22 22 2

Sin ( Y σdt) Cos( Y σdt) Sin ( Y σdt) Cos( Y σdt)

 

 

 

2 T T p p pp p

*ˆ(t) (Y ) (t) [ dt] ..... ..... .........*

Sin ( Y σdt) Cos( Y σdt)

*p T p*

<sup>T</sup> p p

2 Y σdt)

( Y σdt)

*(/) arctan(*

T 1 1

*( /) arctan( )*

( Y σdt) 2 ( Y σdt)

<sup>T</sup> 2 2

T p p <sup>T</sup> <sup>p</sup>

2 Y σdt)

( Y σdt)

*(/) arctan(*

Substitution of Equation (34) into Equation (23) yields

1

*(t) C .....* 

1

<sup>2</sup> ( Y <sup>σ</sup>dt) <sup>2</sup> 2 2 2 Y <sup>σ</sup>dt)

*ˆ(t) (/) arctan( )*

Substituting Equation (29) into (23) yields

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\\hat{\rho}(t)\_2\\\dots\\\hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
\boldsymbol{\theta} \\
1 + \frac{\mathbf{e}^{\left\|(\boldsymbol{\alpha}\_1 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_1}{1 + \mathbf{e}^{\left\|(\boldsymbol{\alpha}\_2 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_2}}(\boldsymbol{Y}^T \boldsymbol{\sigma})\_1\\\boldsymbol{\theta} \frac{\mathbf{e}^{\left\|(\boldsymbol{\alpha}\_2 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_2}{1 + \mathbf{e}^{\left\|(\boldsymbol{\alpha}\_2 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_2}}(\boldsymbol{Y}^T \boldsymbol{\sigma})\_2\\\dots}
\dots
\dots
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\rho}\_1\\\boldsymbol{\rho}\_2\\\vdots\\\boldsymbol{\rho}\_p\\\vdots\\\boldsymbol{\rho}\_p
\end{bmatrix} + \Phi(t)^{-1} \mathbf{C} \begin{bmatrix}
1\\1\\\vdots\\\ldots\\1
\end{bmatrix} \tag{30}
$$

After integration, the result is

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
(\beta\_1 / \alpha\_1) \arctan(\mathbf{e}^{\left[ (a\_1 \mathbf{Y}^\top \text{cdt})\_1 \right]} \\
(\beta\_2 / \alpha\_2) \arctan(\mathbf{e}^{\left[ (a\_2 \mathbf{Y}^\top \text{cdt})\_2 \right]} \\
\dots \\
(\beta\_p / \alpha\_p) \arctan(\mathbf{e}^{\left[ (a\_p \mathbf{Y}^\top \text{cdt})\_p \right]}
\end{bmatrix} + \Phi(t) \begin{bmatrix}
\rho\_1 \\ \rho\_2 \\ \dots \\ \rho\_p
\end{bmatrix} + C \begin{bmatrix} 1 \\ 1 \\ \dots \\ 1 \end{bmatrix} \tag{31}
$$

After multiplication by <sup>1</sup> *(t)* , the result will be

<sup>T</sup> 1 1 <sup>T</sup> 1 1 <sup>T</sup> 1 1 <sup>T</sup> 2 2 <sup>T</sup> 2 2 <sup>T</sup> 2 2 <sup>T</sup> p p <sup>T</sup> <sup>p</sup> ( Y σdt) 2 ( Y σdt) 1 1 2 Y <sup>σ</sup>dt) 1 ( Y σdt) <sup>2</sup> ( Y <sup>σ</sup>dt) <sup>2</sup> 2 2 2 Y <sup>σ</sup>dt) ( Y σdt) 2 2 Y σdt) e e 1 e e e 1 e e e 1 e *<sup>p</sup> ( ( p p p ( ( /) arctan( ) ˆ(t) ˆ(t) (/) arctan( ) ..... ......... ˆ(t) (/) arctan(* <sup>T</sup> 1 1 <sup>T</sup> 1 1 <sup>T</sup> 2 2 <sup>T</sup> 2 2 <sup>T</sup> p p <sup>T</sup> p p <sup>T</sup> <sup>p</sup> ( Y σdt) <sup>1</sup> 2 Y <sup>σ</sup>dt) 1 ( Y σdt) <sup>2</sup> <sup>2</sup> 2 Y <sup>σ</sup>dt) ( Y <sup>σ</sup>dt) ( Y <sup>σ</sup>dt) 2 Y σdt) e 1 e e 1 e e 1 e *<sup>p</sup> ( ( p <sup>p</sup> ( <sup>C</sup> ..... ..... )* (32)

If 0 *ˆ( )* is taken as initial condition, constant C is equivalent to -arctan(1). So, the estimation law for the uncertainty bound is derived as.

<sup>T</sup> 1 1 <sup>T</sup> 1 1 T 1 1 <sup>T</sup> 2 2 <sup>T</sup> 2 2 <sup>T</sup> 2 2 T p p <sup>T</sup> <sup>p</sup> ( Y σdt) 2 ( Y σdt) 1 1 2 Y <sup>σ</sup>dt) 1 ( Y σdt) <sup>2</sup> ( Y <sup>σ</sup>dt) <sup>2</sup> 2 2 2 Y <sup>σ</sup>dt) ( Y σdt) 2 2 Y σdt) e e 1 e e e 1 e e e 1 e *<sup>p</sup> ( ( p p p ( ( /) arctan( ) ˆ(t) ˆ(t) (/) arctan( ) ..... ......... ˆ(t) (/) arctan(* <sup>T</sup> 1 1 T 1 1 <sup>T</sup> 2 2 <sup>T</sup> 2 2 T p p <sup>T</sup> p p <sup>T</sup> <sup>p</sup> ( Y σdt) <sup>1</sup> 2 Y <sup>σ</sup>dt) 1 ( Y σdt) <sup>2</sup> <sup>2</sup> 2 Y <sup>σ</sup>dt) ( Y <sup>σ</sup>dt) ( Y <sup>σ</sup>dt) 2 Y σdt) e 1 e e <sup>1</sup> 1 e e 1 e *<sup>p</sup> ( ( p <sup>p</sup> ( arctan( ) ..... ..... )* (33)

### **2.3 Third choice of (t)-1**

168 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

T T p p

T T p p

 

*sin( )*

1 e

*T*

*(Y )*

 

*T*

<sup>1</sup>

 

*T*

*(Y )*

 

( Y σdt)

<sup>T</sup> p p

( Y σdt)

*<sup>i</sup> <sup>i</sup> ( (t) diag( )*

 

Y σdt) Y σdt)

T T 11 11 T T 22 22

 

*p p*

*ˆ(t)* , <sup>1</sup> *(t)* is defined as

( Y σdt) 1 1 2 Y <sup>σ</sup>dt)

e 1 e e 1 e

*(*

*(*

*ˆ(t) (Y ) (t) [ dt] ..... .....*

e 1 e *<sup>p</sup>*

*p p*

*ˆ(t) ( / )arctan( )*

*( / )arctan( ) ˆ(t)*

 

 

After multiplication by <sup>1</sup> *(t)* , the result will be

 

*......... ˆ(t)*

1 1 ( Y σdt) <sup>2</sup> <sup>1</sup> <sup>2</sup> 2 2 2 Y <sup>σ</sup>dt)

<sup>T</sup> p p

( Y σdt) 2 Y σdt)

<sup>T</sup> <sup>p</sup>

*p p (*

1 1 1 1 ( Y <sup>σ</sup>dt) <sup>2</sup> <sup>1</sup> <sup>2</sup> 2 2

*ˆ(t) ( / )arctan( ) (t) (t) C ..... ..... ..... .........*

e *<sup>p</sup> <sup>p</sup>*

Y σdt) Y σdt)

Y σdt) Y σdt)

 

 

*sin( )*

 

<sup>2</sup> <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>Y</sup> <sup>σ</sup>dt) <sup>2</sup> 2 2

/ )e e e / )e e e

2 Y σdt) Y σdt)

e e

*sin( ) Cos( ) .....*

*( ( ( (*

e e

1

e

*( (*

*ˆ(t) ( sin(* <sup>T</sup> <sup>p</sup>

*( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( )*

/ )e e

*p p*

1

2

e e 1

p

*p ( ( p p*

*..... ................*

2 Y σdt) Y σdt) Y σdt)

*(( ( (( (*

is taken as initial condition, constant C is equivalent to Cos(1). So, the estimation

TT T 11 11 11 TT T 22 22 22

 

 

e

<sup>T</sup> <sup>i</sup> <sup>T</sup> <sup>i</sup>

2 Y σdt)

e 1 e 1

*<sup>p</sup>* 1

*(t) C .....* 

1

 

( Y σdt)

*i*

*p*

*)cos( )*

Y σdt)

*( p*

1 2

*.....*

(29)

1 1

(30)

(31)

(28)

If *ˆ( )* 0 

law for the uncertainty bound is derived as.

**2.2 Second choice of (t)-1** For the second derivation of

1 1 <sup>1</sup>

 

 

 

Substituting Equation (29) into (23) yields

*p*

*ˆ(t)*

After integration, the result is

 For the third derivation of *ˆ(t)* , <sup>1</sup> *(t)* is defined as

$$\left(\Phi(\mathbf{t})\right)^{-1} = \text{diag}\left(\boldsymbol{\beta}\_{\text{i}}\text{Sin}^{2}\left(\mathbf{a}\_{\text{i}}\right)\mathbf{Y}^{\text{T}}\mathbf{c}\mathbf{d}\mathbf{t}\right)\text{Cov}\left(\mathbf{a}\_{\text{i}}\right)\mathbf{Y}^{\text{T}}\mathbf{c}\mathbf{d}\mathbf{t}\mathbf{t}\mathbf{j}\_{\text{i}}\right) \tag{34}$$

Substitution of Equation (34) into Equation (23) yields

$$
\begin{bmatrix}
\hat{\rho}(t)\_1 \\
\hat{\rho}(t)\_2 \\
\dots \\
\hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
\beta\_1 \text{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\text{\\\\\\\\\\\\\\\\\\\\\\\}\ \mathbf{C}^{\text{\text}} \text{\text{\textdth}} \mathbf{t}\_1 (\mathbf{Y}^T \sigma \mathbf{d})\_1 (\mathbf{Y}^T \sigma)\_1 \\
\beta\_2 \text{\\\\\\} \mathbf{C}^{\text{\text\textquag}} \begin{bmatrix}
\mathbf{y}^T \mathbf{d} \mathbf{t} \end{bmatrix} \mathbf{C} \text{\textquag} \begin{bmatrix}
\mathbf{a}\_1 \end{bmatrix} \mathbf{Y}^T \mathbf{d} \mathbf{t} \end{bmatrix} \mathbf{e}\_1 \begin{bmatrix}
\rho\_1 \\
\rho\_2 \\
\vdots \\
\rho\_p \end{bmatrix} + \begin{bmatrix}
\rho\_1 \\
\rho\_2 \\
\vdots \\
\rho\_p \\
\end{bmatrix} \\
\begin{bmatrix}
1 \\
1 \\
\dots \\
1
\end{bmatrix}
\end{bmatrix} + \Phi(t)^{-1} \mathbf{C} \begin{bmatrix}
1 \\
\vdots \\
1 \\
\dots \\
1
\end{bmatrix}
\tag{35}
$$

After integration, the result is

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
(\beta\_1/a\_1)\frac{\mathrm{Sin}^3(a\_1)\mathrm{Y}^1\mathrm{odt})\_1\\ 3\\ (\beta\_2/a\_2)\frac{\mathrm{Sin}^3(a\_2)\mathrm{Y}^1\mathrm{odt})\_2}{3}\\ \dots \\ \cdots \\ (\beta\_p/a\_p)\frac{\mathrm{Sin}^3(a\_p)\mathrm{Y}^1\mathrm{odt})\_p
\end{bmatrix} + \begin{bmatrix}
\rho\_1\\ \rho\_2\\ \dots\\ \dots\\ \rho\_p
\end{bmatrix} + \Phi(t)^{-1}\mathrm{C}\begin{bmatrix}1\\ 1\\ \vdots\\ 1\end{bmatrix} \tag{36}
$$

Modelling of Bound Estimation Laws and

uniformly ultimate bounded.

*ˆ(t)* , and

*V* 0 for ||x||>w where


Note that

function (13) is written as [14, 15].

**Proof** 

If Φ, 

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 171

Where ε>0. If the control input (39) is substituted into the control law (6) for the control of the model manipulator, then, the control law (6) is continuous and the closed-loop system is

It is assumed that there exists an unknown bound on parametric uncertainty such that

π πρ <sup>0</sup> and π π <sup>0</sup>

TT T T T TT T T T


Where *T TT x [ q ,q ]* and Q=diag[TK, K]. Based on the Leitman [1], we can show that

 

*Qx ˆ ................*

Where λmin(Q) denotes the minimum eigenvalue of Q. Second term in Equation (41), if

*u*

V -q Kq-q ΛKΛq σ Y (t) σ Y σ Y

TT T


*Qx ˆ ˆ*

From the Cauchy-Schawartz inequality and our assumption on . If ||YT σ||<ε then

T

Y Y Y (t) - (t) <sup>Y</sup>

 

<sup>T</sup> V - (t) 4 *x Qx /* 

*ˆ(t)* is bounded. The rest of the proof can be seen in [4, 8].

*T*

*T*

*ˆ ˆ () u*

 

<sup>σ</sup> Yu(t) <sup>σ</sup> Y( (t)) Y (t) (t) <sup>Y</sup>

 

 


T T

TT T

This last term achieves a maximum value of

TT T

*Qx u ˆ*


<sup>2</sup> w = (t)/2 (Q) min

 

> 

*ˆ* (42)

*ˆ(t)* are substituded into (13), the time derivative of the Lyapunov

T 1 1 T 1

(Y (t) (Y

*) ˆ )*

T

T

Y

T

 

T

*( ) ˆ ˆ*

T T

 

*ˆ* (45)

ˆ(t) / 4 when ||YT σ||=ε/2. We have that

p T

*) ˆ )*

(Y (t) (Y

*p p*

*u ˆ*

(41)

 (40)

(43)

(44)

If *ˆ( )* 0 is taken as initial condition, constant C is equivalent to zero. So, the estimation law for the uncertainty bound is derived as .

$$\begin{bmatrix} \hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p \end{bmatrix} = \begin{bmatrix} (\boldsymbol{\rho}\_1^2 / \boldsymbol{a}\_1) \frac{\text{Sinc}^5(\boldsymbol{a}\_1 \big[ \mathbf{Y}^\text{-} \text{cdt} \big] \mathbf{C} \text{s}(\boldsymbol{a}\_1 \big[ \mathbf{Y}^\text{-} \text{cdt} \big])}{3} \\\ (\boldsymbol{\rho}\_2^2 / \boldsymbol{a}\_2) \frac{\text{Sinc}^5(\boldsymbol{a}\_2 \big[ \mathbf{Y}^\text{-} \text{cdt} \big] \mathbf{C} \text{s}(\boldsymbol{a}\_2 \big[ \mathbf{Y}^\text{-} \text{cdt} \big])}{3} \\\ \dots \\\ \dots \\\ (\boldsymbol{\rho}\_p^2 / \boldsymbol{a}\_p) \frac{\text{Sinc}^5(\boldsymbol{a}\_p \big[ \mathbf{Y}^\text{-} \text{cdt} \big] \mathbf{C} \text{s}(\boldsymbol{a}\_p \big[ \mathbf{Y}^\text{-} \text{cdt} \big])}{3} \end{bmatrix} + \begin{bmatrix} \rho\_1\\ \rho\_2\\ \dots\\ \dots\\ \rho\_p \end{bmatrix} \end{bmatrix} \tag{37}$$

If we substitute , , and *ˆ(t)* into Equation (16), the right terms of Equation (16) T T <sup>2</sup> [(ρ-ρ*ˆ*(t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 will be always zero and the derivation of the Lyapunov function will become a negative semidefinite function such that

$$\dot{\mathbf{V}} = \dot{\mathbf{q}}^T \mathbf{K} \dot{\mathbf{q}} \cdot \mathbf{\tilde{q}}^T \Lambda \mathbf{K} \Lambda \mathbf{\tilde{q}} \le 0 \tag{38}$$

So, the system is stable for all *ˆ(t)* derived from Equation (23).

### **3. Design of robust contol laws**

Based on the uncertainty bound estimation laws derived in section 2, and in [15], it is possible to develop robust control inputs.

### **3.1 Robust control law 1**

In order to define first robust control input, the following theorem is proposed.

#### **Theorem:**

Additional control input in control law (6) is

$$(u(t))\_i = \begin{cases} -\frac{(Y^T \sigma)\_i}{\left| (Y^T \sigma)\_i \right|} \hat{\rho}(t)\_i & \text{if } \left| (Y^T \sigma)\_i \right| > \varepsilon\_i \\\ -\frac{(Y^T \sigma)\_i}{\varepsilon\_i} \hat{\rho}(t)\_i & \text{if } \left| (Y^T \sigma)\_i \right| \le \varepsilon\_i \end{cases} \tag{39}$$

Where ε>0. If the control input (39) is substituted into the control law (6) for the control of the model manipulator, then, the control law (6) is continuous and the closed-loop system is uniformly ultimate bounded.

### **Proof**

170 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

3 T 1 1

 

Sin ( Y σdt)

<sup>2</sup> 1 1 <sup>2</sup> 2 2

3 T p p

*ˆ(t) (/) (t) (t) C ..... ..... ..... ......... ˆ(t)*

Sin ( Y σdt) 3

3

2 2

is taken as initial condition, constant C is equivalent to zero. So, the estimation

5 T T 1 11 1 2

Sin ( Y σdt) Cos( Y σdt) 3 Sin ( Y σdt) Cos( Y σdt) 3

 

 

 

*ˆ(t)* into Equation (16), the right terms of Equation (16)

T T V -q Kq-q <sup>Λ</sup>KΛq 0 (38)

5 T T 2 p pp p

Sin ( Y σdt) Cos( Y σdt) 3

<sup>1</sup> <sup>1</sup> 5 T <sup>T</sup> 2 22 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> 2 2

 

*p p*

T T <sup>2</sup> [(ρ-ρ*ˆ*(t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 will be always zero and the derivation of the

*ˆ(t)* derived from Equation (23).

Based on the uncertainty bound estimation laws derived in section 2, and in [15], it is

*i T i ii T*

*i T*

*(Y ) ˆ(t) if (Y )*

*i i i*

 

  (39)

In order to define first robust control input, the following theorem is proposed.

*T*

*i*

*<sup>i</sup> <sup>T</sup>*

 

*i*

*(Y ) ˆ(t) if (Y ) (Y ) (u(t))*

*..... ..... ......... ˆ(t)*

<sup>3</sup> <sup>1</sup>

1

(36)

(37)

Sin ( Y <sup>σ</sup>dt) <sup>1</sup>

1 1

*(/)*

 

 

*ˆ(t)*

law for the uncertainty bound is derived as .

*ˆ(t)*

If we substitute , , and

So, the system is stable for all

**3.1 Robust control law 1** 

**Theorem:** 

**3. Design of robust contol laws** 

possible to develop robust control inputs.

Additional control input in control law (6) is

If *ˆ( )* 0  p

 

1 1

*( /)*

 

 

 

*ˆ(t) (/)*

*p p*

Lyapunov function will become a negative semidefinite function such that

*(/)*

*(/)*

<sup>1</sup> <sup>1</sup> 3 T

*p p*

*p*

It is assumed that there exists an unknown bound on parametric uncertainty such that

$$
\|\mathbf{n}\_0 - \mathbf{n} \le \mathbf{p} \qquad \text{and} \qquad \|\mathbf{n}\_0 - \mathbf{n}\| \le \delta \tag{40}
$$

If Φ, *ˆ(t)* , and *ˆ(t)* are substituded into (13), the time derivative of the Lyapunov function (13) is written as [14, 15].

$$\begin{split} \dot{\mathbf{V}} &= -\dot{\mathbf{\tilde{q}}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{\tilde{q}}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \mathbf{A} \mathbf{K} \mathbf{A} \tilde{\mathbf{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\boldsymbol{\pi}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\rho} \\ &= -\dot{\mathbf{\tilde{q}}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{\tilde{q}}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \boldsymbol{\Lambda} \mathbf{K} \boldsymbol{\Lambda} \tilde{\mathbf{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\boldsymbol{\pi}} - \mathbf{o}^{\mathrm{T}} \mathbf{Y} (\rho - \hat{\rho}(\mathbf{t}) \\ &= \boldsymbol{\prec} \mathbf{x}^{\mathrm{T}} \mathbf{Q} \mathbf{x} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \hat{\rho}(\mathbf{t}) \end{split} \tag{41}$$

Where *T TT x [ q ,q ]* and Q=diag[TK, K]. Based on the Leitman [1], we can show that *V* 0 for ||x||>w where

$$\mathbf{w}^2 \hat{=} \hat{\rho}(\mathbf{t}) / 2 \boldsymbol{\lambda}\_{\text{min}}(\mathbf{Q}) \tag{42}$$

Where λmin(Q) denotes the minimum eigenvalue of Q. Second term in Equation (41), if ||YTσ||>ε then

$$\leq \mathbf{x}^{\mathsf{T}} \mathbf{Q} \mathbf{x} + \mathbf{o}^{\mathsf{T}} \mathbf{Y} \hat{\rho}(\mathbf{t}) \mathbf{o}^{\mathsf{T}} \mathbf{Y} \begin{bmatrix} \mathbf{Y}^{\mathsf{T}} \sigma \boldsymbol{\mathsf{t}}\_{1} \\ \hline \lVert \mathbf{Y}^{\mathsf{T}} \sigma \boldsymbol{\mathsf{t}}\_{1} \rVert \\ \hline \end{bmatrix} \begin{aligned} \label{t} \hat{\rho}(\mathbf{t})\_{1} \\ \hline \lVert \mathbf{Y}^{\mathsf{T}} \sigma \boldsymbol{\mathsf{t}}\_{1} \rVert \\ \hline \end{aligned} \tag{43}$$
 
$$\leq \mathbf{x}^{\mathsf{T}} \mathbf{Q} \mathbf{x} + \mathbf{o}^{\mathsf{T}} \mathbf{Y} (\hat{\rho}(\mathbf{t})) \cdot \hat{\rho}(\mathbf{t})) \leq 0$$

From the Cauchy-Schawartz inequality and our assumption on . If ||YT σ||<ε then

$$\begin{split} \boldsymbol{\sigma}^{\mathrm{T}} \mathbf{Y} \mathbf{u}(\mathbf{t}) + \boldsymbol{\sigma}^{\mathrm{T}} \mathbf{Y} (\hat{\boldsymbol{\rho}}(\mathbf{t})) &\leq (\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma})^{\mathrm{T}} \left( \left\lVert \hat{\boldsymbol{\rho}}(\mathbf{t}) \right\rVert \frac{\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma}}{\left\lVert \mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma} \right\rVert} + \boldsymbol{\mu}(\mathbf{t}) \right) \\ &\leq (\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma})^{\mathrm{T}} \left( \left\lVert \hat{\boldsymbol{\rho}}(\mathbf{t}) \right\rVert \frac{\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma}}{\left\lVert \mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma} \right\rVert} \frac{\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma}}{\boldsymbol{\sigma}} \left\lVert \hat{\boldsymbol{\rho}}(\mathbf{t}) \right\rVert \right) \end{split} \tag{44}$$

This last term achieves a maximum value of ˆ(t) / 4 when ||YT σ||=ε/2. We have that

$$\dot{\mathbf{V}} \le \mathbf{-x}^T \mathbf{Q} \mathbf{x} + \varepsilon \|\hat{\boldsymbol{\rho}}(\mathbf{t})\| / \, / \, 4 \tag{45}$$

Note that *ˆ(t)* is bounded. The rest of the proof can be seen in [4, 8].

### **3.2 Robust control law 2**

Based on the bound estimation law *ˆ(t)* derived from general Equation (23), additional control input u(t) are defined [15]. The additional control input in control law (6) is defined as [15]

$$\mathbf{u}(\mathbf{t}) = \begin{bmatrix} -\hat{\rho}(t)\_1 \text{sgn}(\mathbf{Y}^\top \sigma)\_1 \\ -\hat{\rho}(t)\_2 \text{sgn}(\mathbf{Y}^\top \sigma)\_2 \\ \vdots \\ -\hat{\rho}(t)\_p \text{sgn}(\mathbf{Y}^\top \sigma)\_p \end{bmatrix} \tag{46}$$

Modelling of Bound Estimation Laws and

y

l1

lc1

Fig. 1. Two-link planar robot [4].

**4. Simulation results** 

this robot is given by

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 173

For illustration, a two-link robot manipulator is given in Figure 1 [4]. Parameterisation of

m2, I2

q2

1=m11c12+m2l12+I1, *π*2= m2lc22+I2 , *π*3=m2l1lc2 ,

q1

l2

lc2

m1, I1

With this parameterisation, the component yij of Y(q,q,q ) in Equation 2 are given as

y11 1 q ; y12 1 2 q q ; <sup>2</sup> y cos(q )(2q q sin(q )(q 2q q ) 13 2 1 2 2 2 12 *)* ;

y14= gccos(q1); y15= gccos(q1); y16= gccos(q1+q2) ;

y14*=*gccos(q1); y15= gccos(q1) ; y16= gccos(q1+q2)

y21=0; y22 r1 2 q q*<sup>r</sup>* ; ; y23 2 r1 2 1 r1 cos(q )q sin(q )(q q )

Y(q,q,q ,qr r *)* in Equation (4) have the components

y21=0; y22 1 2 q q ; <sup>2</sup> y23 2 1 2 1 cos(q )q sin(q )(q ) ;

*π*4=m1lc1, *π*5=m2l1, *π*6=m2lc2, (51)

x

y24=0 ; y25=0 ; y26= gccos(q1+q2).(52)

y11 r1 q ; y12 r1 2 q q *<sup>r</sup>* ;

y13 2 r1 2 2 1 2 1 r2 2 r2 cos(q )(2q q sin(q )(q q q q q q ) *r r )* ;

y24=0 ; y25=0 ; y26= gccos(q1+q2). (53)

### **3.3 Robust control law 3**

<sup>i</sup> denote the *i*th component of the vector <sup>T</sup> Y , i choose as the ith component of . Then, considering the *ˆ(t)* derived from Equation (23), u(t) for each *ˆ(t)* is defined as as follows: For Equation (28), u(t)i is

$$\mathbf{u}(\mathbf{t}) = \begin{bmatrix} -\frac{\mathsf{u}\_{i}}{\left|\mathsf{o}\_{i}\right|} \left[ -(\boldsymbol{\beta}\_{i}^{2}/\mathsf{a}\_{i}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| + \mathrm{Cost}(\mathbf{1}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right|) + \boldsymbol{\rho}\_{i} \right] & \text{if} \quad \left| \mathbf{\hat{o}}\_{i} \right| > \mathbf{\hat{e}}\_{i} \\\ -\frac{\mathsf{u}\_{i}}{\left|\mathsf{o}\_{i}\right|} \left[ -(\boldsymbol{\beta}\_{i}^{2}/\mathsf{a}\_{i}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right|) \cos(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| + \mathrm{Cost}(\mathbf{1}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right|) + \boldsymbol{\rho}\_{i} \right] & \text{if} \quad \left| \mathbf{\hat{o}}\_{i} \right| \leq \mathbf{\hat{e}}\_{i} \end{bmatrix} \tag{47}$$

For Equation (33),

$$\mathbf{u}(\mathbf{t})\_{i} = \begin{bmatrix} -\frac{\mathsf{v}\_{i}}{|\mathbf{v}\_{i}|} \left[ (\boldsymbol{\beta}\_{i}^{2} \;/\ \ \boldsymbol{\alpha}\_{i}) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} \arctan(\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}) - \arctan(1) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} + \mathbf{p}\_{i} \right] \tag{48}$$
 
$$\begin{bmatrix} -\frac{\mathsf{v}\_{i}}{\mathsf{e}\_{i}} \left[ (\boldsymbol{\beta}\_{i}^{2} \;/\ \boldsymbol{\alpha}\_{i}) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} \arctan(\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}) - \arctan(1) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} + \mathbf{p}\_{i} \right] \quad \text{if} \quad \left| \mathbf{v}\_{i} \right| \leq \mathsf{e}\_{i} \end{bmatrix}$$

For Equation (36):

$$\mathbf{u}(\mathbf{t})\_{\mathbf{i}} = \begin{bmatrix} -\frac{\mathbf{u}\_{\mathrm{i}}}{|\mathbf{u}\_{\mathrm{i}}|} \left[ (\boldsymbol{\beta}\_{\mathrm{i}}^{2} \;/\ \boldsymbol{\alpha}\_{\mathrm{i}}) \frac{\sin^{5} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|) \mathrm{C} \mathbf{s} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|)}{3} + \rho\_{\mathrm{i}} \right] \qquad \text{if} \quad \left| \boldsymbol{\alpha}\_{\mathrm{i}} \right| > \varepsilon\_{\mathrm{i}} \\\ -\frac{\mathbf{u}\_{\mathrm{i}}}{\varepsilon\_{\mathrm{i}}} \left[ (\boldsymbol{\beta}\_{\mathrm{i}}^{2} \;/\ \boldsymbol{\alpha}\_{\mathrm{i}}) \frac{\sin^{5} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|) \mathrm{C} \mathbf{s} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|)}{3} + \rho\_{\mathrm{i}} \right] \quad \text{if} \quad \left| \boldsymbol{\alpha}\_{\mathrm{i}} \right| \leq \varepsilon\_{\mathrm{i}} \end{bmatrix} \tag{49}$$

#### **3.4 Robust control law 4**

From Equations (43) and (44), it is ease to define the following control law.

$$u(t) = \begin{cases} -\frac{Y^T \sigma}{\left\|(Y^T \sigma)\right\|} \left\|\hat{\boldsymbol{\rho}}(t)\right\| & \text{if} \quad \left\|(Y^T \sigma)\right\| > \varepsilon\\ -\frac{Y^T \sigma}{\varepsilon} \left\|\hat{\boldsymbol{\rho}}(t)\right\| & \text{if} \quad \left\|(Y^T \sigma)\right\| \le \varepsilon \end{cases} \tag{50}$$

### **4. Simulation results**

172 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

control input u(t) are defined [15]. The additional control input in control law (6) is defined

sgn(Y u(t)

ν dt ν dt ν dt ν dt ν dt i 2

*ˆ(t)* derived from Equation (23), u(t) for each

T 1 1 T 2 2

 

 

sgn(Y

*ˆ(t) ) ˆ(t) ) ......... ˆ(t) )*

ii i ii

<sup>υ</sup> [ / )e e e 1 e e <sup>ρ</sup> ] if υ ε

ii i ii

i i i i i i i i i i

*i i i*

 

*i i i*

 

 

ν dt ν dt

<sup>υ</sup> e e [( <sup>e</sup> <sup>1</sup> <sup>ρ</sup> )] if υ ε

<sup>υ</sup> e e [( <sup>e</sup> <sup>1</sup> <sup>ρ</sup> )] if υ ε

( <sup>ν</sup> dt ( <sup>ν</sup> dt <sup>υ</sup> [( <sup>ρ</sup> ] if υ ε

*sin )Cos )*

( <sup>ν</sup> dt ( <sup>ν</sup> dt <sup>υ</sup> [( <sup>ρ</sup> ] if υ ε

*sin )Cos )*

i i ii 2 ν dt 2 ν dt

i i ii 2 ν dt 2 ν dt

i i ii

 

i i ii

*T*

 

*T*

 

*<sup>Y</sup> ˆ(t) if (Y )*

 

<sup>υ</sup> [ / )e e e 1 e e <sup>ρ</sup> ] if υ ε

<sup>i</sup> <sup>ν</sup> dt <sup>ν</sup> dt <sup>ν</sup> dt <sup>ν</sup> dt <sup>ν</sup> dt <sup>i</sup> <sup>2</sup> <sup>i</sup> i ii

*ii i ii*

*ii i ii*

*( sin( )cos( Cos( ) sin( )*

 

*( sin( )cos( Cos( ) sin( )*

   , i 

T

 

i i ii

*p p* sgn(Y

*ˆ(t)* derived from general Equation (23), additional

(46)

*ˆ(t)* is defined as as follows:

(47)

(48)

(49)

(50)

choose as the ith component of . Then,

 

> 

**3.2 Robust control law 2** 

**3.3 Robust control law 3** 

For Equation (28), u(t)i is

i

υ

 

For Equation (33),

For Equation (36):

  i

i

i

u(t)

**3.4 Robust control law 4** 

ε

*i*

 

*i*

*i*

*i*

i <sup>i</sup> <sup>5</sup>

 

 

i

 

 

 

considering the

u(t)

<sup>i</sup> denote the *i*th component of the vector <sup>T</sup> Y

2 ν dt i

*i*

2 ν dt i

*i*

<sup>υ</sup> 1 e 1 e u(t)

<sup>i</sup> <sup>ν</sup> dt <sup>ν</sup> dt

5 ii ii <sup>i</sup> <sup>2</sup>

υ 3

*i*

*/ )*

*i*

 

*/ )*

ε 3

From Equations (43) and (44), it is ease to define the following control law.

*T*

*T T*

*<sup>Y</sup> ˆ(t) if (Y ) (Y ) u(t)*

ε 1 e 1 e

*/ ) arctan( ) arctan( )*

*/ ) arctan( ) arctan( )*

ii ii <sup>i</sup> <sup>2</sup>

as [15]

Based on the bound estimation law

For illustration, a two-link robot manipulator is given in Figure 1 [4]. Parameterisation of this robot is given by

$$\text{Fig. 1. Two-link planar robot [4].}$$

$$
\pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{l}} \mathsf{l}\_{\mathsf{c}1} \mathsf{l}\_{\mathsf{c}1} + \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{l}} \mathsf{l} + \mathsf{l}\_{\mathsf{l}} \qquad \qquad \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \mathsf{l} + \mathsf{l}\_{\mathsf{2}} \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{l}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{l}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \tag{51}
$$
 
$$
\pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{i}} \mathsf{l}\_{\mathsf{c}1} \qquad \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{i}} \mathsf{l}\_{\mathsf{$$

x

With this parameterisation, the component yij of Y(q,q,q ) in Equation 2 are given as

$$\mathbf{y}\_{11} = \ddot{\mathbf{q}}\_1; \qquad \mathbf{y}\_{12} = \ddot{\mathbf{q}}\_1 + \ddot{\mathbf{q}}\_2; \quad \mathbf{y}\_{13} = \cos(\mathbf{q}\_2)(2\ddot{\mathbf{q}}\_1 + \ddot{\mathbf{q}}\_2) - \sin(\mathbf{q}\_2)(\dot{\mathbf{q}}\_2^2 + 2\dot{\mathbf{q}}\_1\dot{\mathbf{q}}\_2);$$

$$\mathbf{y}\_{14} = \mathbf{g}\cos(\mathbf{q}); \qquad \mathbf{y}\_{15} = \mathbf{g}\cos(\mathbf{q}); \qquad \mathbf{y}\_{16} = \mathbf{g}\cos(\mathbf{q} + \mathbf{q}\_2) \; ;$$

$$\mathbf{y}\_{21} = \mathbf{0}; \qquad \mathbf{y}\_{22} = \ddot{\mathbf{q}}\_1 + \ddot{\mathbf{q}}\_2 \; ; \qquad \mathbf{y}\_{23} = \cos(\mathbf{q}\_2)\ddot{\mathbf{q}}\_1 + \sin(\mathbf{q}\_2)(\dot{\mathbf{q}}\_1^2) \; ;$$

$$\mathbf{y}\_{24} = \mathbf{0} ; \qquad \mathbf{y}\_{25} = \mathbf{0} \; ; \qquad \mathbf{y}\_{26} = \mathbf{g}\cos(\mathbf{q}\_1 + \mathbf{q}\_2). \tag{52}$$

Y(q,q,q ,qr r *)* in Equation (4) have the components

$$\mathbf{y}\_{11} = \ddot{\mathbf{q}}\_{1r} \ \mathbf{j} \ \mathbf{y}\_{12} = \ddot{\mathbf{q}}\_{r1} + \ddot{\mathbf{q}}\_{r2} \ \ ;$$

$$\mathbf{y}\_{13} = \cos(\mathbf{q}\_{2})(2\ddot{\mathbf{q}}\_{r1} + \ddot{\mathbf{q}}\_{r2}) - \sin(\mathbf{q}\_{2})(\dot{\mathbf{q}}\_{1}\dot{\mathbf{q}}\_{r2} + \dot{\mathbf{q}}\_{1}\dot{\mathbf{q}}\_{r2}) \ ;$$

$$\mathbf{y}\_{14} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1}); \qquad \mathbf{y}\_{15} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1}) \ ; \qquad \mathbf{y}\_{16} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1} + \mathbf{q}\_{2})$$

$$\mathbf{y}\_{21} = 0; \qquad \mathbf{y}\_{22} = \ddot{\mathbf{q}}\_{r1} + \ddot{\mathbf{q}}\_{r2} \ ; \qquad \mathbf{y}\_{23} = \cos(\mathbf{q}\_{2})\ddot{\mathbf{q}}\_{r1} + \sin(\mathbf{q}\_{2})(\dot{\mathbf{q}}\_{1}\dot{\mathbf{q}}\_{r1})$$

$$\mathbf{y}\_{24} = 0 \ ; \qquad \mathbf{y}\_{25} = 0 \ ; \qquad \mathbf{y}\_{26} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1} + \mathbf{q}\_{2}). \tag{53}$$

For illustrated purposes let us assume that the parameters of the unloaded manipulator are known and are given by Table 1. Using these values in Table 1, the ith component of *π* obtained by means of Equation (51) are given in Table 2. It is assumed that the parameters m2, lc2 and I2 are changed in the intervals

$$0 \le \Delta m\_2 \le 10 \quad ; \quad 0 \le \Delta l\_{c2} \le 0.5 \; ; \quad 0 \le l\_2 \le \frac{15}{12} \tag{54}$$

Modelling of Bound Estimation Laws and

vector with components ρi [4].

Table 4. Uncertainty bound [4].


Tracking error(rad)

Tracking error(rad)

Let 

that a measure of uncertainty for each parameter

<sup>i</sup> denote the *i*tch component of the vector <sup>T</sup> Y

 ii i i i i

0 1 2 3 4 5

joint 2

0 1 2 3 4 5

Time(sec)

(28) when =diag(10 10), K=diag(30 30), α=1, β=1.

x 10-3 joint 1

u

 *i i* 

and consequently the *i*tch component of the control input up is defined as [4].

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 175

Having a single number ρ to measure the parameter uncertainty may lead to overly conservative design, higher than necessary gains, etc. For this purpose, different "weights" or gains to the components of u may be assigned. This can be done as follows: Supposing

 , i 

i ii i i ρ υ / υ if υ ε

(ρ /ε )υ if υ ε 

Since extended algorithm (56) is used, the uncertainty bounds for each parameter are shown separately in Table 4. The uncertainty bounds i in Table 4 are simply the difference between values given in Table 3 and Table 2, and the value of ρ is the Euclidean norm of the

For computer simulation, a fifth order polynomial function is used as a reference trajectory for both joints. In order to analyse performance of the proposed controllers, each control law with the same control parameters K and Ʌ is applied to the same model system using same trajectory. The control matrices Ʌ and K are chosen to be identical as Ʌ=diag(10 10) and


Tork input: Joint 2

Fig. 2. Response using the robust control law (39) with uncertainty bound estimation law

0

200

Tork input:Joint 1

400

 ρ1 ρ2 ρ3 ρ4 ρ5 ρ<sup>6</sup> 5 7.29 6. 25 0 5 6.25

K=diag(30 30) for all controllers. The obtained results are plotted in Figures 2-4.

*<sup>i</sup>* can be defined separately as:

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

I=1,2,……….,p (58)

choose as the itch component of ,

(59)

Choosing the mean value for the range of possible i in Equation (54) yields the nominal parameter vector and the computed values for ith component of *π*0 is shown in Table 3 [4].


Table 1. Parameters of the unloaded arm [4].


Table 2. *π*i for the unloaded arm [4]


Table 3. Nominal parameter vector *π*0 [4].

With this choice of nominal parameter vector 0 and uncertainty range given by (54), it is an easy matter to calculate the uncertainty bound as follows:

$$\left\|\tilde{\boldsymbol{\pi}}\right\|^2 = \sum\_{i=1}^6 (\pi\_{i0} - \pi\_i)^2 \le 181.26\tag{55}$$

and thus 181 26 13 46 *. .* .

For explanation, Spong's algorithms are given.

$$\mathbf{u}(\mathbf{t}) = \begin{cases} -\delta \frac{\mathbf{Y}^{\mathrm{T}}\mathbf{o}}{\left\| \mathbf{Y}^{\mathrm{T}}\mathbf{o} \right\|} & \text{if} \quad \left\| \mathbf{Y}^{\mathrm{T}}\mathbf{o} \right\| > \varepsilon \\\\ -\delta \frac{\mathbf{Y}^{\mathrm{T}}\mathbf{o}}{\varepsilon} & \text{if} \quad \left\| \mathbf{Y}^{\mathrm{T}}\mathbf{o} \right\| \le \varepsilon \end{cases} \tag{56}$$

As a measure of parameter uncertainty on which the additional control input is based, ρ can be defined as

$$\mathcal{S} = \left(\sum\_{i=1}^{\mathcal{P}} \rho\_i^{\mathcal{Q}}\right)^{1/2} \tag{57}$$

Having a single number ρ to measure the parameter uncertainty may lead to overly conservative design, higher than necessary gains, etc. For this purpose, different "weights" or gains to the components of u may be assigned. This can be done as follows: Supposing that a measure of uncertainty for each parameter *<sup>i</sup>* can be defined separately as:

$$\left\lfloor \tilde{\pi}\_{i} \right\rfloor \leq \rho\_{i} \qquad \text{I} = \text{1}, \text{2}, \dots, \text{...}, \text{p} \tag{58}$$

Let <sup>i</sup> denote the *i*tch component of the vector <sup>T</sup> Y , i choose as the itch component of , and consequently the *i*tch component of the control input up is defined as [4].

$$\mathbf{u}\_{i} = \begin{cases} -\mathfrak{p}\_{i}\mathfrak{v}\_{i}/\left|\mathfrak{v}\_{i}\right| & \text{if} \quad \left|\mathfrak{v}\_{i}\right| > \mathfrak{e}\_{i} \\ -(\mathfrak{p}\_{i}/\mathfrak{e}\_{i})\mathfrak{v}\_{i} & \text{if} \quad \left|\mathfrak{v}\_{i}\right| \le \mathfrak{e}\_{i} \end{cases} \tag{59}$$

Since extended algorithm (56) is used, the uncertainty bounds for each parameter are shown separately in Table 4. The uncertainty bounds i in Table 4 are simply the difference between values given in Table 3 and Table 2, and the value of ρ is the Euclidean norm of the vector with components ρi [4].


Table 4. Uncertainty bound [4].

174 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

For illustrated purposes let us assume that the parameters of the unloaded manipulator are known and are given by Table 1. Using these values in Table 1, the ith component of *π* obtained by means of Equation (51) are given in Table 2. It is assumed that the parameters

> <sup>15</sup> <sup>0</sup> 12

*I* (54)

0 10 *m*2 ; 0 05 *<sup>c</sup>*<sup>2</sup> *l .* ; 2

Choosing the mean value for the range of possible i in Equation (54) yields the nominal parameter vector and the computed values for ith component of *π*0 is shown in Table 3 [4].

m1 m2 l1 l2 lc1 lc2 I1 I2 10 5 1 1 0.5 0.5 10/12 5/12

*π*<sup>1</sup> *π*<sup>2</sup> *π*<sup>3</sup> *π*<sup>4</sup> *π*<sup>5</sup> *π*<sup>6</sup> 8.33 1.67 2.5 5 5 2.5

13.33 8.96 8.75 5 10 8.75

With this choice of nominal parameter vector 0 and uncertainty range given by (54), it is an

04

181 26 *i i*

<sup>Y</sup> <sup>σ</sup> if Y σ ε

1 2 <sup>p</sup> 2 i i 1 ρ

*/*

As a measure of parameter uncertainty on which the additional control input is based, ρ can

 

<sup>Y</sup> <sup>σ</sup> if Y σ ε

T

T

05

(57)

(55)

06

(56)

03

6 2 2 0 1

 *( ).*

T

T T

ε

*i*

<sup>Y</sup> <sup>σ</sup> u(t)

  m2, lc2 and I2 are changed in the intervals

Table 1. Parameters of the unloaded arm [4].

02

Table 3. Nominal parameter vector *π*0 [4].

 181 26 13 46 *. .* . For explanation, Spong's algorithms are given.

easy matter to calculate the uncertainty bound as follows:

Table 2. *π*i for the unloaded arm [4]

01

and thus

be defined as

For computer simulation, a fifth order polynomial function is used as a reference trajectory for both joints. In order to analyse performance of the proposed controllers, each control law with the same control parameters K and Ʌ is applied to the same model system using same trajectory. The control matrices Ʌ and K are chosen to be identical as Ʌ=diag(10 10) and K=diag(30 30) for all controllers. The obtained results are plotted in Figures 2-4.

Fig. 2. Response using the robust control law (39) with uncertainty bound estimation law (28) when =diag(10 10), K=diag(30 30), α=1, β=1.

Modelling of Bound Estimation Laws and

**5. Conclusion** 

diffucult.

(t) as a derivtion of

618, 1989.

**6. References** 

Control Vol*.*26, pp. 1139-1144, 1981.

according to uncertainty bound estimation laws.

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 177

As shown in Figures 2-4, tracking error is small and tracking performance changes

In the past, some robust controllers are developed for robot manipulators. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. Spong [4] proposed a new robust controller for robot manipulators and Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. In [4], uncertainty bound on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. However, constant uncertainty bound parameters cause pure tracking performances. In order to increase tracking performance of the uncetain system, uncertainty bound estimation laws are developed [11-13]. Uncertainty bound estimation laws are updated as a function of exponential [11, 12], logarithmic [13] and trigonometric [14] functions depending on robot kinematics parameters and tracking error. A first order differential equation function is developed for derivation of control parameters and only a single derivation of uncertainty bound estimation law is possible. A new method for derivation of a bound estimation law is not proposed in [11-13], because, definition of a new variable function for other derivation is

In the study [14], a general equation is developed from Lyapunov function and uncertainty bound estimation laws depending on trigonometric functions are developed. However, a general method for derivation of uncertainty bound estimation laws is not proposed. In a recent study [15], a general method for derivation of bound estimation laws based on the Lyapunov theory is proposed. In this method, functions and integration techniques are used for derivation of uncertainty bound estimation laws. Then, relations between the bound estimation laws and robust control inputs are established and four new robust control inputs are designed depending on each bound estimation law. It is possible to derive other different uncertainty bound estimation laws from general equation (23) if appropriate functions and integration techniques are defined. In this work, three different variable

functions are defined and integration techniques are used in order to derive

distinct rule for definition of the (t) and integration techniques in order to derive

Equation (23) if appropriate function (t) and integration techniques are chosen.

relations between the uncertainty bouns and robust control laws are established. There is no

use system state parameters and mathematical insight to search for appropriate function of

with these derivations. It will be also possible to derive another bound estimation laws from

[1] M. Corless, G. Leitmann, "Continuous feedback guaranteeing uniform ultimate

[2] M. Corless, "Tracking controllers for uncertain systems: Applications to a manutec R3

boundedness for uncertain dynamic systems", IEEE Transactions Automatic

robot. Journal of Dynamic Systems Measurement and Control", Vol.111, pp. 609-

*ˆ(t)* . This study also shows that robust controllers are not limited

*ˆ(t)* and

*ˆ(t)* . We

Fig. 3. Response using the robust control law (39) with uncertainty bound estimation law (35) when =diag(10 10), K=diag(30 30), α=0.8, β=0.4.

Fig. 4. Response using the robust control law (39) with uncertainty bound estimation law (37) when =diag(10 10), K=diag(30 30), α=0.5, β=2.

As shown in Figures 2-4, tracking error is small and tracking performance changes according to uncertainty bound estimation laws.

### **5. Conclusion**

176 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics





Tork input: Joint 2

Fig. 4. Response using the robust control law (39) with uncertainty bound estimation law

0

200

Tork input:Joint 1

400

Tork input: Joint 2

Fig. 3. Response using the robust control law (39) with uncertainty bound estimation law

0

200

Tork input:Joint 1

400

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

0 1 2 3 4 5

Time(sec)

(37) when =diag(10 10), K=diag(30 30), α=0.5, β=2.

(35) when =diag(10 10), K=diag(30 30), α=0.8, β=0.4.

x 10-3 joint 1

x 10-3 joint 2

x 10-3 joint 1

x 10-3 joint 2



Tracking error(rad)


Tracking error(rad)

0

2

Tracking error(rad)

Tracking error(rad)

In the past, some robust controllers are developed for robot manipulators. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. Spong [4] proposed a new robust controller for robot manipulators and Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. In [4], uncertainty bound on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. However, constant uncertainty bound parameters cause pure tracking performances. In order to increase tracking performance of the uncetain system, uncertainty bound estimation laws are developed [11-13]. Uncertainty bound estimation laws are updated as a function of exponential [11, 12], logarithmic [13] and trigonometric [14] functions depending on robot kinematics parameters and tracking error. A first order differential equation function is developed for derivation of control parameters and only a single derivation of uncertainty bound estimation law is possible. A new method for derivation of a bound estimation law is not proposed in [11-13], because, definition of a new variable function for other derivation is diffucult.

In the study [14], a general equation is developed from Lyapunov function and uncertainty bound estimation laws depending on trigonometric functions are developed. However, a general method for derivation of uncertainty bound estimation laws is not proposed. In a recent study [15], a general method for derivation of bound estimation laws based on the Lyapunov theory is proposed. In this method, functions and integration techniques are used for derivation of uncertainty bound estimation laws. Then, relations between the bound estimation laws and robust control inputs are established and four new robust control inputs are designed depending on each bound estimation law. It is possible to derive other different uncertainty bound estimation laws from general equation (23) if appropriate functions and integration techniques are defined. In this work, three different variable functions are defined and integration techniques are used in order to derive *ˆ(t)* and relations between the uncertainty bouns and robust control laws are established. There is no distinct rule for definition of the (t) and integration techniques in order to derive *ˆ(t)* . We use system state parameters and mathematical insight to search for appropriate function of (t) as a derivtion of *ˆ(t)* . This study also shows that robust controllers are not limited with these derivations. It will be also possible to derive another bound estimation laws from Equation (23) if appropriate function (t) and integration techniques are chosen.

### **6. References**


[3] M.W. Spong, M. Vidyasagar, "Robust linear compensator design for nonlinear robotic control", IEEE Journal of Robotic and Automation, Vol.3(4), pp. 345-350, 1987.

**11** 

*1,3China* 

*2Republic of Korea* 

**Missile Cooperative Engagement Formation** 

In traditional engagement model, there is only one single weapon or platform which resists to another single one, and the connections between members are few, so the complete and global information in engagement space can't be utilized sufficiently, which directly results in traditional model is confronting with more and more disadvantages especially as the high-tech and information-tech are developing faster and faster. In view of those issues, the concepts and technology of missile formation cooperative are presented, developed and expanded recently(Cui et al., 2009). Compared to traditional model, weapon or platform with cooperative manner manifests great advantages in aspects of ability of penetration, electronic countermeasures and ability of searching moving targets etc, furthermore the

Many new cooperative weapon systems are established and developed fast recently, such as Cooperative Engagement Capability (CEC) system, Net Fire System and LOw Cost Autonomous Attack System (LOCAAS) etc, wherein the LOCAAS is most relevant to our

In order to meet the requirements of future aerial warfare, United States Force has developed a series of high technical and high accurate airborne guided weapon systems, such as Joint Common Missile (JCM), Joint Direct Attack Missile (JDAM), Wind Corrected Munitions Dispenser (WCMD) and LOw Cost Autonomous Attack System (LOCAAS). These weapons are paid much attention because their great capabilities of high-precision, all-weather engagement and attacking beyond defence area etc. Especially, LOCAAS also has other great advantages besides those aspects mentioned above, such as low cost, general utilization and attacking multi-targets simultaneously, so it is paid more attention than other weapons, and it becomes an outstanding representative of high-tech weapon and new

LOCAAS is developed from an previous weapon named as Low Cost Anti-Armor Submunitions (also called LOCAAS for short) which was a kind of short-range unpowered airborne and air-to-ground guided weapon developed in 1998. At a later time, researchers added thrust system to this old LOCAAS, so it had capability of launching beyond defence

**1. Introduction**

engagement model.

synthetical engagement efficacy is developed greatly.

topic, so we will introduce it more detailed.

**Configuration Control Method** 

Changzhu Wei1,2, Yi Shen1, Xiaoxiao Ma3,

Naigang Cui1 and Jifeng Guo1 *1Harbin Institute of Technology, Harbin, 2Yonsei University Observatory, Seoul, 3Xuzhou Air Force College, Xuzhou,* 


## **Missile Cooperative Engagement Formation Configuration Control Method**

Changzhu Wei1,2, Yi Shen1, Xiaoxiao Ma3, Naigang Cui1 and Jifeng Guo1 *1Harbin Institute of Technology, Harbin, 2Yonsei University Observatory, Seoul, 3Xuzhou Air Force College, Xuzhou, 1,3China 2Republic of Korea* 

### **1. Introduction**

178 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

[3] M.W. Spong, M. Vidyasagar, "Robust linear compensator design for nonlinear robotic control", IEEE Journal of Robotic and Automation, Vol.3(4), pp. 345-350, 1987. [4] M.W. Spong, "On the robust control of robot manipulators", IEEE Transactions on

[5] G. Leitmann, "On the efficiency of nonlinear control in uncertain linear system", Journal of Dynamic Systems Measurement and Control, Vol.102, pp. 95-102, 1981. [6] E. Yaz, "Comments on the robust control of robot manipulators", IEEE Transactions on

[7] M. Danesh, M. Keshmiri and F. Sheikholeslam, "Developing a robust control scheme

[8] K.M. Koo, J.H. Kim, "Robust control of robot manipulators with parametric

[9] G. Liu, A.A. Goldenberg, "Uncertainty decomposition-based robust control of robot

[10]. G. Liu, Decomposition-based control of mechanical systems Electrical and Computer

[11] R. Burkan, New approaches in controlling robot manipulators with parametric uncertainty, Ph.D. Thesis, Erciyes University, Institute of Science, Turkey, 2002. [12] R. Burkan, İ. Uzmay, "Upper bounding estimation for robustness to the parameter

[13] R. Burkan, İ. Uzmay, "Logarithmic based robust approach to parametric uncertainty for

[14] R. Burkan, "Upper bounding estimation for robustness to the parameter uncertainty

[15] R. Burkan, "Modelling of bound estimation laws and robust controllers for robustness

[16] M.W. Spong, S. Hutchinson, M. Vidyasagar, Robot Dynamics and Control, John

[17] J.J. Slotine, W. Li, "On the adaptive control of robotic manipulator", The International

[18] L. Sciavicco, B. Siciliano, Modeling and Control of Robot Manipulators, The McGraw-

Intelligent & Robotic Systems, Vol.46, pp.263-283, 2006.

Journal of Robotics Research, Vol.6(3), pp. 49-59, 1987.

and Robotic Systems, Vol.60, pp. 365-394, 2010.

robust to uncertain model parameters and unmodeled dynamics" Proceeding of IEEE Conference on Industrial Electronics and Applications (ICIEA), May 24-26,

uncertainty", IEEE Transactions Automatic Control, Vol.39(6), pp. 1230-1233, 1994.

manipulators", IEEE Transactions on Control Systems Technology, Vol.4, pp. 384-

Engineering, 2000 Canadian Conference on Volume: 2 Publication Year: 2000 ,

uncertainty in trajectory control of robot arm", Robotics and Autonomous Systems

control of robot manipulators", International Journal of Robust and Nonlinear

with trigonometric function in trajectory control of robot arms"*.* Journal of

to parametric uncertainty for control of robot manipulators", Journal of Intelligent

Automatic Control, Vol.37, pp.1782-1786, 1992.

Automatic Control, Vol. 38(38), pp. 511-512, 1993.

2006, Singapore.

Page(s): 966 - 970 vol.2

Vol.45, pp.99-110, 2003.

Willey&Sons, Inc., 2006.

Hill Companies, 1996.

Control, Vol.15, pp. 427-436, 2005.

393, 1996.

In traditional engagement model, there is only one single weapon or platform which resists to another single one, and the connections between members are few, so the complete and global information in engagement space can't be utilized sufficiently, which directly results in traditional model is confronting with more and more disadvantages especially as the high-tech and information-tech are developing faster and faster. In view of those issues, the concepts and technology of missile formation cooperative are presented, developed and expanded recently(Cui et al., 2009). Compared to traditional model, weapon or platform with cooperative manner manifests great advantages in aspects of ability of penetration, electronic countermeasures and ability of searching moving targets etc, furthermore the synthetical engagement efficacy is developed greatly.

Many new cooperative weapon systems are established and developed fast recently, such as Cooperative Engagement Capability (CEC) system, Net Fire System and LOw Cost Autonomous Attack System (LOCAAS) etc, wherein the LOCAAS is most relevant to our topic, so we will introduce it more detailed.

In order to meet the requirements of future aerial warfare, United States Force has developed a series of high technical and high accurate airborne guided weapon systems, such as Joint Common Missile (JCM), Joint Direct Attack Missile (JDAM), Wind Corrected Munitions Dispenser (WCMD) and LOw Cost Autonomous Attack System (LOCAAS). These weapons are paid much attention because their great capabilities of high-precision, all-weather engagement and attacking beyond defence area etc. Especially, LOCAAS also has other great advantages besides those aspects mentioned above, such as low cost, general utilization and attacking multi-targets simultaneously, so it is paid more attention than other weapons, and it becomes an outstanding representative of high-tech weapon and new engagement model.

LOCAAS is developed from an previous weapon named as Low Cost Anti-Armor Submunitions (also called LOCAAS for short) which was a kind of short-range unpowered airborne and air-to-ground guided weapon developed in 1998. At a later time, researchers added thrust system to this old LOCAAS, so it had capability of launching beyond defence

Missile Cooperative Engagement Formation Configuration Control Method 181

The missile formation control system mentioned in this chapter only has one leader missile (showed as Fig.2), and it mainly consists of the following subsystems: cooperative engagement mission planning subsystem, formation configuration describing subsystem, leader inner-loop control subsystem, follower inner-loop control subsystem and formation configuration control subsystem. The sketch map illustrates relationships between each

*I x*

, , *Vfi vfi fi* 

*r x*

*r o*

*r z*

, , *Vfci vfci fci* 

*<sup>r</sup> <sup>y</sup>* <sup>1</sup> *<sup>o</sup> <sup>r</sup>* 

**2. Frame of missile formation control system** 

*I z*

  *I o*

Fig. 2. Sketch map of missile formation consisted of two followers and only one leader.

 

, , , , *Vli vli li*

Fig. 3. Relationships between subsystems of missile formation control system.

The cooperative mission planning subsystem supplies real-time status of mission space and distribution situation of targets, and restricts the flight status of leader at the same time; the formation configuration describing system receives the information supplied by cooperative engagement subsystem and establishes associated function of engagement efficacy according to different missile formation configurations, and then makes the decision set about formation configurations based on the returned value of this efficacy function. After optimizing the values of efficacy function according to the specific mission space, the optimal formation configuration can be calculated within this decision set. The optimal

*Vlci vlci lci*

subsystem in one missile formation system is showed in Fig.3.

*I y*

area and searching moving targets within large scope. This weapon was added bidirectional data link once again in 2003, so it can implement Man-in-the-Loop control and command, which meant that this weapon system can attack targets autonomously and manifest a kind of smart engagement capability. After those improvements, this old anti-armor weapon was re-named to LOw Cost Autonomous Attack System (LOCAAS). LOCAAS adopts INS and mid-course guidance, and it also equips a smart fuze and a sensor that can be used to search moving targets, so LOCAAS can not only monitor targets within large range but also can relocate, recognize and aim at them autonomously. The most outstanding character is that LOCAAS can also make intelligent decision for choosing the optimal orientation and sequence to achieve optimal attack, and it also has the capability of on-line planning mission.

At present, United State Force sets the engagement schemes for LOCAAS as: there will be some LOCAASs flying in the battle field to cruise or put on standby, and they can connect with each other by data link. While one of them finding the targets and can't destroy them alone, it will send out the signals to require for cooperative attacking, however if it can destroy the targets by itself, this LOCAAS will attack the targets and the other LOCAASs will continue searching targets after receiving the instruction signals from the mentioned LOCAAS which has gone into the battle. The sketch map Fig.1 shows the main concepts of how LOCAASs take part in engagement cooperatively. Authors who are interested in LOCAAS can get more detailed information in the website of Lockheed Martin.

Fig. 1. The sketch map of LOCAAS engagement.

The background and significance were presented first, and then the technique frame of missile formation cooperative control system is showed, which can clearly elaborate how missiles in a formation work together in a cooperative manner. The specific relationships between each loop are analyzed, which are very necessary to clearly explain how to design missile formation control system. Followed are the main contents of this chapter, which are the detailed processes of establishing and designing missile formation control system. In this part, we will only consider the external loop of missile formation control system because the design method on individual missile inner controller is easy to be found in many literatures, and then we just assumed it is closed-loop and stable. The detailed process is divided into two steps, first is using proportion-differential control method to design missile formation keeping controller, the other explains how to design optimal keeping controller of missile formation. Some simulations are made to compare these two formation control systems proposed in this chapter at last.

### **2. Frame of missile formation control system**

180 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

area and searching moving targets within large scope. This weapon was added bidirectional data link once again in 2003, so it can implement Man-in-the-Loop control and command, which meant that this weapon system can attack targets autonomously and manifest a kind of smart engagement capability. After those improvements, this old anti-armor weapon was re-named to LOw Cost Autonomous Attack System (LOCAAS). LOCAAS adopts INS and mid-course guidance, and it also equips a smart fuze and a sensor that can be used to search moving targets, so LOCAAS can not only monitor targets within large range but also can relocate, recognize and aim at them autonomously. The most outstanding character is that LOCAAS can also make intelligent decision for choosing the optimal orientation and sequence to achieve optimal attack, and it also has the capability of on-line planning

At present, United State Force sets the engagement schemes for LOCAAS as: there will be some LOCAASs flying in the battle field to cruise or put on standby, and they can connect with each other by data link. While one of them finding the targets and can't destroy them alone, it will send out the signals to require for cooperative attacking, however if it can destroy the targets by itself, this LOCAAS will attack the targets and the other LOCAASs will continue searching targets after receiving the instruction signals from the mentioned LOCAAS which has gone into the battle. The sketch map Fig.1 shows the main concepts of how LOCAASs take part in engagement cooperatively. Authors who are interested in

The background and significance were presented first, and then the technique frame of missile formation cooperative control system is showed, which can clearly elaborate how missiles in a formation work together in a cooperative manner. The specific relationships between each loop are analyzed, which are very necessary to clearly explain how to design missile formation control system. Followed are the main contents of this chapter, which are the detailed processes of establishing and designing missile formation control system. In this part, we will only consider the external loop of missile formation control system because the design method on individual missile inner controller is easy to be found in many literatures, and then we just assumed it is closed-loop and stable. The detailed process is divided into two steps, first is using proportion-differential control method to design missile formation keeping controller, the other explains how to design optimal keeping controller of missile formation. Some simulations are made to compare these two formation control systems

LOCAAS can get more detailed information in the website of Lockheed Martin.

Fig. 1. The sketch map of LOCAAS engagement.

proposed in this chapter at last.

mission.

The missile formation control system mentioned in this chapter only has one leader missile (showed as Fig.2), and it mainly consists of the following subsystems: cooperative engagement mission planning subsystem, formation configuration describing subsystem, leader inner-loop control subsystem, follower inner-loop control subsystem and formation configuration control subsystem. The sketch map illustrates relationships between each subsystem in one missile formation system is showed in Fig.3.

Fig. 2. Sketch map of missile formation consisted of two followers and only one leader.

Fig. 3. Relationships between subsystems of missile formation control system.

The cooperative mission planning subsystem supplies real-time status of mission space and distribution situation of targets, and restricts the flight status of leader at the same time; the formation configuration describing system receives the information supplied by cooperative engagement subsystem and establishes associated function of engagement efficacy according to different missile formation configurations, and then makes the decision set about formation configurations based on the returned value of this efficacy function. After optimizing the values of efficacy function according to the specific mission space, the optimal formation configuration can be calculated within this decision set. The optimal formation configuration will be different when the missile formation is flying in different segments of mission space, that is, the optimal formation configuration will be decided by specific engagement requirement. The formation configuration describing subsystem will also restrict flight status of leader together with cooperative mission planning subsystem at the same time; The leader inner-loop control subsystem receives the information sent from the cooperative engagement planning subsystem and controls the leader to fly stably, and then the real flight states of leader can be obtained; The required states of follower missile can be calculated through the states in relative spatial dimension i.e. the distance between missiles which is obtained from formation configuration describing subsystem; The follower inner-loop control subsystem receives the command states, which is similar to that of leader mentioned above, and then control follower to fly stably. The actual flight states of follower missiles are fed back to the formation configuration control subsystem and used to achieve the expected formation configuration which is decided by the specific cooperative engagement mission.

This chapter focus on missile formation keeping control problems. First, we consider the actual flight states of leader as perturbation variables acting on the controller, and assume the follower inner-controller is closed and stable loop, that is, followers can track the required commands of velocity, flight path angle and flight deflection angle rapidly and stably. We will make further assumption that those three channels referred above are oneorder systems expressed as (Wei et al., 2010):

$$\begin{aligned} \dot{V}\_f &= -\frac{1}{\tau\_{vf}}(V\_f - V\_{fc}) \\ \dot{\theta}\_f &= -\frac{1}{\tau\_{\theta f}}(\theta\_f - \theta\_{fc}) \\ \dot{\nu}\_{vf} &= -\frac{1}{\tau\_{\nu\_{vf}}}(\nu\_{vf} - \nu\_{vf}) \end{aligned} \tag{1}$$

Missile Cooperative Engagement Formation Configuration Control Method 183

dimensional states for some periods of time. Next sections will present some further and

The main thought of the following parts is showed as below: basing on the kinematics relationships between missiles in inertial coordinate frame, the missile formation proportional-derivative (PD) controller via feeding back full states will be present firstly; Second, the relative motion will be established in relative coordinates frame, which can indicate the characteristics of relative motion directly. Based on this direct relative motion, the optimal controller that has non-zero given point and restrains slowly variant

The origin *<sup>r</sup> o* of this coordinate coincides with the mass centre of leader, and axis *r r o x* points to the velocity direction of leader, *r r o y* is perpendicular to *r r o x* and points to up direction, *r r o z* composes right-hand coordinate frame together with other two axes mentioned above.

The origin *OI* of this coordinate frame is fixed to an arbitrary point on ground, axis *O XI I* lies in horizontal plane and points to target, axis*O YI I* is perpendicular to *O XI I* and points to up direction, the last axis*O ZI I* also composes right-hand coordinate frame system together with

yr o1

*r*

The origin of this coordinate 1*o* coincides with the mass centre of missile, and axis 1 2 *o x* points to the velocity direction of missile, 1 2 *o y* is perpendicular to 1 2 *o x* and points to up

XI

r x

or r z

The relationship between these two coordinate frames is showed in Fig.4.

OI

Fig. 4. Relationship between relative and inertial coordinate frame.

YI

detailed researches on how to keep missile formation optimally and robustly.

perturbations is designed in the third part.

**3.1 Definition of coordinate frame system**  1. Relative coordinate frame *r r rr o xyz*

2. Inertial coordinate frame system *O XYZ I II I*

those two axes mentioned above.

ZI

3. Trajectory coordinate frame system 1 2 22 *o xyz*

**3. PD controller of missile formation keeping** 

where , *Vf f* and *vf* is the velocity, flight path angle and flight deflection angle of follower respectively; , *Vfc fc* and *vfc* are commands of velocity, flight path angle and flight deflection angle of follower respectively; , *vf f* and *vf* are the inertial time constants of velocity, flight path angle and flight deflection angle channel of follower respectively. These values can be calculated by analyzing four-dimensional guidance and control system referrd in the literature of Cui et al., 2010, they are:

$$\begin{aligned} \tau\_v &= 3.51 \text{s} \\ \tau\_\theta &= 2.25 \text{s} \\ \tau\_{\varphi\_v} &= 2.37 \text{s} \end{aligned} \tag{2}$$

These inertial time constants will be used in the subsequent sections for designing and simulating missile formation keeping controller.

After missile formation achieving four-dimensionally rendezvous, this formation should carry out some subsequent missions, such as relative navigation and location, cooperative searching targets, locating and recognizing targets and cooperative penetration etc., however, these missions require missile formation keeping its configuration or relative dimensional states for some periods of time. Next sections will present some further and detailed researches on how to keep missile formation optimally and robustly.

The main thought of the following parts is showed as below: basing on the kinematics relationships between missiles in inertial coordinate frame, the missile formation proportional-derivative (PD) controller via feeding back full states will be present firstly; Second, the relative motion will be established in relative coordinates frame, which can indicate the characteristics of relative motion directly. Based on this direct relative motion, the optimal controller that has non-zero given point and restrains slowly variant perturbations is designed in the third part.

### **3. PD controller of missile formation keeping**

### **3.1 Definition of coordinate frame system**

1. Relative coordinate frame *r r rr o xyz*

182 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

formation configuration will be different when the missile formation is flying in different segments of mission space, that is, the optimal formation configuration will be decided by specific engagement requirement. The formation configuration describing subsystem will also restrict flight status of leader together with cooperative mission planning subsystem at the same time; The leader inner-loop control subsystem receives the information sent from the cooperative engagement planning subsystem and controls the leader to fly stably, and then the real flight states of leader can be obtained; The required states of follower missile can be calculated through the states in relative spatial dimension i.e. the distance between missiles which is obtained from formation configuration describing subsystem; The follower inner-loop control subsystem receives the command states, which is similar to that of leader mentioned above, and then control follower to fly stably. The actual flight states of follower missiles are fed back to the formation configuration control subsystem and used to achieve the expected formation configuration which is decided by the specific cooperative

This chapter focus on missile formation keeping control problems. First, we consider the actual flight states of leader as perturbation variables acting on the controller, and assume the follower inner-controller is closed and stable loop, that is, followers can track the required commands of velocity, flight path angle and flight deflection angle rapidly and stably. We will make further assumption that those three channels referred above are one-

<sup>1</sup> ( )

<sup>1</sup> ( )

*vf*

 *vf* 

flight path angle and flight deflection angle channel of follower respectively. These values can be calculated by analyzing four-dimensional guidance and control system referrd in the

> 3.51s 2.25s 2.37s *<sup>v</sup>*

These inertial time constants will be used in the subsequent sections for designing and

After missile formation achieving four-dimensionally rendezvous, this formation should carry out some subsequent missions, such as relative navigation and location, cooperative searching targets, locating and recognizing targets and cooperative penetration etc., however, these missions require missile formation keeping its configuration or relative

*v*   

*vf vf vfc*

 

*f f fc vf*

*V VV*

 and

*f f fc f*

 

<sup>1</sup> ( )

*vf* is the velocity, flight path angle and flight deflection angle of follower

*vfc* are commands of velocity, flight path angle and flight deflection

are the inertial time constants of velocity,

(2)

(1)

engagement mission.

where , *Vf f*

respectively; , *Vfc*

 and

> *fc* and

angle of follower respectively; , *vf f*

literature of Cui et al., 2010, they are:

simulating missile formation keeping controller.

order systems expressed as (Wei et al., 2010):

The origin *<sup>r</sup> o* of this coordinate coincides with the mass centre of leader, and axis *r r o x* points to the velocity direction of leader, *r r o y* is perpendicular to *r r o x* and points to up direction, *r r o z* composes right-hand coordinate frame together with other two axes mentioned above. 2. Inertial coordinate frame system *O XYZ I II I*

The origin *OI* of this coordinate frame is fixed to an arbitrary point on ground, axis *O XI I* lies in horizontal plane and points to target, axis*O YI I* is perpendicular to *O XI I* and points to up direction, the last axis*O ZI I* also composes right-hand coordinate frame system together with those two axes mentioned above.

The relationship between these two coordinate frames is showed in Fig.4.

3. Trajectory coordinate frame system 1 2 22 *o xyz*

The origin of this coordinate 1*o* coincides with the mass centre of missile, and axis 1 2 *o x* points to the velocity direction of missile, 1 2 *o y* is perpendicular to 1 2 *o x* and points to up direction, 1 2 *o z* composes right-hand coordinate frame together with other two axes mentioned above. The spatial directions are parallel to those in relative coordinate frame system.

### **3.2 Model of relative motion between missiles**

#### **3.2.1 Basic assumptions**

It is assumed that the control system of missile self is closed-loop and stable, that is, missile can track the required velocity, flight path angle and flight deflection angle rapidly and stably. Further, we consider these tracking channels are one-order inertial loops described as section 2. In order to express the relationships conveniently, we will change the one-order inertial loops showed in Equ.(1) to the following forms:

$$\begin{cases} \dot{V}\_i = -\mathcal{\lambda}\_v (V\_i - V\_{ic}) \\ \dot{\theta}\_i = -\mathcal{\lambda}\_\theta (\theta\_i - \theta\_{ic}) \\ \dot{\nu}\_{vi} = -\mathcal{\lambda}\_{\psi\_v} (\nu\_{vi} - \nu\_{vic}) \end{cases} \tag{3}$$

Missile Cooperative Engagement Formation Configuration Control Method 185

*<sup>r</sup> <sup>o</sup> <sup>V</sup>*<sup>2</sup>

*X*2

*V*2

2

*X*2

2 1 2 111 \* 2 1

2 1 11 1 2 1 1 1 1 \* \* 2 1

*<sup>d</sup> <sup>d</sup> Y Y <sup>y</sup> <sup>y</sup> d d Z Z z z*

*X X x x*

 

*T T*

*X X x Y Y y Z Z z* 

( )() *<sup>v</sup>*

 

2 1 \* \* 1 1

*e T T* (7)

 

( ) ( ) () ( ) *<sup>v</sup> v v*

Further, the deviations of relative positions can be expressed from Equ.(5) to:

2 1

*v*

 *y*

*v*2

2

*X*1

*z*

*X*1

Fig. 5. Relative position relations between two missiles.

2

*YI*

*Y*1

*ZI*

*Z*1

and making derivatives to last equation yields:

making further derivatives gets:

2 1

*Z*2

*Y*2

*V*1

*v*1

*XI*

*XI*

*r x*

1

*x*

*r z*

1

*r o*

*V*1

\*

\*

*e TT* (6)

\* \*

   

*x*

1

*<sup>r</sup> x <sup>r</sup> y*

where i is the number of missile,*Vi* ,*<sup>i</sup>* and *vi* represent the actual velocity, flight path angle and flight deflection angle of ith missile respectively;*Vic* ,*ic* and *vic* are desired velocity, desired flight path angle and desired flight deflection angle of ith missile respectively; *v* , and *v* are the reciprocals of inertial time constants *<sup>v</sup>* , and *v* in channels of veloctiy, flight path angle and flight deflection angle respectively.

### **3.2.2 Establishing the control model**

The kinematics equations of missile in inertial coordinate frame can be expressed as:

$$\begin{cases} \dot{X}\_i = V\_i \cos \theta\_i \cos \varphi\_{vi} \\ \dot{Y}\_i = V\_i \sin \theta\_i \\ \dot{Z} = -V\_i \cos \theta\_i \sin \varphi\_{vi} \end{cases} \tag{4}$$

The kinematics relationship of two missiles in inertial and relative coordinate frame is showed in Fig.5, so the relative positions of two missiles can be obtained from these sketch maps:

$$
\begin{bmatrix} X\_2 \\ Y\_2 \\ Z\_2 \end{bmatrix} = \begin{bmatrix} X\_1 \\ Y\_1 \\ Z\_1 \end{bmatrix} + \mathcal{T}\_2(\boldsymbol{\psi}\_{v1})\mathcal{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} \tag{5}
$$

where \* \* *x* ,*y* and \* *z* are the distances relative to leader in relative coordinate frame, and the transformation matrices in Equ.(5) are:

$$\begin{aligned} \mathbf{T}\_1(.) = \begin{bmatrix} \cos(.) & -\sin(.) & 0\\ \sin(.) & \cos(.) & 0\\ 0 & 0 & 1 \end{bmatrix}, \ \mathbf{T}\_2(.) = \begin{bmatrix} \cos(.) & 0 & \sin(.) \\ 0 & 1 & 0\\ -\sin(.) & 0 & \cos(.) \end{bmatrix} \end{aligned}$$

184 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

direction, 1 2 *o z* composes right-hand coordinate frame together with other two axes mentioned above. The spatial directions are parallel to those in relative coordinate frame

It is assumed that the control system of missile self is closed-loop and stable, that is, missile can track the required velocity, flight path angle and flight deflection angle rapidly and stably. Further, we consider these tracking channels are one-order inertial loops described as section 2. In order to express the relationships conveniently, we will change the one-order

> ( ) ( ) ( )

  (3)

 *v* 

(4)

*vic* are desired velocity,

in channels of

*vi* represent the actual velocity, flight path angle

*ic* and

> , and

*v*

desired flight path angle and desired flight deflection angle of ith missile respectively;

cos cos

 

*i i i vi*

The kinematics relationship of two missiles in inertial and relative coordinate frame is showed in Fig.5, so the relative positions of two missiles can be obtained from these sketch

> 2 1 2 111 \* 2 1

*X X x Y Y y Z Z z* 

where \* \* *x* ,*y* and \* *z* are the distances relative to leader in relative coordinate frame, and the

cos sin

*i i vi*

( )() *<sup>v</sup>*

 

\*

\*

cos(.) 0 sin(.)

sin(.) 0 cos(.) *T*

(.) 0 1 0

*T T* (5)

sin

*ii i*

*i v i ic i i ic vi vi vic*

*V VV* 

 

 

 

 

*<sup>i</sup>* and

are the reciprocals of inertial time constants *<sup>v</sup>*

The kinematics equations of missile in inertial coordinate frame can be expressed as:

*X V Y V Z V*

 

 

2 1

cos(.) sin(.) 0 (.) sin(.) cos(.) 0

0 01 *T* , 2

system.

*v* , and *v* 

maps:

**3.2.1 Basic assumptions** 

where i is the number of missile,*Vi* ,

**3.2.2 Establishing the control model** 

transformation matrices in Equ.(5) are:

1

**3.2 Model of relative motion between missiles** 

inertial loops showed in Equ.(1) to the following forms:

and flight deflection angle of ith missile respectively;*Vic* ,

veloctiy, flight path angle and flight deflection angle respectively.

Fig. 5. Relative position relations between two missiles.

Further, the deviations of relative positions can be expressed from Equ.(5) to:

$$\mathbf{e} = \begin{bmatrix} X\_2 - X\_1 \\ Y\_2 - Y\_1 \\ Z\_2 - Z\_1 \end{bmatrix} - T\_2(\boldsymbol{\psi}\_{v1})T\_1(\theta\_1) \begin{bmatrix} x^\* \\ y^\* \\ z^\* \end{bmatrix} \tag{6}$$

and making derivatives to last equation yields:

$$\dot{\mathbf{e}} = \begin{bmatrix} \dot{X}\_2 - \dot{X}\_1 \\ \dot{Y}\_2 - \dot{Y}\_1 \\ \dot{Z}\_2 - \dot{Z}\_1 \end{bmatrix} - \frac{d\mathbf{T}\_2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}} \dot{\boldsymbol{\nu}}\_{v1} \mathbf{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \mathbf{x}^\* \\ \mathbf{y}^\* \\ \mathbf{z}^\* \end{bmatrix} - \mathbf{T}\_2(\boldsymbol{\nu}\_{v1}) \frac{d\mathbf{T}\_1(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1} \dot{\boldsymbol{\theta}}\_1 \begin{bmatrix} \mathbf{x}^\* \\ \mathbf{y}^\* \\ \mathbf{z}^\* \end{bmatrix} \tag{7}$$

making further derivatives gets:

$$
\begin{split}
\dot{\boldsymbol{\theta}} &= \begin{bmatrix} \ddot{X}\_2 - \ddot{X}\_1 \\ \ddot{Y}\_2 - \ddot{Y}\_1 \\ \ddot{Z}\_2 - \ddot{Z}\_1 \end{bmatrix} - \frac{d\mathbf{T}\_2^2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}^2} \dot{\boldsymbol{\nu}}\_{v1}^2 \mathbf{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} - \frac{d\mathbf{T}\_2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}} \dot{\boldsymbol{\nu}}\_{v1} \mathbf{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} \\ &- 2\frac{d\mathbf{T}\_2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}} \dot{\boldsymbol{\nu}}\_{v1}^2 \frac{d\mathbf{T}\_1(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1} \dot{\boldsymbol{\theta}}\_1 \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} - \mathbf{T}\_2(\boldsymbol{\nu}\_{v1}) \frac{d\mathbf{T}\_1^2(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1^2} \dot{\boldsymbol{\theta}}\_1^2 \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} - \mathbf{T}\_2(\boldsymbol{\nu}\_{v1}) \frac{d\mathbf{T}\_1(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1} \dot{\boldsymbol{\theta}}\_1 \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} \end{split} \tag{8}
$$

where

$$\begin{aligned} \ddot{\theta}\_1 &= -\dot{\lambda}\_\theta (\dot{\theta}\_1 - \dot{\theta}\_{1c}), \quad \ddot{\psi}\_{v1} = -\dot{\lambda}\_{\psi\_v} (\dot{\psi}\_{v1} - \dot{\psi}\_{v1c}) \\\\ \frac{dT\_1(.)}{d(.)} &= \begin{bmatrix} -\sin(.) & -\cos(.) & 0 \\ \cos(.) & -\sin(.) & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \frac{dT\_2(.)}{d(.)} = \begin{bmatrix} -\sin(.) & 0 & \cos(.) \\ 0 & 0 & 0 \\ -\cos(.) & 0 & -\sin(.) \end{bmatrix} \\\\ \frac{d^2T\_1(.)}{d^2(.)} &= \begin{bmatrix} -\cos(.) & \sin(.) & 0 \\ -\sin(.) & -\cos(.) & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \frac{d^2T\_2(.)}{d^2(.)} &= \begin{bmatrix} -\cos(.) & 0 & -\sin(.) \\ 0 & 0 & 0 \\ \sin(.) & 0 & -\cos(.) \end{bmatrix} \end{aligned}$$

besides, there is:

$$\begin{aligned} \ddot{X}\_i &= \dot{V}\_i \cos\theta\_i \cos\nu\_{vi} - V\_i \sin\theta\_i \dot{\theta}\_i \cos\nu\_{vi} - V\_i \cos\theta\_i \sin\nu\_{vi} \dot{\nu}\_{vi} \\ \ddot{Y}\_i &= \dot{V}\_i \sin\theta\_i + V\_i \cos\theta\_i \dot{\theta}\_i \\ \ddot{Z}\_i &= -\dot{V}\_i \cos\theta\_i \sin\nu\_{vi} + V\_i \sin\theta\_i \dot{\theta}\_i \sin\nu\_{vi} - V\_i \cos\theta\_i \cos\nu\_{vi} \dot{\nu}\_{vi} \end{aligned} \tag{9}$$

so we can get the following vector equation:

$$
\ddot{\mathbf{e}} = f\_1 + \mathbf{G}\_r \mu \tag{10}
$$

Missile Cooperative Engagement Formation Configuration Control Method 187

*V*

2 2 2

 

 

1 1 1 1 11

*ZV V V*

asymptotically equivalent to zero, we will choose the following PD control laws:

*Y V V*

further, the required control quantity can be expressed as below:

 

 

*X VV V*

 

*r v*

As for the expression of <sup>1</sup>*f* , we also need to know:

The control variables of follower are:

*G<sup>r</sup>* can be calculated by Equ.(11), it is:

expression (13) to be converged, which is lim*t*

**4.1 Establishing model of relative motion** 

**4. Optimal controller of missile formation keeping** 

so there will be:

derivatives is:

2 2 22 2 2 2 2

*V V*

*v*

 

 

*u* (12)

 

 

*v*

 

 

1 2 *eee k k* **0** (13)

11 2 ( ) *<sup>r</sup> k k u G <sup>f</sup> e e* (14)

*V* (15)

   

 

 

cos cos sin cos cos sin sin cos 0 cos sin sin sin cos cos

 *G* (11)

 

 

*v v v v*

*vv v v*

2 2 2

 

*V* 

1 1 1 1 1 11 1 1 1 1 1

 

 

cos cos sin cos cos sin sin cos cos sin sin sin cos cos

1 1 1 1 1 11 1 1 1 1 1

In order to eliminate the deviations of relative positions, that is, make the deviation *e* be

12 1 *<sup>r</sup> k k e e f G u*

What should be done next is to analyze the existence conditions of expression (14). The condition is: iff matrix *G<sup>r</sup>* is non-singular, the control quantity exists. The determinant of

> <sup>2</sup> det cos <sup>2</sup> *<sup>v</sup> <sup>G</sup>r v*

Through checking this expression, it is obvious that *G<sup>r</sup>* is non-singular when the missile formation is flying, that is, <sup>2</sup> *V* 0 , so we can choose proper coefficients 1 2 *k k*, to guarantee

*e* **0** .

Because relative coordinate frame is rotating, the relationship between relative derivative and absolute derivative should be considered during the process of establishing relative motion equations in relative coordinate frame. The relationship between these two

1

 

*c c v c*

 

2 2 22 2 2 2 2

*v v v v*

 

 

> 

*v v v v*

*V V*

where

\* \* <sup>1</sup> <sup>2</sup> <sup>2</sup> 2 1 2 \* 2 1 \* 11 2 2 11 1 11 1 <sup>1</sup> <sup>1</sup> \* \* 1 2 ( ) ( ) ( ) ( ) *v v rv v v v v X V x x d d <sup>Y</sup> y y <sup>d</sup> <sup>d</sup> Z z z T T fG T T* \* \*\* 2 2 1 1 1 \* 1 1 2 \* 1 1 \* 1 1 21 1 21 1 2 1 1 1 1 \* \*\* ( ) ( ) ( ) ( ) <sup>2</sup> ( ) ( ) *<sup>v</sup> vv v v x xx d dd d y yy d d <sup>d</sup> <sup>d</sup> z zz <sup>T</sup> TT T T T*

$$\mathbf{G}\_r = \begin{bmatrix} \lambda\_v \cos \theta\_2 \cos \psi\_{v2} & -\lambda\_\theta V\_2 \sin \theta\_2 \cos \psi\_{v2} & -\lambda\_{\varphi\_v} V\_2 \cos \theta\_2 \sin \psi\_{v2} \\ \lambda\_v \sin \theta\_2 & \lambda\_\theta V\_2 \cos \theta\_2 & 0 \\ -\lambda\_v \cos \theta\_2 \sin \psi\_{v2} & \lambda\_\theta V\_2 \sin \theta\_2 \sin \psi\_{v2} & -\lambda\_{\varphi\_v} V\_2 \cos \theta\_2 \cos \psi\_{v2} \end{bmatrix} \tag{11}$$

The control variables of follower are:

$$\mathbf{u} = \begin{bmatrix} V\_{2c} \\ \theta\_{2c} \\ \boldsymbol{\nu}\_{v2c} \end{bmatrix} \tag{12}$$

As for the expression of <sup>1</sup>*f* , we also need to know:

$$\begin{bmatrix} \ddot{X}\_1\\ \ddot{Y}\_1\\ \ddot{Z}\_1 \end{bmatrix} = \begin{bmatrix} \dot{V}\_1 \cos \theta\_1 \cos \nu\_{v1} - V\_1 \sin \theta\_1 \dot{\theta}\_1 \cos \nu\_{v1} - V\_1 \cos \theta\_1 \sin \nu\_{v1} \dot{\nu}\_{v1} \\\ \dot{V}\_1 \sin \theta\_1 + V\_1 \cos \theta\_1 \dot{\theta}\_1 \\\ -\dot{V}\_1 \cos \theta\_1 \sin \nu\_{v1} + V\_1 \sin \theta\_1 \dot{\theta}\_1 \sin \nu\_{v1} - V\_1 \cos \theta\_1 \cos \nu\_{v1} \dot{\nu}\_{v1} \end{bmatrix}.$$

In order to eliminate the deviations of relative positions, that is, make the deviation *e* be asymptotically equivalent to zero, we will choose the following PD control laws:

$$
\ddot{\mathbf{e}} + k\_1 \dot{\mathbf{e}} + k\_2 \mathbf{e} = \mathbf{0} \tag{13}
$$

so there will be:

186 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

2 1 2 \* 2 1 \*

1 11 1 1 1 ( ), ( )

*c v v vc*

*d d*

> 2 2 2

*d d*

cos cos sin cos cos sin

 

*i i i vi i i i vi i i vi vi*

 

> 

cos sin sin sin cos cos

*i i i vi i i i vi i i vi vi*

 

11 2 2 11 1 11 1

 

<sup>1</sup> <sup>1</sup> \* \* 1 2 ( ) ( ) ( ) ( ) *v v rv v v v*

*y yy d d <sup>d</sup> <sup>d</sup>*

 *<sup>T</sup> TT T T T*

2 2 1 1 1 \* 1 1 2 \* 1 1 \* 1 1 21 1 21 1 2 1 1 1 1 \* \*\*

 

*X V x x d d <sup>Y</sup> y y <sup>d</sup> <sup>d</sup> Z z z*

*T T fG T T*

( ) ( ) ( ) ( ) <sup>2</sup> ( ) ( ) *<sup>v</sup> vv v*

*d dd d*

 

 

( ) ( ) ( ) ( )

*v v v v v v*

2

 

 

*v*

 

 

 

2 1 2 11 1 11 1 <sup>1</sup> <sup>1</sup> \* \* 2 1

> 

 

 

*X X x x d d Y Y <sup>y</sup> <sup>y</sup> <sup>d</sup> <sup>d</sup> Z Z z z*

 

\*

*x*

 

2 1 1 1 \* 1 1 1 1 21 2 1 1 \* 1

*T T e TT*

( ) ( ) ( ) <sup>2</sup> ( )

 

sin(.) cos(.) 0 (.) cos(.) sin(.) 0 (.) 0 00

cos(.) sin(.) 0 (.) sin(.) cos(.) 0 (.) 0 00

sin cos

*i i i i ii*

 

<sup>1</sup> <sup>2</sup> <sup>2</sup>

*v*

 

*YV V*

so we can get the following vector equation:

*v*

  *<sup>T</sup>* ,

*XV V V*

 

*ZV V V*

 *<sup>T</sup>* , <sup>2</sup>

*<sup>v</sup> v v*

*<sup>T</sup> T T <sup>T</sup>*

*d d d <sup>y</sup> d d <sup>d</sup> z*

 

2 1 <sup>2</sup>

*v*

1

*d d*

2 1 2

*d d*

besides, there is:

where

where

\* \*

 

sin(.) 0 cos(.) (.) <sup>000</sup> (.) cos(.) 0 sin(.)

 *<sup>T</sup>*

cos(.) 0 sin(.) (.) <sup>000</sup> (.) sin(.) 0 cos(.)

> 

 

2 1 2 \* 2 1 \*

 

\* \*\*

 

*x xx*

*z zz*

<sup>1</sup> *<sup>r</sup> e f G u* (10)

\* \*

 

 

 *<sup>T</sup>*

\* \* 2 \* 1 1 \* 1 21 1 1 \* \* ( ) ( ) *<sup>v</sup> x x d y y <sup>d</sup> z z*

 *<sup>T</sup> <sup>T</sup>*

 

(9)

(8)

$$-k\_1 \dot{e} - k\_2 e = f\_1 + G\_r u$$

further, the required control quantity can be expressed as below:

$$\mathbf{u} = \mathbf{G}\_r^{-1}(-f\_1 - k\_1\dot{\mathbf{e}} - k\_2\mathbf{e}) \tag{14}$$

What should be done next is to analyze the existence conditions of expression (14). The condition is: iff matrix *G<sup>r</sup>* is non-singular, the control quantity exists. The determinant of *G<sup>r</sup>* can be calculated by Equ.(11), it is:

$$\det \mathbf{G}\_r = -\mathbb{A}\_\vartheta \mathbb{A}\_{\boldsymbol{\psi}\_v} \mathbb{A}\_{\boldsymbol{\psi}\_2} V\_2^2 \cos \theta \tag{15}$$

Through checking this expression, it is obvious that *G<sup>r</sup>* is non-singular when the missile formation is flying, that is, <sup>2</sup> *V* 0 , so we can choose proper coefficients 1 2 *k k*, to guarantee expression (13) to be converged, which is lim*t e* **0** .

### **4. Optimal controller of missile formation keeping**

### **4.1 Establishing model of relative motion**

Because relative coordinate frame is rotating, the relationship between relative derivative and absolute derivative should be considered during the process of establishing relative motion equations in relative coordinate frame. The relationship between these two derivatives is:

$$\mathbf{V}\_{fr} - \mathbf{V}\_{lr} = \mathbf{V}\_r + \mathfrak{op} \times \mathbf{r} \tag{16}$$

Missile Cooperative Engagement Formation Configuration Control Method 189

where , *x y* and *z* are components of relative position vector *r* in relative coordinate frame, and the rotating angular velocity of relative coordinate frame with respect to inertial space

can be expressed as:

so, as for the expression

[,,]*<sup>T</sup>*

sin cos *vl l vl l l*

cos

After expanding all the terms from Equ.(17) to Equ.(20) finally, we can obtain the missile

cos cos cos sin sin cos cos sin cos sin cos sin

The formation motion equations (22) are nonlinear, we can treat them by linearized method to get linear forms that can be utilized and analyzed more conveniently. During the process of missile formation flight, some variables can be considered as small quantities, such

> *f f f l l vl l f l f f l vl l f vl vf vl l vl*

 

 

 

*xV V V z y yV V x z z V y x*

 

(1 ) ( ) ( )

 

 

 

 

*x y z VV V*

*l vl fb l f fb l f l l l vl fb l f fb f vl vl l vl vfb f fb vf*

( )

*y x z VV z x y VV*

 

Dealing with this expression by small perturbation linearized method yields:

 

 

 

   

*x V V Vz y*

 

*f f l e f f l l vl l l f f l e f f l l vl l*

 

 

*z y x z y x*

*vl l l l vl l vl l vl l*

 

 

 

sin sin cos

 

> 

 

*<sup>f</sup>* , and the states of leader can be considered as inputs, then Equ.(22)

 

 

> 

> >

 

> 

*r* (20)

   

> 

(22)

(23)

cos sin sin cos

**4.2 Establishing the optimal control model of missile formation keeping** 

 

*z V y x*

 

 

*f f e vl l vl l*

*y V V x z*

*r* , there will be:

further, we can get the following expression directly from equation (16):

() *VVV r r fr lr*

relative kinematics model in 3-dimension space:

 

*e vl vf*

**4.2.1 Linearized method** 

can be transformed to:

as *<sup>f</sup>* , *<sup>l</sup>* and *e l*  *<sup>r</sup> V xyz* (18)

(19)

(21)

where*Vfr* and*Vlr* are the absolute velocities of follower and leader in relative coordinate frame respectively;*V<sup>r</sup>* is the relative velocity from follower to leader in relative coordinate frame; represents the rotating angular velocity of relative coordinate frame relative with respect to inertial space, and it is described in relative coordinate frame;*r* is the position vector of follower relative to leader in relative coordinate frame.

We can get the absolute velocities of follower and leader in relative coordinate frame by the transformations as below:

$$\begin{aligned} \mathcal{V}\_{fr} &= \boldsymbol{\Phi}\_{l}^{r} \boldsymbol{\Phi}\_{2}^{l} \boldsymbol{\mathcal{V}}\_{f2} \\ \mathcal{V}\_{lr} &= \mathcal{V}\_{l2} \end{aligned}$$

where *<sup>r</sup>* **Φ***<sup>I</sup>* is the transformation matrix from inertial coordinate frame to relative coordinate frame:

$$\mathbf{d}\Phi\_l^r = \begin{bmatrix} \cos\theta\_l \cos\psi\_{vl} & \sin\theta\_l & -\cos\theta\_l \sin\psi\_{vl} \\ -\sin\theta\_l \cos\psi\_{vl} & \cos\theta\_l & \sin\theta\_l \sin\psi\_{vl} \\ \sin\psi\_{vl} & 0 & \cos\psi\_{vl} \end{bmatrix}$$

<sup>2</sup> **<sup>Φ</sup>***<sup>I</sup>* is the transformation matrix from trajectory coordinate frame of follower *O xyz* 1 222 *<sup>f</sup>* to inertial coordinate frame:

$$\mathbf{O}\_2^I = \begin{vmatrix} \cos \theta\_f \cos \psi\_{vf} & -\sin \theta\_f \cos \psi\_{vf} & \sin \psi\_{vf} \\ \sin \theta\_f & \cos \theta\_f & 0 \\ -\cos \theta\_f \sin \psi\_{vf} & \sin \theta\_f \sin \psi\_{vf} & \cos \psi\_{vf} \end{vmatrix}$$

*V<sup>f</sup>* <sup>2</sup> and*Vl*<sup>2</sup> represent the velocity vectors of follower and leader in themselves trajectory coordinate frames respectively, and the components of them are:

$$\mathcal{V}\_{f2} = \begin{bmatrix} V\_f \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix}, \mathcal{V}\_{l2} = \begin{bmatrix} V\_l \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix}$$

Besides,*Vl* is the velocity of leader in inertial coordinate frame;*<sup>l</sup>* is the flight path angle of leader;*vl* is flight deflection angle of leader.

Further, the difference of absolute velocities between two missiles in the relative coordinate frame is:

$$\mathbf{V}\_{fr} - \mathbf{V}\_{lr} = \begin{bmatrix} V\_f \cos \theta\_f \cos \theta\_l \cos(\nu\_{vl} - \nu\_{vf}) + V\_f \sin \theta\_f \sin \theta\_l - V\_l \\ -V\_f \cos \theta\_f \sin \theta\_l \cos(\nu\_{vl} - \nu\_{vf}) + V\_f \sin \theta\_f \cos \theta\_l \\ V\_f \cos \theta\_f \sin(\nu\_{vl} - \nu\_{vf}) \end{bmatrix} \tag{17}$$

at the same time, the relative velocity between two missiles in relative coordinate frame:

$$\mathcal{V}\_r = [\dot{\mathbf{x}}, \dot{\mathbf{y}}, \dot{\mathbf{z}}]^T \tag{18}$$

where , *x y* and *z* are components of relative position vector *r* in relative coordinate frame, and the rotating angular velocity of relative coordinate frame with respect to inertial space can be expressed as:

$$\mathbf{d}\mathbf{p} = \begin{bmatrix} \dot{\psi}\_{vl}\sin\theta\_l\\ \psi\_{vl}\cos\theta\_l\\ \dot{\theta}\_l \end{bmatrix} \tag{19}$$

so, as for the expression*r* , there will be:

188 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

*VVV r fr lr r*

where*Vfr* and*Vlr* are the absolute velocities of follower and leader in relative coordinate frame respectively;*V<sup>r</sup>* is the relative velocity from follower to leader in relative coordinate

We can get the absolute velocities of follower and leader in relative coordinate frame by the

2

where *<sup>r</sup>* **Φ***<sup>I</sup>* is the transformation matrix from inertial coordinate frame to relative coordinate

cos cos sin cos sin sin cos cos sin sin sin 0 cos

 **Φ**

*I l vl l l vl*

<sup>2</sup> **<sup>Φ</sup>***<sup>I</sup>* is the transformation matrix from trajectory coordinate frame of follower *O xyz* 1 222 *<sup>f</sup>* to

cos cos sin cos sin sin cos 0 cos sin sin sin cos

 **Φ**

*f f*

*V<sup>f</sup>* <sup>2</sup> and*Vl*<sup>2</sup> represent the velocity vectors of follower and leader in themselves trajectory

2 2 0, 0 0 0

Further, the difference of absolute velocities between two missiles in the relative coordinate

*fr lr f f l vl vf f f l*

at the same time, the relative velocity between two missiles in relative coordinate frame:

*V*

 

 

*V V*

 

cos cos cos( ) sin sin cos sin cos( ) sin cos cos sin( )

 

*f f l vl vff f l l*

*V V V*

*f f vl vf*

 

*V V* (17)

 *<sup>V</sup> <sup>V</sup> V V*

*f l*

*lr l*

coordinate frames respectively, and the components of them are:

Besides,*Vl* is the velocity of leader in inertial coordinate frame;

*vl* is flight deflection angle of leader.

 *V V* **Φ Φ** *V V*

*r I fr I f*

vector of follower relative to leader in relative coordinate frame.

*r*

2

*I*

frame;

frame:

leader;

frame is:

transformations as below:

inertial coordinate frame:

 represents the rotating angular velocity of relative coordinate frame relative with respect to inertial space, and it is described in relative coordinate frame;*r* is the position

2 2

*l vl l l vl*

 

> 

> >

 

 

> 

 

> 

*<sup>l</sup>* is the flight path angle of

 

 

 

*vl vl*

*f vf f vf vf*

 

 

*f l*

*f vf f vf vf*

  (16)

$$\mathbf{p} \clubsuit r = \begin{bmatrix} z\dot{\boldsymbol{\nu}}\_{vl}\cos\theta\_l - y\dot{\theta}\_l \\ x\dot{\theta}\_l - z\dot{\boldsymbol{\nu}}\_{vl}\sin\theta\_l \\ y\dot{\boldsymbol{\nu}}\_{vl}\sin\theta\_l - x\dot{\boldsymbol{\nu}}\_{vl}\cos\theta\_l \end{bmatrix} \tag{20}$$

further, we can get the following expression directly from equation (16):

$$\mathbf{V}\_r = (\mathbf{V}\_{fr} - \mathbf{V}\_{lr}) - \boldsymbol{\mathfrak{op}} \times \mathbf{r} \tag{21}$$

After expanding all the terms from Equ.(17) to Equ.(20) finally, we can obtain the missile relative kinematics model in 3-dimension space:

$$\begin{cases} \dot{x} = V\_f \cos \theta\_f \cos \theta\_l \cos \psi\_e + V\_f \sin \theta\_f \sin \theta\_l - V\_l - z \dot{\nu}\_{vl} \cos \theta\_l + y \dot{\theta}\_l \\ \dot{y} = -V\_f \cos \theta\_f \sin \theta\_l \cos \psi\_e + V\_f \sin \theta\_f \cos \theta\_l - x \dot{\theta}\_l + z \dot{\nu}\_{vl} \sin \theta\_l \\ \dot{z} = V\_f \cos \theta\_f \sin \psi\_e - y \dot{\nu}\_{vl} \sin \theta\_l + x \dot{\nu}\_{vl} \cos \theta\_l \\ \nu\_e = \nu\_{vl} - \nu\_{vf} \end{cases} \tag{22}$$

### **4.2 Establishing the optimal control model of missile formation keeping 4.2.1 Linearized method**

The formation motion equations (22) are nonlinear, we can treat them by linearized method to get linear forms that can be utilized and analyzed more conveniently. During the process of missile formation flight, some variables can be considered as small quantities, such as *<sup>f</sup>* , *<sup>l</sup>* and *e l <sup>f</sup>* , and the states of leader can be considered as inputs, then Equ.(22) can be transformed to:

$$\begin{cases} \dot{\mathbf{x}} = V\_f + V\_f \theta\_f \theta\_l - V\_l - z \dot{\boldsymbol{\nu}}\_{vl} + y \dot{\theta}\_l \\ \dot{\mathbf{y}} = -V\_f \theta\_l + V\_f \theta\_f - \mathbf{x} \dot{\theta}\_l + z \dot{\boldsymbol{\nu}}\_{vl} \theta\_l \\ \dot{z} = V\_f \left( \boldsymbol{\nu}\_{vl} - \boldsymbol{\nu}\_{vf} \right) - y \dot{\boldsymbol{\nu}}\_{vl} \theta\_l + \boldsymbol{\kappa} \dot{\boldsymbol{\nu}}\_{vl} \end{cases}$$

Dealing with this expression by small perturbation linearized method yields:

$$\begin{cases} \dot{\mathbf{x}} = \dot{\theta}\_l \mathbf{y} - \dot{\nu}\_{vl} \mathbf{z} + (\mathbf{1} + \theta\_{fb} \theta\_l) V\_f + V\_{fb} \theta\_l \theta\_f - V\_l\\ \dot{\mathbf{y}} = -\dot{\theta}\_l \mathbf{x} + \theta\_l \dot{\nu}\_{vl} \mathbf{z} + (\theta\_{fb} - \theta\_l) V\_f + V\_{fb} \theta\_f \\ \dot{\mathbf{z}} = \dot{\nu}\_{vl} \mathbf{x} - \dot{\nu}\_{vl} \theta\_l \mathbf{y} + (\mathbf{y}\_{vl} - \mathbf{y}\_{vfb}) V\_f - V\_{fb} \mathbf{y}\_{vf} \end{cases} \tag{23}$$

where , *Vfb fb* and *vfb* are feature points of linearized equations. We can describe this model by the following states space form:

$$\begin{aligned} \dot{X} &= AX + BU + \bar{B}W \\ Y &= CX \end{aligned} \tag{24}$$

Missile Cooperative Engagement Formation Configuration Control Method 191

cos cos sin cos sin

1 3

 

1

*u* 

cos( ) arctan *vf*

*u*

0 cos

cos cos sin cos sin sin cos cos sin sin sin 0 cos

It can be stated through analyzing this section that the indirect control model of relative motion based on the relationships of relative motion between missiles has a direct mapping relationship between inputs and outputs. However, this mapping just can be indicated by a mathematical form rather than some intuitive physical meanings, so we can't give the linear expressions just through modelling directly. As mentioned above, we should transform the

After the missile formation controller system is described by the Equ.(27), we also need to consider the completely controllable ability of those indirect variables*U* , that is, we should analyze the relationships between rank of <sup>2</sup> [ ] *B AB A B* and the dimensions of this system. Because matrix *B* is full rank in the feasible flight scope, that is, <sup>2</sup> rank[ ]=3 *B AB A B* ,

coordinate space of input variables and then establish the indirect linear model (27).

 *B*

*A*

*l vl l*

 

*l vl l l vl l vl l l vl vl vl*

 

 

cos sin 0

*vl l vl l*

0 sin

*l vl l*

 

 

> 

 

 

cos cos cos sin cos sin sin sin cos cos sin sin

*x y z uu u V yx z u u u*

further, the direct control variables *U* which act on followers can be calculated by the

 

, <sup>2</sup>

*X AX BU BW*

*l vl l l vl l l vl l l vl l l vl l l vl*

*f f vf f f f f vf*

 

*U* (25)

 

12 3 12 3

 

> 

, <sup>2</sup> sin *<sup>f</sup>*

*Y CX* (27)

 

 

*<sup>u</sup> <sup>V</sup>*

*f*

 (26)

1 2 3

then an indirectly linear control model of relation motion can be obtained:

 

> 

3 1

*u u*

 

following expressions:

where system matrix *A* is:

control matrix *B* is:

 

*vf* arctan

cos sin sin cos

 

*z x yu u*

 

*vl l vl l vl vl*

*f*

at last, we can tranform equation (26) to the states space form showed as below:

> 

the meanings of other variables are same as those in Equ.(24).

system(27) should be completely controllable.

 

*u V u V u V*

where [ , , ]*<sup>T</sup> X x y z* are the state variables; the control variables of formation controller are motion states of follower, that is, [ , , ]*<sup>T</sup> U Vf f vf* ; outputs are [ , , ]*<sup>T</sup> Y x y z* ; perturbation variables are the velocities of leader [ ,0,0]*<sup>T</sup> W Vl* , which can be considered as slowly variant variables; the system matrix *A* is described as:

$$\mathcal{A} = \begin{bmatrix} 0 & \dot{\theta}\_l & -\dot{\psi}\_{\text{vl}} \\ -\dot{\theta}\_l & 0 & \theta\_l \dot{\nu}\_{\text{vl}} \\ \dot{\psi}\_{\text{vl}} & -\dot{\psi}\_{\text{vl}} \theta\_l & 0 \end{bmatrix}$$

control matrix *B* is:

$$\mathbf{B} = \begin{bmatrix} 1 + \theta\_{f0}\theta\_l & V\_{f0}\theta\_l & 0 \\ \theta\_{f0} - \theta\_l & V\_{f0} & 0 \\ \boldsymbol{\nu}\_{vl} - \boldsymbol{\nu}\_{vf0} & 0 & -V\_{f0} \end{bmatrix}$$

and output matrix *C* is:

$$C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

effect matrix of perturbations *B* is:

$$
\tilde{B} = \begin{bmatrix} -1 \\ 0 \\ 0 \end{bmatrix}
$$

### **4.2.2 Method of substituting variables**

As analyzed in last section, we can get linear control model showed as Equ.(24) via small perturbation linearized method, however, this linear model requires the actual flight path is closely around feature points. Although we can design the missile formation keeping controller at those points within the whole flight scope by gain-scheduled method, it will increase amount of work greatly. Next we will re-deal with the nonlinear equations of relative kinematics showed in Equ.(22). If we transform the control variables from [ , , ]*<sup>T</sup> U Vf f vf* to *U* with the following form:

$$\begin{aligned} \boldsymbol{\overline{U}} = \begin{bmatrix} \boldsymbol{u}\_1 \\ \boldsymbol{u}\_2 \\ \boldsymbol{u}\_3 \end{bmatrix} = \begin{bmatrix} \boldsymbol{V}\_f \cos \theta\_f \cos \psi\_{vf} \\ \boldsymbol{V}\_f \sin \theta\_f \\ \boldsymbol{V}\_f \cos \theta\_f \sin \psi\_{vf} \end{bmatrix} \end{aligned} \tag{25}$$

then an indirectly linear control model of relation motion can be obtained:

$$\begin{cases} \dot{x} = \dot{\theta}\_l \dot{y} - \dot{\nu}\_{vl} \cos \theta\_l z + \cos \theta\_l \cos \nu\_{vl} \mu\_1 + \sin \theta\_l \mu\_2 + \cos \theta\_l \sin \nu\_{vl} \mu\_3 - V\_l\\ \dot{y} = -\dot{\theta}\_l x + \dot{\nu}\_{vl} \sin \theta\_l z - \sin \theta\_l \cos \nu\_{vl} \mu\_1 + \cos \theta\_l \mu\_2 - \sin \theta\_l \sin \nu\_{vl} \mu\_3\\ \dot{z} = \dot{\nu}\_{vl} \cos \theta\_l x - \dot{\nu}\_{vl} \sin \theta\_l y + \sin \nu\_{vl} \mu\_1 - \cos \nu\_{vl} \mu\_3 \end{cases} \tag{26}$$

further, the direct control variables *U* which act on followers can be calculated by the following expressions:

$$\left|\boldsymbol{\nu}\_{vf} = \arctan\left(\frac{\boldsymbol{\mu}\_{3}}{\boldsymbol{\mu}\_{1}}\right), \boldsymbol{\theta}\_{f} = \arctan\left[\frac{\boldsymbol{\mu}\_{2}\cos(\boldsymbol{\nu}\_{vf})}{\boldsymbol{\mu}\_{1}}\right], V\_{f} = \frac{\boldsymbol{\mu}\_{2}}{\sin\theta\_{f}}$$

at last, we can tranform equation (26) to the states space form showed as below:

$$\begin{aligned} \dot{X} &= AX + B\overline{U} + \overline{B}W \\ Y &= CX \end{aligned} \tag{27}$$

where system matrix *A* is:

190 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

*X AX BU BW*

where [ , , ]*<sup>T</sup> X x y z* are the state variables; the control variables of formation controller are

variables are the velocities of leader [ ,0,0]*<sup>T</sup> W Vl* , which can be considered as slowly

0

*vl vl l*

 *A*

 

0 0 0 0

*fl fl fl f*

 *B*

1 0

*V V* 0

0

*V*

*l vl l l vl*

 

 

0 0

0

 

*vl vf f*

100 010 001 *C*

As analyzed in last section, we can get linear control model showed as Equ.(24) via small perturbation linearized method, however, this linear model requires the actual flight path is closely around feature points. Although we can design the missile formation keeping controller at those points within the whole flight scope by gain-scheduled method, it will increase amount of work greatly. Next we will re-deal with the nonlinear equations of relative kinematics showed in Equ.(22). If we transform the control variables

to *U* with the following form:

 

 

0

 

 

*vfb* are feature points of linearized equations. We can describe this model

*Y CX* (24)

; outputs are [ , , ]*<sup>T</sup> Y x y z* ; perturbation

where , *Vfb*

 *fb* and

control matrix *B* is:

and output matrix *C* is:

from [ , , ]*<sup>T</sup> U Vf f vf* 

effect matrix of perturbations *B* is:

**4.2.2 Method of substituting variables** 

by the following states space form:

motion states of follower, that is, [ , , ]*<sup>T</sup> U Vf f vf*

variant variables; the system matrix *A* is described as:

$$\mathcal{A} = \begin{bmatrix} 0 & \dot{\theta}\_l & -\dot{\psi}\_{\upsilon l}\cos\theta\_l \\ -\dot{\theta}\_l & 0 & \dot{\psi}\_{\upsilon l}\sin\theta\_l \\ \dot{\psi}\_{\upsilon l}\cos\theta\_l & -\dot{\psi}\_{\upsilon l}\sin\theta\_l & 0 \end{bmatrix}$$

control matrix *B* is:

$$\mathbf{B} = \begin{bmatrix} \cos\theta\_l \cos\psi\_{vl} & \sin\theta\_l & \cos\theta\_l \sin\psi\_{vl} \\ -\sin\theta\_l \cos\psi\_{vl} & \cos\theta\_l & -\sin\theta\_l \sin\psi\_{vl} \\ \sin\psi\_{vl} & 0 & -\cos\psi\_{vl} \end{bmatrix}$$

the meanings of other variables are same as those in Equ.(24).

It can be stated through analyzing this section that the indirect control model of relative motion based on the relationships of relative motion between missiles has a direct mapping relationship between inputs and outputs. However, this mapping just can be indicated by a mathematical form rather than some intuitive physical meanings, so we can't give the linear expressions just through modelling directly. As mentioned above, we should transform the coordinate space of input variables and then establish the indirect linear model (27).

After the missile formation controller system is described by the Equ.(27), we also need to consider the completely controllable ability of those indirect variables*U* , that is, we should analyze the relationships between rank of <sup>2</sup> [ ] *B AB A B* and the dimensions of this system. Because matrix *B* is full rank in the feasible flight scope, that is, <sup>2</sup> rank[ ]=3 *B AB A B* , system(27) should be completely controllable.

### **4.3 Establishing the optimal control model of missile formation keeping**

As for the missile formation keeping control problem, the aim is to keep the distances between members within one formation on a non-zero states. After considering equation (27), we can further describe this formation keeping problem to another problem that how to regulate non-zero given value of output affected by slowly variant perturbations. So we can design this optimal controller by two steps: first, we need to design an optimal output regulator that can overcome slowly variant perturbations; second, we should further design optimal controller based on the first step which can maintain the missile formation on a nonzero desired relative states.

### **4.3.1 Optimal proportional-integral (PI) controller of missile formation**

As for the system with invariant or slowly variant perturbations showed in expression (27), we can choose PI control law to overcome these perturbations, which is similar to classical control method.

In order to clearly explain the principle of how this integral feed-back controller eliminates stable errors, we will make a hypothesis that the stable outputs of system are zeros. Because there has an integrator in the controller, outputs can be constants although inputs are zeros, if these values are just rightly equivalent to the perturbations that are effecting on the input points but the signs are inverse, then the inputs of this control system will be eliminated to zeros, so the final outputs of system can be kept to zeros (Xie, 1986).

First, we transform the perturbations of system to control inputting ports, and then system (27) can be changed to:

$$
\dot{X} = AX + B(U + \tilde{W}) \tag{28}
$$

Missile Cooperative Engagement Formation Configuration Control Method 193

control matrix of this augmented system is 1 33 33 [,]*<sup>T</sup> B I* **0** , and output matrix is

<sup>0</sup> 1 1 1 1 11 [ ] *<sup>f</sup> <sup>t</sup> T T*

where *Q*<sup>1</sup> is state regulating weight matrix of augmented system; *R*<sup>1</sup> is control energy weight matrix of augmented system. When augmented system (30) is controllable, the optimal

> 1 1 1 11 1 1

<sup>111</sup> ( )( ) *TT T X Q X X QX U W R U W* (33)

*<sup>Y</sup> <sup>t</sup> <sup>J</sup> dt Y Q Y U W RU W U RU* (34)

<sup>1</sup> *U U* (35)

\* 1 1 11 1

*T T P PA A P PB R B P Q*

decomposed to the following form based on the expansions of state variables *X*<sup>1</sup> :

1

Here we have to make further analysis on performance index expression(31). *Q*<sup>1</sup> can be

 **0**

**0** *<sup>Q</sup> <sup>Q</sup> <sup>R</sup>*

where *Q* is the states regulating weight matrix of original system(27), *R* is the control energy weight matrix of original system. From original system(27), there will be expression showed

*T TT <sup>T</sup> X QX X C Q CX Y Q Y Y Y*

where *Q<sup>Y</sup>* is the outputs regulating weight matrix, then expression(31) that describes the problem of quadratic optimal states regulating can be transformed to the problem of

<sup>0</sup> 1 11 [ ( )( ) ] *<sup>f</sup> <sup>t</sup> T TT*

besides, because perturbations*W* is slowly variant variables, following expression can be

 <sup>0</sup> *<sup>W</sup>*

*<sup>t</sup> <sup>J</sup> dt X Q X U RU* (31)

*<sup>T</sup> U R B PX* (32)

The quadratic optimal performance index for system (30) is appointed to:

control quantities for minimizing the performance index (31) should be:

where *P* is the solution of Riccati Equation:

quadratic optimal output regulating, which is:

1 33 33 [,] *C I* **0** .

and then, there will be:

as below:

noted:

and then

where*W* is transformation of initial perturbations, it has following form after comparing to the Equ.(27):

$$
\vec{W} = \mathbf{B}^{-1} \vec{B} W
$$

Until now, the problem has been transformed to how to design PI optimal controller for system(28). This new system is augmented to:

$$\begin{aligned} \dot{X} &= AX + B(U + \tilde{W}) \\ \dot{U} + \dot{\tilde{W}} &= U\_1 \qquad U(\mathbf{t}\_0) + \tilde{W}(\mathbf{t}\_0) \text{is given} \end{aligned} \tag{29}$$

This augmented system can be further marked as:

$$\begin{aligned} \dot{X}\_1 &= A\_1 X\_1 + B\_1 U\_1 \\ Y &= C\_1 X\_1 \end{aligned} \tag{30}$$

where states variables of augmented system are <sup>T</sup> <sup>1</sup> [, ] *X XU W* ; system matrix *<sup>A</sup>*<sup>1</sup> of this augmented system is:

$$\mathcal{A}\_1 = \begin{bmatrix} \mathcal{A} & \mathcal{B} \\ \mathbf{0}\_{3 \times 3} & \mathbf{0}\_{3 \times 3} \end{bmatrix}$$

control matrix of this augmented system is 1 33 33 [,]*<sup>T</sup> B I* **0** , and output matrix is 1 33 33 [,] *C I* **0** .

The quadratic optimal performance index for system (30) is appointed to:

$$J = \int\_{t\_0}^{t\_f} [X\_1^T \underline{\mathbf{Q}}\_1 X\_1 + U\_1^T \underline{\mathbf{R}}\_1 U\_1] dt \tag{31}$$

where *Q*<sup>1</sup> is state regulating weight matrix of augmented system; *R*<sup>1</sup> is control energy weight matrix of augmented system. When augmented system (30) is controllable, the optimal control quantities for minimizing the performance index (31) should be:

$$\mathbf{U}\_1^\* = -\mathbf{R}\_1^{-1} \mathbf{B}\_1^T \overline{\mathbf{P}} \mathbf{X}\_1 \tag{32}$$

where *P* is the solution of Riccati Equation:

$$\dot{\overline{P}} = -\overline{P}A\_1 - \mathcal{A}\_1^T \overline{P} + \overline{P} \mathcal{B}\_1 \mathcal{R}\_1^{-1} \mathcal{B}\_1^T \overline{P} - \mathcal{Q}\_1$$

Here we have to make further analysis on performance index expression(31). *Q*<sup>1</sup> can be decomposed to the following form based on the expansions of state variables *X*<sup>1</sup> :

$$\mathcal{Q}\_1 = \begin{bmatrix} \mathcal{Q} & \mathbf{0} \\ \mathbf{0} & \mathbf{k} \end{bmatrix}$$

and then, there will be:

192 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

As for the missile formation keeping control problem, the aim is to keep the distances between members within one formation on a non-zero states. After considering equation (27), we can further describe this formation keeping problem to another problem that how to regulate non-zero given value of output affected by slowly variant perturbations. So we can design this optimal controller by two steps: first, we need to design an optimal output regulator that can overcome slowly variant perturbations; second, we should further design optimal controller based on the first step which can maintain the missile formation on a non-

As for the system with invariant or slowly variant perturbations showed in expression (27), we can choose PI control law to overcome these perturbations, which is similar to classical

In order to clearly explain the principle of how this integral feed-back controller eliminates stable errors, we will make a hypothesis that the stable outputs of system are zeros. Because there has an integrator in the controller, outputs can be constants although inputs are zeros, if these values are just rightly equivalent to the perturbations that are effecting on the input points but the signs are inverse, then the inputs of this control system will be eliminated to

First, we transform the perturbations of system to control inputting ports, and then system

where*W* is transformation of initial perturbations, it has following form after comparing to

<sup>1</sup> *W B BW* Until now, the problem has been transformed to how to design PI optimal controller for

1 00

1 1 1 11 1 1 

> 33 33 *A B*

**0 0**

*X AX BU*

(t ) (t )is given

( )

*X AX B U W*

*UW U U W*

1

*A*

where states variables of augmented system are <sup>T</sup>

( ) *X AX B U W* (28)

*Y CX* (30)

<sup>1</sup> [, ] *X XU W* ; system matrix *<sup>A</sup>*<sup>1</sup> of this

(29)

**4.3 Establishing the optimal control model of missile formation keeping** 

**4.3.1 Optimal proportional-integral (PI) controller of missile formation** 

zeros, so the final outputs of system can be kept to zeros (Xie, 1986).

zero desired relative states.

control method.

(27) can be changed to:

augmented system is:

system(28). This new system is augmented to:

This augmented system can be further marked as:

the Equ.(27):

$$X\_1^T \underline{\mathbf{Q}}\_1 X\_1 = X^T \underline{\mathbf{Q}} X + \left(U + \tilde{W}\right)^T \mathbf{R} (U + \tilde{W}) \tag{33}$$

where *Q* is the states regulating weight matrix of original system(27), *R* is the control energy weight matrix of original system. From original system(27), there will be expression showed as below:

$$\mathbf{X}^{\top} \mathbf{Q} \mathbf{X} = \mathbf{X}^{\top} \mathbf{C}^{\top} \mathbf{Q}\_{\mathbf{Y}} \mathbf{C} \mathbf{X} = \mathbf{Y}^{\top} \mathbf{Q}\_{\mathbf{Y}} \mathbf{Y}$$

where *Q<sup>Y</sup>* is the outputs regulating weight matrix, then expression(31) that describes the problem of quadratic optimal states regulating can be transformed to the problem of quadratic optimal output regulating, which is:

$$J = \int\_{t\_0}^{t\_f} [Y^T \underline{\mathbf{Q}}\_Y Y + (U + \tilde{W})^T \mathbf{R} (U + \tilde{W}) + U\_1^T \mathbf{R}\_1 U\_1] dt \tag{34}$$

besides, because perturbations*W* is slowly variant variables, following expression can be noted:

$$
\dot{\hat{W}} = 0
$$

and then

$$U\_1 = \dot{\tilde{U}} \tag{35}$$

At last the quadratic optimal performance index of outputs regulating can be expressed as:

$$\mathbf{J} = \int\_{t\_0}^{t\_l} [\mathbf{Y}^T \underline{\mathbf{Q}}\_Y \mathbf{Y} + (\tilde{U} + \tilde{W})^T \mathbf{R}(\tilde{U} + \tilde{W}) + \dot{\tilde{U}}^T \mathbf{R}\_1 \dot{\tilde{U}}] \mathrm{d}t \tag{36}$$

As for the optimal control variables(32), it can be transformed from Equ.(35) to:

$$\boldsymbol{U}\_{1}\stackrel{\star}{\ }^{\star} = \dot{\tilde{U}}^{\star} = -\mathbf{R}\_{1}^{-1}\mathbf{B}\_{1}^{\mathrm{T}}\overline{\mathbf{P}}\mathbf{X}\_{1} \tag{37}$$

Missile Cooperative Engagement Formation Configuration Control Method 195

<sup>1</sup> lim ( ) 0

where *X*<sup>10</sup> is the stable value of state *X*<sup>1</sup> . If all the eigenvalues of matrix *A BK* 1 1 lie in the left complex plane, then matrix is *A BK* 1 1 non-singular, and further we can get the

\* *Y CX* 10 1 10

further the relationship required between non-zero states \* *Y*<sup>10</sup> and the stable value *<sup>X</sup>*<sup>10</sup> of

At last we can implement optimal control to system(27) that describes the relative

\* 1 1 1 1\* 1 1 1 1 1 1 1 1 1 10 [ ( ) ] *T T*

and this control quantity can keep missile formation on the desired relative dimensional

In order to analyze the stability of optimal controller of missile formation, we need to transform the model of missile formation relative motion to tracking error model. The

where is *X* the actual flight states of missile formation; *X<sup>d</sup>* is the expected states. Substituting

ˆˆ ˆ

where*W*ˆ is the invariant perturbations of tracking error state equation, the value of this

<sup>ˆ</sup> *W W AX <sup>d</sup>*

Because invariant perturbations do not affect the dynamic performance of control system no more than affect the stable tracking performance of this system, we can ignore this term

ˆ

1

1 1\*

*U R B PX C B R B P A B Y* (45)

1 1 10 1 0 **0** ( ) *A BK X BU* (42)

10 1 1 1 0 ( ) *<sup>X</sup> A BK BU* (43)

0 1 1 1 1 10 [( ) ] *U C BK A B Y* (44)

*X X Xd* (46)

*X AX BU +W* (47)

*t <sup>t</sup> X*

and the relationship between non-zero points and stable value of states will meet:

and then the asymptotically stable system can be expressed as:

kinematics of missiles by the following optimal control quantity:

tracking error of state is chosen by state equation(27) as:

last equation into Equ.(27) obtains the error state equation:

expression showed as below from Equ.(42):

state *X*<sup>1</sup> should be expressed as:

states.

term is:

**4.4 Stability analysis** 

Expanding equation(37) yields:

$$\begin{split} \dot{\tilde{U}}^{\*} &= -\mathcal{R}\_{1}^{-1} \begin{bmatrix} \mathbf{0} \\ I \end{bmatrix} \begin{bmatrix} \overline{P}\_{11} & \overline{P}\_{12} \\ \overline{P}\_{21} & \overline{P}\_{22} \end{bmatrix} \begin{bmatrix} X \\ \dot{U}^{\*} + \tilde{W} \end{bmatrix} \\ &= -\mathcal{R}\_{1}^{-1} \overline{P}\_{21} X - \mathcal{R}\_{1}^{-1} \overline{P}\_{22} (\tilde{U}^{\*} + \tilde{W}) \end{split} \tag{38}$$

which is the expression of optimal control quantities that can minimize the performance index (34) for zero-given points.

### **4.3.2 Optimal controller for non-zero given points**

In order to keep the output variables [ , , ]*<sup>T</sup> Y xyz* on non-zero points, the final system states and control inputs should also be non-zero, and then the optimal control quantities will be transformed from Equ.(37) to the following form:

$$
\dot{\tilde{U}}^\* = -\mathbf{R}\_1^{-1} \mathbf{B}\_1^T \overline{\mathbf{P}} \mathbf{X}\_1 + \mathbf{U}\_0' = -\mathbf{K} \mathbf{X}\_1 + \mathbf{U}\_0' \tag{39}
$$

where <sup>0</sup> *U* is additional control quantities for non-zero states. Considering the output equations of augmented system:

$$Y = \mathbf{C}\_1 X\_1 \tag{40}$$

and the expansions of augmented states *X*<sup>1</sup> :

$$X\_1 = [x, y, z, V\_f + \tilde{W}(1), \theta\_f + \tilde{W}(2), \tilde{\nu}\_{vf} + \tilde{W}(3)]^T$$

we should choose the following output matrix of augmented system to make sure the outputs of augmented system accord with those of original system:

$$\mathbf{C}\_1 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{bmatrix}$$

Substituting the control values(39) into the state equation(30) of augmented system yields:

$$
\dot{X}\_1 = (A\_1 - B\_1 K)X\_1 + B\_1 U\_0' \tag{41}
$$

Because the closed-loop system(41) is asymptotically stable, there will be:

<sup>1</sup> lim ( ) 0 *t <sup>t</sup> X*

and then the asymptotically stable system can be expressed as:

$$\mathbf{0} = (\mathcal{A}\_1 - \mathcal{B}\_1 \mathbf{K})X\_{10} + \mathcal{B}\_1 U\_0' \tag{42}$$

where *X*<sup>10</sup> is the stable value of state *X*<sup>1</sup> . If all the eigenvalues of matrix *A BK* 1 1 lie in the left complex plane, then matrix is *A BK* 1 1 non-singular, and further we can get the expression showed as below from Equ.(42):

$$X\_{10} = -(A\_1 - B\_1 K)^{-1} B\_1 U\_0' \tag{43}$$

and the relationship between non-zero points and stable value of states will meet:

$$\mathbf{Y}\_{10}^\* = \mathbf{C}\_1 \mathbf{X}\_{10}$$

further the relationship required between non-zero states \* *Y*<sup>10</sup> and the stable value *<sup>X</sup>*<sup>10</sup> of state *X*<sup>1</sup> should be expressed as:

$$U\_0' = \left[\mathbf{C}\_1 \left(\mathbf{B}\_1 \mathbf{K} - \mathbf{A}\_1\right)^{-1} \mathbf{B}\_1\right]^{-1} \mathbf{Y}\_{10}^\* \tag{44}$$

At last we can implement optimal control to system(27) that describes the relative kinematics of missiles by the following optimal control quantity:

$$\dot{\tilde{U}}^{\*} = -\mathbf{R}\_{1}^{-1}\mathbf{B}\_{1}^{T}\overline{\mathbf{P}}\mathbf{X}\_{1} + \left[\mathbf{C}\_{1}(-\mathbf{B}\_{1}\mathbf{R}\_{1}^{-1}\mathbf{B}\_{1}^{T}\overline{\mathbf{P}} - \mathbf{A}\_{1})^{-1}\mathbf{B}\_{1}\right]^{-1}\mathbf{Y}\_{10}^{\*}\tag{45}$$

and this control quantity can keep missile formation on the desired relative dimensional states.

### **4.4 Stability analysis**

194 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

At last the quadratic optimal performance index of outputs regulating can be expressed as:

t T TT

\* \* 1T

<sup>1</sup> \* 21 22 1 1\* 1 21 1 22( )

which is the expression of optimal control quantities that can minimize the performance

In order to keep the output variables [ , , ]*<sup>T</sup> Y xyz* on non-zero points, the final system states and control inputs should also be non-zero, and then the optimal control quantities will be

> 11 1 0 1 0 *<sup>T</sup>*

<sup>1</sup> [ , , , (1), (2), (3)]*<sup>T</sup> <sup>f</sup> <sup>f</sup> vf X WW W xy zV* 

we should choose the following output matrix of augmented system to make sure the

Substituting the control values(39) into the state equation(30) of augmented system yields:

100000 010000 001000 *C*

 

**0** *P P X*

*R PX R P U W*

*I P P U W*

As for the optimal control variables(32), it can be transformed from Equ.(35) to:

\* 1 11 12

*U R*

\* 1

outputs of augmented system accord with those of original system:

1

Because the closed-loop system(41) is asymptotically stable, there will be:

where <sup>0</sup> *U* is additional control quantities for non-zero states. Considering the output equations of augmented system:

**4.3.2 Optimal controller for non-zero given points** 

transformed from Equ.(37) to the following form:

and the expansions of augmented states *X*<sup>1</sup> :

<sup>t</sup> Y 1 J [ ( ) ( ) ]dt *Y Q Y U W RU W U RU* (36)

*U R B PX U KX U* (39)

*Y CX* 1 1 (40)

1 1 1 1 10 ( ) *X A BK X BU* (41)

1 11 1 *U U R B PX* (37)

(38)

f 0

Expanding equation(37) yields:

index (34) for zero-given points.

In order to analyze the stability of optimal controller of missile formation, we need to transform the model of missile formation relative motion to tracking error model. The tracking error of state is chosen by state equation(27) as:

$$
\hat{X} = X - X\_d \tag{46}
$$

where is *X* the actual flight states of missile formation; *X<sup>d</sup>* is the expected states. Substituting last equation into Equ.(27) obtains the error state equation:

$$
\dot{\hat{X}} = A\hat{X} + BU + \hat{W} \tag{47}
$$

where*W*ˆ is the invariant perturbations of tracking error state equation, the value of this term is:

$$
\hat{W} = W + AX\_d
$$

Because invariant perturbations do not affect the dynamic performance of control system no more than affect the stable tracking performance of this system, we can ignore this term when just analyzing the stability of control system, that is, we can just analyze the following control system:

$$
\dot{\hat{X}} = A\hat{X} + \mathbf{B}U\tag{48}
$$

Missile Cooperative Engagement Formation Configuration Control Method 197

After analyzing that Lyapunov function appointed in the previous section, we can find that this function satisfies the former two aspects of Barbalat Lemma. If we want to prove expression(52) also satisfies the third point, we just need to prove exists and is bounded.

ˆ ˆˆ ˆ *TT TT <sup>X</sup> PX XPX*

 0 so, according to Equ.(49), it can be concluded that system state *X*ˆ is bounded, further exists and is bounded, and then Lyapunov function meets Barbalat Lamme, that is, when *t* ,there will be ( , ) 0 *x t* ,and then further *X*ˆ **0** , which indicates that the

initial positions in inertial coordinate frame are: <sup>0</sup> *Xl* 500m , 0 *Yl* 800m and

the change rules of veloctity, fligh path angle and flight deflection angle of leader are:

*t t*

500 400 550 m, 450 m 350 350

240 20sin(0.15 ) 0.2sin(0.05 0.4515) 0.4sin(0.06 0.5511)

*V t*

1 2 1 2 1 2

*f f f f f f*

*x x y y z z*  , 2.25s 

and

10 ;initial flight

The expression of is:

**5. Simulations** 

**5.1.1 Initial conditions** 

2.37s

4. the flight states of leader are:

deflection angle is <sup>0</sup> 20

;

 *v* 

at the same time, from Equ.(52) we can know:

**5.1 Simulations for PD controller of missile formation keeping** 

1. missile formation is consisted of three missiles;

2. flight time of missile formation is 150s ;

*<sup>v</sup>* .

We will choose the following conditions for the simulations of PD controller:

3. three inertial time constants of each channel of follower are 3.51s *<sup>v</sup>*

<sup>0</sup> *Zl* 0m ; intial velocitiy is <sup>0</sup> 240m/s *Vl* ; initial flight path angle is <sup>0</sup>

*l l vl*

initial distances between leader and follower 1 and follower 2 are:

desired distances between leader and follower 1 and follower 2 are:

system(48) is asymptotically stable.

Taking Lyapunov function to this system gets:

$$\Phi = \frac{1}{2} (\hat{\mathbf{x}}^2 + \hat{y}^2 + \hat{z}^2) \tag{49}$$

The derivative of this Lyapunov function is:

$$
\dot{\Phi} = \hat{\mathbf{x}}\dot{\hat{\mathbf{x}}} + \hat{\mathbf{y}}\dot{\hat{\mathbf{y}}} + \hat{\mathbf{z}}\dot{\hat{\mathbf{z}}}\tag{50}
$$

Substituting Equ.(26) to expression(50) yields:

$$
\dot{\Phi} = (\mathbf{B}U)^T \hat{X} \tag{51}
$$

and the optimal control quantity *U* for this error state equation is:

$$U = -\boldsymbol{\mathcal{R}}^{-1} \boldsymbol{\mathcal{B}}^{T} \boldsymbol{\mathcal{P}} \hat{\boldsymbol{\mathcal{Y}}}$$

substituting it into Equ.(50) gets:

$$\dot{\Phi} = (-\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^T\mathbf{P}\hat{\mathbf{X}}^\dagger)^T\hat{\mathbf{X}}$$

Because the control energy weight matrix *R* in this chapter is unit matrix, there should have:

$$\mathbf{B} \mathbf{R}^{-1} \mathbf{B}^{T} P \hat{\mathbf{X}} = P \hat{\mathbf{X}}$$

and then the derivative of Lyapunov function can be changed to:

$$
\dot{\Phi} = -\hat{X}^T \mathbf{P}^T \hat{X} \tag{52}
$$

where *P* is the solution of Riccati Equation, which has the following characters (Xie, 1986):

1. for every <sup>0</sup> *t tT* [,], *P* is symmetrical matrix;

2. for every <sup>0</sup> *t tT* [,], *P* is non-negative matrix.

So Equ.(52) is non-negative, and it also states that we can't get the conclusion that this system is asymptotical until now, and need some more stronger conditions. Original system is time-variant system, so we can complement the stability conditions of system(48) through Barbalat Lemma.

The contents of Barbalat Lemma are: if a scalar function ( ,) *x t* is satisfied with the following series of conditions:


then ( , ) 0 *x t* when *t* (Slotine et al, 1991).

After analyzing that Lyapunov function appointed in the previous section, we can find that this function satisfies the former two aspects of Barbalat Lemma. If we want to prove expression(52) also satisfies the third point, we just need to prove exists and is bounded. The expression of is:

$$\ddot{\Phi} = -\dot{\hat{X}}^T \mathbf{P}^T \hat{X} - \hat{X}^T \mathbf{P}^T \dot{\hat{X}}$$

at the same time, from Equ.(52) we can know:

0

so, according to Equ.(49), it can be concluded that system state *X*ˆ is bounded, further exists and is bounded, and then Lyapunov function meets Barbalat Lamme, that is, when *t* ,there will be ( , ) 0 *x t* ,and then further *X*ˆ **0** , which indicates that the system(48) is asymptotically stable.

### **5. Simulations**

196 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

when just analyzing the stability of control system, that is, we can just analyze the following

<sup>1</sup> 2 22 ( ) ˆˆˆ <sup>2</sup>

<sup>1</sup> *<sup>T</sup>* ˆ *U R B PX*

<sup>1</sup> ˆ ˆ ( ) *T T BR B PX X*

Because the control energy weight matrix *R* in this chapter is unit matrix, there should have:

*BR B PX = PX* <sup>1</sup> *<sup>T</sup>* ˆ ˆ

where *P* is the solution of Riccati Equation, which has the following characters (Xie, 1986):

So Equ.(52) is non-negative, and it also states that we can't get the conclusion that this system is asymptotical until now, and need some more stronger conditions. Original system is time-variant system, so we can complement the stability conditions of system(48) through

The contents of Barbalat Lemma are: if a scalar function ( ,) *x t* is satisfied with the following

*X AX BU* (48)

*xyz* (49)

*xx*ˆˆ ˆˆ ˆˆ *yy zz* (50)

ˆ ( )*<sup>T</sup> BU X* (51)

ˆ ˆ *T T XPX* (52)

ˆ ˆ

control system:

Taking Lyapunov function to this system gets:

The derivative of this Lyapunov function is:

Substituting Equ.(26) to expression(50) yields:

substituting it into Equ.(50) gets:

Barbalat Lemma.

series of conditions:

1. ( ,) *x t* has lower bounded; 2. ( ,) *x t* is semi-negative;

and the optimal control quantity *U* for this error state equation is:

and then the derivative of Lyapunov function can be changed to:

1. for every <sup>0</sup> *t tT* [,], *P* is symmetrical matrix; 2. for every <sup>0</sup> *t tT* [,], *P* is non-negative matrix.

3. ( ,) *x t* is uniformly continuous with respect to time.

then ( , ) 0 *x t* when *t* (Slotine et al, 1991).

### **5.1 Simulations for PD controller of missile formation keeping 5.1.1 Initial conditions**

We will choose the following conditions for the simulations of PD controller:


 initial positions in inertial coordinate frame are: <sup>0</sup> *Xl* 500m , 0 *Yl* 800m and <sup>0</sup> *Zl* 0m ; intial velocitiy is <sup>0</sup> 240m/s *Vl* ; initial flight path angle is <sup>0</sup> 10 ;initial flight deflection angle is <sup>0</sup> 20 *<sup>v</sup>* .

the change rules of veloctity, fligh path angle and flight deflection angle of leader are:

$$\begin{aligned} V\_l &= 240 + 20 \sin(0.15t) \\ \theta\_l &= 0.2 \sin(0.05t + 0.4515) \\ \nu\_{vl} &= 0.4 \sin(0.06t + 0.5511) \end{aligned}$$

initial distances between leader and follower 1 and follower 2 are:

$$
\begin{bmatrix} \mathbf{x}\_{f1} \\ \mathbf{y}\_{f1} \\ \mathbf{z}\_{f1} \end{bmatrix} = \begin{bmatrix} -500 \\ -550 \\ -350 \end{bmatrix} \mathbf{m} \,, \begin{bmatrix} \mathbf{x}\_{f2} \\ \mathbf{y}\_{f2} \\ \mathbf{z}\_{f2} \end{bmatrix} = \begin{bmatrix} -400 \\ -450 \\ 350 \end{bmatrix} \mathbf{m} \,\mathbf{x}
$$

desired distances between leader and follower 1 and follower 2 are:

Missile Cooperative Engagement Formation Configuration Control Method 199

0 50 100 150

0 50 100 150

t(s)

Fig. 8. Distances between follower 2 and leader in relative coordinate frame.

t(s)

Fig. 7. Distances between follower 1 and leader in relative coordinate frame.

X: 91.78 Y: -444.8 X: 145 Y: -393.4

> xf1 yf1 zf1

X: 141.7 Y: 405.9

> xf2 yf2 zf2

X: 143.7 Y: -444.2

> X: 150 Y: -501.1

X: 89.3 Y: -498.8





0

x,y,z(m)

200

400

600




x,y,z(m)




$$
\begin{bmatrix} \boldsymbol{x}\_{f1}^\* \\ \boldsymbol{y}\_{f1}^\* \\ \boldsymbol{z}\_{f1}^\* \end{bmatrix} = \begin{bmatrix} -450 \\ -500 \\ -400 \end{bmatrix} \mathbf{m} \prime \begin{bmatrix} \boldsymbol{x}\_{f2}^\* \\ \boldsymbol{y}\_{f2}^\* \\ \boldsymbol{z}\_{f2}^\* \end{bmatrix} = \begin{bmatrix} -450 \\ -500 \\ 400 \end{bmatrix} \mathbf{m} \prime$$

intial states of follower 1 and follower 2 are:


satuaration of control quantities

Considering the feasible fight scope of missile formation, we should limit the control quantities of formation controller, that is, the command states of follower should be limilted. Here we set the satuatations are:

$$\text{150m/s} \le V\_{fc} \le \text{300m/s} \text{ -15}^\circ \le \theta\_{fc} \le \text{35}^\circ \text{ -40}^\circ \le \nu\_{\eta \circ \text{c}} \le 40^\circ$$

#### **5.1.2 Simulation results and analysis**

After simulating, we can get the result curves Fig.6-Fig.12, where Fig.6 is the 3-dimensional motions of missile formation; Fig.7 and Fig.8 are the position components of follower 1 and follower 2 in relative coordinate frame respectively; Fig.9 is the distances between members in missile formation; Fig.10-Fig.11 are the control quantities in velocity, flight path angle and flight deflection angle channels of followers respectively.

Fig. 6. Three-dimensional trajectories of missile formation under PD controlling.

198 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

450 450 500 m, 500 m 400 400

*vf* .

Considering the feasible fight scope of missile formation, we should limit the control quantities of formation controller, that is, the command states of follower should be limilted.

After simulating, we can get the result curves Fig.6-Fig.12, where Fig.6 is the 3-dimensional motions of missile formation; Fig.7 and Fig.8 are the position components of follower 1 and follower 2 in relative coordinate frame respectively; Fig.9 is the distances between members in missile formation; Fig.10-Fig.11 are the control quantities in velocity, flight path angle and

*fc*

, -40 40

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5

X(m)

x 104

 *vfc* 

> follower1 leader follower2

\* \* 1 2 \* \* 1 2 \* \* 1 2

*x x y y z z*

*<sup>f</sup>* ;

150m/s 300m/s *Vfc* , -15 35

 *vf* , <sup>20</sup> 10 

intial states of follower 1 and follower 2 are:

flight deflection angle: <sup>10</sup> 10

flight path angle: <sup>10</sup> 5

satuaration of control quantities

Here we set the satuatations are:

**5.1.2 Simulation results and analysis** 

velocity: <sup>10</sup> 180m/s *Vf* , <sup>20</sup> 240m/s *Vf* ;

 *<sup>f</sup>* , <sup>20</sup> 5 

flight deflection angle channels of followers respectively.


Fig. 6. Three-dimensional trajectories of missile formation under PD controlling.




Z(m)

0

1000 0

1000

Y(m)

2000

3000

*f f f f f f*

Fig. 7. Distances between follower 1 and leader in relative coordinate frame.

Fig. 8. Distances between follower 2 and leader in relative coordinate frame.

Missile Cooperative Engagement Formation Configuration Control Method 201

follower1 follower2

follower1 follower2

0 50 100 150

0 50 100 150

t(s)

t(s)



Fig. 12. Flight deflection angle commands of followers.





0

ψvc(°)

5

10

15

20

25

Fig. 11. Flight path angle commands of followers.



0

θc(°)

5

10

15

Fig. 9. Distances between members of missile formation in relative coordinate frame.

Fig. 10. Velocity commands of followers.

200 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

follower1,follower2 follower1,leader follower2,leader

> follower1 follower2

0 50 100 150

0 50 100 150

t(s)

t(s)

Fig. 9. Distances between members of missile formation in relative coordinate frame.

650

180

Fig. 10. Velocity commands of followers.

190

200

210

220

Vc/(m/s)

230

240

250

260

700

X: 2.75 Y: 659

750

ρ(m)

800

850

X: 2.75 Y: 840.3

Fig. 11. Flight path angle commands of followers.

Fig. 12. Flight deflection angle commands of followers.

Missile Cooperative Engagement Formation Configuration Control Method 203


x(m)

0 50 100 150

t(s)

Fig. 14. Distances between follower 1 and leader in relative coordinate frame.

Fig. 13. Three-dimensional trajectories of missile formation under optimal controlling.





z(m)




X: 3.048 Y: -485.9

X: 1.865 Y: -431

xr,yr,zr(m)




0

1000 0

1000

y(m)

2000

3000

x 104

xrf1 yrf1 zrf1

X: 111 Y: -396

X: 131.2 Y: -502.8

X: 115.7 Y: -450.5

leader follower1 follower2

It can be found from those result curves showed above that PD keeping controller of missile formation can implement keeping control at about 25s, and control quantities are feasible; the stable error in *<sup>r</sup> x* direction is about 5m, and those in *<sup>r</sup> y* and *<sup>r</sup> z* directions are 1m and 6m respectively.

Because three channels of PD controller are coupled seriously, it is very difficult to find the obvious relationships between the inertial time constants of each channel and control performance; From the change of each curve, we can find that the maneuver of leader affect the formation keeping control distinctly; Besides, we can find that there are evident regulating processes from initial states to expected states, and the maximum control quantity exists in two followers, its value is about 60m.

### **5.2 Simulations for Optimal Controller of missile formation keeping 5.2.1 Initial Conditions**

We will choose the following conditions for the simulations to optimal controller:


### **5.2.2 Simulation results and analysis**

After simulating, we can get the result curves Fig.13-Fig.18, where Fig.13 is the 3 dimensional motion of missile formation; Fig.14 and Fig.15 are the position components of follower 1 and follower 2 in relative coordinate frame respectively; Fig.16-Fig.18 are the control quantities in velocity, flight path angle and flight deflection angle channels of followers respectively.

There are following conclusions after analyzing above result curves:


202 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

It can be found from those result curves showed above that PD keeping controller of missile formation can implement keeping control at about 25s, and control quantities are feasible; the stable error in *<sup>r</sup> x* direction is about 5m, and those in *<sup>r</sup> y* and *<sup>r</sup> z* directions are 1m and 6m

Because three channels of PD controller are coupled seriously, it is very difficult to find the obvious relationships between the inertial time constants of each channel and control performance; From the change of each curve, we can find that the maneuver of leader affect the formation keeping control distinctly; Besides, we can find that there are evident regulating processes from initial states to expected states, and the maximum control

quantity exists in two followers, its value is about 60m.

 output regulating weight matrix: diag(4.0,6.0,6.0) *Q<sup>y</sup>* ; control energy weight matrix: *R* diag(1.0,1.0,1.0) ;

1. weight matrices of optimal controller are:

2. other conditions are same as section 5.1.

**5.2.2 Simulation results and analysis** 

**5.2 Simulations for Optimal Controller of missile formation keeping** 

weight matrix of control energy changing: *R*<sup>1</sup> diag(1.0,1.0,1.0).

There are following conclusions after analyzing above result curves:

quantities, and achieve the minimum stable tracking error.

We will choose the following conditions for the simulations to optimal controller:

After simulating, we can get the result curves Fig.13-Fig.18, where Fig.13 is the 3 dimensional motion of missile formation; Fig.14 and Fig.15 are the position components of follower 1 and follower 2 in relative coordinate frame respectively; Fig.16-Fig.18 are the control quantities in velocity, flight path angle and flight deflection angle channels of

1. Optimal keeping controller of missile formation can implement keeping on desired

2. The maneuver motion of leader also disturbs missile formation keeping control, and there also exists regulation process when the states of missile formation changing from initial states to final states, but the disturbance amplitude is smaller that of PD

3. The stable error of relative motion in *<sup>r</sup> x* direction is about 0.5m, and those in other two directions are 3m and 4m respectively, which are decreased by comparing to PD controller. Especially the improvement in *<sup>r</sup> x* direction is more evident, which is because the velocity of leader is considered as slowly variant perturbation that will affect evidently to the relative motion in *<sup>r</sup> x* direction, however, the optimal controller designed in this chapter can restrict this perturbation well, so there will be more higher

4. We can further decrease the stable errors by enhancing output regulating weight matrix *Q<sup>y</sup>* , but this manner will increase the changing rate at the same time, so we should coordinate the values of weight matrix *Q<sup>y</sup>* , *R* and *R*<sup>1</sup> to obtain proper control

relative states at about 20s, it has faster response speed than PD controller;

respectively.

**5.2.1 Initial Conditions** 

followers respectively.

controller;

tracking precision;

Fig. 13. Three-dimensional trajectories of missile formation under optimal controlling.

Fig. 14. Distances between follower 1 and leader in relative coordinate frame.

Missile Cooperative Engagement Formation Configuration Control Method 205

follower1 follower2

follower1 follower2

0 50 100 150

0 50 100 150

t(s)

t(s)



Fig. 18. Flight deflection angle commands of followers.





0

ψvf(°)

5

10

15

20

25

Fig. 17. Flight path angle commands of followers.



0

θf(°)

5

10

15

Fig. 15. Distances between follower 2 and leader in relative coordinate frame.

Fig. 16. Velocity commands of followers.

204 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

X: 105.4 Y: 404.3

> X: 88.85 Y: -450.6

> > X: 131.6

follower1 follower2

xrf2 yrf2 zrf2

0 50 100 150

0 50 100 150

t(s)

t(s)

Y: -502.9 X: 3.733

Fig. 15. Distances between follower 2 and leader in relative coordinate frame.


180

Fig. 16. Velocity commands of followers.

190

200

210

220

230

Vfc(m/s)

240

250

260

270

280

Y: -533



0

xr,yr,zr(m)

200

400

600

Fig. 17. Flight path angle commands of followers.

Fig. 18. Flight deflection angle commands of followers.

### **6. Conclusion**

In this chapter, we mainly focus on some control problems of missile formation engagement. Significance of cooperative engagement was first presented by taking LOCAAS as instance, which showed the synthetical efficacy can be greatly increased by adopting cooperative engagement manner. Following the significance was analysis of frame of cooperative engagement system, which supplied the train of thought of how to research on missile formation control problem. Missile formation keeping control system design is the main content of this chapter. In this part, we established the model of relative motion in two ways firstly, and then designed missile formation control system based on PD control law and optimal control method respectively. Finally, the comparisons were made by a series of simulations, the conclusion that optimal controller is better from the points of view of stability and rapidity can be obtained from the result curves.

### **7. Acknowledgment**

Thank Prof. Sang-Young Park and Korean Industry-Academic Cooperation Foundation very much. They supply plenteous financial support and good working conditions.

**1. Introduction**

(1999); Shin et al. (2011); Wu et al. (2009).

University.

Parallel manipulators, often called parallel kinematics machines (PKM), are controlled non-linear dynamical systems. From a mechanical point of view PKM are (holonomically) constrained mechanical systems characterized by a power transmission between input and output. A so-called end-effector (EE), representing the mechanical output, is connected to a fixed platform by several (often identical) serial linkages, and the constraints reflect the existence of closed loops formed by these chains. Each chain is equipped with one or more actuators, representing the mechanical inputs. The modeling, identification, and control of PKM have advanced in the last decades culminating in successful industrial implementations. Yet the acceptance of PKM is far beyond that of the well-established serial manipulators. This is mainly due to the limited workspace, drastically varying static and dynamic properties, the

**Robust Modeling and Control Issues of Parallel** 

**Manipulators with Actuation Redundancy** 

*University Duisburg-Essen, Chair of Mechanics and Robotics* 

Andreas Mueller \*

*Germany* 

**12**

abundance of singularities within the workspace, and the seemingly complex control. Traditionally the number of mechanical inputs of a PKM equals its mechanical degree-of-freedom (DOF) so that the PKM is non-redundantly actuated. A means to overcome the aforementioned mechanical limitations is the inclusion of additional actuators, commonly by adding further limbs to the moving platform without increasing the DOF of the PKM. As a simple example consider the PKM in figure 1. The EE can be positioned in the plane thus possesses 2 degrees of freedom. Also the PKM as a whole has the DOF 2 so that two actuators would be sufficient for controlling this PKM. Yet the PKM is actuated by 3 actuators, which gives rise to actuation redundancy in the sense that the actuator forces are not independent. Such actuation redundancy has the potential to increase the EE-acceleration, to homogenize stiffness and manipulability, and to eliminate input singularities (where the motion of the moving platform is not controllable by the actuators), and thus to increase the usable workspace. The design of RA-PKM, and the possible dexterity improvement were addressed in several publications as for instance Garg et al. (2009); Gogu (2007); Krut et al. (2004); Kurtz & Hayward (1992); Lee et al. (1998); Nahon & Angeles (1989); O'Brien & Wen

The existence of redundant actuators allows for control forces that have no effect on the PKM motion but rather lead to mechanical prestress within the PKM. This effect can be exploited for different second-level control tasks such as backlash avoidance and stiffness control. In

\*All control concepts proposed in this chapter were implemented by Timo Hufnagel at the Heilbronn

### **8. References**


## **Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy**

Andreas Mueller \*

*University Duisburg-Essen, Chair of Mechanics and Robotics Germany* 

### **1. Introduction**

206 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

In this chapter, we mainly focus on some control problems of missile formation engagement. Significance of cooperative engagement was first presented by taking LOCAAS as instance, which showed the synthetical efficacy can be greatly increased by adopting cooperative engagement manner. Following the significance was analysis of frame of cooperative engagement system, which supplied the train of thought of how to research on missile formation control problem. Missile formation keeping control system design is the main content of this chapter. In this part, we established the model of relative motion in two ways firstly, and then designed missile formation control system based on PD control law and optimal control method respectively. Finally, the comparisons were made by a series of simulations, the conclusion that optimal controller is better from the points of view of

Thank Prof. Sang-Young Park and Korean Industry-Academic Cooperation Foundation very

Cui, Naigang ; Wei, Changzhu; Guo, Jifeng & Zhao, Biao (2010). Research on the missile

Cui, Naigang; Wei, Changzhu & Guo, Jifeng (2010). Research on four-dimensional guidance

Xie, Xueshu (1986). *Optimal control theroty and application* (1St edition), Tsinghua University

Slotine, Jean Jacques & Li, Weiping (1991). *Applied nonlinear control* (1St edition), Prentice-

Hall International (UK) Limited, London, ISBN: 0130408905

formation control system. *Proceeding of the 2009 IEEE International Conference on Mechatronics and Automation*, pp. 4197-4198, Changchun, China, August 9-12, 2009 Wei, Changzhu; Guo, Jifeng & Cui, Naigang (2010). Research on the missile formation

keeping optimal control for cooerative engagement. *Journal of Astronautics*, Vol.31,

and control for missile cooperative engagement. *Flight Dynamics*, Vol.28, No.2,

much. They supply plenteous financial support and good working conditions.

stability and rapidity can be obtained from the result curves.

No.4, (2010), pp. 1043-1044, ISBN 1000-1328

(2010), pp. 63-66, ISBN 1002-0853

Press, Beijing

**6. Conclusion** 

**7. Acknowledgment** 

**8. References** 

Parallel manipulators, often called parallel kinematics machines (PKM), are controlled non-linear dynamical systems. From a mechanical point of view PKM are (holonomically) constrained mechanical systems characterized by a power transmission between input and output. A so-called end-effector (EE), representing the mechanical output, is connected to a fixed platform by several (often identical) serial linkages, and the constraints reflect the existence of closed loops formed by these chains. Each chain is equipped with one or more actuators, representing the mechanical inputs. The modeling, identification, and control of PKM have advanced in the last decades culminating in successful industrial implementations. Yet the acceptance of PKM is far beyond that of the well-established serial manipulators. This is mainly due to the limited workspace, drastically varying static and dynamic properties, the abundance of singularities within the workspace, and the seemingly complex control.

Traditionally the number of mechanical inputs of a PKM equals its mechanical degree-of-freedom (DOF) so that the PKM is non-redundantly actuated. A means to overcome the aforementioned mechanical limitations is the inclusion of additional actuators, commonly by adding further limbs to the moving platform without increasing the DOF of the PKM. As a simple example consider the PKM in figure 1. The EE can be positioned in the plane thus possesses 2 degrees of freedom. Also the PKM as a whole has the DOF 2 so that two actuators would be sufficient for controlling this PKM. Yet the PKM is actuated by 3 actuators, which gives rise to actuation redundancy in the sense that the actuator forces are not independent. Such actuation redundancy has the potential to increase the EE-acceleration, to homogenize stiffness and manipulability, and to eliminate input singularities (where the motion of the moving platform is not controllable by the actuators), and thus to increase the usable workspace. The design of RA-PKM, and the possible dexterity improvement were addressed in several publications as for instance Garg et al. (2009); Gogu (2007); Krut et al. (2004); Kurtz & Hayward (1992); Lee et al. (1998); Nahon & Angeles (1989); O'Brien & Wen (1999); Shin et al. (2011); Wu et al. (2009).

The existence of redundant actuators allows for control forces that have no effect on the PKM motion but rather lead to mechanical prestress within the PKM. This effect can be exploited for different second-level control tasks such as backlash avoidance and stiffness control. In

<sup>\*</sup>All control concepts proposed in this chapter were implemented by Timo Hufnagel at the Heilbronn University.

coordinates are not globally valid on the entire configuration space. Configurations where these coordinates become invalid are called parameterization singularities. That is, it is not possible to uniquely determine any PKM configuration by one specific set of minimal coordinates. The most natural and practicable choice of minimal coordinates is to use the actuator (input) coordinates as they can be measured. Then, the parameterization singularities are also input singularities. An ad hoc method to cope with this phenomenon is to switch between different minimal coordinates as proposed in Hufnagel & Müller (2011). This is a computationally expensive approach since it requires monitoring the numerical conditioning of the constraint equations, and the entire set of motion equation must be changed accordingly. To avoid such switching a novel formulation of motion equations that does use minimal coordinates was proposed in Müller (2011). This formulation is robust with respect to input respectively parameterization singularities and hence does not require switching between different parameterizations. Exponentially stable trajectory tracking can be shown when this

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 209

formulation is employed in a computed torque and augmented PD control scheme.

recent developments in modeling and control of RA-PKM.

**2. The PKM control problem**

geometric and kinematic loop constraints

**2.1 Manipulator dynamics**

These observations call for robust modeling and control concepts. This chapter reports some

Aiming for efficient real-time implementations model-based control schemes for PKM are based on a dynamic model in terms of minimal coordinates as pursued in Cheng et al. (2003); Müller (2005); Nakamura & Ghodoussi (1989); Thanh et al. (2009). A PKM is a force-controlled mechanism with kinematic loops. Following the standard approach in multibody dynamics the Lagrangian motion equations of first kind are first derived for the unconstrained system (opening kinematic loops), and the Lagrange multipliers are eliminated by projecting these equations to the configuration space defined by the (holonomic) geometric constraints. This

Denote with **<sup>q</sup>** <sup>∈</sup> **<sup>V</sup>***<sup>n</sup>* the vector of joint variables *<sup>q</sup>a*, *<sup>a</sup>* <sup>=</sup> 1, . . . , *<sup>n</sup>* of the unconstrained system, obtained by opening the kinematic loops, where in each kinematic loop one joint is removed. The loop closure is enforced by the corresponding loop constraints giving rise to a set of *r*

**<sup>0</sup>** <sup>=</sup> **<sup>h</sup>** (**q**), **<sup>h</sup>** (**q**) <sup>∈</sup> **<sup>R</sup>***<sup>r</sup>*

The PKM in figure 1, for instance, comprises two independent kinematic loops that can be opened by removing the two joints at the EE connecting the kinematic chain formed by joints 2 and 5 with that formed by joints 1 and 4, and joints 3 and 6, respectively. The resulting unconstrained tree-system in figure 2 has *n* = 6 generalized coordinates *qa*, *a* = 1, . . . , 6.

**G** is the generalized mass matrix of the tree-system, **Cq**˙ represents generalized Coriolis and centrifugal forces, **Q** represents all remaining forces, including EE loads, and **u** are the generalized control forces. The Lagrange multipliers λ can be identified with the

With the constraints (1) the Lagrangian motion equations of the PKM are

**<sup>0</sup>** <sup>=</sup> **<sup>J</sup>** (**q**) **<sup>q</sup>**˙ , **<sup>J</sup>**(**q**) <sup>∈</sup> **<sup>R</sup>***r*,*n*. (1)

**G** (**q**) **q**¨ + **C** (**q**, ˙**q**) **q**˙ + **Q** (**q**, ˙**q**, *t*) + **J***<sup>T</sup>* (**q**)λ = **u**. (2)

approach is know as the coordinate partitioning method, see Wehage & Haug (1982).

Fig. 1. Multibody model of a 2 DOF planar RA-PKM.

a power port representation of the PKM control system this means that control forces are possible without a (mechanical) power inflow into the system. This also means that in the non-linear control system there are more control vector fields than the dimension of the state space manifold, and that the control forces are not unique. Several strategies for redundancy resolution were proposed exploiting redundancy for second-level control tasks, see Lee et al. (2005); Müller (2005; 2006).

While the described advantages make the redundant actuation scheme attractive its implementation raises several challenges, however. These challenges are due to model uncertainties, synchronization errors in decentralized control schemes, as well to the lack of a globally valid dynamics model. The effect of uncertainties has been analyzed in Müller (2008) and Müller (2010) and it was show that geometric imperfections cause a qualitative change of the way in which actuation forces act upon the PKM. This is in sharp contrast to non-redundantly actuated manipulators where geometric uncertainties simply cause quantitative control errors and so impair the control performance. Moreover, geometric uncertainties of redundantly actuated PKM (RA-PKM) lead to antagonistic control forces proportional to the linear feedback gains. It turns out that this can severely deteriorate the integrity of the controlled RA-PKM.

Although model-based control concepts have been proposed for a long time and implemented recently, robotic manipulators are dominantly controlled by means of decentralized control schemes in practice. Now the indiscriminate application of decentralized control methods to RA-PKM leads again to the problem of antagonistic control forces. In contrast to intentionally generated counteracting control forces, generating desired prestresses, the latter are uncontrolled parasitic control forces. This is an inherent problem of the decentralized control of RA-PKM that can be observed even for a perfect matching of model and plant. To eliminate such antagonistic control forces a so-called antagonism filter (AF) was proposed in Müller & Hufnagel (2011).

The motion equations governing the PKM dynamics form the basis for any model-based control. Aiming on an efficient formulation, the motion equations are usually derived in terms of a minimal number of generalized coordinates that constitute a local parameterization of the configuration space. A well-known problem of this formulation is that such minimal

coordinates are not globally valid on the entire configuration space. Configurations where these coordinates become invalid are called parameterization singularities. That is, it is not possible to uniquely determine any PKM configuration by one specific set of minimal coordinates. The most natural and practicable choice of minimal coordinates is to use the actuator (input) coordinates as they can be measured. Then, the parameterization singularities are also input singularities. An ad hoc method to cope with this phenomenon is to switch between different minimal coordinates as proposed in Hufnagel & Müller (2011). This is a computationally expensive approach since it requires monitoring the numerical conditioning of the constraint equations, and the entire set of motion equation must be changed accordingly. To avoid such switching a novel formulation of motion equations that does use minimal coordinates was proposed in Müller (2011). This formulation is robust with respect to input respectively parameterization singularities and hence does not require switching between different parameterizations. Exponentially stable trajectory tracking can be shown when this formulation is employed in a computed torque and augmented PD control scheme.

These observations call for robust modeling and control concepts. This chapter reports some recent developments in modeling and control of RA-PKM.

#### **2. The PKM control problem**

#### **2.1 Manipulator dynamics**

2 Will-be-set-by-IN-TECH

a power port representation of the PKM control system this means that control forces are possible without a (mechanical) power inflow into the system. This also means that in the non-linear control system there are more control vector fields than the dimension of the state space manifold, and that the control forces are not unique. Several strategies for redundancy resolution were proposed exploiting redundancy for second-level control tasks, see Lee et al.

While the described advantages make the redundant actuation scheme attractive its implementation raises several challenges, however. These challenges are due to model uncertainties, synchronization errors in decentralized control schemes, as well to the lack of a globally valid dynamics model. The effect of uncertainties has been analyzed in Müller (2008) and Müller (2010) and it was show that geometric imperfections cause a qualitative change of the way in which actuation forces act upon the PKM. This is in sharp contrast to non-redundantly actuated manipulators where geometric uncertainties simply cause quantitative control errors and so impair the control performance. Moreover, geometric uncertainties of redundantly actuated PKM (RA-PKM) lead to antagonistic control forces proportional to the linear feedback gains. It turns out that this can severely deteriorate the

Although model-based control concepts have been proposed for a long time and implemented recently, robotic manipulators are dominantly controlled by means of decentralized control schemes in practice. Now the indiscriminate application of decentralized control methods to RA-PKM leads again to the problem of antagonistic control forces. In contrast to intentionally generated counteracting control forces, generating desired prestresses, the latter are uncontrolled parasitic control forces. This is an inherent problem of the decentralized control of RA-PKM that can be observed even for a perfect matching of model and plant. To eliminate such antagonistic control forces a so-called antagonism filter (AF) was proposed in

The motion equations governing the PKM dynamics form the basis for any model-based control. Aiming on an efficient formulation, the motion equations are usually derived in terms of a minimal number of generalized coordinates that constitute a local parameterization of the configuration space. A well-known problem of this formulation is that such minimal

End Effector

3

2

1

Fig. 1. Multibody model of a 2 DOF planar RA-PKM.

(2005); Müller (2005; 2006).

integrity of the controlled RA-PKM.

Müller & Hufnagel (2011).

Aiming for efficient real-time implementations model-based control schemes for PKM are based on a dynamic model in terms of minimal coordinates as pursued in Cheng et al. (2003); Müller (2005); Nakamura & Ghodoussi (1989); Thanh et al. (2009). A PKM is a force-controlled mechanism with kinematic loops. Following the standard approach in multibody dynamics the Lagrangian motion equations of first kind are first derived for the unconstrained system (opening kinematic loops), and the Lagrange multipliers are eliminated by projecting these equations to the configuration space defined by the (holonomic) geometric constraints. This approach is know as the coordinate partitioning method, see Wehage & Haug (1982).

Denote with **<sup>q</sup>** <sup>∈</sup> **<sup>V</sup>***<sup>n</sup>* the vector of joint variables *<sup>q</sup>a*, *<sup>a</sup>* <sup>=</sup> 1, . . . , *<sup>n</sup>* of the unconstrained system, obtained by opening the kinematic loops, where in each kinematic loop one joint is removed. The loop closure is enforced by the corresponding loop constraints giving rise to a set of *r* geometric and kinematic loop constraints

$$\mathbf{0} = \mathbf{h}\left(\mathbf{q}\right), \ \mathbf{h}\left(\mathbf{q}\right) \in \mathbb{R}^r$$

$$\mathbf{0} = \mathbf{J}\left(\mathbf{q}\right)\dot{\mathbf{q}}, \ \mathbf{J}\left(\mathbf{q}\right) \in \mathbb{R}^{r, n}.\tag{1}$$

The PKM in figure 1, for instance, comprises two independent kinematic loops that can be opened by removing the two joints at the EE connecting the kinematic chain formed by joints 2 and 5 with that formed by joints 1 and 4, and joints 3 and 6, respectively. The resulting unconstrained tree-system in figure 2 has *n* = 6 generalized coordinates *qa*, *a* = 1, . . . , 6. With the constraints (1) the Lagrangian motion equations of the PKM are

$$\mathbf{G}\left(\mathbf{q}\right)\ddot{\mathbf{q}} + \mathbf{C}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}} + \mathbf{Q}\left(\mathbf{q}, \dot{\mathbf{q}}, t\right) + \mathbf{J}^{T}\left(\mathbf{q}\right)\lambda = \mathbf{u}.\tag{2}$$

**G** is the generalized mass matrix of the tree-system, **Cq**˙ represents generalized Coriolis and centrifugal forces, **Q** represents all remaining forces, including EE loads, and **u** are the generalized control forces. The Lagrange multipliers λ can be identified with the

Fig. 2. Definition of loop constraints for the planar 2 DOF RA-PKM.

constraint reactions in cut-joints. The vector **<sup>q</sup>** <sup>∈</sup> **<sup>V</sup>***<sup>n</sup>* represents the PKM configuration. The configuration space (c-space) of the PKM model is defined by the geometric constraints:

$$V := \{ \mathbf{q} \in \mathbb{V}^n | \mathbf{h} \, (\mathbf{q}) = \mathbf{0} \}. \tag{3}$$

Projecting the motion equations (2) of the tree-system to the configuration space *V*, with the

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 211

system of *n* ODE's in the *n* generalized coordinates **q** that govern the PKM dynamics when controlled via the generalized control forces **c**. The equations (6) have been first proposed by Voronets (1901) and are a special kind of Maggi's equations, Maggi (1901). They have been proposed for use in multibody dynamics in Angles & Lee (1988); Wehage & Haug (1982), and were put forward for PKM modeling in Cheng et al. (2003); Müller (2005); Thanh et al. (2009). The PKM control problem can now be represented as the control-affine control system

> *m* ∑ *i*=1

Based on the control system (8) different actuation schemes can be distinguished. Actuation refers to the immediate effect of control forces in a given state of the PKM. Apparently the degree of actuation has to do with the number of independent control vector fields. The degree of actuation (DOA) can be defined as the number of independent input vector fields **g***<sup>i</sup>* in the

If *α* (*q*) < *δ*, the system is called *underactuated*, and if *α* (*q*) = *δ*, it is called *full-actuated* at *q*. The system is called *redundantly actuated* at *q*, if *m* − *α* (*q*) > 0 and *non-redundantly actuated* at *q*, if *m* = *α* (*q*). Apparently a system can be redundantly underactuated. Configurations *q* where the DOA changes, i.e. when *α* is not constant in a neighborhood of *q*, are called *input*

The inverse dynamics problem is to find the control forces **c** required for controlling the PKM along a prescribed target trajectory **q** (*t*). The system (6) has no unique solution for **c**. Moreover, it is clear that only those **c** in the range of the control matrix **A***<sup>T</sup>* are effective control forces, and that arbitrary forces **c**<sup>0</sup> (prestress) in the null-space of **A***<sup>T</sup>* can be superposed. If **c**<sup>0</sup> is the vector of desired prestress, for given **q**, ˙**q**, ¨**q**, a solution such that

, **g** :=

 **0 <sup>G</sup>**−<sup>1</sup> **A***<sup>T</sup>*

**G** := **F***T***GF**, **C** :=**F***T*(**CF** + **GF**˙), **Q** := **F***T***Q**. (7)

(**q**) **c** (6)

**g***<sup>i</sup>* (**x**) *ci* (8)

, *i* = 1, . . . , *m* are the control vector fields that

*α* (*q*) := rank *M* (*q*). (10)

**Jq**˙ = **0** yield a

. (9)

**G**(**q**) **q**¨ <sup>2</sup> + **C** (**q**, ˙**q**) **q**˙ <sup>2</sup> + **Q**(**q**, ˙**q**, *t*) = **A***<sup>T</sup>*

The *<sup>δ</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> *<sup>m</sup>* equations (6) together with the *<sup>r</sup>* dynamic constraints **Jq**¨ <sup>+</sup> ˙

**x**˙ = **f**(**x**) +

**Cq**˙ <sup>2</sup> + **Q**

 **<sup>q</sup>**˙ <sup>2</sup> −**G**−<sup>1</sup>

help of the orthogonal complement **F**, and with (5), yields

with state vector **x** := (**q**2, ˙**q**2), where

**2.2 Actuation concepts**

**2.3 Inverse dynamics solution**

**f** :=

determine how the control forces affect the system's state.

*singularities*, see Müller (2009); Zlatanov & Fenton (1998).

**f** is the drift vector field, and the columns **g***<sup>i</sup>*

control system (8). With regular *G* the DOA is

where

If **J** has locally full rank *r*, one can select *δ* := *n* − *r* joint variables, called independent coordinates, such that the admissible configurations **q** ∈ *V* are functions of these independent coordinates. This induces a coordinate partitioning. If the rank of **J** is constant, the c-space is smooth *δ*-dimensional manifold and *δ* is the DOF of the PKM, see Müller (2009). A configuration **q** where the rank of **J** changes is called a c-space singularity since then *V* is not a smooth manifold in **q**.

Denote with **q**<sup>1</sup> and **q**<sup>2</sup> respectively the vector of dependent and independent coordinates, the velocity constraints can be written as

$$\mathbf{J}\_1 \dot{\mathbf{q}}\_1 + \mathbf{J}\_2 \dot{\mathbf{q}}\_2 = \mathbf{0},\tag{4}$$

where **<sup>J</sup>** <sup>=</sup> (**J**1,**J**2), with **<sup>J</sup>**<sup>1</sup> (**q**) <sup>∈</sup> **<sup>R</sup>***r*,*<sup>r</sup>* , **<sup>J</sup>**<sup>2</sup> (**q**) <sup>∈</sup> **<sup>R</sup>***r*,*δ*. By definition of independent coordinates **J**<sup>1</sup> has full rank, and the generalized velocities can be expressed as

$$\dot{\mathbf{q}} = \mathbf{F}\dot{\mathbf{q}}\_{2'} \text{ where } \mathbf{F} := \begin{pmatrix} -\mathbf{J}\_1^{-1}\mathbf{J}\_2 \\ \mathbf{I}\_\delta \end{pmatrix}. \tag{5}$$

where the matrix **F** is an orthogonal complement of **J** because **JF** ≡ **0** . The time derivative of (5) yields the accelerations ¨**q** = **Fq**¨ <sup>2</sup> + **F**˙ **q**˙ 2.

Due to the existence of kinematic loops, PKM comprise passive joints and only the *m* control forces corresponding to the active joints are present in **u**. Denote with **c** ≡ (*c*1,..., *cm*) the vector of generalized control forces in the actuated joints, and let **A** be that part of **F** so that **F***T***u** = **A***T***c**. This means that if **q**<sup>a</sup> denote the vector of *m* actuator coordinates, then ˙**q**<sup>a</sup> = **Aq**˙ 2. Projecting the motion equations (2) of the tree-system to the configuration space *V*, with the help of the orthogonal complement **F**, and with (5), yields

$$\mathbf{C}(\mathbf{q})\,\ddot{\mathbf{q}}\_2 + \overline{\mathbf{C}}\,(\mathbf{q}\_\prime\dot{\mathbf{q}})\,\dot{\mathbf{q}}\_2 + \overline{\mathbf{Q}}(\mathbf{q}\_\prime\dot{\mathbf{q}}\_\prime t) = \mathbf{A}^T(\mathbf{q})\,\mathbf{c} \tag{6}$$

where

4 Will-be-set-by-IN-TECH

cuts

q 5

constraint reactions in cut-joints. The vector **<sup>q</sup>** <sup>∈</sup> **<sup>V</sup>***<sup>n</sup>* represents the PKM configuration. The configuration space (c-space) of the PKM model is defined by the geometric constraints:

If **J** has locally full rank *r*, one can select *δ* := *n* − *r* joint variables, called independent coordinates, such that the admissible configurations **q** ∈ *V* are functions of these independent coordinates. This induces a coordinate partitioning. If the rank of **J** is constant, the c-space is smooth *δ*-dimensional manifold and *δ* is the DOF of the PKM, see Müller (2009). A configuration **q** where the rank of **J** changes is called a c-space singularity since then *V* is

Denote with **q**<sup>1</sup> and **q**<sup>2</sup> respectively the vector of dependent and independent coordinates, the

where the matrix **F** is an orthogonal complement of **J** because **JF** ≡ **0** . The time derivative of

Due to the existence of kinematic loops, PKM comprise passive joints and only the *m* control forces corresponding to the active joints are present in **u**. Denote with **c** ≡ (*c*1,..., *cm*) the vector of generalized control forces in the actuated joints, and let **A** be that part of **F** so that **F***T***u** = **A***T***c**. This means that if **q**<sup>a</sup> denote the vector of *m* actuator coordinates, then ˙**q**<sup>a</sup> = **Aq**˙ 2.

 <sup>−</sup>**<sup>J</sup>** −1 <sup>1</sup> **J**<sup>2</sup> **I***δ*

Fig. 2. Definition of loop constraints for the planar 2 DOF RA-PKM.

**J**<sup>1</sup> has full rank, and the generalized velocities can be expressed as

**q**˙ = **Fq**˙ 2, where **F** :=

q 4

q 1

not a smooth manifold in **q**.

velocity constraints can be written as

where **<sup>J</sup>** <sup>=</sup> (**J**1,**J**2), with **<sup>J</sup>**<sup>1</sup> (**q**) <sup>∈</sup> **<sup>R</sup>***r*,*<sup>r</sup>*

(5) yields the accelerations ¨**q** = **Fq**¨ <sup>2</sup> + **F**˙ **q**˙ 2.

1

q 3

3

q 2

2

*<sup>V</sup>* :<sup>=</sup> {**<sup>q</sup>** <sup>∈</sup> **<sup>V</sup>***n*|**<sup>h</sup>** (**q**) <sup>=</sup> **<sup>0</sup>**} . (3)

**J**1**q**˙ <sup>1</sup> + **J**2**q**˙ <sup>2</sup> = **0**, (4)

, **<sup>J</sup>**<sup>2</sup> (**q**) <sup>∈</sup> **<sup>R</sup>***r*,*δ*. By definition of independent coordinates

. (5)

q 6

$$
\overline{\mathbf{G}} := \mathbf{F}^T \mathbf{G} \mathbf{F}, \; \overline{\mathbf{C}} := \mathbf{F}^T (\mathbf{C} \mathbf{F} + \mathbf{G} \mathbf{F}), \; \overline{\mathbf{Q}} := \mathbf{F}^T \mathbf{Q}. \tag{7}
$$

The *<sup>δ</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> *<sup>m</sup>* equations (6) together with the *<sup>r</sup>* dynamic constraints **Jq**¨ <sup>+</sup> ˙ **Jq**˙ = **0** yield a system of *n* ODE's in the *n* generalized coordinates **q** that govern the PKM dynamics when controlled via the generalized control forces **c**. The equations (6) have been first proposed by Voronets (1901) and are a special kind of Maggi's equations, Maggi (1901). They have been proposed for use in multibody dynamics in Angles & Lee (1988); Wehage & Haug (1982), and were put forward for PKM modeling in Cheng et al. (2003); Müller (2005); Thanh et al. (2009). The PKM control problem can now be represented as the control-affine control system

$$\dot{\mathbf{x}} = \mathbf{f}\left(\mathbf{x}\right) + \sum\_{i=1}^{m} \mathbf{g}^{i}\left(\mathbf{x}\right) c\_{i} \tag{8}$$

with state vector **x** := (**q**2, ˙**q**2), where

$$\mathbf{f} := \begin{pmatrix} \dot{\mathbf{q}}\_2 \\ -\overline{\mathbf{G}}^{-1} (\overline{\mathbf{C}} \dot{\mathbf{q}}\_2 + \overline{\mathbf{Q}}) \end{pmatrix}, \quad \mathbf{g} := \begin{pmatrix} \mathbf{0} \\ \overline{\mathbf{G}}^{-1} \mathbf{A}^T \end{pmatrix}. \tag{9}$$

**f** is the drift vector field, and the columns **g***<sup>i</sup>* , *i* = 1, . . . , *m* are the control vector fields that determine how the control forces affect the system's state.

#### **2.2 Actuation concepts**

Based on the control system (8) different actuation schemes can be distinguished. Actuation refers to the immediate effect of control forces in a given state of the PKM. Apparently the degree of actuation has to do with the number of independent control vector fields. The degree of actuation (DOA) can be defined as the number of independent input vector fields **g***<sup>i</sup>* in the control system (8). With regular *G* the DOA is

$$\mathfrak{a}\left(\mathfrak{q}\right) := \text{rank}\,\overline{M}\left(\mathfrak{q}\right). \tag{10}$$

If *α* (*q*) < *δ*, the system is called *underactuated*, and if *α* (*q*) = *δ*, it is called *full-actuated* at *q*. The system is called *redundantly actuated* at *q*, if *m* − *α* (*q*) > 0 and *non-redundantly actuated* at *q*, if *m* = *α* (*q*). Apparently a system can be redundantly underactuated. Configurations *q* where the DOA changes, i.e. when *α* is not constant in a neighborhood of *q*, are called *input singularities*, see Müller (2009); Zlatanov & Fenton (1998).

#### **2.3 Inverse dynamics solution**

The inverse dynamics problem is to find the control forces **c** required for controlling the PKM along a prescribed target trajectory **q** (*t*). The system (6) has no unique solution for **c**. Moreover, it is clear that only those **c** in the range of the control matrix **A***<sup>T</sup>* are effective control forces, and that arbitrary forces **c**<sup>0</sup> (prestress) in the null-space of **A***<sup>T</sup>* can be superposed. If **c**<sup>0</sup> is the vector of desired prestress, for given **q**, ˙**q**, ¨**q**, a solution such that  **<sup>c</sup>** <sup>−</sup> **<sup>c</sup>**<sup>0</sup>*<sup>T</sup>* **<sup>W</sup> <sup>c</sup>** <sup>−</sup> **<sup>c</sup>**<sup>0</sup> → min is given by **c** = **A***<sup>T</sup>* + **W <sup>F</sup>***T*(**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ + **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ + **<sup>Q</sup>** (**q**, ˙**q**)) + **NA***<sup>T</sup>*,**Wc**<sup>0</sup> = **A***<sup>T</sup>* + **<sup>W</sup>** (**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ <sup>2</sup> <sup>+</sup> **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>2</sup> <sup>+</sup> **<sup>Q</sup>** (**q**, ˙**q**)) + **NA***<sup>T</sup>*,**Wc**<sup>0</sup> (11)

**K**<sup>P</sup> and **K**<sup>D</sup> are diagonal positive definite gain matrices. The effect of measurement errors can be understood considering the pose **<sup>q</sup>**<sup>a</sup> that is attained as result of a constant setpoint

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 213

If there are no measurement errors (and no model uncertainties) it is Δ**q**<sup>a</sup> = **0**, and the PKM

not comply with the geometric constraints the PKM converges to a **q**<sup>a</sup> such that **e**<sup>a</sup> + Δ**q**<sup>a</sup> is in the null-space of **A***T***K**P. The exact value of **q** depends on external forces, e.g. gravity and

the PD controllers yield the steady state actuator forces **c**<sup>0</sup> = **K**<sup>P</sup> (**e**<sup>a</sup> + Δ**q**a). Since they are in

The consideration so far applies to constant calibration offsets. A further source of systematic measurement errors is the encoder resolution within the actuators. Assuming exact calibration, for simplicity, the error measurement, and thus Δ**q**a, change discontinuously according to the resolution. Then the solution for the steady state error (14) is only piecewise valid and may not be unique. Moreover the interplay of antagonistic forces and the quantization of **<sup>q</sup>**<sup>a</sup> can cause alternating control forces, and hence excited vibrations. This is clear by noting that Δ**q**a changes in discrete steps so that during the settling process the components of **e**<sup>a</sup> + Δ**q**<sup>a</sup> in the null-space of **A***T***K**<sup>P</sup> are changing discontinuously and thus

The crucial point is that the decentralized control scheme (13) is not restricted to the range of **A***T*, and so yields antagonistic control forces in the null-space of **A***T*. Such contradictory control forces are on the one hand due to measurement errors but are on the other hand inevitably caused by the uncoordinated control of individual actuators with (12). In the decentralized PD control the individual controller for each kinematic chain, connecting the EE to the base, acts independently without respecting the coordination within the closed kinematic loops of the RA-PKM. Consider this phenomenon for the 2 DOF RA-PKM in figure 1. The manipulator can be viewed as the cooperation of two (virtually independent) non-redundantly actuated 5-bar linkages as shown in figure 3. Consider the situation when the EE is to follow a straight line depicted in figure 3. The three base joints, with coordinates *q*1, *q*2, and *q*3, are actuated. Two of these actuators are sufficient for this 2 DOF system, however. For instance, the PKM can be controlled by joints 1 and 2, or joints 2 and 3. Consequently the motion control of the 5-bar loop consisting of joints 1 and 2, and the other 5-bar loop with joints 2 and 3 must be synchronized. Controlling the loops independently, the control commands for steering the EE along the straight line, determined from (12), drive either 5-bar loop along a straight line in the *<sup>q</sup>*<sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> and *<sup>q</sup>*<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>3</sup> joint subspace, respectively. The latter correspond to EE-curves depicted in figure 3. It is apparent that the control commands of the 5-bar linkages are contradicting due to the missing synchronization of the two loops. This is an inherent problem of the decentralized control, which does exist for the model-based

**<sup>e</sup>**<sup>a</sup> <sup>=</sup> <sup>−</sup>**K**−<sup>1</sup> P 

**<sup>A</sup>***<sup>T</sup>*<sup>+</sup> = **<sup>A</sup>**

control schemes that a priori respects such interdependencies.

<sup>a</sup> . The stationary control forces, not producing any motion, are those **c**<sup>0</sup> in the

**<sup>0</sup>** <sup>=</sup> **<sup>A</sup>***<sup>T</sup>* (**q**) **<sup>c</sup>**<sup>0</sup> <sup>=</sup> <sup>−</sup>**A***<sup>T</sup>* (**q**) **<sup>K</sup>**<sup>P</sup>**<sup>e</sup>**<sup>a</sup> <sup>=</sup> <sup>−</sup>**A***<sup>T</sup>* (**q**) **<sup>K</sup>**<sup>P</sup> (**e**<sup>a</sup> <sup>+</sup> <sup>Δ</sup>**q**a). (13)

<sup>a</sup> satisfy the geometric constraints. Since generally Δ**q**<sup>a</sup> do

**<sup>A</sup>***<sup>T</sup>*<sup>+</sup>**K**PΔ**q**<sup>a</sup> (14)

**<sup>A</sup>***T***<sup>A</sup>**<sup>−</sup><sup>1</sup> of **<sup>A</sup>***T*. In this equilibrium configuration

command **q**<sup>d</sup>

null-space of **A***<sup>T</sup>* (**q**), i.e.

converges to **e**<sup>a</sup> = 0 since **q**<sup>a</sup> and **q**<sup>d</sup>

with the right pseudoinverse

cause discontinuous control forces.

process loads. The attained steady state error is

the null-space of **A***<sup>T</sup>* the RA-PKM stays at rest.

where **<sup>A</sup>***<sup>T</sup>*<sup>+</sup> **<sup>W</sup>** :<sup>=</sup> **<sup>W</sup>**−1**<sup>A</sup> A***T***W**−1**A** <sup>−</sup><sup>1</sup> is the weighted right pseudoinverse, and **NA***<sup>T</sup>*,**<sup>W</sup>** :<sup>=</sup> (**I***<sup>m</sup>* <sup>−</sup> **<sup>A</sup>***<sup>T</sup>*<sup>+</sup> **<sup>W</sup> <sup>A</sup>***T*) is a projector to the null-space of **<sup>A</sup>***T*. **<sup>W</sup>** is a symmetric positive definite weighting matrix for the drive forces in accordance with the drive capabilities. The pseudoinverse solution in (11) delivers the controls that produce the desired motion, where the drive load is balanced between the individual drives according to the weights. The second part of the control vector **c** is the null-space component generating prestress that is closest to the desired **c**0. The possibility of generating control forces in the null-space has been used for backlash avoidance and stiffness control Müller (2005; 2006); Valasek et al. (2002); Yi et al. (1989).

#### **3. Decentralized control schemes**

#### **3.1 Peculiarities of PKM with actuation redundancy**

Decentralized control of individual actuators without taking into account the dynamics of the controlled system is still the standard control method in industrial applications. Moreover the majority of contemporary robotic manipulators are controlled by a decentralized PD law in favor of its simple computation and low-cost setup. In contrast to model-based control, the actuators are controlled independently, without reference to the dynamics of the non-linear control system, exclusively upon the individual commands obtained from motion planning and inverse kinematics. In other words, it is assumed that all actuators can be independently controlled without mutual interference. This applies to serial manipulators as well as to PKM without actuation redundancy as summarized in Paccot et al. (2009) and Thanh et al. (2009). However, since in case of RA-PKM more actuators are activated than required, decentralized control of RA-PKM naturally leads to conflicting control forces, reflected by undesired prestresses and an increased power consumption as observed in Saglia et al. (2009),Valasek et al. (2005), Wang et al. (2009). Hence antagonistic control forces cannot be attributed to model uncertainties alone, as analyzed in Müller (2010), which could be minimized using model identification methods. Even more such counteraction is inherent to the decentralized control method. In order to eliminate contradicting control forces the actuation redundancy must be resolved also within decentralized control, which requires a kinematic model. In the following the interplay of measurement errors and decentralized control is discussed.

As above denote with **q** and **q**a the actual generalized coordinates of the plant (which is unknown). Further a measurement error is assumed caused by a constant (calibration) offset <sup>Δ</sup>**q**a. The vector of measured actuator coordinates is then introduced as **<sup>q</sup>**<sup>a</sup> :<sup>=</sup> **<sup>q</sup>**<sup>a</sup> <sup>+</sup> <sup>Δ</sup>**q**a. If **q**d <sup>a</sup> (*t*) denotes the desired actuator motion, **<sup>e</sup>**<sup>a</sup> :<sup>=</sup> **<sup>q</sup>**<sup>a</sup> <sup>−</sup> **<sup>q</sup>**<sup>d</sup> <sup>a</sup> is the actual tracking error. Due to the calibration offset the measured tracking error is **e**<sup>a</sup> <sup>=</sup> **<sup>q</sup>**<sup>a</sup> <sup>−</sup> **<sup>q</sup>**<sup>d</sup> <sup>a</sup> = **e**<sup>a</sup> + Δ**q**a.

The simple PD control law that independently regulates the *m* actuator positions upon these measurements is

$$\mathbf{c} = -\mathbf{K}\_{\mathbf{P}} \tilde{\mathbf{e}}\_{\mathbf{a}} - \mathbf{K}\_{\mathbf{D}} \check{\mathbf{e}}\_{\mathbf{a}}.\tag{12}$$

6 Will-be-set-by-IN-TECH

definite weighting matrix for the drive forces in accordance with the drive capabilities. The pseudoinverse solution in (11) delivers the controls that produce the desired motion, where the drive load is balanced between the individual drives according to the weights. The second part of the control vector **c** is the null-space component generating prestress that is closest to the desired **c**0. The possibility of generating control forces in the null-space has been used for backlash avoidance and stiffness control Müller (2005; 2006); Valasek et al. (2002); Yi et al.

Decentralized control of individual actuators without taking into account the dynamics of the controlled system is still the standard control method in industrial applications. Moreover the majority of contemporary robotic manipulators are controlled by a decentralized PD law in favor of its simple computation and low-cost setup. In contrast to model-based control, the actuators are controlled independently, without reference to the dynamics of the non-linear control system, exclusively upon the individual commands obtained from motion planning and inverse kinematics. In other words, it is assumed that all actuators can be independently controlled without mutual interference. This applies to serial manipulators as well as to PKM without actuation redundancy as summarized in Paccot et al. (2009) and Thanh et al. (2009). However, since in case of RA-PKM more actuators are activated than required, decentralized control of RA-PKM naturally leads to conflicting control forces, reflected by undesired prestresses and an increased power consumption as observed in Saglia et al. (2009),Valasek et al. (2005), Wang et al. (2009). Hence antagonistic control forces cannot be attributed to model uncertainties alone, as analyzed in Müller (2010), which could be minimized using model identification methods. Even more such counteraction is inherent to the decentralized control method. In order to eliminate contradicting control forces the actuation redundancy must be resolved also within decentralized control, which requires a kinematic model. In the following the interplay of measurement errors and decentralized

As above denote with **q** and **q**a the actual generalized coordinates of the plant (which is unknown). Further a measurement error is assumed caused by a constant (calibration) offset <sup>Δ</sup>**q**a. The vector of measured actuator coordinates is then introduced as **<sup>q</sup>**<sup>a</sup> :<sup>=</sup> **<sup>q</sup>**<sup>a</sup> <sup>+</sup> <sup>Δ</sup>**q**a. If

The simple PD control law that independently regulates the *m* actuator positions upon these

**<sup>c</sup>** <sup>=</sup> <sup>−</sup>**K**P**e**<sup>a</sup> <sup>−</sup> **<sup>K</sup>**D˙

<sup>a</sup> is the actual tracking error. Due to

**e**a. (12)

<sup>a</sup> = **e**<sup>a</sup> + Δ**q**a.

**<sup>F</sup>***T*(**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ + **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ + **<sup>Q</sup>** (**q**, ˙**q**)) + **NA***<sup>T</sup>*,**Wc**<sup>0</sup>

**<sup>W</sup> <sup>A</sup>***T*) is a projector to the null-space of **<sup>A</sup>***T*. **<sup>W</sup>** is a symmetric positive

**<sup>W</sup>** (**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ <sup>2</sup> <sup>+</sup> **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>2</sup> <sup>+</sup> **<sup>Q</sup>** (**q**, ˙**q**)) + **NA***<sup>T</sup>*,**Wc**<sup>0</sup> (11)

<sup>−</sup><sup>1</sup> is the weighted right pseudoinverse, and **NA***<sup>T</sup>*,**<sup>W</sup>** :<sup>=</sup>

**<sup>c</sup>** <sup>−</sup> **<sup>c</sup>**<sup>0</sup>*<sup>T</sup>* **<sup>W</sup>**

where

(**I***<sup>m</sup>* <sup>−</sup>

(1989).

**<sup>A</sup>***<sup>T</sup>*<sup>+</sup>

**<sup>A</sup>***<sup>T</sup>*<sup>+</sup>

control is discussed.

measurements is

**q**d

**<sup>c</sup>** <sup>−</sup> **<sup>c</sup>**<sup>0</sup>

**c** = **A***<sup>T</sup>* + **W**

> = **A***<sup>T</sup>* +

**<sup>W</sup>** :<sup>=</sup> **<sup>W</sup>**−1**<sup>A</sup>**

**3. Decentralized control schemes**

**3.1 Peculiarities of PKM with actuation redundancy**

<sup>a</sup> (*t*) denotes the desired actuator motion, **<sup>e</sup>**<sup>a</sup> :<sup>=</sup> **<sup>q</sup>**<sup>a</sup> <sup>−</sup> **<sup>q</sup>**<sup>d</sup>

the calibration offset the measured tracking error is **e**<sup>a</sup> <sup>=</sup> **<sup>q</sup>**<sup>a</sup> <sup>−</sup> **<sup>q</sup>**<sup>d</sup>

→ min is given by

**A***T***W**−1**A**

**K**<sup>P</sup> and **K**<sup>D</sup> are diagonal positive definite gain matrices. The effect of measurement errors can be understood considering the pose **<sup>q</sup>**<sup>a</sup> that is attained as result of a constant setpoint command **q**<sup>d</sup> <sup>a</sup> . The stationary control forces, not producing any motion, are those **c**<sup>0</sup> in the null-space of **A***<sup>T</sup>* (**q**), i.e.

$$\mathbf{0} = \mathbf{A}^T(\mathbf{q})\,\mathbf{c}^0 = -\mathbf{A}^T(\mathbf{q})\,\mathbf{K}\_\mathbf{P}\tilde{\mathbf{e}}\_\mathbf{a} = -\mathbf{A}^T(\mathbf{q})\,\mathbf{K}\_\mathbf{P}\left(\mathbf{e}\_\mathbf{a} + \Delta\mathbf{q}\_\mathbf{a}\right). \tag{13}$$

If there are no measurement errors (and no model uncertainties) it is Δ**q**<sup>a</sup> = **0**, and the PKM converges to **e**<sup>a</sup> = 0 since **q**<sup>a</sup> and **q**<sup>d</sup> <sup>a</sup> satisfy the geometric constraints. Since generally Δ**q**<sup>a</sup> do not comply with the geometric constraints the PKM converges to a **q**<sup>a</sup> such that **e**<sup>a</sup> + Δ**q**<sup>a</sup> is in the null-space of **A***T***K**P. The exact value of **q** depends on external forces, e.g. gravity and process loads. The attained steady state error is

$$\mathbf{e}\_{\mathbf{a}} = -\mathbf{K}\_{\mathbf{p}}^{-1} \left(\mathbf{A}^{T}\right)^{+} \mathbf{K}\_{\mathbf{P}} \Delta \mathbf{q}\_{\mathbf{a}} \tag{14}$$

with the right pseudoinverse **<sup>A</sup>***<sup>T</sup>*<sup>+</sup> = **<sup>A</sup> <sup>A</sup>***T***<sup>A</sup>**<sup>−</sup><sup>1</sup> of **<sup>A</sup>***T*. In this equilibrium configuration the PD controllers yield the steady state actuator forces **c**<sup>0</sup> = **K**<sup>P</sup> (**e**<sup>a</sup> + Δ**q**a). Since they are in the null-space of **A***<sup>T</sup>* the RA-PKM stays at rest.

The consideration so far applies to constant calibration offsets. A further source of systematic measurement errors is the encoder resolution within the actuators. Assuming exact calibration, for simplicity, the error measurement, and thus Δ**q**a, change discontinuously according to the resolution. Then the solution for the steady state error (14) is only piecewise valid and may not be unique. Moreover the interplay of antagonistic forces and the quantization of **<sup>q</sup>**<sup>a</sup> can cause alternating control forces, and hence excited vibrations. This is clear by noting that Δ**q**a changes in discrete steps so that during the settling process the components of **e**<sup>a</sup> + Δ**q**<sup>a</sup> in the null-space of **A***T***K**<sup>P</sup> are changing discontinuously and thus cause discontinuous control forces.

The crucial point is that the decentralized control scheme (13) is not restricted to the range of **A***T*, and so yields antagonistic control forces in the null-space of **A***T*. Such contradictory control forces are on the one hand due to measurement errors but are on the other hand inevitably caused by the uncoordinated control of individual actuators with (12). In the decentralized PD control the individual controller for each kinematic chain, connecting the EE to the base, acts independently without respecting the coordination within the closed kinematic loops of the RA-PKM. Consider this phenomenon for the 2 DOF RA-PKM in figure 1. The manipulator can be viewed as the cooperation of two (virtually independent) non-redundantly actuated 5-bar linkages as shown in figure 3. Consider the situation when the EE is to follow a straight line depicted in figure 3. The three base joints, with coordinates *q*1, *q*2, and *q*3, are actuated. Two of these actuators are sufficient for this 2 DOF system, however. For instance, the PKM can be controlled by joints 1 and 2, or joints 2 and 3. Consequently the motion control of the 5-bar loop consisting of joints 1 and 2, and the other 5-bar loop with joints 2 and 3 must be synchronized. Controlling the loops independently, the control commands for steering the EE along the straight line, determined from (12), drive either 5-bar loop along a straight line in the *<sup>q</sup>*<sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> and *<sup>q</sup>*<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>3</sup> joint subspace, respectively. The latter correspond to EE-curves depicted in figure 3. It is apparent that the control commands of the 5-bar linkages are contradicting due to the missing synchronization of the two loops. This is an inherent problem of the decentralized control, which does exist for the model-based control schemes that a priori respects such interdependencies.

Fig. 3. Explanation of the synchronization error in decentralized control.

#### **3.2 Antagonism filter - a method for reducing counteraction**

Apparently the antagonistic actuator forces are those control commands that are (unintentionally) in the null-space of **A***T*. Elimination of these antagonistic control forces is hence equivalent to the removal of those components of the control forces **c** that are in the null-space of **A***T*. This is readily achieved using the projector

$$\mathbf{R\_{A}}^{\tau} := \mathbf{I} - \mathbf{N\_{A}}^{\tau} = \left(\mathbf{A}^{T}\right)^{+} \mathbf{A}^{T} \tag{15}$$

onto the range of **A***T*, where **NA***<sup>T</sup>* is the null-space projector. This projector can be applied to any actuator force commands (not necessarily computed from (12)), and return the effective control forces

$$\mathbf{c}\_{\rm eff} = \mathbf{R}\_{\mathbf{A}} \tau \mathbf{c}.\tag{16}$$

joints are actuated with DC motors (Maxon Re30). The RA-PKM can be controlled either with individual PD controllers or a model-base control scheme (see section 4.5). The PD controllers are designed for a load corresponding to the weight of one arm (134 g). In the following results

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 215

As example the PKM is controlled within 5 s along the triangular EE trajectory in figure 4.

actuator motions are determined from the inverse kinematics that are the target trajectories of the PD controllers. The manipulator was calibrated manually in order to reduce calibration errors. The required joint torques are shown in figure 5 and the joint errors in figure 6. The

*<sup>s</sup>*<sup>2</sup> . The required

2

EE-Path

The trajectory is planned according to the maximal acceleration of about 0.5 *<sup>m</sup>*

3

Start/Terminal

1 1.5 2 2.5 3 3.5 4 4.5 5 t [s]

Fig. 5. Joint torques when the RA-PKM is controlled by decentralized PD controllers.

1 c

3 c

Fig. 4. Triangular EE-path along which the RA-PKM is controlled with the CTC.

2 c

for the PD controller are reported.

1




Motor Torques

 [Nm] ci

0

0.01

0.02

0.03

0.04

These control forces can be applied to the PKM without changing the drive action since **A***<sup>T</sup>***RA***<sup>T</sup>* = **A***T*. Because **RA***<sup>T</sup>* eliminates the antagonistic actuator forces it is called the *antagonism filter* (AF) in Müller & Hufnagel (2011).

In practice the individual actuators are position/velocity controlled rather than force controlled. Since this splitting concerns the actuator forces it needs to be transformed to the position and velocity command. Therefore the error vector in (12) is projected to the range of **A***T***K**<sup>P</sup> and **A***T***K**D, respectively, so that

$$\mathbf{e}\_{\mathrm{eff}} = \mathbf{R}\_{\mathbf{A}^{\mathsf{T}} \mathbf{K}\_{\mathsf{P}}} \tilde{\mathbf{e}}\_{\mathbf{a} \mathsf{\prime}} \quad \dot{\mathbf{e}}\_{\mathrm{eff}} = \mathbf{R}\_{\mathbf{A}^{\mathsf{T}} \mathbf{K}\_{\mathsf{D}}} \acute{\mathbf{e}}\_{\mathbf{a}}.\tag{17}$$

Then the corrected command sent to the individual PD controllers is

$$\mathbf{q}\_{\rm eff}^{\rm d} = \mathbf{q}\_{\rm a} - \mathbf{e}\_{\rm eff} \quad \dot{\mathbf{q}}\_{\rm eff}^{\rm d} = \dot{\mathbf{q}}\_{\rm a} - \dot{\mathbf{e}}\_{\rm eff}.\tag{18}$$

#### **3.3 Case study: Planar 2 DOF RA-PKM**

The AF has been applied to decentralized control of the planar 2 DOF redundantly full-actuated PKM in figure 1. The testbed was developed at the Heilbronn University as reported in Hufnagel & Müller (2011) and Müller & Hufnagel (2011), where all experiments were carried out. The prototype consists of arm segments with a length of 200 mm. The three revolute joints are located at an equilateral triangle with lateral length 400 mm. The base 8 Will-be-set-by-IN-TECH

Apparently the antagonistic actuator forces are those control commands that are (unintentionally) in the null-space of **A***T*. Elimination of these antagonistic control forces is hence equivalent to the removal of those components of the control forces **c** that are in the

onto the range of **A***T*, where **NA***<sup>T</sup>* is the null-space projector. This projector can be applied to any actuator force commands (not necessarily computed from (12)), and return the effective

These control forces can be applied to the PKM without changing the drive action since **A***<sup>T</sup>***RA***<sup>T</sup>* = **A***T*. Because **RA***<sup>T</sup>* eliminates the antagonistic actuator forces it is called the

In practice the individual actuators are position/velocity controlled rather than force controlled. Since this splitting concerns the actuator forces it needs to be transformed to the position and velocity command. Therefore the error vector in (12) is projected to the range of

**<sup>e</sup>**eff <sup>=</sup> **RA***<sup>T</sup>***K**<sup>P</sup> **<sup>e</sup>**a, ˙**e**eff <sup>=</sup> **RA***<sup>T</sup>***K**<sup>D</sup> ˙

The AF has been applied to decentralized control of the planar 2 DOF redundantly full-actuated PKM in figure 1. The testbed was developed at the Heilbronn University as reported in Hufnagel & Müller (2011) and Müller & Hufnagel (2011), where all experiments were carried out. The prototype consists of arm segments with a length of 200 mm. The three revolute joints are located at an equilateral triangle with lateral length 400 mm. The base

eff <sup>=</sup> **<sup>q</sup>**<sup>a</sup> <sup>−</sup> **<sup>e</sup>**eff, ˙**q**<sup>d</sup>

Then the corrected command sent to the individual PD controllers is

**q**d

**<sup>A</sup>***<sup>T</sup>*<sup>+</sup>**A***<sup>T</sup>* (15)

**e**a. (17)

eff = **q**˙ <sup>a</sup> − **e**˙ eff. (18)

**c**eff = **RA***<sup>T</sup>* **c**. (16)

**RA***<sup>T</sup>* :<sup>=</sup> **<sup>I</sup>** <sup>−</sup> **NA***<sup>T</sup>* <sup>=</sup>

Fig. 3. Explanation of the synchronization error in decentralized control.

**3.2 Antagonism filter - a method for reducing counteraction**

null-space of **A***T*. This is readily achieved using the projector

*antagonism filter* (AF) in Müller & Hufnagel (2011).

**A***T***K**<sup>P</sup> and **A***T***K**D, respectively, so that

**3.3 Case study: Planar 2 DOF RA-PKM**

control forces

joints are actuated with DC motors (Maxon Re30). The RA-PKM can be controlled either with individual PD controllers or a model-base control scheme (see section 4.5). The PD controllers are designed for a load corresponding to the weight of one arm (134 g). In the following results for the PD controller are reported.

As example the PKM is controlled within 5 s along the triangular EE trajectory in figure 4. The trajectory is planned according to the maximal acceleration of about 0.5 *<sup>m</sup> <sup>s</sup>*<sup>2</sup> . The required actuator motions are determined from the inverse kinematics that are the target trajectories of the PD controllers. The manipulator was calibrated manually in order to reduce calibration errors. The required joint torques are shown in figure 5 and the joint errors in figure 6. The

Fig. 4. Triangular EE-path along which the RA-PKM is controlled with the CTC.

Fig. 5. Joint torques when the RA-PKM is controlled by decentralized PD controllers.

**Motor** *i* **Electric Energy Consumption** *E*el

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 217

This can be explained by noting that the control matrix **A***<sup>T</sup>* and the AF are computed upon the measurement **<sup>q</sup>***<sup>a</sup>*, and that there are always model imperfections. A reduction of electric energy translates directly to the reduction of actuator torques, and the reduced actuator

the energy consumptions when the RA-PKM is controlled with and without application of the AF. A significant reduction of electrical energy consumption and thus of the drive torques is

Two accepted methods for the model-based control of robotic manipulators are the augmented PD (APD) and computed torque control (CTC) schemes Asada & Slotine (1986); Murray et al. (1993). Both schemes consist of a non-linear feedforward term, that delivers the control forces required for steering the PKM along the desired trajectory, and a linear feedback term to compensate drifts from the desired motion. Now the feedforward term requires the inverse dynamics solution (11). These control methods, originally derived for non-redundantly actuated systems, can be directly adopted for RA-PKM as in Cheng et al. (2003); Müller (2005); Paccot et al. (2009). The APD can be used in the form (omitting null-space components)

<sup>2</sup> <sup>+</sup> **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>d</sup>

independent coordinates. The gain matrices **K**<sup>D</sup> and **K**<sup>P</sup> are diagonal and positive definite.

laws applied to (6) result in exponentially stable trajectory tracking for sufficiently large gains **K**<sup>D</sup> and **K**P, provided **G** is regular. The latter assumption only fails in configuration space singularities of the PKM and in singularities of the parameterization of the model. This is in

It is well-known that there is generally no choice of minimal coordinates that is valid for the entire motion range of the manipulator. Parameterization-singularities refer to configurations

**G** (**q**) **v**<sup>2</sup> + **C** (**q**, ˙**q**) **q**˙ <sup>2</sup>+**Q** (**q**, ˙**q**)

<sup>2</sup> − **K**D**e**˙ <sup>2</sup> − **K**P**e**2. Perfect matching of model and plant presumed, both control

<sup>2</sup>+**Q** (**q**, ˙**q**) − **K**D**e**˙ <sup>2</sup> − **K**P**e**<sup>2</sup>

**without AF with AF** 1.3347 Ws 0.4574 Ws 1.5414 Ws 0.7010 Ws 1.6844 Ws 1.0801 Ws

1 2 3

**4. Model-based control schemes**

**c** = **A***<sup>T</sup>* (**q**) +

with **v**<sup>2</sup> := **q**¨ <sup>d</sup>

**4.1 Augmented PD and computed torque controller**

Table 1. Energy consumption if the PKM is manually recalibrated.

torques are reflected by a reduced electrical power consumption *E*el

apparent. The performed mechanical work is indeed not altered by AF.

**G** (**q**) **q**¨ <sup>d</sup>

wherein **<sup>q</sup>**<sup>d</sup> (*t*) is the desired path, and **<sup>e</sup>**<sup>2</sup> (*t*) :<sup>=</sup> **<sup>q</sup>**<sup>2</sup> (*t*) <sup>−</sup> **<sup>q</sup>**<sup>d</sup>

particular critical for RA-PKM as explained in the next section.

**4.2 Parameterization-singularities of the dynamic model**

An adopted form of the CTC law for RA-PKM is

**c** = **A***<sup>T</sup>* (**q**) + *i*

*<sup>i</sup>* of motor *i*. Table 1 shows

, (20)

<sup>2</sup> (*t*) is the tracking error of the

, (19)

Fig. 6. Actuator tracking errors when the RA-PKM is controlled by decentralized PD controllers.

Fig. 7. Joint torques when the AF is applied to decentralized PD control of the RA-PKM.

initial and final torques and errors in the actuator joint angles when the PKM is at rest are due to the remaining calibration uncertainty and encoder quantization. These steady state drive torques are in accordance with (13) and (14). Application of the AF (15) reduces the overall drive torques as shown in figure 7. The antagonistic actuator torques are almost removed by the AF. A small part of the constant initial and final actuator torques remain, however.


Table 1. Energy consumption if the PKM is manually recalibrated.

This can be explained by noting that the control matrix **A***<sup>T</sup>* and the AF are computed upon the measurement **<sup>q</sup>***<sup>a</sup>*, and that there are always model imperfections. A reduction of electric energy translates directly to the reduction of actuator torques, and the reduced actuator torques are reflected by a reduced electrical power consumption *E*el *<sup>i</sup>* of motor *i*. Table 1 shows the energy consumptions when the RA-PKM is controlled with and without application of the AF. A significant reduction of electrical energy consumption and thus of the drive torques is apparent. The performed mechanical work is indeed not altered by AF.

### **4. Model-based control schemes**

10 Will-be-set-by-IN-TECH

1e

<sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> -0.02

Fig. 6. Actuator tracking errors when the RA-PKM is controlled by decentralized PD

3 c

2 c

<sup>3</sup><sup>e</sup> <sup>2</sup><sup>e</sup>

t [s]

1 c

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

t [s]

Fig. 7. Joint torques when the AF is applied to decentralized PD control of the RA-PKM.

initial and final torques and errors in the actuator joint angles when the PKM is at rest are due to the remaining calibration uncertainty and encoder quantization. These steady state drive torques are in accordance with (13) and (14). Application of the AF (15) reduces the overall drive torques as shown in figure 7. The antagonistic actuator torques are almost removed by the AF. A small part of the constant initial and final actuator torques remain, however.




Motor Torques

 [Nm] ci

0

0.01

0.02

0.03



Joint Errors [rad] ei

controllers.

0

0.005

0.01

#### **4.1 Augmented PD and computed torque controller**

Two accepted methods for the model-based control of robotic manipulators are the augmented PD (APD) and computed torque control (CTC) schemes Asada & Slotine (1986); Murray et al. (1993). Both schemes consist of a non-linear feedforward term, that delivers the control forces required for steering the PKM along the desired trajectory, and a linear feedback term to compensate drifts from the desired motion. Now the feedforward term requires the inverse dynamics solution (11). These control methods, originally derived for non-redundantly actuated systems, can be directly adopted for RA-PKM as in Cheng et al. (2003); Müller (2005); Paccot et al. (2009). The APD can be used in the form (omitting null-space components)

$$\mathbf{c} = \left(\mathbf{A}^{\mathrm{T}}(\mathbf{q})\right)^{+} \left[\overline{\mathbf{G}}\left(\mathbf{q}\right)\ddot{\mathbf{q}}\_{2}^{\mathrm{d}} + \overline{\mathbf{C}}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}}\_{2}^{\mathrm{d}} + \overline{\mathbf{Q}}\left(\mathbf{q}, \dot{\mathbf{q}}\right) - \mathbf{K}\_{\mathrm{D}}\dot{\mathbf{e}}\_{2} - \mathbf{K}\_{\mathrm{P}}\mathbf{e}\_{2}\right],\tag{19}$$

wherein **<sup>q</sup>**<sup>d</sup> (*t*) is the desired path, and **<sup>e</sup>**<sup>2</sup> (*t*) :<sup>=</sup> **<sup>q</sup>**<sup>2</sup> (*t*) <sup>−</sup> **<sup>q</sup>**<sup>d</sup> <sup>2</sup> (*t*) is the tracking error of the independent coordinates. The gain matrices **K**<sup>D</sup> and **K**<sup>P</sup> are diagonal and positive definite. An adopted form of the CTC law for RA-PKM is

$$\mathbf{c} = \left(\mathbf{A}^T(\mathbf{q})\right)^+ \left[\overline{\mathbf{C}}(\mathbf{q})\,\mathbf{v}\_2 + \overline{\mathbf{C}}(\mathbf{q}, \dot{\mathbf{q}})\,\dot{\mathbf{q}}\_2 + \overline{\mathbf{Q}}(\mathbf{q}, \dot{\mathbf{q}})\right],\tag{20}$$

with **v**<sup>2</sup> := **q**¨ <sup>d</sup> <sup>2</sup> − **K**D**e**˙ <sup>2</sup> − **K**P**e**2. Perfect matching of model and plant presumed, both control laws applied to (6) result in exponentially stable trajectory tracking for sufficiently large gains **K**<sup>D</sup> and **K**P, provided **G** is regular. The latter assumption only fails in configuration space singularities of the PKM and in singularities of the parameterization of the model. This is in particular critical for RA-PKM as explained in the next section.

#### **4.2 Parameterization-singularities of the dynamic model**

It is well-known that there is generally no choice of minimal coordinates that is valid for the entire motion range of the manipulator. Parameterization-singularities refer to configurations

a) 1 b)

Fig. 8. Different parameterization singularities if the 2 DOF planer PKM is non-redundantly actuated. The mechanism shown in color is the equivalent non-redundantly actuated

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 219

require the selection of minimal coordinates, it is a rank deficient *n* × *n* matrix with constant

As with the orthogonal complement the null-space projector must be partitioned according to the coordinates of the passive and actuated joints. To this end the coordinate vector is

of passive joints and *δ* coordinates of actuator joints, respectively. With the assumption that the PKM configuration is determined by the *m* actuator coordinates, *δ* actuator coordinates serve as minimal coordinates, so that **q**<sup>a</sup> = (..., **q**2). Then the projector can be partitioned as

where **A** corresponds to the actuator coordinates so that ˙**q**<sup>a</sup> = **Aq**˙ . Premultiplication of the

Only *δ* = *n* − *r* of these *n* equations are independent, however. The ˙**q** and ¨**q** in (22) must satisfy the constraints. If this cannot be ensured, in particular if **q** are determined from measured values, they must be projected according to ˙**q**proj = **NJq**˙ and ¨**q**proj = **NJq**¨ + **N**˙ **Jq**˙ . This leads

<sup>−</sup> (*<sup>n</sup>* <sup>−</sup> *<sup>m</sup>*) <sup>×</sup> *<sup>n</sup>*

, Denote with **q**p and **q**a the vector consisting of the *m* coordinates

**<sup>J</sup>** = **NJ** and **JNJ** = **0** yields a system of *n* motion equations

**NJ**(**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ <sup>+</sup> **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>+</sup> **<sup>Q</sup>** (**q**, ˙**q**, *<sup>t</sup>*)) = **<sup>A</sup>** *<sup>T</sup>***c**. (22)

<sup>−</sup> *<sup>m</sup>* <sup>×</sup> *<sup>n</sup>* (21)

2 3

mechanisms being instantaneously in an input-singularity.

**q**p, **q**a  **<sup>J</sup>** = **NJNJ**.

**NJ** =

 **<sup>P</sup> A**

EE

rank *<sup>δ</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> *<sup>r</sup>*, and **NJ** <sup>=</sup> **<sup>N</sup>***<sup>T</sup>*

motion equations (2) with **N***<sup>T</sup>*

rearranged as **q** =

where a selected set of independent coordinates becomes invalid. This problem is usually solved ad-hoc by switching between different mathematical models as proposed in Hufnagel & Müller (2011). There is, however, no general approach to cope with this problem. While this is essentially a numerical problem of the particular PKM model it does have a great significance for PKM control. In particular the improper selection of generalized coordinates can severely deteriorate the stability of model-based control schemes.

The inherent problem of the minimal coordinate formulation is the need for selecting independent minimal coordinates **q**2. Since they are local coordinates on the c-space *V* the PKM configuration cannot be expressed globally in terms of these minimal coordinates. That is, any such minimal coordinates are only valid in a limited range of motion, and a collection of different sets of minimal coordinates is necessary to cover the entire c-space. The switching method proposed in Hufnagel & Müller (2011) switches between such local coordinates.

From a practical point of view it makes sense to use *δ* actuator coordinates as independent coordinates in the motion equations (6). That is, **q**<sup>2</sup> is a subset of **q**a. In other words the PKM is considered as non-redundantly actuated and its motion equations are parameterized in terms of *δ* actuator coordinates. Consequently, parameterization-singularities are exactly the input-singularities of the non-redundantly actuated PKM. For example, the planar 2 DOF RA-PKM in figure 1 is naturally parameterized in terms of *δ* = 2 out of the *m* = 3 actuator coordinates. This leads to parameterization-singularities shown in figure 8. Figure 8 a) shows two configurations where the actuator coordinates *q*<sup>1</sup> and *q*<sup>2</sup> are not valid as independent coordinates for the PKM model. In these configurations the PKM configuration is not uniquely determined by the motion joints of 1 and 2 so that *q*<sup>1</sup> and *q*<sup>2</sup> fail as independent coordinates. Alternatively joints 1 and 3 could be used to control the PKM. That is, *q*<sup>1</sup> and *q*<sup>3</sup> would constitute independent coordinates of the minimal coordinate model (6). This parameterization exhibits the singular configurations in figure 8 b), however. Similar singularities exist when *q*<sup>2</sup> and *q*<sup>3</sup> are used as independent, and moreover there are parameterization-singularities for any choice of two actuator angles.

Now it is important to notice that any switching to different independent coordinates **q**<sup>2</sup> causes a complete change of the motion equations (6). Such a switching method is thus computationally rather complex and accompanied by a high implementation effort. Its application to general RA-PKM requires monitoring the numerical conditioning of the orthogonal complement **F** in (5) in order to detect switching points. Only for simple mechanisms, such as the reported 2 DOF RA-PKM in figure 1, the switching points can be determined explicitly giving rise to a switching map.

#### **4.3 A robust formulation of the dynamic model in redundant coordinates**

The minimal coordinate formulation (6) is prone to parameterization-singularities. Coordinate switching methods, introduced to cope with this problem, are computationally rather complex for general PKM. An approach that completely avoids the use of independent coordinates was proposed in Müller (2011) where the motion equations are expressed in terms of *n* redundant coordinates. The idea is to eliminate the Lagrange multipliers from (2) by means of a projector to the null-space of **J**. That is, instead of premultiplication with **F***T*, (2) is premultiplied with a null-space projector determined from the pseudoinverse of **J**. As long as the PKM does not encounter c-space singularities, where rank **J** drops, the constraint Jacobian **J** is always full rank *r*, and its right pseudoinverse is given by **J**<sup>+</sup> = **J***T*(**JJ***T*)−1. The corresponding projector to the null-space of **<sup>J</sup>** is then **NJ** <sup>=</sup> **<sup>I</sup>***<sup>n</sup>* <sup>−</sup> **<sup>J</sup>**+**J**. This projector does not 12 Will-be-set-by-IN-TECH

where a selected set of independent coordinates becomes invalid. This problem is usually solved ad-hoc by switching between different mathematical models as proposed in Hufnagel & Müller (2011). There is, however, no general approach to cope with this problem. While this is essentially a numerical problem of the particular PKM model it does have a great significance for PKM control. In particular the improper selection of generalized coordinates

The inherent problem of the minimal coordinate formulation is the need for selecting independent minimal coordinates **q**2. Since they are local coordinates on the c-space *V* the PKM configuration cannot be expressed globally in terms of these minimal coordinates. That is, any such minimal coordinates are only valid in a limited range of motion, and a collection of different sets of minimal coordinates is necessary to cover the entire c-space. The switching method proposed in Hufnagel & Müller (2011) switches between such local coordinates. From a practical point of view it makes sense to use *δ* actuator coordinates as independent coordinates in the motion equations (6). That is, **q**<sup>2</sup> is a subset of **q**a. In other words the PKM is considered as non-redundantly actuated and its motion equations are parameterized in terms of *δ* actuator coordinates. Consequently, parameterization-singularities are exactly the input-singularities of the non-redundantly actuated PKM. For example, the planar 2 DOF RA-PKM in figure 1 is naturally parameterized in terms of *δ* = 2 out of the *m* = 3 actuator coordinates. This leads to parameterization-singularities shown in figure 8. Figure 8 a) shows two configurations where the actuator coordinates *q*<sup>1</sup> and *q*<sup>2</sup> are not valid as independent coordinates for the PKM model. In these configurations the PKM configuration is not uniquely determined by the motion joints of 1 and 2 so that *q*<sup>1</sup> and *q*<sup>2</sup> fail as independent coordinates. Alternatively joints 1 and 3 could be used to control the PKM. That is, *q*<sup>1</sup> and *q*<sup>3</sup> would constitute independent coordinates of the minimal coordinate model (6). This parameterization exhibits the singular configurations in figure 8 b), however. Similar singularities exist when *q*<sup>2</sup> and *q*<sup>3</sup> are used as independent, and moreover there are

Now it is important to notice that any switching to different independent coordinates **q**<sup>2</sup> causes a complete change of the motion equations (6). Such a switching method is thus computationally rather complex and accompanied by a high implementation effort. Its application to general RA-PKM requires monitoring the numerical conditioning of the orthogonal complement **F** in (5) in order to detect switching points. Only for simple mechanisms, such as the reported 2 DOF RA-PKM in figure 1, the switching points can be

The minimal coordinate formulation (6) is prone to parameterization-singularities. Coordinate switching methods, introduced to cope with this problem, are computationally rather complex for general PKM. An approach that completely avoids the use of independent coordinates was proposed in Müller (2011) where the motion equations are expressed in terms of *n* redundant coordinates. The idea is to eliminate the Lagrange multipliers from (2) by means of a projector to the null-space of **J**. That is, instead of premultiplication with **F***T*, (2) is premultiplied with a null-space projector determined from the pseudoinverse of **J**. As long as the PKM does not encounter c-space singularities, where rank **J** drops, the constraint Jacobian **J** is always full rank *r*, and its right pseudoinverse is given by **J**<sup>+</sup> = **J***T*(**JJ***T*)−1. The corresponding projector to the null-space of **<sup>J</sup>** is then **NJ** <sup>=</sup> **<sup>I</sup>***<sup>n</sup>* <sup>−</sup> **<sup>J</sup>**+**J**. This projector does not

can severely deteriorate the stability of model-based control schemes.

parameterization-singularities for any choice of two actuator angles.

**4.3 A robust formulation of the dynamic model in redundant coordinates**

determined explicitly giving rise to a switching map.

Fig. 8. Different parameterization singularities if the 2 DOF planer PKM is non-redundantly actuated. The mechanism shown in color is the equivalent non-redundantly actuated mechanisms being instantaneously in an input-singularity.

require the selection of minimal coordinates, it is a rank deficient *n* × *n* matrix with constant rank *<sup>δ</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> *<sup>r</sup>*, and **NJ** <sup>=</sup> **<sup>N</sup>***<sup>T</sup>* **<sup>J</sup>** = **NJNJ**.

As with the orthogonal complement the null-space projector must be partitioned according to the coordinates of the passive and actuated joints. To this end the coordinate vector is rearranged as **q** = **q**p, **q**a , Denote with **q**p and **q**a the vector consisting of the *m* coordinates of passive joints and *δ* coordinates of actuator joints, respectively. With the assumption that the PKM configuration is determined by the *m* actuator coordinates, *δ* actuator coordinates serve as minimal coordinates, so that **q**<sup>a</sup> = (..., **q**2). Then the projector can be partitioned as

$$\mathbf{N}\_{\mathbf{J}} = \begin{pmatrix} \tilde{\mathbf{P}} \\ \tilde{\mathbf{A}} \end{pmatrix} - \begin{pmatrix} n - m \end{pmatrix} \times n \tag{21}$$

where **A** corresponds to the actuator coordinates so that ˙**q**<sup>a</sup> = **Aq**˙ . Premultiplication of the motion equations (2) with **N***<sup>T</sup>* **<sup>J</sup>** = **NJ** and **JNJ** = **0** yields a system of *n* motion equations

$$\mathbf{N}\_{\mathbf{J}}(\mathbf{G}\left(\mathbf{q}\right)\ddot{\mathbf{q}} + \mathbf{C}\left(\mathbf{q}\_{\prime}\dot{\mathbf{q}}\right)\dot{\mathbf{q}} + \mathbf{Q}\left(\mathbf{q}\_{\prime}\dot{\mathbf{q}}\_{\prime}t\right)) = \tilde{\mathbf{A}}^{T}\mathbf{c}.\tag{22}$$

Only *δ* = *n* − *r* of these *n* equations are independent, however. The ˙**q** and ¨**q** in (22) must satisfy the constraints. If this cannot be ensured, in particular if **q** are determined from measured values, they must be projected according to ˙**q**proj = **NJq**˙ and ¨**q**proj = **NJq**¨ + **N**˙ **Jq**˙ . This leads to

$$
\tilde{\mathbf{G}}\left(\mathbf{q}\right)\ddot{\mathbf{q}} + \tilde{\mathbf{C}}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}} + \tilde{\mathbf{Q}}\left(\mathbf{q}\right) = \tilde{\mathbf{A}}^T \mathbf{c} \tag{23}
$$

**4.4.2 Augmented PD control**

following APD control scheme

**c** =

=

the c-space *V* using the Lyapunov function

projection onto *V* with the projector **NJ**.

**c** =

=

**<sup>A</sup>** *<sup>T</sup>*<sup>+</sup>

**4.4.3 Computed torque control**

<sup>V</sup> (**e**˙ , **<sup>e</sup>**, *<sup>t</sup>*) <sup>=</sup> <sup>1</sup>

2 **e**˙

**<sup>A</sup>** *<sup>T</sup>*<sup>+</sup>

coordinates. That is, assuming the coordinate partitioning **<sup>q</sup>**<sup>≡</sup>

trajectory with **q**<sup>d</sup>

with **<sup>D</sup>** <sup>=</sup> **<sup>A</sup>** *<sup>T</sup>*

The control task is to minimize the tracking error of the actuator coordinates given a target

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 221

<sup>a</sup> (*t*). With the above formulation it is straightforward to introduce the

**<sup>G</sup> <sup>q</sup>**¨ <sup>−</sup> **DG <sup>q</sup>**¨ <sup>d</sup> <sup>+</sup> **<sup>C</sup><sup>q</sup>**˙ <sup>−</sup> **DC<sup>q</sup>**˙ <sup>d</sup> <sup>+</sup> **<sup>Q</sup>** <sup>−</sup> **DQ** <sup>+</sup> **DK**D**e**˙ <sup>+</sup> **DK**P**<sup>e</sup>** <sup>=</sup> **<sup>0</sup>** (30)

**e***T***K**P**e** +

**G** (**q**) **v** + **C** (**q**˙ , **q**) **q**˙ + **Q** (**q**, ˙**q**, *t*)

1 2

**<sup>A</sup>** *<sup>T</sup>*<sup>+</sup>**NJ** (**<sup>G</sup>** (**q**) **<sup>v</sup>** <sup>+</sup> **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>+</sup> **<sup>Q</sup>** (**q**, ˙**q**, *<sup>t</sup>*)) (32)

**P**2**G** (**e**¨ + **K**D**e**˙ + **K**P**e**) = **0**. (34)

, they have the form

*<sup>ε</sup>***e***T***<sup>G</sup>** (**q**) **<sup>e</sup>**˙ (31)

= **0** (33)

**q**p, **q**a 

**K** = diag (0, . . . , 0, *K*1,..., *Km*). (29)

(28)

**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ <sup>d</sup>+**<sup>C</sup>** (**q**˙ , **<sup>q</sup>**) **<sup>q</sup>**˙ <sup>d</sup>+**<sup>Q</sup>** (**q**, ˙**q**, *<sup>t</sup>*) <sup>−</sup> **<sup>K</sup>**P**<sup>e</sup>** <sup>−</sup> **<sup>K</sup>**D**e**˙

**<sup>A</sup>** *<sup>T</sup>*<sup>+</sup>**NJ**(**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ <sup>d</sup>+**<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>d</sup>+**<sup>Q</sup>** (**q**, ˙**q**, *<sup>t</sup>*) <sup>−</sup> **<sup>K</sup>**P**<sup>e</sup>** <sup>−</sup> **<sup>K</sup>**D**e**˙)

with error vector **<sup>e</sup>** :<sup>=</sup> **<sup>q</sup>** <sup>−</sup> **<sup>q</sup>**d. Now the gain matrices measure the errors in the *<sup>m</sup>* actuator

The closed loop dynamics, when (28) is applied to the model (23), is governed by

**<sup>A</sup>** *<sup>T</sup>*<sup>+</sup> � **<sup>I</sup>***n*. The rank deficiency of **<sup>A</sup>** implies that **<sup>D</sup>** � **<sup>I</sup>***n*. It can be shown that the APD control scheme (28) achieves exponential trajectory tracking on

*<sup>T</sup>***<sup>G</sup>** (**q**) **<sup>e</sup>**˙ <sup>+</sup>

1 2

with *<sup>ε</sup>* <sup>&</sup>gt; 0. <sup>V</sup> (**e**˙ , **<sup>e</sup>**, *<sup>t</sup>*) is positive definite, and <sup>V</sup>˙ is negative definite for all trajectories in *V*. These properties are directly inherited from the minimal coordinate formulation via the

The standard CTC scheme is easily adapted to the redundant coordinate formulation as

with **v** = **q**¨ − **K**P**e** − **K**D**e**˙ and the gain matrices in (29). It is readily shown that the CTC (32) achieves exponential trajectory tracking on *V*. Choose local coordinates **q**<sup>2</sup> on *V*, and let **P**<sup>1</sup> be a projector to the vector space of dependent velocities ˙**q**1, and **P**<sup>2</sup> a projector to the vector

**G q**¨ + **Cq**˙ + **Q**

The first *n* − *δ* equations are automatically satisfied for trajectories in *V* if the second system is satisfied. The second system, consisting of *δ* equations, governs the error dynamics in term of

space of independent velocities ˙**q**2. Then the dynamics of the closed loop splits into

**P**1 

with

$$
\tilde{\mathbf{G}} \coloneqq \mathbf{N}\_{\mathbf{J}}^{T} \mathbf{G} \mathbf{N}\_{\mathbf{J}} \; \tilde{\mathbf{C}} := \mathbf{N}\_{\mathbf{J}}^{T} (\mathbf{C} \mathbf{N}\_{\mathbf{J}} + \mathbf{G} \dot{\mathbf{N}}\_{\mathbf{J}}) , \; \tilde{\mathbf{Q}} \coloneqq \mathbf{N}\_{\mathbf{J}}^{T} \mathbf{Q} . \tag{24}
$$

The time derivative in (21) is readily found to be **<sup>N</sup>**˙ **<sup>J</sup>** <sup>=</sup> <sup>−</sup> � **B**+**BN**˙ **<sup>J</sup>** � − � **B**+**BN**˙ **<sup>J</sup>** �*T* .

The dynamics model (22) in redundant coordinates is globally valid in all regular configurations as it does not involve any minimal coordinates.

#### **4.4 Model-based control in redundant coordinates**

#### **4.4.1 Inverse dynamics**

In order to use the dynamics formulation in redundant coordinates within a model-based controller (22), respectively (23), must be solved for **c**. The general inverse dynamics solution (neglecting prestress forces) is

$$\mathbf{c} = \left(\tilde{\mathbf{A}}^{T}\right)^{+} \mathbf{N}\_{\mathbf{J}}(\mathbf{G}\left(\mathbf{q}\right)\ddot{\mathbf{q}} + \mathbf{C}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}} + \mathbf{Q}\left(\mathbf{q}, \dot{\mathbf{q}}, t\right)). \tag{25}$$

The crucial point here is the computation of the pseudoinverse. The *<sup>n</sup>* <sup>×</sup> *<sup>m</sup>* matrix **<sup>A</sup>**� *<sup>T</sup>* is not regular since rank **<sup>A</sup>**� *<sup>T</sup>* <sup>=</sup> *<sup>δ</sup>* <sup>&</sup>lt; *<sup>m</sup>* <sup>&</sup>lt; *<sup>n</sup>*. Hence the closed form Moore-Penrose pseudoinverse is not applicable. The singular value decomposition (SVD) can always be used to iteratively determine the pseudoinverse. This is generally not applicable for real-time applications due to its numerical complexity. Now if the redundantly actuated PKM does not poses input-singularities, the PKM configuration is uniquely determined by the *m* > *δ* input coordinates. Hence at any time rank **A**� = *δ*, and a full-rank *δ* × *n* submatrix **A**�<sup>1</sup> can be separated so that

$$
\tilde{\mathbf{A}} = \begin{pmatrix} \tilde{\mathbf{A}}\_1 \\ \tilde{\mathbf{A}}\_2 \end{pmatrix} \tag{26}
$$

with the remaining (*m* − *δ*) × *n* matrix **A**�2. Upon this partitioning the following explicit expression for the pseudoinverse was presented in Müller (2011):

$$\left(\tilde{\mathbf{A}}^{T}\right)^{+} = \begin{pmatrix} \left(\tilde{\mathbf{A}}\_{1}^{T}\right)^{+} \left(\mathbf{I}\_{n} - \tilde{\mathbf{A}}\_{2}^{T} \left(\mathbf{I}\_{m-\delta} + \mathbf{B}^{T}\mathbf{B}\right)^{-1} \mathbf{B}^{T} \left(\tilde{\mathbf{A}}\_{1}^{T}\right)^{+}\right) \\\\ \left(\mathbf{I}\_{m-\delta} + \mathbf{B}^{T}\mathbf{B}\right)^{-1} \mathbf{B}^{T} \left(\tilde{\mathbf{A}}\_{1}^{T}\right)^{+} \end{pmatrix} \tag{27}$$

with **B** = � **A**� *T* 1 �+**A**� *<sup>T</sup>* <sup>2</sup> and the left pseudoinverse � **A**� *T* 1 �<sup>+</sup> = � **<sup>A</sup>**�1**A**� *<sup>T</sup>* 1 �−<sup>1</sup> **A**�1.

The partitioning (26) is equivalent to selecting *δ* independent coordinates. As already discussed such a selection, and thus the submatrix **A**�1, is not unique, which rises again the problem of selecting *δ* independent coordinates out of the *m* actuator coordinates. Consequently the full-rank *δ* × *n* matrix **A**�<sup>1</sup> must be selected depending on the actual PKM pose. The important difference to the minimal coordinate formulation is that only the submatrix **A**� must be selected whereas the motion equations (22) and (23) are globally valid and remain unaltered in the entire motion range. Switching is only performed within the pseudoinverse computation for **<sup>A</sup>**� *<sup>T</sup>*. The selection of a proper submatrix requires monitoring the rank of the *<sup>δ</sup>* <sup>×</sup> *<sup>δ</sup>* matrix **<sup>A</sup>**�1**A**� *<sup>T</sup>* 1 .

#### **4.4.2 Augmented PD control**

14 Will-be-set-by-IN-TECH

The dynamics model (22) in redundant coordinates is globally valid in all regular

In order to use the dynamics formulation in redundant coordinates within a model-based controller (22), respectively (23), must be solved for **c**. The general inverse dynamics solution

The crucial point here is the computation of the pseudoinverse. The *<sup>n</sup>* <sup>×</sup> *<sup>m</sup>* matrix **<sup>A</sup>**� *<sup>T</sup>* is not regular since rank **<sup>A</sup>**� *<sup>T</sup>* <sup>=</sup> *<sup>δ</sup>* <sup>&</sup>lt; *<sup>m</sup>* <sup>&</sup>lt; *<sup>n</sup>*. Hence the closed form Moore-Penrose pseudoinverse is not applicable. The singular value decomposition (SVD) can always be used to iteratively determine the pseudoinverse. This is generally not applicable for real-time applications due to its numerical complexity. Now if the redundantly actuated PKM does not poses input-singularities, the PKM configuration is uniquely determined by the *m* > *δ* input coordinates. Hence at any time rank **A**� = *δ*, and a full-rank *δ* × *n* submatrix **A**�<sup>1</sup> can be

**A**� =

**<sup>I</sup>***<sup>n</sup>* <sup>−</sup> **<sup>A</sup>**� *<sup>T</sup>* 2 �

�

expression for the pseudoinverse was presented in Müller (2011):

<sup>2</sup> and the left pseudoinverse �

1 .

⎛ ⎝ � **A**� *T* 1 �<sup>+</sup> � � **<sup>A</sup>**�<sup>1</sup> **A**�2

with the remaining (*m* − *δ*) × *n* matrix **A**�2. Upon this partitioning the following explicit

**<sup>I</sup>***m*−*<sup>δ</sup>* <sup>+</sup> **<sup>B</sup>***T***<sup>B</sup>**

The partitioning (26) is equivalent to selecting *δ* independent coordinates. As already discussed such a selection, and thus the submatrix **A**�1, is not unique, which rises again the problem of selecting *δ* independent coordinates out of the *m* actuator coordinates. Consequently the full-rank *δ* × *n* matrix **A**�<sup>1</sup> must be selected depending on the actual PKM pose. The important difference to the minimal coordinate formulation is that only the submatrix **A**� must be selected whereas the motion equations (22) and (23) are globally valid and remain unaltered in the entire motion range. Switching is only performed within the pseudoinverse computation for **<sup>A</sup>**� *<sup>T</sup>*. The selection of a proper submatrix requires monitoring

�

**<sup>I</sup>***m*−*<sup>δ</sup>* <sup>+</sup> **<sup>B</sup>***T***<sup>B</sup>**

**A**� *T* 1 �<sup>+</sup> = �

�−<sup>1</sup> **<sup>B</sup>***T*�

�−<sup>1</sup> **<sup>B</sup>***T*(**A**� *<sup>T</sup>*

**A**� *T* 1 �+

**<sup>A</sup>**�1**A**� *<sup>T</sup>* 1 �−<sup>1</sup> **A**�1.

1 �+� ⎞

⎠ (27)

**<sup>G</sup>**� :=**N***<sup>T</sup>*

**4.4 Model-based control in redundant coordinates**

**c** = �

The time derivative in (21) is readily found to be **<sup>N</sup>**˙ **<sup>J</sup>** <sup>=</sup> <sup>−</sup> �

configurations as it does not involve any minimal coordinates.

**<sup>J</sup> GNJ**, **<sup>C</sup>**� :=**N***<sup>T</sup>*

**<sup>G</sup>**� (**q**) **<sup>q</sup>**¨ <sup>+</sup> **<sup>C</sup>**� (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>+</sup> **<sup>Q</sup>**� (**q**) <sup>=</sup> **<sup>A</sup>**� *<sup>T</sup>***<sup>c</sup>** (23)

**B**+**BN**˙ **<sup>J</sup>** � − �

**<sup>A</sup>**� *<sup>T</sup>*�+**NJ**(**<sup>G</sup>** (**q**) **<sup>q</sup>**¨ <sup>+</sup> **<sup>C</sup>** (**q**, ˙**q**) **<sup>q</sup>**˙ <sup>+</sup> **<sup>Q</sup>** (**q**, ˙**q**, *<sup>t</sup>*)). (25)

**<sup>J</sup> Q**. (24)

�*T* .

(26)

**B**+**BN**˙ **<sup>J</sup>**

**<sup>J</sup>** (**CNJ** <sup>+</sup> **GN**˙ **<sup>J</sup>**), **<sup>Q</sup>**� :=**N***<sup>T</sup>*

to

with

**4.4.1 Inverse dynamics**

separated so that

with **B** = �

(neglecting prestress forces) is

� **<sup>A</sup>**� *<sup>T</sup>*�<sup>+</sup> <sup>=</sup>

the rank of the *<sup>δ</sup>* <sup>×</sup> *<sup>δ</sup>* matrix **<sup>A</sup>**�1**A**� *<sup>T</sup>*

**A**� *T* 1 �+**A**� *<sup>T</sup>* The control task is to minimize the tracking error of the actuator coordinates given a target trajectory with **q**<sup>d</sup> <sup>a</sup> (*t*). With the above formulation it is straightforward to introduce the following APD control scheme

$$\mathbf{c} = \left(\tilde{\mathbf{A}}^{T}\right)^{+} \left(\tilde{\mathbf{G}}\left(\mathbf{q}\right)\ddot{\mathbf{q}}^{\mathrm{d}} + \tilde{\mathbf{C}}\left(\dot{\mathbf{q}}\_{\nu}\mathbf{q}\right)\dot{\mathbf{q}}^{\mathrm{d}} + \tilde{\mathbf{Q}}\left(\mathbf{q}\_{\nu}\dot{\mathbf{q}}\_{\nu}t\right) - \mathbf{K}\mathbf{e} - \mathbf{K}\_{\mathrm{D}}\dot{\mathbf{e}}\right) \tag{28}$$
 
$$= \left(\tilde{\mathbf{A}}^{T}\right)^{+} \mathbf{N}\_{\mathrm{I}}(\mathbf{G}\left(\mathbf{q}\right)\ddot{\mathbf{q}}^{\mathrm{d}} + \mathbf{C}\left(\mathbf{q}\_{\nu}\dot{\mathbf{q}}\right)\dot{\mathbf{q}}^{\mathrm{d}} + \mathbf{Q}\left(\mathbf{q}\_{\nu}\dot{\mathbf{q}}\_{\nu}t\right) - \mathbf{K}\_{\mathrm{P}}\mathbf{e} - \mathbf{K}\_{\mathrm{D}}\dot{\mathbf{e}}\right)$$

with error vector **<sup>e</sup>** :<sup>=</sup> **<sup>q</sup>** <sup>−</sup> **<sup>q</sup>**d. Now the gain matrices measure the errors in the *<sup>m</sup>* actuator coordinates. That is, assuming the coordinate partitioning **<sup>q</sup>**<sup>≡</sup> **q**p, **q**a , they have the form

$$\mathbf{K} = \text{diag}\left(0, \dots, 0, \mathbf{K}\_1, \dots, \mathbf{K}\_{\mathfrak{m}}\right). \tag{29}$$

The closed loop dynamics, when (28) is applied to the model (23), is governed by

$$
\tilde{\mathbf{G}}\tilde{\mathbf{q}} - \mathbf{D}\tilde{\mathbf{G}}\tilde{\mathbf{q}}^{\mathrm{d}} + \tilde{\mathbf{C}}\dot{\mathbf{q}} - \mathbf{D}\tilde{\mathbf{C}}\dot{\mathbf{q}}^{\mathrm{d}} + \tilde{\mathbf{Q}} - \mathbf{D}\tilde{\mathbf{Q}} + \mathbf{D}\mathbf{K}\_{\mathrm{D}}\dot{\mathbf{e}} + \mathbf{D}\mathbf{K}\_{\mathrm{P}}\mathbf{e} = \mathbf{0} \tag{30}
$$

with **<sup>D</sup>** <sup>=</sup> **<sup>A</sup>** *<sup>T</sup>* **<sup>A</sup>** *<sup>T</sup>*<sup>+</sup> � **<sup>I</sup>***n*. The rank deficiency of **<sup>A</sup>** implies that **<sup>D</sup>** � **<sup>I</sup>***n*. It can be shown that the APD control scheme (28) achieves exponential trajectory tracking on

the c-space *V* using the Lyapunov function

$$\mathcal{V}\left(\dot{\mathbf{e}},\mathbf{e},t\right) = \frac{1}{2}\dot{\mathbf{e}}^T\widetilde{\mathbf{G}}\left(\mathbf{q}\right)\dot{\mathbf{e}} + \frac{1}{2}\mathbf{e}^T\mathbf{K}\_\mathbf{P}\mathbf{e} + \frac{1}{2}\varepsilon\mathbf{e}^T\widetilde{\mathbf{G}}\left(\mathbf{q}\right)\dot{\mathbf{e}}\tag{31}$$

with *<sup>ε</sup>* <sup>&</sup>gt; 0. <sup>V</sup> (**e**˙ , **<sup>e</sup>**, *<sup>t</sup>*) is positive definite, and <sup>V</sup>˙ is negative definite for all trajectories in *V*. These properties are directly inherited from the minimal coordinate formulation via the projection onto *V* with the projector **NJ**.

#### **4.4.3 Computed torque control**

The standard CTC scheme is easily adapted to the redundant coordinate formulation as

$$\mathbf{c} = \left(\tilde{\mathbf{A}}^T\right)^+ \left(\tilde{\mathbf{G}}\left(\mathbf{q}\right)\mathbf{v} + \tilde{\mathbf{C}}\left(\dot{\mathbf{q}}\_\prime \mathbf{q}\right)\dot{\mathbf{q}} + \tilde{\mathbf{Q}}\left(\mathbf{q}\_\prime \dot{\mathbf{q}}\_\prime t\right)\right)$$

$$= \left(\tilde{\mathbf{A}}^T\right)^+ \mathbf{N}\_\mathbf{J}\left(\mathbf{G}\left(\mathbf{q}\right)\mathbf{v} + \mathbf{C}\left(\mathbf{q}\_\prime \dot{\mathbf{q}}\right)\dot{\mathbf{q}} + \mathbf{Q}\left(\mathbf{q}\_\prime \dot{\mathbf{q}}\_\prime t\right)\right) \tag{32}$$

with **v** = **q**¨ − **K**P**e** − **K**D**e**˙ and the gain matrices in (29). It is readily shown that the CTC (32) achieves exponential trajectory tracking on *V*. Choose local coordinates **q**<sup>2</sup> on *V*, and let **P**<sup>1</sup> be a projector to the vector space of dependent velocities ˙**q**1, and **P**<sup>2</sup> a projector to the vector space of independent velocities ˙**q**2. Then the dynamics of the closed loop splits into

$$\mathbf{P}\_1 \left( \tilde{\mathbf{G}} \ddot{\mathbf{q}} + \tilde{\mathbf{C}} \dot{\mathbf{q}} + \tilde{\mathbf{Q}} \right) = \mathbf{0} \tag{33}$$

$$\mathbf{P}\_2 \mathbf{\bar{G}} \left( \ddot{\mathbf{e}} + \mathbf{K}\_D \dot{\mathbf{e}} + \mathbf{K}\_P \mathbf{e} \right) = \mathbf{0}. \tag{34}$$

The first *n* − *δ* equations are automatically satisfied for trajectories in *V* if the second system is satisfied. The second system, consisting of *δ* equations, governs the error dynamics in term of independent coordinates. For trajectories in *V* this is equivalent to the system of *δ* equations

$$\mathbf{G}\left(\ddot{\mathbf{e}}\_2 + \mathbf{F}^T \mathbf{K}\_\mathbf{D} \mathbf{F} \dot{\mathbf{e}}\_2 + \mathbf{F}^T \mathbf{K}\_\mathbf{P} \mathbf{F} \mathbf{e}\_2\right) = \mathbf{0} \tag{35}$$

t [s]

0 2 4 6 8 10

Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy 223

Fig. 10. Joint tracking error when the RA-PKM is controlled by a CTC model in redundant

<sup>2</sup> = (*q*2, *<sup>q</sup>*3) based on the infinity norm of the corresponding **<sup>A</sup>**1. That is, the submatrix **<sup>A</sup>**1,

Redundant actuation has the potential to improve the kinematic and dynamic performance of PKM. This redundancy is easily taken into account within the design process so to optimize dexterity. It turns out, however, that the control of RA-PKM poses several challenges. A problem that is peculiar to RA-PKM is the existence of antagonistic control forces. Such forces can be employed purposefully to avoid backlash or to modulate the EE stiffness, but impair the performance and stability of decentralized control schemes for RA-PKM. In this chapter the applicability of decentralized control schemes is analyzed, and it is shown that they inherently cause antagonistic control forces. As remedy a so-called antagonism filter is proposed that eliminates antagonistic control forces. Another problem arises in the model-based control of RA-PKM. Since the dynamics model is commonly formulated in terms of a set of independent actuator coordinates the model of the RA-PKM becomes invalid at the input singularities of the non-redundantly actuated PKM. This would limit the controllable motion range to that of the non-redundant PKM. To overcome this problem a formulation in redundant coordinates was presented that does not require the selection of independent actuator coordinates. This formulation is valid in the entire motion range. Thereupon an augmented PD and computed torque controller was proposed. Experimental results are reported for the planar 2 DOF RA-PKM that confirmed the robustness of these

The author is indebted to Timo Hufnagel from Heilbronn University who implemented the proposed control methods. Support from the company Schunk GmbH is gratefully

<sup>2</sup> , is selected for which ||**A**<sup>1</sup>||<sup>∞</sup> is minimal.

1e


e [rad] i

corresponding to a certain **q**(*i*)

coordinates.

**5. Summary**

control schemes.

acknowledged.

**6. Acknowledgment**

**q**(3)

0

0.005

0.01

0.015

<sup>3</sup>e <sup>2</sup>e

with positive definite **<sup>G</sup>** <sup>=</sup> **<sup>F</sup>***T***GF** in (7). The stability of the controller is ensured with positive definite gains.

#### **4.5 Case study: Planar 2 DOF RA-PKM**

The proposed CTC control scheme in redundant coordinates was implemented in a prototype of the planar 2 DOF PKM in figure 1, which is briefly described in section 3.3. As discussed in section 2.1 the dynamic model is given in terms of *n* = 6 joint angles, giving rise to a system of *n* = 6 motion equations (22). The projected 6 × 3 control matrix *M* has rank 2.

The manipulator is controlled along the EE-path in figure 4. If two actuator joint angles are used as independent coordinates the minimal coordinate model (6) exhibits parameterization singularities as described in section 4.2. Moreover the EE-path passes such singularities several times and the minimal coordinate model is not valid. The redundant coordinate formulation does not suffer from such singularities.

Fig. 9. Actuator torques when the RA-PKM is controlled along the EE-path of figure 4 by a CTC model in terms of redundant coordinates.

Figure 9 shows the actuator torques when the RA-PKM is controlled by the CTC scheme (32). Figure 10 shows the corresponding actuator tracking errors. Apparently the motion and the torque evolution is smooth and unaffected by any singularities thanks to the redundant coordinate formulation. Notice that also for this CTC method there are non-zero drive torques even if the RA-PKM is at rest. This is again due to measurement errors in conjunction with actuation redundancy. The crucial point in the inverse dynamics formulation (25) is the computation of the pseudoinverse (27). This requires identification of a full rank submatrix **A**1, i.e. *δ* actuator coordinates representing valid local coordinates on *V*. A straightforward implementation is to select one of the three combinations **q**(1) <sup>2</sup> = (*q*1, *<sup>q</sup>*2), **<sup>q</sup>**(2) <sup>2</sup> = (*q*1, *<sup>q</sup>*3), and

Fig. 10. Joint tracking error when the RA-PKM is controlled by a CTC model in redundant coordinates.

**q**(3) <sup>2</sup> = (*q*2, *<sup>q</sup>*3) based on the infinity norm of the corresponding **<sup>A</sup>**1. That is, the submatrix **<sup>A</sup>**1, corresponding to a certain **q**(*i*) <sup>2</sup> , is selected for which ||**A**<sup>1</sup>||<sup>∞</sup> is minimal.

### **5. Summary**

16 Will-be-set-by-IN-TECH

independent coordinates. For trajectories in *V* this is equivalent to the system of *δ* equations

**e**¨2 + **F***T***K**D**Fe**˙ <sup>2</sup> + **F***T***K**P**Fe**<sup>2</sup>

with positive definite **<sup>G</sup>** <sup>=</sup> **<sup>F</sup>***T***GF** in (7). The stability of the controller is ensured with positive

The proposed CTC control scheme in redundant coordinates was implemented in a prototype of the planar 2 DOF PKM in figure 1, which is briefly described in section 3.3. As discussed in section 2.1 the dynamic model is given in terms of *n* = 6 joint angles, giving rise to a system

The manipulator is controlled along the EE-path in figure 4. If two actuator joint angles are used as independent coordinates the minimal coordinate model (6) exhibits parameterization singularities as described in section 4.2. Moreover the EE-path passes such singularities several times and the minimal coordinate model is not valid. The redundant coordinate

0 2 4 6 8 10

2 c

Fig. 9. Actuator torques when the RA-PKM is controlled along the EE-path of figure 4 by a

Figure 9 shows the actuator torques when the RA-PKM is controlled by the CTC scheme (32). Figure 10 shows the corresponding actuator tracking errors. Apparently the motion and the torque evolution is smooth and unaffected by any singularities thanks to the redundant coordinate formulation. Notice that also for this CTC method there are non-zero drive torques even if the RA-PKM is at rest. This is again due to measurement errors in conjunction with actuation redundancy. The crucial point in the inverse dynamics formulation (25) is the computation of the pseudoinverse (27). This requires identification of a full rank submatrix **A**1, i.e. *δ* actuator coordinates representing valid local coordinates on *V*. A straightforward

1

c

3 c

<sup>2</sup> = (*q*1, *<sup>q</sup>*2), **<sup>q</sup>**(2)

<sup>2</sup> = (*q*1, *<sup>q</sup>*3), and

t [s]

of *n* = 6 motion equations (22). The projected 6 × 3 control matrix *M* has rank 2.

= **0** (35)

**G** 

**4.5 Case study: Planar 2 DOF RA-PKM**

formulation does not suffer from such singularities.


CTC model in terms of redundant coordinates.

implementation is to select one of the three combinations **q**(1)

c

[Nm] i

0

0.01

0.02

definite gains.

Redundant actuation has the potential to improve the kinematic and dynamic performance of PKM. This redundancy is easily taken into account within the design process so to optimize dexterity. It turns out, however, that the control of RA-PKM poses several challenges. A problem that is peculiar to RA-PKM is the existence of antagonistic control forces. Such forces can be employed purposefully to avoid backlash or to modulate the EE stiffness, but impair the performance and stability of decentralized control schemes for RA-PKM. In this chapter the applicability of decentralized control schemes is analyzed, and it is shown that they inherently cause antagonistic control forces. As remedy a so-called antagonism filter is proposed that eliminates antagonistic control forces. Another problem arises in the model-based control of RA-PKM. Since the dynamics model is commonly formulated in terms of a set of independent actuator coordinates the model of the RA-PKM becomes invalid at the input singularities of the non-redundantly actuated PKM. This would limit the controllable motion range to that of the non-redundant PKM. To overcome this problem a formulation in redundant coordinates was presented that does not require the selection of independent actuator coordinates. This formulation is valid in the entire motion range. Thereupon an augmented PD and computed torque controller was proposed. Experimental results are reported for the planar 2 DOF RA-PKM that confirmed the robustness of these control schemes.

### **6. Acknowledgment**

The author is indebted to Timo Hufnagel from Heilbronn University who implemented the proposed control methods. Support from the company Schunk GmbH is gratefully acknowledged.

Müller, A. (2010). Consequences of Geometric Imperfections for the Control of Redundantly Actuated Parallel Manipulators, *IEEE Trans. Robotics*, Vol. 26, No. 1, 2010, pp. 21-31 Müller, A.; T. Hufnagel (2011). A Projection Method for the Elimination of Contradicting

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## **Part 3**

## **Robust Control of Electromechanical Systems**

20 Will-be-set-by-IN-TECH

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mechanisms/robot linkage systems, *Proc. IEEE Int. Conf. Robotics Automation*,

**13** 

*1China 2France* 

**Robust High Order Sliding Mode Control of** 

**Permanent Magnet Synchronous Motors** 

Huangfu Yigeng1, S. Laghrouche2, Liu Weiguo1 and A. Miraoui2

Nonlinear system control has been widely concern of the research. At present, the nonlinear system decoupling control and static feedback linearization that based on the theory of differential geometry brought the research getting rid of limitation for local linearization and small scale motion. However, differential geometry control must depend on precise mathematical model. As a matter of fact, the control system usually is with parameters uncertainties and output disturbance. Considering sliding mode variable structure control with good robust, which was not sensitive for parameters perturbation and external disturbance, the combination idea of nonlinear system and sliding mode controls was obtained by reference to the large number of documents. Thus, it not only can improve system robustness but solve the difficulties problem of nonlinear sliding mode surface structure. As known to all, traditional sliding mode had a defect that is chattering phenomenon. A plenty of research papers focus on elimination/avoidance chattering by using different methods. By comparing, the chapter is concerned with novel design method for high order sliding mode control, which can eliminate chattering fundamentally. Especially, the approach and realization of nonlinear system high order sliding mode

High order sliding mode technique is the latest study. This chapter from the theory analysis to the simulation and experiment deeply study high order sliding mode control principle and its applications. The arbitrary order sliding mode controller is employed, whose relative

In addition, the control systems design is very often to differentiate the variables. Through the derivation of sliding mode, the expression of sliding mode differential value is obtained. At the same time, the differentiator for arbitrary sliding mode is given to avoiding complex numerical calculation. It not only remains the precision of variables differential value, but

Due to its inherent advantages, the permanent magnet synchronous motor (PMSM) deserves attention and is the most used drive in machine tool servos and modern speed control applications. For improving performance, this chapter will apply nonlinear high order sliding mode research achievement to MIMO permanent magnet synchronous motor. It changes the coupling nonlinear PMSM to single input single output (SISO) linear

**1. Introduction**

control is presented.

also obtains the robustness.

degree can equal any values instead of one.

*1Northwestern Polytechnical University* 

*2University of Technology of Belfort-Montbéliard* 

## **Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors**

Huangfu Yigeng1, S. Laghrouche2, Liu Weiguo1 and A. Miraoui2 *1Northwestern Polytechnical University* 

*2University of Technology of Belfort-Montbéliard 1China 2France* 

### **1. Introduction**

Nonlinear system control has been widely concern of the research. At present, the nonlinear system decoupling control and static feedback linearization that based on the theory of differential geometry brought the research getting rid of limitation for local linearization and small scale motion. However, differential geometry control must depend on precise mathematical model. As a matter of fact, the control system usually is with parameters uncertainties and output disturbance. Considering sliding mode variable structure control with good robust, which was not sensitive for parameters perturbation and external disturbance, the combination idea of nonlinear system and sliding mode controls was obtained by reference to the large number of documents. Thus, it not only can improve system robustness but solve the difficulties problem of nonlinear sliding mode surface structure. As known to all, traditional sliding mode had a defect that is chattering phenomenon. A plenty of research papers focus on elimination/avoidance chattering by using different methods. By comparing, the chapter is concerned with novel design method for high order sliding mode control, which can eliminate chattering fundamentally. Especially, the approach and realization of nonlinear system high order sliding mode control is presented.

High order sliding mode technique is the latest study. This chapter from the theory analysis to the simulation and experiment deeply study high order sliding mode control principle and its applications. The arbitrary order sliding mode controller is employed, whose relative degree can equal any values instead of one.

In addition, the control systems design is very often to differentiate the variables. Through the derivation of sliding mode, the expression of sliding mode differential value is obtained. At the same time, the differentiator for arbitrary sliding mode is given to avoiding complex numerical calculation. It not only remains the precision of variables differential value, but also obtains the robustness.

Due to its inherent advantages, the permanent magnet synchronous motor (PMSM) deserves attention and is the most used drive in machine tool servos and modern speed control applications. For improving performance, this chapter will apply nonlinear high order sliding mode research achievement to MIMO permanent magnet synchronous motor. It changes the coupling nonlinear PMSM to single input single output (SISO) linear

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 231

applied the sub-optimal algorithm (Bartolini et al., 1997, 1999). After the concept of high order sliding mode control was applied to bound operator in (Bartolini et al., 2000). Levant used high order sliding mode control in aircraft pitch control (Levant, 2000) as well as the exact robust differentiator (Levant, 1998). About the summary of high order sliding mode

In recent years, because arbitrary order sliding mode control technique not only retains the traditional sliding mode control simple structure with strong robustness, but also eliminates the chattering phenomenon in the traditional sliding mode, at the same time, gets rid of the constraints of system relative degree. Therefore theoretical research and engineering

Without losing generality, considering a state equation of single input nonlinear system as

() () ( ,) *x f x g x u*

Where, *<sup>n</sup> x R* is system state variable, *t* is time, *y* is output, *u* is control input. Here, *f* ( ) *x* , *g x*( ) and *s x*( ) are smooth functions. The control objective is making output function 0 *s* . Differentiate the output variables continuously, we can get every order derivative of *s*.

In arbitrary order sliding mode control, its core idea is the discrete function acts on a higher

Suppose the relative degree of system (1) equals to *r* , generally speaking, when the control input *u* first time appears in *r* -order derivative of *s*, that is ( ) 0 *<sup>r</sup> ds du* , then we take *r* order derivative of *s* for the output of system (1), ( 1) ,,, *<sup>r</sup> sss s* can be obtained. They are continuous function for all the *x* and *t* . However, corresponding discrete control law *u*

1 2 ( , , ) (,, ) *<sup>r</sup>*

Therefore, high order sliding mode control is transformed to stability of *r* order dynamic

*r* 1 ( )*r*

*b L L s ds du*

system (2), (4). Through the Lie derivative calculation, it is very easy to verify that

*g f r f*

*a Ls*

(1)

( 1) ( ,) ( ,) ( ,) ( ,) 0 *<sup>r</sup> sxt sxt sxt s xt* (2)

( 1)

*<sup>r</sup> y y y y ss s* (3)

( ) ( , ) ( , ) , ( , ) 0 *<sup>r</sup> s ayt bytu byt* (4)

(5)

applications has caused widespread concern and has been constant development.

*y sxt* 

According to the conception of system relative degree, there are two conditions.

ii. Relative degree 2 *r* , if ( ) 0 ( 1,2, 1) *<sup>i</sup> su i r* , and ( ) 0 *<sup>r</sup> s u*

control is also described in the literature (Fridman & Levant, 2002).

**2.1 Review of high order sliding mode control** 

i. Relative degree 1 *r* , if and only if *s u* 0

acts on ( )*<sup>r</sup> s* . Selecting a new local coordinate, then

So, the following expression can be obtained

order sliding mode surface, making

subsystem control problem instead of near equilibrium point linearization. Thereby, the problem of nonlinear and coupling for PMSM has been solved. In addition, Uncertainty nonlinear robust control system has been well-received study of attention. Because the robust control theory is essentially at the expense of certain performance. This kind of robust control strategy often limits bandwidth of closed loop, so that system tracking performance and robustness will be decreased. So, sliding mode control is an effective approach for improving system robust. This chapter first proposed a robust high order sliding mode controller for PMSM. The system has good position servo tracking precision in spite of parameters uncertainties and external torque disturbance.

On this basis, according to the principle of high order sliding mode, as well as differentiator, the state variables of PMSM are identified online firstly and successfully. The results of simulation indicate observe value has high precision when sliding mode variable and its differentials are convergent into zero. The same theory is used in external unknown torque disturbance estimation online for PMSM. As if, load torque will no longer be unknown disturbance. System performance can be improved greatly. It establishes theoretical foundation for the future applications.

At the end of chapter, using advanced half-physical platform controller dSPACE to drive a PMSM, hardware experiment implement is structured completely. The experiment results illustrate that PMSM adopting precious feedback linearization decoupling and high order sliding mode controller can realize system servo tracking control with good dynamic and steady character.

### **2. Robust high order sliding mode control**

As known to all, the sliding mode control with the strong robustness for the internal parameters and external disturbances. In addition, the appropriate sliding surface can be selected to reduce order for control system. However, due to the chattering phenomena of sliding mode control, the high frequency oscillation of control system brings challenge for the application of sliding mode control. On the other hand, the choice of sliding surface strictly requires system relative degree to equal to 1, which limits the choice of sliding surface.

In order to solve the above problems, this chapter focuses on a new type of sliding mode control, that is, higher order sliding mode control. The technology not only retains advantage of strong robustness in the traditional sliding mode control, but also enables discontinuous items transmit into the first order or higher order sliding mode derivative to eliminate the chattering. Besides, the design of the controller no longer must require relative degree to be 1. Therefore, it is greatly simplified to design parameters of sliding mode surface.

Emelyanov and others first time propose the concept of high order differentiation of sliding mode variable, but also provide a second order sliding mode twisting algorithm, and prove its convergence (Emelyanov et al., 1996). Another algorithm is super twisting, which can completely eliminate chattering (Emelyanov et al., 1990), although the relative degree of sliding mode variable is required to equal to 1. In the second order sliding mode control, Levant proved sliding mode accuracy is proportional to ( ) <sup>2</sup> *o* the square of the switching delay time. It has also become one of the merits of high order sliding mode control (Levant, 1993). Since then, the high order sliding mode controller has been developed and applied rapidly. For example, Bartolini and others propose a second order sliding mode control applied the sub-optimal algorithm (Bartolini et al., 1997, 1999). After the concept of high order sliding mode control was applied to bound operator in (Bartolini et al., 2000). Levant used high order sliding mode control in aircraft pitch control (Levant, 2000) as well as the exact robust differentiator (Levant, 1998). About the summary of high order sliding mode control is also described in the literature (Fridman & Levant, 2002).

### **2.1 Review of high order sliding mode control**

230 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

subsystem control problem instead of near equilibrium point linearization. Thereby, the problem of nonlinear and coupling for PMSM has been solved. In addition, Uncertainty nonlinear robust control system has been well-received study of attention. Because the robust control theory is essentially at the expense of certain performance. This kind of robust control strategy often limits bandwidth of closed loop, so that system tracking performance and robustness will be decreased. So, sliding mode control is an effective approach for improving system robust. This chapter first proposed a robust high order sliding mode controller for PMSM. The system has good position servo tracking precision in

On this basis, according to the principle of high order sliding mode, as well as differentiator, the state variables of PMSM are identified online firstly and successfully. The results of simulation indicate observe value has high precision when sliding mode variable and its differentials are convergent into zero. The same theory is used in external unknown torque disturbance estimation online for PMSM. As if, load torque will no longer be unknown disturbance. System performance can be improved greatly. It establishes theoretical

At the end of chapter, using advanced half-physical platform controller dSPACE to drive a PMSM, hardware experiment implement is structured completely. The experiment results illustrate that PMSM adopting precious feedback linearization decoupling and high order sliding mode controller can realize system servo tracking control with good dynamic and

As known to all, the sliding mode control with the strong robustness for the internal parameters and external disturbances. In addition, the appropriate sliding surface can be selected to reduce order for control system. However, due to the chattering phenomena of sliding mode control, the high frequency oscillation of control system brings challenge for the application of sliding mode control. On the other hand, the choice of sliding surface strictly requires system relative degree to equal to 1, which limits the choice of sliding

In order to solve the above problems, this chapter focuses on a new type of sliding mode control, that is, higher order sliding mode control. The technology not only retains advantage of strong robustness in the traditional sliding mode control, but also enables discontinuous items transmit into the first order or higher order sliding mode derivative to eliminate the chattering. Besides, the design of the controller no longer must require relative degree to be 1.

Emelyanov and others first time propose the concept of high order differentiation of sliding mode variable, but also provide a second order sliding mode twisting algorithm, and prove its convergence (Emelyanov et al., 1996). Another algorithm is super twisting, which can completely eliminate chattering (Emelyanov et al., 1990), although the relative degree of sliding mode variable is required to equal to 1. In the second order sliding mode control,

delay time. It has also become one of the merits of high order sliding mode control (Levant, 1993). Since then, the high order sliding mode controller has been developed and applied rapidly. For example, Bartolini and others propose a second order sliding mode control

<sup>2</sup> *o* 

the square of the switching

Therefore, it is greatly simplified to design parameters of sliding mode surface.

Levant proved sliding mode accuracy is proportional to ( )

spite of parameters uncertainties and external torque disturbance.

foundation for the future applications.

**2. Robust high order sliding mode control** 

steady character.

surface.

In recent years, because arbitrary order sliding mode control technique not only retains the traditional sliding mode control simple structure with strong robustness, but also eliminates the chattering phenomenon in the traditional sliding mode, at the same time, gets rid of the constraints of system relative degree. Therefore theoretical research and engineering applications has caused widespread concern and has been constant development.

Without losing generality, considering a state equation of single input nonlinear system as

$$\begin{aligned} \dot{\mathbf{x}} &= f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\mu \\ y &= \mathbf{s}(\mathbf{x}, t) \end{aligned} \tag{1}$$

Where, *<sup>n</sup> x R* is system state variable, *t* is time, *y* is output, *u* is control input. Here, *f* ( ) *x* , *g x*( ) and *s x*( ) are smooth functions. The control objective is making output function 0 *s* . Differentiate the output variables continuously, we can get every order derivative of *s*. According to the conception of system relative degree, there are two conditions.

i. Relative degree 1 *r* , if and only if *s u* 0

ii. Relative degree 2 *r* , if ( ) 0 ( 1,2, 1) *<sup>i</sup> su i r* , and ( ) 0 *<sup>r</sup> s u*

In arbitrary order sliding mode control, its core idea is the discrete function acts on a higher order sliding mode surface, making

$$\dot{s}\mathbf{s}(\mathbf{x},t) = \dot{\mathbf{s}}(\mathbf{x},t) = \ddot{\mathbf{s}}(\mathbf{x},t) = \cdots \mathbf{s}^{(r-1)}(\mathbf{x},t) = \mathbf{0} \tag{2}$$

Suppose the relative degree of system (1) equals to *r* , generally speaking, when the control input *u* first time appears in *r* -order derivative of *s*, that is ( ) 0 *<sup>r</sup> ds du* , then we take *r* order derivative of *s* for the output of system (1), ( 1) ,,, *<sup>r</sup> sss s* can be obtained. They are continuous function for all the *x* and *t* . However, corresponding discrete control law *u* acts on ( )*<sup>r</sup> s* . Selecting a new local coordinate, then

$$\mathbf{y} = (y\_1, y\_2, \cdots, y\_r) = (\mathbf{s}\_r \dot{\mathbf{s}}\_r \cdots \mathbf{s}^{(r-1)}) \tag{3}$$

So, the following expression can be obtained

$$s^{(r)} = a(y, t) + b(y, t)u, \quad b(y, t) \neq 0 \tag{4}$$

Therefore, high order sliding mode control is transformed to stability of *r* order dynamic system (2), (4). Through the Lie derivative calculation, it is very easy to verify that

$$\begin{aligned} b &= L\_g L\_f^{r-1} \mathbf{s} = d \mathbf{s}^{(r)} \Big/ \text{d}u \\ a &= L\_f^r \mathbf{s} \end{aligned} \tag{5}$$

Suppose 1 2 (,, ) *rr n yy y* , then

$$
\mu = \xi(t, s, \dot{s}, \dots, s^{(r-1)}, \eta) + \chi(t, s, \dot{s}, \dots, s^{(r-1)}, \eta)\mu \tag{6}
$$

Now, equation (3), (4) and (6) are transformed to Isidori-Brunowsky canonical form. The sliding mode equivalent control is ( ,) ( ,) *u a eq y t b y t* (Utkin,1992). At present, the aim of control is to design a discrete feedback control *u Uxt* ( ,) , so that new system converge into origin on the *r* order sliding mode surface within limited time. Therefore, in equation (4), both *ayt* ( ,) and *byt* ( ,) are bounded function. There are positive constants *Km* , *KM* and *C* so that

$$\begin{aligned} 0 &< K\_m \le b(y, t) \le K\_M\\ |a(y, t)| &\le C \end{aligned} \tag{7}$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 233

To get the differentiation of a given signal is always essential in automatic control systems. We often need derivative a variable or function. So there are a lot of numerical algorithms for this issue. The same situation also appears in the design of high order sliding mode controller (10) that needs to calculate the derivative values of sliding mode variable. In order to be able to accurately calculate, at the same time simplifying the algorithm, this chapter directly uses own advantages of high order sliding mode control due to high accuracy and robustness. We can design a high order sliding mode differentiator used to calculate the

Presentation above in the previous has been explained in detail the principles of high order sliding mode control and sliding mode controller design method. This part focuses on how to take use of high order sliding mode technique to solve the differentiation of a given signal

At this moment, according to the principle of sliding mode control, a proper controller is designed. When the system enter into sliding mode, *s x ft* () 0 . Derivative of sliding

() () *s x ft u ft* (14)

Because control input *u* first time appears in the derivative of sliding mode surface *s*, the relative degree of system is 1 *r* . It satisfies the requirement about relative degree of second order sliding mode. So the super twisting algorithm (Fridman & Levant, 2002) is adopted.

1/2

*u x ft x ft u*

Lipschitz constant about derivative of *f x*( ) . ( ( ), ( )) *t t* is the solution of equation of (16),

*C*

*C*

sgn( ( ))

*u x ft* 

1/2


 

2

 

 

2


are positive constant. Definite a function as ( , , ) | ( )|

<sup>1</sup> ( ), | | 0

<sup>1</sup> ( ), | | 0

1/2

1/2

*x u* (11)

*x ft* ( ) (12)

*s x ft* ( ) (13)

1

 

*C t* , *C* is

(15)

(16)

numerical derivative of the variables.

Therefore, sliding mode surface is selected as

mode surface (13),

Thus,

Where, 0 

 , 0 

the initial value are (0) 0 , (0) 1

or variable function. And their simulation results are verified Suppose given signal is *f t*( ) , now set a dynamic system as

1

The control object is to make the variable *x* follow given signal *f t*( ) , that is

**Theorem 1:** (Levant, 1998, 2003) Suppose the relative degree of nonlinear system (1) to output function *sxt* ( ,) is *r* , and satisfying the condition (7), the arbitrary order sliding mode controller has following expression

$$
\mu = -a \text{sgn}(\psi\_{r-1,r}(\mathbf{s}\_r \dot{\mathbf{s}}\_{r} \cdots \mathbf{s}^{(r-1)})) \tag{8}
$$

Where,

$$\begin{aligned} \boldsymbol{\wp}\_{0,r} &= \boldsymbol{s} \\ \boldsymbol{\wp}\_{1,r} &= \boldsymbol{s} + \beta\_1 \boldsymbol{N}\_{1,r} \operatorname{sgn}(\mathbf{s}) \\ \boldsymbol{\wp}\_{i,r} &= \mathbf{s}^{(r)} + \beta\_i \boldsymbol{N}\_{i,r} \operatorname{sgn}(\mathbf{y}\_{i-1,r}), \qquad i = \mathbf{1}\_r \cdots, r - 1 \\ \boldsymbol{N}\_{1,r} &= \|\,\,\mathbf{s}^{\left\|{r-1}\right\|/r} \\ \boldsymbol{N}\_{i,r} &= \{\,\,\|\,\,\mathbf{s}^{\left\|{r}\right\|/r} + \left\|\,\,\dot{\mathbf{s}}^{\left\|{r}\right\|} \boldsymbol{\wp}^{\left\|{r}/r - \dot{\mathbf{1}}\right\|} \}^{p/(r-i+1)}\boldsymbol{\wp}^{\left\|{r}/r - i+1\right\|}\}^{(r-i)/p} \quad \boldsymbol{i} = \mathbf{1}\_r \cdots \boldsymbol{\jmath}\_{r} \boldsymbol{r} - \mathbf{1} \\ \boldsymbol{N}\_{r-1,r} &= \{\,\|\,\,\mathbf{s}^{\left\|{r}\right\|^{p/r} + \left\|\,\,\mathbf{s}^{\left\|{r}\right\|^{p}} \boldsymbol{\wp}^{\left\|{r}/r - \mathbf{1}\right\|} \}^{p/2} \}^{1/p} \end{aligned} \tag{9}$$

Properly choose positive parameters 12 1 , , *<sup>r</sup>* , the system converge into origin on the *r* order sliding mode surface within limited time. Finally, when 0 *s* , it achieves control object. The choice of positive parameters 12 1 , , *<sup>r</sup>* is not unique. Here, 4 *r* order sliding mode controller is given, which is also tested.

$$\begin{aligned} \text{1. } u &= -\alpha \text{ sgn(s)}\\ \text{2. } u &= -\alpha \text{ sgn(s)} + |s|^{1/2} \text{ sgn(s)}\\ \text{3. } u &= -\alpha \text{ sgn(\ddot{s} + 2\{|\dot{s}|^{\frac{3}{2}} + |s|^{2}\}^{1/6} \text{sgn(\dot{s} + |s|^{2/3} \text{sgn(s)})})} \\ \text{4. } u &= -\alpha \text{ sgn}[\ddot{s} + 3\{\ddot{s}\}^{6} + (\dot{s})^{4} + |s|^{3}]^{1/12} \text{sgn}[\ddot{s} + (\dot{s})^{4} \\ &+ |s|^{3}\}^{1/6} \text{sgn(\dot{s} + 0.5 \mid s\}^{3/4} \text{sgn(s)})]) \\ &\vdots \end{aligned} \tag{10}$$

From the above equation (10) we can also see that, when 1 *r* , the controller is traditional relay sliding mode control; when 2 *r* , in fact, the controller is super twisting algorithm of second order sliding mode.

To get the differentiation of a given signal is always essential in automatic control systems. We often need derivative a variable or function. So there are a lot of numerical algorithms for this issue. The same situation also appears in the design of high order sliding mode controller (10) that needs to calculate the derivative values of sliding mode variable. In order to be able to accurately calculate, at the same time simplifying the algorithm, this chapter directly uses own advantages of high order sliding mode control due to high accuracy and robustness. We can design a high order sliding mode differentiator used to calculate the numerical derivative of the variables.

Presentation above in the previous has been explained in detail the principles of high order sliding mode control and sliding mode controller design method. This part focuses on how to take use of high order sliding mode technique to solve the differentiation of a given signal or variable function. And their simulation results are verified

Suppose given signal is *f t*( ) , now set a dynamic system as

$$
\dot{\mathbf{x}} = \mathbf{u} \tag{11}
$$

The control object is to make the variable *x* follow given signal *f t*( ) , that is

$$\mathbf{x} = f(t) \tag{12}$$

Therefore, sliding mode surface is selected as

232 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

( 1) ( 1) (,,, , , ) (,,, , , ) *r r*

 *tss s tss s u* 

Now, equation (3), (4) and (6) are transformed to Isidori-Brunowsky canonical form. The sliding mode equivalent control is ( ,) ( ,) *u a eq y t b y t* (Utkin,1992). At present, the aim of control is to design a discrete feedback control *u Uxt* ( ,) , so that new system converge into origin on the *r* order sliding mode surface within limited time. Therefore, in equation (4), both *ayt* ( ,) and *byt* ( ,) are bounded function. There are positive constants *Km* , *KM* and *C*

0 ( ,)

*K byt K m M*

**Theorem 1:** (Levant, 1998, 2003) Suppose the relative degree of nonlinear system (1) to output function *sxt* ( ,) is *r* , and satisfying the condition (7), the arbitrary order sliding

1, sgn( ( , , , )) *<sup>r</sup> <sup>u</sup> r r*

/ /( 1) ( 1) /( 1) ( )/

*Ns s s i r*

*pr p r i p ri ri p*

/2 1/ ) *<sup>p</sup>*

sgn( ), 1, , 1

(| | | | | | ) 1, , 1

 

3 2 1/6 2/3 6 4 3 1/12 4

 

*r* order sliding mode surface within limited time. Finally, when 0 *s* , it achieves control

3. sgn( 2(| | | | ) sgn( | | sgn( ))) 4. sgn{ 3[( ) ( ) | | ] sgn[ (( )

From the above equation (10) we can also see that, when 1 *r* , the controller is traditional relay sliding mode control; when 2 *r* , in fact, the controller is super twisting algorithm of

*u s s s ss s u s s s s ss*

 


sgn( )

/ /( 1) ( 2)

*pr p r r p*

 

1/2

3 1/6 3/4


*s ss s*

2. sgn( | | sgn( ))

*u ss s*

*s N i r*

(| | | | | |

*ayt C*

  (7)

*<sup>r</sup>* , the system converge into origin on the

*<sup>r</sup>* is not unique. Here, 4 *r* order

(9)

(10)

(6)

( 1)

*ss s* (8)

Suppose 1 2 (,, ) *rr n* 

so that

Where,

*yy y* , then

mode controller has following expression

0,

*r*

1,

*r*

*N s*

,

*i r*

1,

*r r*

second order sliding mode.

1, 1 1, ( )

*s*

 

*r r r*

 


Properly choose positive parameters 12 1 , ,

, , 1, ( 1)/

*N ss s*

object. The choice of positive parameters 12 1 , ,

sliding mode controller is given, which is also tested.

1. sgn( )

*u s*

*sN s*

*i r i ir i r r r*

 

$$s = \mathbf{x} - f(t) \tag{13}$$

At this moment, according to the principle of sliding mode control, a proper controller is designed. When the system enter into sliding mode, *s x ft* () 0 . Derivative of sliding mode surface (13),

$$
\dot{s} = \dot{\mathbf{x}} - \dot{f}(t) = \boldsymbol{\mu} - \dot{f}(t) \tag{14}
$$

Because control input *u* first time appears in the derivative of sliding mode surface *s*, the relative degree of system is 1 *r* . It satisfies the requirement about relative degree of second order sliding mode. So the super twisting algorithm (Fridman & Levant, 2002) is adopted. Thus,

$$\begin{aligned} \dot{u} &= -\mathcal{A} \|\mathbf{x} - f(t)\|^{1/2} \operatorname{sgn}(\mathbf{x} - f(t)) + u\_1 \\ \dot{u}\_1 &= -\alpha \operatorname{sgn}(\mathbf{x} - f(t)) \end{aligned} \tag{15}$$

Where, 0 , 0 are positive constant. Definite a function as ( , , ) | ( )| *C t* , *C* is Lipschitz constant about derivative of *f x*( ) . ( ( ), ( )) *t t* is the solution of equation of (16), the initial value are (0) 0 , (0) 1

$$\begin{aligned} \dot{\Sigma} &= -\left| \begin{array}{c} \Sigma \end{array} \right|^{1/2} + \Psi \\ \dot{\Psi} &= \begin{cases} -\frac{1}{\lambda^2} \{\alpha - \mathcal{C}\}, & -\left| \begin{array}{c} \Sigma \end{array} \right|^{1/2} + \Psi > 0 \\ -\frac{1}{\lambda^2} \{\alpha + \mathcal{C}\}, & -\left| \begin{array}{c} \Sigma \end{array} \right|^{1/2} + \Psi \le 0 \end{cases} \end{aligned} \tag{16}$$

**Theorem 2:** (Levant, 1998) Let 0 *C* , 0 , function (,,) 1 *C* . Then, provided *f* ( )*t* has a derivative with Lipschitz's constant *C* , the equality ( ) *u ft* is fulfilled identically after finite time transient process. And the smaller value of , faster convergence; If (,,) 1 *C* , control input *u* will not converge into ( ) *f t* . Observer parameters should meet the following sufficient condition for convergence of the secondorder sliding mode control,

$$a > \mathbb{C}$$

$$\lambda^2 \ge 4\mathbb{C} \frac{a+\mathbb{C}}{a-\mathbb{C}} \tag{17}$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 235

recorded as 0 *L* . Now, the object of sliding mode differentiator is estimating the value of

Arbitrary order sliding mode differentiator has the following recursive form as equation

It can be verified, When 1 *n* , it is first order differentiator. Suppose 0*f* ( )*t* is basic value of

equalities are true in the absence of input noise after a finite time of a transient process.

( ) 0

**Theorem 5:** (Levant, 2003) Let the input noise satisfy the inequality 0

( ) ( 1)/( 1)

*i ni n*

 

0

*z s*

*i*

The sliding mode controller (8) can be rewritten by

*i ni n*

 


*zft i n vf t i n*

By Theorem 5, we can see that the arbitrary order sliding mode differentiator has

The arbitrary order sliding mode differentiator can accurately estimate any order derivative of a given input. If this differentiator can be used in high order sliding mode controller (10), any order derivative of sliding mode variable *s* can be accurately estimated avoiding the complicated calculation, which greatly simplifies the controller design. Adopting the differentiator, consider *s t*( ) in high order sliding mode controller as given input for differentiator. Then the output of differentiator (0 ) *<sup>i</sup> z in* can substitute any order

( ) 1, *<sup>i</sup>*

*zs i n*

1, 0 1 ( 1) sgn( ( , , , )) *u r r <sup>r</sup>*

The expression from this controller can also be clearly seen, with high order sliding mode differentiator, the differentiation of arbitrary order sliding mode variable will not be difficult to solve, which makes the high order sliding mode controller design has been

( 1) ( )/( 1)

 

Then the following inequality are established in finite time for some positive constants

*z v f ti n*

The theorem 4 illustrates that arbitrary order sliding mode differentiator can use differentiation (0 ) *<sup>i</sup> z in* to estimate any order derivative of input function *f t*( ) online

( )

**Theorem 4:** (Levant, 2003) If properly choose parameter (0 ) *<sup>i</sup>*

0 0

*z ft*

*i i*

depending exclusively on the parameters of the differentiator.

*i i*

*i i*

0

0

( )*t* is uncertain part, but bounded, satisfying | ( )|

( ), 1, , *<sup>i</sup>*

(21)

(23)

*zz z* (24)

 *t* , then

*i n* , the following

( ) | ( ) ( )| *t ft f t*

(22)

.

*i* ,

( ) ( ), ( ), , ( ) *<sup>n</sup> f t f t f t* in real time.

(20).

*i* 

robustness.

derivative of *s t*( ) , that is

simplified greatly.

given signal *f t*( ) ,

within limited time.

.

<sup>0</sup> *f* () () () *t ft t*

According to the principle of second order sliding mode, after a finite time, the system will converge into the origin, that is,

$$\mathbf{s}(\mathbf{x},t) = \dot{\mathbf{s}}(\mathbf{x},t) = \mathbf{0} \tag{18}$$

Then,

$$
\mu = \dot{f}(t) \tag{19}
$$

Now, observer input *u* is the estimation of derivative of given signal *f t*( ) . Using a sliding mode controller achieve differentiation of variable function.

Let input signal be presented in the form 0 *f* () () () *t f t nt* , where 0*f* ( )*t* is a differentiable base signal, 0*f* ( )*t* has a derivative with Lipschitz's constant 0 *C* , and *n t*( ) is a noise, | ( )| *n t* . Then, there exists such a constant 0 *b* depend on <sup>2</sup> ( )/ *C* and <sup>2</sup> ( )/ *C* that after a finite time, the inequality 1/2 | ( ) ( )| *ut f t b* <sup>0</sup> holds. (Levant, 1998)

Through the first order sliding mode differentiator description of the working principle, it will naturally think, whether can design a sliding mode differentiator to obtain the arbitrary order derivative of given signal. Well, the design of high order sliding mode controller (10) needs to know all sliding mode variables and their corresponding differentiation.

**Theorem 3:** Design an arbitrary order sliding mode differentiator, which can be used to estimate the derivative value of sliding mode variables, so as to achieve a simplified numerical differential purposes as following.

$$\begin{aligned} \dot{z}\_0 &= v\_0\\ v\_0 &= -\dot{\lambda}\_0 \left| z\_0 - f(t) \right|^{n/(n+1)} \operatorname{sgn}(z\_0 - f(t)) + z\_1\\ \dot{z}\_1 &= v\_1\\ v\_1 &= -\dot{\lambda}\_1 \left| z\_1 - v\_0 \right|^{(n-1)/n} \operatorname{sgn}(z\_1 - v\_0) + z\_{2'}\\ &\vdots\\ \dot{z}\_{n-1} &= v\_{n-1}\\ v\_{n-1} &= -\dot{\lambda}\_{n-1} \left| z\_{n-1} - v\_{n-2} \right|^{1/2} \operatorname{sgn}(z\_{n-1} - v\_{n-2}) + z\_{n'}\\ \dot{z}\_n &= -\dot{\lambda}\_n \operatorname{sgn}(z\_n - v\_{n-1}) \end{aligned} \tag{20}$$

The same with first order sliding mode differentiator, suppose given signal is *f* ( )*t* , *t* [0, ) . It has been known that the *n* order derivative of *f t*( ) has Lipschitz constant, 234 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

*f* ( )*t* has a derivative with Lipschitz's constant *C* , the equality ( ) *u ft* is fulfilled identically after finite time transient process. And the smaller value of , faster

parameters should meet the following sufficient condition for convergence of the second-

*<sup>C</sup> <sup>C</sup> C*

 

According to the principle of second order sliding mode, after a finite time, the system will

() *u ft* (19)

Now, observer input *u* is the estimation of derivative of given signal *f t*( ) . Using a sliding

Let input signal be presented in the form 0 *f* () () () *t f t nt* , where 0*f* ( )*t* is a differentiable base signal, 0*f* ( )*t* has a derivative with Lipschitz's constant 0 *C* , and *n t*( ) is a noise,

Through the first order sliding mode differentiator description of the working principle, it will naturally think, whether can design a sliding mode differentiator to obtain the arbitrary order derivative of given signal. Well, the design of high order sliding mode controller (10)

**Theorem 3:** Design an arbitrary order sliding mode differentiator, which can be used to estimate the derivative value of sliding mode variables, so as to achieve a simplified

> /( 1) 0 00 0 1

*n n*

*v z ft z ft z*


1/2 1 11 2 1 2 1

*n nn n n n n*

The same with first order sliding mode differentiator, suppose given signal is *f* ( )*t* , *t* [0, ) . It has been known that the *n* order derivative of *f t*( ) has Lipschitz constant,

*v zv zv z*

 



( 1)/ 1 11 0 10 2

*v zv zv z*

*n n*

sgn( )

*n n nn*

*z zv*

 

. Then, there exists such a constant 0 *b* depend on <sup>2</sup> ( )/

needs to know all sliding mode variables and their corresponding differentiation.

<sup>2</sup> 4

*C*

, function (,,) 1

*C* , control input *u* will not converge into ( ) *f t* . Observer

 

*sxt sxt* ( ,) ( ,) 0 (18)

*C*

holds. (Levant, 1998)

 and <sup>2</sup> ( )/ *C*

(20)

*C* . Then, provided

(17)

 *C* , 0 

**Theorem 2:** (Levant, 1998) Let 0

 

convergence; If (,,) 1

order sliding mode control,

converge into the origin, that is,

Then,


that after a finite time, the inequality 1/2 | ( ) ( )| *ut f t b* <sup>0</sup>

0 0

*z v*

1 1

*z v*

1 1

*n n*

*z v*

numerical differential purposes as following.

recorded as 0 *L* . Now, the object of sliding mode differentiator is estimating the value of ( ) ( ), ( ), , ( ) *<sup>n</sup> f t f t f t* in real time.

Arbitrary order sliding mode differentiator has the following recursive form as equation (20).

It can be verified, When 1 *n* , it is first order differentiator. Suppose 0*f* ( )*t* is basic value of given signal *f t*( ) , ( )*t* is uncertain part, but bounded, satisfying | ( )| *t* , then <sup>0</sup> *f* () () () *t ft t* .

**Theorem 4:** (Levant, 2003) If properly choose parameter (0 ) *<sup>i</sup> i n* , the following equalities are true in the absence of input noise after a finite time of a transient process.

$$\begin{aligned} z\_0 &= f\_0(t) \\ z\_i &= \upsilon\_i = f\_0^{(i)}(t), \quad i = 1, \dots, n \end{aligned} \tag{21}$$

The theorem 4 illustrates that arbitrary order sliding mode differentiator can use differentiation (0 ) *<sup>i</sup> z in* to estimate any order derivative of input function *f t*( ) online within limited time.

**Theorem 5:** (Levant, 2003) Let the input noise satisfy the inequality 0 ( ) | ( ) ( )| *t ft f t* . Then the following inequality are established in finite time for some positive constants *i* , *i* depending exclusively on the parameters of the differentiator.

$$\begin{aligned} |z\_i - f\_0^{(i)}(t)| &\le \mu\_i \varepsilon^{(n-i+1)/(n+1)} & i = 0, \cdots, n\\ |v\_i - f\_0^{(i+1)}(t)| &\le \tau\_i \varepsilon^{(n-i)/(n+1)} & i = 0, \cdots, n-1 \end{aligned} \tag{22}$$

By Theorem 5, we can see that the arbitrary order sliding mode differentiator has robustness.

The arbitrary order sliding mode differentiator can accurately estimate any order derivative of a given input. If this differentiator can be used in high order sliding mode controller (10), any order derivative of sliding mode variable *s* can be accurately estimated avoiding the complicated calculation, which greatly simplifies the controller design. Adopting the differentiator, consider *s t*( ) in high order sliding mode controller as given input for differentiator. Then the output of differentiator (0 ) *<sup>i</sup> z in* can substitute any order derivative of *s t*( ) , that is

$$\begin{aligned} z\_0 &= s\\ z\_i &= s^{(i)} \quad i = 1, \cdots n \end{aligned} \tag{23}$$

The sliding mode controller (8) can be rewritten by

$$
\mu = -\alpha \text{sgn}(\psi\_{r-1,r}(z\_0, z\_1, \dots, z\_{(r-1)})) \tag{24}
$$

The expression from this controller can also be clearly seen, with high order sliding mode differentiator, the differentiation of arbitrary order sliding mode variable will not be difficult to solve, which makes the high order sliding mode controller design has been simplified greatly.

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 237

*d dd f q qq*

*q q*

*em l*

Where: *ud* , *uq* are d-q axis stator voltage; *di* , *qi* are d-q axis stator current; *Ld* , *Lq* are d-q axis stator inductance, as *L L <sup>d</sup> <sup>q</sup>* , motor is non-salient pole; as *L L <sup>d</sup> <sup>q</sup>* , motor is salient pole;

number of motor pole pairs; *Tem* is electromagnetic torque; *Tl* is load torque; is motor's

Set of equations (25), (26), (27) and (28), we can get the state equation expression of PMSM

[( ) ]

*P B <sup>T</sup> <sup>d</sup> L Li i J J J dt*

*di <sup>L</sup> <sup>R</sup> P Pi i u dt L L LL*

 *P*

[( ) ]

*dt P B <sup>T</sup> L Li i <sup>d</sup> <sup>J</sup> J J*

*d d qd*

*di iP i u*

*d qd fq*

*<sup>f</sup> <sup>d</sup> <sup>q</sup> dq q q q qq*

*di <sup>L</sup> <sup>R</sup> P Pi i u L L LL dt*

*q*

*d dd*

*L LL*

 

*d qd fq*

is motor's electrical angular velocity; *R* is stator phase resistance; *P* is

1

 

*dq q*

1

 

1

*l*

*d qd d dd*

*iP i u*

*q q qq*

0

0 is rotor initial angular position. Considering position control, equation (29) can

 

*<sup>e</sup>* is the electrical angle between rotor axis and stator *A* phase axis,

1

*<sup>e</sup>* , and following equality is set up.

(30)

*l*

(29)

is

(31)

*J T TB*

(26)

*d d <sup>q</sup> <sup>d</sup> q fq* (27)

(28)

*<sup>f</sup>* is magnetic potential generated by permanent

; *J* is total inertia of rotor and load; *B* is viscous

 

*L i L i*

*d*

*dt*

*q d*

*dt LLL*

*di R L*

*<sup>q</sup> <sup>f</sup> <sup>d</sup>*

*dt*

*d P*

*dt <sup>R</sup> <sup>L</sup>*

 

*dt*

 

 ( ) [( ) ] *T P i i P L L ii i em d* 

The flux linkage equation:

The motor motion equation:

 *<sup>d</sup>* ,

magnet rotor;

as following.

Suppose

Where,

be rewritten by

friction coefficient.

The electromagnetic torque equation:

*<sup>q</sup>* are d-q axis stator flux linkage;

mechanical angular velocity, with *P*

mechanical angular position of motor, with

### **2.2 Applications for permanent magnet synchronous motor**

Permanent magnet synchronous motors (PMSM) are receiving increased attention for electric drive applications due to their high power density, large torque to inertia ratio and high efficiency over other kinds of motors (Glumineau et al, 1993; Ziribi et al, 2001; Caravani et al, 1998).

But the dynamic model of a PMSM is highly nonlinear because of the coupling between the motor speed and the electrical quantities, such as the d, q axis currents. In last years, many different control algorithms have been used to improve the performance of the magnet motor. For example, as the dynamic model of the machine is nonlinear, a natural approach is the exact feedback linearization control method, by which the original nonlinear model can be transformed into a linear model through proper coordinate transformation. However, in general, the dynamics of the synchronous motors may not be fully known, since some of parameters appearing in the equations will vary. For instance, the resistance and inductance will be changed when the temperature alters. As a consequence, nonlinearities can only be partially cancelled by the feedback linearization technique, and parameters uncertainties act on the equations of the motion. Then an important aim of the control design is to develop a robust controller which ensures good dynamic performances in spite of parameters uncertainties and perturbation.

The sliding mode control is known to be a robust approach to solve the control problems of nonlinear systems. Robustness properties against various kinds of uncertainties such as parameter perturbations and external disturbances can be guaranteed. However, this control strategy has a main drawback: the well known chattering phenomenon. In order to reduce the chattering, the sign function can be replaced by a smooth approximation. However, this technique induces deterioration in accuracy and robustness. In last decade, another approach called higher order sliding mode (HOSM) has been proposed and developed. It is the generalization of classical sliding mode control and can be applied to control systems with arbitrary relative degree r respecting to the considered output. In HOSM control, the main objective is to obtain a finite time convergence in the non empty manifold ( 1) { | 0} *<sup>r</sup> S x Xs s s s* by acting discontinuously on r order derivatives of the sliding variable s. Advantageous properties of HOSM are: the chattering effect is eliminated, higher order precision is provided whereas all the qualities of standard sliding mode are kept, and control law is not limited by relative degree of the output.

The common analysis of permanent magnet synchronous motor is d-q axis mathematical model. It can be used to analyze not only the permanent magnet synchronous motor steady state operating characteristics, but also can be used to analyze the transient performance motor. In order to establish sinusoidal PMSM d-q axis mathematical model, firstly assume:

i. Motor core saturation neglected;

ii. Excluding the eddy current and magnetic hysteresis loss of motor;

iii. The motor current is symmetrical three phase sine wave current.

Thereby, the following voltage, flux linkage, electromagnetic torque and mechanical motion equations can be obtained, where all the values in equations are transient.

The voltage equation:

$$\begin{aligned} \mu\_d &= \frac{d\nu\_d}{dt} - \alpha \nu \nu\_q + \text{Ri}\_d\\ \mu\_q &= \frac{d\nu\_q}{dt} + \alpha \nu \nu\_d + \text{Ri}\_q \end{aligned} \tag{25}$$

The flux linkage equation:

236 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Permanent magnet synchronous motors (PMSM) are receiving increased attention for electric drive applications due to their high power density, large torque to inertia ratio and high efficiency over other kinds of motors (Glumineau et al, 1993; Ziribi et al, 2001; Caravani

But the dynamic model of a PMSM is highly nonlinear because of the coupling between the motor speed and the electrical quantities, such as the d, q axis currents. In last years, many different control algorithms have been used to improve the performance of the magnet motor. For example, as the dynamic model of the machine is nonlinear, a natural approach is the exact feedback linearization control method, by which the original nonlinear model can be transformed into a linear model through proper coordinate transformation. However, in general, the dynamics of the synchronous motors may not be fully known, since some of parameters appearing in the equations will vary. For instance, the resistance and inductance will be changed when the temperature alters. As a consequence, nonlinearities can only be partially cancelled by the feedback linearization technique, and parameters uncertainties act on the equations of the motion. Then an important aim of the control design is to develop a robust controller which ensures good dynamic performances

The sliding mode control is known to be a robust approach to solve the control problems of nonlinear systems. Robustness properties against various kinds of uncertainties such as parameter perturbations and external disturbances can be guaranteed. However, this control strategy has a main drawback: the well known chattering phenomenon. In order to reduce the chattering, the sign function can be replaced by a smooth approximation. However, this technique induces deterioration in accuracy and robustness. In last decade, another approach called higher order sliding mode (HOSM) has been proposed and developed. It is the generalization of classical sliding mode control and can be applied to control systems with arbitrary relative degree r respecting to the considered output. In HOSM control, the main objective is to obtain a finite time convergence in the non empty manifold ( 1) { | 0} *<sup>r</sup> S x Xs s s s* by acting discontinuously on r order derivatives of the sliding variable s. Advantageous properties of HOSM are: the chattering effect is eliminated, higher order precision is provided whereas all the qualities of standard sliding mode are

The common analysis of permanent magnet synchronous motor is d-q axis mathematical model. It can be used to analyze not only the permanent magnet synchronous motor steady state operating characteristics, but also can be used to analyze the transient performance motor. In order to establish sinusoidal PMSM d-q axis mathematical model, firstly assume:

Thereby, the following voltage, flux linkage, electromagnetic torque and mechanical motion

*d d q d*

*d u Ri dt d*

*q q d q*

*u Ri dt*

(25)

**2.2 Applications for permanent magnet synchronous motor** 

in spite of parameters uncertainties and perturbation.

kept, and control law is not limited by relative degree of the output.

ii. Excluding the eddy current and magnetic hysteresis loss of motor; iii. The motor current is symmetrical three phase sine wave current.

equations can be obtained, where all the values in equations are transient.

i. Motor core saturation neglected;

The voltage equation:

et al, 1998).

$$\begin{aligned} \, \nu \nu\_d &= \mathbf{L}\_d \mathbf{i}\_d + \mathcal{W}\_f \\ \, \nu \nu\_q &= \mathbf{L}\_q \mathbf{i}\_q \end{aligned} \tag{26}$$

The electromagnetic torque equation:

$$T\_{em} = P(\nu\_d i\_q - \nu\_q i\_d) = P[(L\_d - L\_q)i\_d i\_q + \nu\_f i\_q] \tag{27}$$

The motor motion equation:

$$J\frac{d\Omega}{dt} = T\_{em} - T\_l - B\Omega\tag{28}$$

Where: *ud* , *uq* are d-q axis stator voltage; *di* , *qi* are d-q axis stator current; *Ld* , *Lq* are d-q axis stator inductance, as *L L <sup>d</sup> <sup>q</sup>* , motor is non-salient pole; as *L L <sup>d</sup> <sup>q</sup>* , motor is salient pole; *<sup>d</sup>* , *<sup>q</sup>* are d-q axis stator flux linkage; *<sup>f</sup>* is magnetic potential generated by permanent magnet rotor; is motor's electrical angular velocity; *R* is stator phase resistance; *P* is number of motor pole pairs; *Tem* is electromagnetic torque; *Tl* is load torque; is motor's mechanical angular velocity, with *P* ; *J* is total inertia of rotor and load; *B* is viscous friction coefficient.

Set of equations (25), (26), (27) and (28), we can get the state equation expression of PMSM as following.

$$\begin{bmatrix} \frac{d\boldsymbol{\alpha}}{dt} \\ \frac{d\boldsymbol{i}\_d}{dt} \\ \frac{d\boldsymbol{i}\_q}{dt} \\ \frac{d\boldsymbol{i}\_q}{dt} \end{bmatrix} = \begin{bmatrix} \frac{P}{I} [(L\_d - L\_q)\boldsymbol{i}\_d + \boldsymbol{\nu}\boldsymbol{\nu}\_f] \mathbf{i}\_q - \frac{B}{I} \boldsymbol{\alpha} - \frac{T\_l}{I} \\ -\frac{R}{L\_d} \boldsymbol{i}\_d + P \frac{L\_q}{L\_d} \boldsymbol{\alpha} \mathbf{i}\_q + \frac{\mathbf{1}}{L\_d} \boldsymbol{\mu}\_d \\ -P \frac{\boldsymbol{\nu}\boldsymbol{\nu}\_f}{L\_q} \boldsymbol{\alpha} - P \frac{L\_d}{L\_q} \boldsymbol{\alpha} \mathbf{i}\_d - \frac{R}{L\_q} \boldsymbol{i}\_q + \frac{\mathbf{1}}{L\_q} \boldsymbol{\mu}\_q \end{bmatrix} \tag{29}$$

Suppose *<sup>e</sup>* is the electrical angle between rotor axis and stator *A* phase axis, is mechanical angular position of motor, with *P<sup>e</sup>* , and following equality is set up.

$$
\partial \!\!\/= \left[\Omega dt + \theta\_0\right] \tag{30}
$$

Where, 0 is rotor initial angular position. Considering position control, equation (29) can be rewritten by

$$\begin{aligned} \left[ \frac{d\theta}{dt} \right] &= \left[ \frac{P\alpha}{I} \right] = \left[ \frac{P\alpha}{I} \right] \left[ (L\_d - L\_q)i\_d + \nu\_f \right] \mathbf{i}\_q - \frac{B}{I} \alpha - \frac{T\_l}{I} \right] \\ \left[ \frac{d\mathbf{i}\_d}{dt} \right] &= \left[ -\frac{R}{L\_d} \mathbf{i}\_d + P \frac{L\_q}{L\_d} \alpha \mathbf{i}\_q + \frac{1}{L\_d} \mathbf{u}\_d \right] \\ \left[ \frac{d\mathbf{i}\_q}{dt} \right] &- P \frac{\nu\_f}{L\_q} \alpha - P \frac{L\_d}{L\_q} \alpha \mathbf{i}\_d - \frac{R}{L\_q} \mathbf{i}\_q + \frac{1}{L\_q} \mathbf{u}\_q \end{aligned} \tag{31}$$

From the equation (31) we can see that PMSM is a multi-variable, coupling, nonlinear time varying systems. In addition, the variables in d-q axis can be changed to three phase *abc* axis by coordinate transformation.

### **2.2.1 Robust control for PMSM**

This section will use the high order sliding mode control algorithm with differentiator, in spite of system parameter uncertainties, external disturbances and other factors, to design a robust controller for nonlinear multi-input multi-output permanent magnet synchronous motor. The advantage of this controller is the elimination of the chattering in standard sliding mode. At the same time, it is still with precision and robustness of the standard sliding mode control. And its control law no longer subjects to relative degree constraints.

Firstly, let *x* denotes the motor state variable 1234 [ , , , ] [, , , ] *T T d q x xxxx ii* , and control input 1 2 [,] [,] *T T u uu uu d q* . The parameters *<sup>R</sup>* , *Ld* , *Lq* and *B* are considered as uncertain parameters, such as *R* will change with the temperature rise of the synchronous motor. Therefore, use *R*<sup>0</sup> , *Ld*<sup>0</sup> , *Lq*0 and *B*0 to express their nominal value part of *R* , *Ld* , *Lq* and *B* respectively.

$$\begin{aligned} k\_1 &= k\_{01} + \delta k\_1 = P(L\_d - L\_q) / f \\ k\_2 &= k\_{02} + \delta k\_2 = P\nu\_f / f \\ k\_3 &= k\_{03} + \delta k\_3 = -B \ / f \\ k\_4 &= k\_{04} + \delta k\_4 = -R \ / L\_d \\ k\_5 &= k\_{05} + \delta k\_5 = P L\_q / L\_d \\ k\_6 &= k\_{06} + \delta k\_6 = 1 / L\_d \\ k\_7 &= k\_{07} + \delta k\_7 = -P\nu\_f / L\_q \\ k\_8 &= k\_{08} + \delta k\_8 = -P L\_d / L\_q \\ k\_9 &= k\_{09} + \delta k\_9 = -R \ / L\_q \\ k\_{10} &= k\_{010} + \delta k\_{10} = 1 / L\_q \end{aligned} \tag{32}$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 239

The aim is to design an appropriate control which guarantees robust performance in presence of parameters and load variations. The control objective is double aspect. First, the

Second, the nonlinear electromagnetic torque must be linearized to avoid reluctance effects and torque ripple. This objective is equivalent to constrain 3 *<sup>d</sup> x i* to track a constant direct

As we known that PMSM is a multi-input multi-output nonlinear dynamic system. It is assumed that the position and current are available for measurement. A first sliding variable *s* for the tracking of direct current 3 *x* towards its equilibrium point 3*ref x* f is defined from

Derivative of 1*s* , we can see that the relative degree of sliding mode variable 1*s* equals 1,

*ref*

43 524 61 3

*kx kxx ku x*

dynamics follows a desired third order dynamic. Denoting 1*ref x* the desired trajectory,

Considering load torque as external disturbance, derivative of 2*s* continuously until control

2 14 33 52 4 16 41 3 13 2 4 32 1 3 2 7 2 8 2 3 9 4 1 3 2 10 2 1

The control input *u* appears in the 3 order derivative of 2*s* , so the relative degree of 2*s* equals 3. Considering sliding mode variable 1 2 [,]*<sup>T</sup> s ss* as a new dynamci system, the space

> 1 1 11 1 2 2 21 22 2 *s AB u* 0 *s A BBu*

1 4 3 5 2 4 10 1 *A kx kxx A A* :

2 1 3 2 7 2 8 2 3 9 4 3 1 3 2 4 3 2 1 4 3 3 5 2 4 1 20 2 ( )( ) [( ) ] ( ) : *A kx k kx kxx kx k kx k x kx kx kx kxx x A A ref*

*s kx kx kxx kkxu k kx k x kx kx k kx kxx kx kx k k u x*

 

( )( ) ( )

*ref ref*

( ) [( ) ]

133

*sxx*

2 2 1 13 2 4 32 1

*s x x kx k x kx x*

*ref ref*

( )

2 11 21

*sxx xx*

must track a reference trajectory angular position 1*ref x* .

11 33 ( ) *ref s hx x x* (34)

(35)

*ref*

(37)

*ref*

, another sliding manifold is proposed so that the error

2 2 11 ( ) *ref s hx x x* (36)

(38)

rotor angular position 1 *x*

current reference 3 0 *ref x* .

To track the angular position 1 *x*

following form can be obtained.

state express can be writtern by

that is

input appears.

Where,

the direct current error. So, the first sliding mode variable is

In order to facilitate the calculation, the coefficient (1 10) *<sup>i</sup> k i* is used to plan these variable expressions, Where, 0 (1 10) *<sup>i</sup> k i* is the nominal value of the concerned parameter, *i k* is the uncertainty on the concerned parameter such that | | || *i ii* 0 0 *k kk* , with 0*<sup>i</sup> k* a known positive bound. The state variable <sup>4</sup> *x R* , such that | | (2 4) *i iMAX xx i* , 2*MAX x* is the maximum values of the angular velocity, 3*MAX x* and 4*MAX x* are the maximum values of the current *di* and *qi* respectively. And control input <sup>2</sup> *u R* such that | | ,(1 2) *uu i i iMAX* . Where, *u*1*MAX* and *u*2*MAX* are the maximum values of the voltage input *<sup>d</sup> v* and *<sup>q</sup> v* respectively.

Then the state space model of the synchronous motor can be changed as following nonlinear system.

$$
\begin{bmatrix}
\dot{\boldsymbol{x}}\_{1} \\
\dot{\boldsymbol{x}}\_{2} \\
\dot{\boldsymbol{x}}\_{3} \\
\dot{\boldsymbol{x}}\_{4}
\end{bmatrix} = \underbrace{\begin{bmatrix}
\mathbf{x}\_{2} \\
(\mathbf{k}\_{1}\mathbf{x}\_{3} + \mathbf{k}\_{2})\mathbf{x}\_{4} + \mathbf{k}\_{3}\mathbf{x}\_{2} - T\_{1}/f \\
\mathbf{k}\_{4}\mathbf{x}\_{3} + \mathbf{k}\_{5}\mathbf{x}\_{2}\mathbf{x}\_{4} \\
\mathbf{k}\_{7}\mathbf{x}\_{2} + \mathbf{k}\_{8}\mathbf{x}\_{2}\mathbf{x}\_{3} + \mathbf{k}\_{9}\mathbf{x}\_{4}
\end{bmatrix}}\_{f(\mathbf{x},t)} + \begin{bmatrix}
0 & 0 \\
0 & 0 \\
\mathbf{k}\_{6} & 0 \\
0 & \mathbf{k}\_{10} \\
\hline
\end{bmatrix} \underbrace{\begin{bmatrix}
\boldsymbol{u}\_{1} \\
\boldsymbol{u}\_{2}
\end{bmatrix}}\_{g(\mathbf{x},t)}
\tag{33}
$$

The aim is to design an appropriate control which guarantees robust performance in presence of parameters and load variations. The control objective is double aspect. First, the rotor angular position 1 *x* must track a reference trajectory angular position 1*ref x* . Second, the nonlinear electromagnetic torque must be linearized to avoid reluctance effects and torque ripple. This objective is equivalent to constrain 3 *<sup>d</sup> x i* to track a constant direct current reference 3 0 *ref x* .

As we known that PMSM is a multi-input multi-output nonlinear dynamic system. It is assumed that the position and current are available for measurement. A first sliding variable *s* for the tracking of direct current 3 *x* towards its equilibrium point 3*ref x* f is defined from the direct current error. So, the first sliding mode variable is

$$\mathbf{x}\_1 = h\_1(\mathbf{x}) = \mathbf{x}\_3 - \mathbf{x}\_{3ref} \tag{34}$$

Derivative of 1*s* , we can see that the relative degree of sliding mode variable 1*s* equals 1, that is

$$\begin{aligned} \dot{\mathbf{s}}\_1 &= \dot{\mathbf{x}}\_3 - \dot{\mathbf{x}}\_{3ref} \\ &= k\_4 \mathbf{x}\_3 + k\_5 \mathbf{x}\_2 \mathbf{x}\_4 + k\_6 \boldsymbol{\mu}\_1 - \dot{\mathbf{x}}\_{3ref} \end{aligned} \tag{35}$$

To track the angular position 1 *x* , another sliding manifold is proposed so that the error dynamics follows a desired third order dynamic. Denoting 1*ref x* the desired trajectory, following form can be obtained.

$$\mathbf{s}\_2 = h\_2(\mathbf{x}) = \mathbf{x}\_1 - \mathbf{x}\_{1ref} \tag{36}$$

Considering load torque as external disturbance, derivative of 2*s* continuously until control input appears.

$$\begin{aligned} \dot{s}\_2 &= \dot{\mathbf{x}}\_1 - \dot{\mathbf{x}}\_{1ref} = \mathbf{x}\_2 - \dot{\mathbf{x}}\_{1ref} \\ \dot{s}\_2 &= \dot{\mathbf{x}}\_2 - \ddot{\mathbf{x}}\_{1ref} = (k\_1 \mathbf{x}\_3 + k\_2)\mathbf{x}\_4 + k\_3 \mathbf{x}\_2 - \ddot{\mathbf{x}}\_{1ref} \\ \dot{s}\_2 &= k\_1 \mathbf{x}\_4 (k\_3 \mathbf{x}\_3 + k\_5 \mathbf{x}\_2 \mathbf{x}\_4) + k\_1 k\_6 \mathbf{x}\_4 \mu\_1 + k\_3 [(k\_1 \mathbf{x}\_3 + k\_2)\mathbf{x}\_4 + k\_3 \mathbf{x}\_2] \\ &+ (k\_1 \mathbf{x}\_3 + k\_2)(k\_7 \mathbf{x}\_2 + k\_8 \mathbf{x}\_2 \mathbf{x}\_3 + k\_9 \mathbf{x}\_4) + (k\_1 \mathbf{x}\_3 + k\_2)k\_{10} \mu\_2 - \dddot{\mathbf{x}}\_{1ref} \end{aligned} \tag{37}$$

The control input *u* appears in the 3 order derivative of 2*s* , so the relative degree of 2*s* equals 3. Considering sliding mode variable 1 2 [,]*<sup>T</sup> s ss* as a new dynamci system, the space state express can be writtern by

$$
\begin{bmatrix}
\dot{\mathbf{s}}\_1 \\
\ddot{\mathbf{s}}\_2
\end{bmatrix} = \begin{bmatrix}
A\_1 \\ A\_2
\end{bmatrix} + \begin{bmatrix}
B\_{11} & 0 \\ B\_{21} & B\_{22}
\end{bmatrix} \begin{bmatrix}
u\_1 \\ \mu\_2
\end{bmatrix} \tag{38}
$$

Where,

238 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

From the equation (31) we can see that PMSM is a multi-variable, coupling, nonlinear time varying systems. In addition, the variables in d-q axis can be changed to three phase *abc*

This section will use the high order sliding mode control algorithm with differentiator, in spite of system parameter uncertainties, external disturbances and other factors, to design a robust controller for nonlinear multi-input multi-output permanent magnet synchronous motor. The advantage of this controller is the elimination of the chattering in standard sliding mode. At the same time, it is still with precision and robustness of the standard sliding mode control. And its control law no longer subjects to relative degree constraints.

input 1 2 [,] [,] *T T u uu uu d q* . The parameters *<sup>R</sup>* , *Ld* , *Lq* and *B* are considered as uncertain parameters, such as *R* will change with the temperature rise of the synchronous motor. Therefore, use *R*<sup>0</sup> , *Ld*<sup>0</sup> , *Lq*0 and *B*0 to express their nominal value part of *R* , *Ld* , *Lq* and

> *k k k PL L J kk kP J k k k BJ k k k RL k k k PL L kk k L kk k P L k k k PL L k k k RL kk k L*

( )/ / / / /

*d q f*

/ / / 1 /

*d q d d f q d q q q*

1 /

 

 

*d q x xxxx ii*

0 0

0

0

 *k kk* 

, with 0*<sup>i</sup>*

*k* a

(33)

, and control

(32)

Firstly, let *x* denotes the motor state variable 1234 [ , , , ] [, , , ] *T T*

 

*k* is the uncertainty on the concerned parameter such that | | || *i ii* 0 0

1 2

*x x*

  In order to facilitate the calculation, the coefficient (1 10) *<sup>i</sup> k i* is used to plan these variable expressions, Where, 0 (1 10) *<sup>i</sup> k i* is the nominal value of the concerned parameter,

known positive bound. The state variable <sup>4</sup> *x R* , such that | | (2 4) *i iMAX xx i* , 2*MAX x* is the maximum values of the angular velocity, 3*MAX x* and 4*MAX x* are the maximum values of the current *di* and *qi* respectively. And control input <sup>2</sup> *u R* such that | | ,(1 2) *uu i i iMAX* . Where, *u*1*MAX* and *u*2*MAX* are the maximum values of the voltage

Then the state space model of the synchronous motor can be changed as following nonlinear

2 13 2 4 32 1 3 43 524 6 2 4 72 823 94 10

( ) / 00

*x kx k x kx T J u x kx kxx k u x kx kxx kx k*

( ,) ( ,)

*f xt gxt*

*l*

axis by coordinate transformation.

**2.2.1 Robust control for PMSM** 

*B* respectively.

*i* 

system.

input *<sup>d</sup> v* and *<sup>q</sup> v* respectively.

$$A\_1 = k\_4 \mathfrak{x}\_3 + k\_5 \mathfrak{x}\_2 \mathfrak{x}\_4 =: A\_{10} + \delta A\_{11}$$

$$A\_2 = (k\_1 \mathbf{x}\_3 + k\_2)(k\_7 \mathbf{x}\_2 + k\_8 \mathbf{x}\_2 \mathbf{x}\_3 + k\_9 \mathbf{x}\_4) + k\_3[(k\_1 \mathbf{x}\_3 + k\_2)\mathbf{x}\_4 + k\_3 \mathbf{x}\_2] + k\_1 \mathbf{x}\_4(k\_3 \mathbf{x}\_3 + k\_5 \mathbf{x}\_2 \mathbf{x}\_4) - \dddot{\mathbf{x}}\_{1 \text{ref}} =: A\_{20} + \delta A\_{20}$$

$$\begin{aligned} B\_{11} &= k\_6 =: B\_{110} + \delta B\_{11} \\\\ B\_{21} &= k\_1 k\_6 \ge\_4 =: B\_{210} + \delta B\_{21} \\\\ B\_{22} &= (k\_1 \ge\_3 + k\_2) k\_{10} =: B\_{220} + \delta B\_{22} \end{aligned}$$

*A*<sup>10</sup> , *A*<sup>20</sup> , *B*<sup>110</sup> , *B*210 and *B*220 are the known nominal expressions whereas the expressions of *A*<sup>1</sup> , *A*<sup>2</sup> , *B*<sup>11</sup> , *B*<sup>21</sup> and *B*<sup>22</sup> contain all the uncertainties due to parameters and load torque variations.

Next, controller should be designed so that sliding mode variable 1*s* achieves to zero in finite time. Another sliding mode variable 2*s* and its first and second derivative likewise achieve to zero in finite time. When the sliding mode happens, then

$$\begin{aligned} S\_1 &= \{ \mathbf{x} \in X \mid \mathbf{s\_1(x,t)} = \mathbf{0} \} \\ S\_2 &= \{ \mathbf{x} \in X \mid \mathbf{s\_2(x,t)} = \dot{\mathbf{s\_2(x,t)}} = \ddot{\mathbf{s\_2(x,t)}} = \mathbf{0} \} \end{aligned} \tag{39}$$

The control problem is equivalent to the finite time stabilization of the following MIMO system.

$$
\begin{bmatrix}
\dot{\mathbf{s}}\_1 \\
\ddot{\mathbf{s}}\_2
\end{bmatrix} = A + B \begin{bmatrix}
\boldsymbol{\mu}\_1 \\
\boldsymbol{\mu}\_2
\end{bmatrix} \tag{40}
$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 241

1 1 10 110

11

110 110 220

of (43) is the so-called equivalent control in the sliding mode

*m M m M*

220

*B* 

110

<sup>ˆ</sup> *<sup>B</sup> B B <sup>B</sup> B BB*

2 2 10 20 110 110 220 220

sgn( ) (47)

1 properly to satisfy convergence. For the motor angular

1 0 2

0 1

 *A* and 0 

*B* . Then

(46)

(49)

<sup>ˆ</sup> [ ] *B B B B A A A A B BB B*

<sup>ˆ</sup> *<sup>B</sup> AA A B* 

 

<sup>ˆ</sup> <sup>1</sup> *<sup>B</sup> <sup>B</sup> B* 

 <sup>21</sup> 22 210 21

<sup>ˆ</sup> <sup>1</sup> *<sup>B</sup> <sup>B</sup>*

In the new dynamic system with 1 2 [,]*<sup>T</sup> w ww* , it leads to 1*s* equals integrator of *w*1 and 2*<sup>s</sup>*

context. In this new system, due to state variable (2 4) *<sup>i</sup> x x* , there exist three positive

1 1 11 11 11 2 2 22 22 22

Then, owing to the relative degree of 1*s* equals 1, the first order sliding mode algorithm

11 1 *w s* 

position control, a 3 order sliding mode control law is used. In this case, only a single scalar

2 2 2 2 2 22 2 *w s s s ss s*

According to the principle of sliding mode differentiator, the arbitrary order derivative of

, 160 | | sgn( ) , 150 | | sgn( )

 

*z z z z z z s z s z*

 1/ 2

2 / 3

1 is positive constant. In the actual system, due to all the state variables have the

2 is to be adjusted. Actually, the control input *w*2 can be chosen as following.

3 2 1/6 2/3

sgn( 2(| | | | ) s gn( | | s gn( ))) (48)

*AC K B K AC K B K*

 

ˆ ˆ | | , 0 ˆ ˆ | | , 0

11

21 22 210 <sup>22</sup>

11

22

equals three time integrators of *w*<sup>2</sup> , if the part of uncertainties 0

21 21

*B K*

ˆ | |

<sup>2</sup>*s* can be estimated by the output of differentiator 0*z* , 1*z* and 2 *z* .

400 | |

0 0 0 0

1 1 1 1 0

2 2 1

 

 

*z z*

 

<sup>22</sup>

*w*1 and *w*2 are designed to stabilize in this new system.

constants *C*<sup>1</sup> , *C*<sup>2</sup> , *K*11*<sup>m</sup>* , *K*22*<sup>m</sup>* , *K*11*<sup>M</sup>* , *K*22*<sup>M</sup>* and *K*<sup>21</sup> , so that

In fact, the term <sup>1</sup> *B A*0 0

Where

parameter

bound, selecting parameter

previously presented with control law

Where,

Where,

$$A = \begin{bmatrix} A\_{10} \\ A\_{20} \end{bmatrix} + \begin{bmatrix} \delta A\_1 \\ \delta A\_2 \end{bmatrix} =: A\_0 + \delta A \tag{41}$$

$$B = \begin{bmatrix} B\_{110} & 0 \\ B\_{210} & B\_{220} \end{bmatrix} + \begin{bmatrix} \delta B\_{11} & 0 \\ \delta B\_{21} & \delta B\_{22} \end{bmatrix} =: B\_0 + \delta B \tag{42}$$

From the equation (37), the outputs of this MIMO system are coupled since 2 *s* is affected by *u*1 and *u*<sup>2</sup> . So an input-output feedback linearization technique can be used, here *w* is new control input.

$$
\mu = B\_0^{-1} \cdot \left[ -A\_0 + w \right] \tag{43}
$$

Now, if considering influence of external disturbance and parameter uncertainties, equation (40) can be rewritten by

$$\begin{aligned} \mathbf{s} &= A + Bu\\ \mathbf{b} &= A\_0 + \delta A + \{B\_0 + \delta B\} u\\ \mathbf{b} &= A\_0 + \delta A + \{B\_0 + \delta B\} [B\_0^{-1}(-A\_0 + \mathbf{w})] \end{aligned} \tag{44}$$

Evolution and ordinate, then

$$
\Longrightarrow \begin{bmatrix} \dot{\mathbf{s}}\_{1} \\ \ddot{\mathbf{s}}\_{2} \end{bmatrix} = \begin{bmatrix} \hat{\boldsymbol{A}}\_{1} \\ \hat{\boldsymbol{A}}\_{2} \end{bmatrix} + \begin{bmatrix} \hat{\boldsymbol{B}}\_{11} & \mathbf{0} \\ \hat{\boldsymbol{B}}\_{21} & \hat{\boldsymbol{B}}\_{22} \end{bmatrix} \begin{bmatrix} \boldsymbol{w}\_{1} \\ \boldsymbol{w}\_{2} \end{bmatrix} \tag{45}
$$

Where,

240 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

11 6 110 11 *BkB B* :

21 1 6 4 210 21 *B kkx B B* :

22 1 3 2 10 220 22 *B kx k k B B* ( ):

*A*<sup>10</sup> , *A*<sup>20</sup> , *B*<sup>110</sup> , *B*210 and *B*220 are the known nominal expressions whereas the expressions

Next, controller should be designed so that sliding mode variable 1*s* achieves to zero in finite time. Another sliding mode variable 2*s* and its first and second derivative likewise

{ | ( , ) ( , ) ( , ) 0}

The control problem is equivalent to the finite time stabilization of the following MIMO

1 1 2 2 *s u A B s u* 

0

:

0

0 0 *uB A w* [ ] (43)

:

 

1

 

(45)

2 1

*w w*

( )[ ( )]

21 22 11

ˆ ˆ 0 ˆ

*B B B*

10 1

*A A A A A A A* 

110 11

*B B*

0 0

2 1

*s s* 

*A A B Bu*

 

2 1

ˆ ˆ

*A A*

*s A Bu*

 

20 2

210 220 21 22

1

 

0 0

*B B B BB B B*

From the equation (37), the outputs of this MIMO system are coupled since 2 *s* is affected by *u*1 and *u*<sup>2</sup> . So an input-output feedback linearization technique can be used, here *w* is new

Now, if considering influence of external disturbance and parameter uncertainties, equation

0 0 00 ( )

*A A B BB A w*

 

 

 

2 222

*S x X s xt s xt s xt*

{ | ( , ) 0}

 and *B*<sup>22</sup> 

achieve to zero in finite time. When the sliding mode happens, then

1 1

*S x X s xt*

of *A*<sup>1</sup> , *A*<sup>2</sup> , *B*<sup>11</sup> , *B*<sup>21</sup> 

system.

Where,

control input.

(40) can be rewritten by

Evolution and ordinate, then

torque variations.

contain all the uncertainties due to parameters and load

(39)

(40)

(41)

(42)

(44)

$$\begin{aligned} \hat{A}\_1 &= \delta A\_1 - \frac{\delta B\_{11}}{B\_{110}} A\_{10} \\\\ \hat{A}\_2 &= \delta A\_2 - [\frac{\delta B\_{21}}{B\_{110}} - \frac{\delta B\_{22} B\_{210}}{B\_{110} B\_{220}}] A\_{10} - \frac{\delta B\_{22}}{B\_{220}} A\_{20} \\\\ \hat{B}\_{11} &= 1 + \frac{\delta B\_{11}}{B\_{110}} \\\\ \hat{B}\_{21} &= \frac{\delta B\_{21}}{B\_{110}} - \frac{\delta B\_{22} B\_{210}}{B\_{110} B\_{220}} \\\\ \hat{B}\_{22} &= 1 + \frac{\delta B\_{22}}{B\_{220}} \end{aligned}$$

In the new dynamic system with 1 2 [,]*<sup>T</sup> w ww* , it leads to 1*s* equals integrator of *w*1 and 2*<sup>s</sup>* equals three time integrators of *w*<sup>2</sup> , if the part of uncertainties 0 *A* and 0 *B* . Then *w*1 and *w*2 are designed to stabilize in this new system.

In fact, the term <sup>1</sup> *B A*0 0 of (43) is the so-called equivalent control in the sliding mode context. In this new system, due to state variable (2 4) *<sup>i</sup> x x* , there exist three positive constants *C*<sup>1</sup> , *C*<sup>2</sup> , *K*11*<sup>m</sup>* , *K*22*<sup>m</sup>* , *K*11*<sup>M</sup>* , *K*22*<sup>M</sup>* and *K*<sup>21</sup> , so that

$$\begin{aligned} \mid \hat{A}\_{1} \mid \leq \mathcal{C}\_{1'} \quad 0 < \mathcal{K}\_{11m} \leq \hat{B}\_{11} \leq \mathcal{K}\_{11M} \\\mid \hat{A}\_{2} \mid \leq \mathcal{C}\_{2} \quad 0 < \mathcal{K}\_{22m} \leq \hat{B}\_{22} \leq \mathcal{K}\_{22M} \\\mid \hat{B}\_{21} \mid \leq \mathcal{K}\_{21} \end{aligned} \tag{46}$$

Then, owing to the relative degree of 1*s* equals 1, the first order sliding mode algorithm previously presented with control law

$$w\_1 = -\alpha\_1 \operatorname{sgn}(s\_1) \tag{47}$$

Where 1 is positive constant. In the actual system, due to all the state variables have the bound, selecting parameter 1 properly to satisfy convergence. For the motor angular position control, a 3 order sliding mode control law is used. In this case, only a single scalar parameter 2 is to be adjusted. Actually, the control input *w*2 can be chosen as following.

$$\mathbf{w}\_2 = -\alpha\_2 \operatorname{sgn}(\ddot{\mathbf{s}}\_2 + \mathcal{Q}(|\dot{\mathbf{s}}\_2|^3 + |\mathbf{s}\_2|^2)^{1/6} \operatorname{sgn}(\dot{\mathbf{s}}\_2 + |\mathbf{s}\_2|^{2/3} \operatorname{sgn}(\mathbf{s}\_2)))\tag{48}$$

According to the principle of sliding mode differentiator, the arbitrary order derivative of <sup>2</sup>*s* can be estimated by the output of differentiator 0*z* , 1*z* and 2 *z* .

$$\begin{aligned} \dot{z}\_0 &= \nu\_0, \ \nu\_0 = -160 \mid z\_0 - s \mid^{2/3} \operatorname{sgn}(z\_0 - s) + z\_1 \\ \dot{z}\_1 &= \nu\_1, \ \nu\_1 = -160 \mid z\_1 - \nu\_0 \mid^{1/2} \operatorname{sgn}(z\_1 - \nu\_0) + z\_2 \\ \dot{z}\_2 &= -400 \mid z\_2 - \nu\_1 \mid \end{aligned} \tag{49}$$

Then, substituting <sup>0</sup>*z* , 1*z* and 2 *z* for 2*s* , 2*s* and 2 *s* respectively in equation (48), that is

$$aw\_2 = -\alpha\_2 \operatorname{sgn}(z\_2 + 2(\lfloor z\_1 \rfloor^3 + \lfloor z\_0 \rfloor^2)^{1/6} \operatorname{sgn}(z\_1 + \lfloor z\_0 \rfloor^{2/3} \operatorname{sgn}(z\_0)))\tag{50}$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 243

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -0.08

*time (second)*

0 0.5 1 1.5 2 2.5 3

*time (second)*

real reference

> *ud uq*

10


Fig. 3. The curve of input *ud* and *uq* .

*ud and uq (V)*

Fig. 2. Reference angle positon and actual angle position.

*error (rad)*

20

*position (rad)*

30

40

The figure 1 is the block graph of control system. The first sliding mode variable 1*s* is given by the error between direct axis current reference and feedback. And the second variable 2*s* is identified by the error between the motor reference position and actual feedback. According to the Theorem 4 after finite time, 0*z* , 1*z* and 2 *z* can be used to estimate 2*s* , 2*s* and 2 *s* . In system, the state variable speed of motor *w*2 is obtained by differentiator of angle position signal . Finally, the nonlinear dynamic system must be linearized by inputoutput feedback linearization, then the control input *u*1 and *u*2 are used to drive the synchronous motor.

Fig. 1. The block graph of dynamic system structure.

In the simulation, The PMSM is a DutymAx 95DSC060300 (Leroy Somer Co.) drive. Two sensors give measurements of phase currents, a optical encoder is used to measure the position of the motor. The parameters of synchronous motor are *P* 3 , *R* 3 3. , 0 027 . *L H <sup>d</sup>* , 0 0034 . *L H <sup>q</sup>* , *B Nms* 0 0034 . , 0 341 . *<sup>f</sup> Wb* , <sup>2</sup> *J k* 0 00037 . *g m* . A phase current of the maximum accepted value is 6 0. *A* , the load torque maximum value is 6*N m* , and angular velocity is 3000*rpm* .To achieve the efficiency of controller, the parameter in (47) and (50) are chosen by 5 <sup>1</sup> , 3300 <sup>2</sup> . In the differentiator, the coefficient of (49) are selected by 150 <sup>0</sup> , 160 <sup>1</sup> , 400 <sup>2</sup> in order to allow the convergence of the differentiator. The system sampling frequency is 8000*Hz* . To show the system robustness of the controller, consider permanent magnet synchronous motor parameters uncertainties (with*R* 50%*R* , with *Ld* %*Ld* 25 <sup>0</sup> and *Lq* %*Lq* 25 <sup>0</sup> and with 20%*B* ).

The trajectory of motor angular position reference and feedback are shown in figure 2 above in spit of PMSM parameters uncertainty. From this figure, we can see that the servo system track trajectory has good performance. The precision can achieve 10-3. In addition, using high order sliding mode control, the chattering is eliminated in lower sliding mode surface so that the track trajectory becomes smoother.

Figure 2 below shows position tracking error, which does not exceed 0.09 rad. It means that the controller has high robust capability versus the parameters variations.

The figure 3 shows the curve of input *<sup>d</sup> u* and *<sup>q</sup> u* for PMSM using the high order sliding mode observer.

242 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

2 2 2 1 0 10 0 *w z z z zz z*

The figure 1 is the block graph of control system. The first sliding mode variable 1*s* is given by the error between direct axis current reference and feedback. And the second variable 2*s* is identified by the error between the motor reference position and actual feedback. According to the Theorem 4 after finite time, 0*z* , 1*z* and 2 *z* can be used to estimate 2*s* , 2*s*

output feedback linearization, then the control input *u*1 and *u*2 are used to drive the

1 *z* 2 *z*

*s* . In system, the state variable speed of motor *w*2 is obtained by differentiator of

3 2 1/6 2/3

sgn( 2(| | | | ) sgn( | | sgn( ))) (50)

. Finally, the nonlinear dynamic system must be linearized by input-

<sup>2</sup> *<sup>u</sup>* <sup>0</sup> *<sup>z</sup>*

*w*1

*w*2

*z ref* 

In the simulation, The PMSM is a DutymAx 95DSC060300 (Leroy Somer Co.) drive. Two sensors give measurements of phase currents, a optical encoder is used to measure the position of the motor. The parameters of synchronous motor are *P* 3 , *R* 3 3. ,

current of the maximum accepted value is 6 0. *A* , the load torque maximum value is 6*N m* , and angular velocity is 3000*rpm* .To achieve the efficiency of controller, the parameter in (47)

differentiator. The system sampling frequency is 8000*Hz* . To show the system robustness of the controller, consider permanent magnet synchronous motor parameters uncertainties

The trajectory of motor angular position reference and feedback are shown in figure 2 above in spit of PMSM parameters uncertainty. From this figure, we can see that the servo system track trajectory has good performance. The precision can achieve 10-3. In addition, using high order sliding mode control, the chattering is eliminated in lower sliding mode surface

Figure 2 below shows position tracking error, which does not exceed 0.09 rad. It means that

The figure 3 shows the curve of input *<sup>d</sup> u* and *<sup>q</sup> u* for PMSM using the high order sliding

 <sup>0</sup> and *Lq* %*Lq* 25 

the controller has high robust capability versus the parameters variations.

 <sup>1</sup> 

*s* respectively in equation (48), that is

1 *u*

[ ] <sup>0</sup> <sup>1</sup> *u B*<sup>0</sup> *A w*

*qi*

*di*

*<sup>f</sup> Wb* , <sup>2</sup> *J k* 0 00037 . *g m* . A phase

<sup>2</sup> . In the differentiator, the coefficient of (49) are

<sup>0</sup> and with 20%*B* ).

<sup>2</sup> in order to allow the convergence of the

Then, substituting <sup>0</sup>*z* , 1*z* and 2 *z* for 2*s* , 2*s* and 2

and 2

angle position signal

synchronous motor.

*dref i ref* 1 *s*

and (50) are chosen by 5

selected by 150 

mode observer.

(with 2 *s*

Fig. 1. The block graph of dynamic system structure.

<sup>0</sup> , 160 

*R* 50%*R* , with *Ld* %*Ld* 25 

so that the track trajectory becomes smoother.

0 027 . *L H <sup>d</sup>* , 0 0034 . *L H <sup>q</sup>* , *B Nms* 0 0034 . , 0 341 .

<sup>1</sup> , 3300 

<sup>1</sup> , 400 

Fig. 2. Reference angle positon and actual angle position.

Fig. 3. The curve of input *ud* and *uq* .

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 245

The figure 4 is the speed and d-q axis current of synchronous motor. The motor in the fourquadrant operation, with acceleration, deceleration, has good dynamic performance. In this

The figure 5 shows that sliding mode variable 2*s* converge into origin in three dimensional

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -0.1

The Figure 6 shows the controller is strong robustness versus the load torque variations. The

To sum up, this section takes the multiple-input multiple-output nonlinear permanent magnet synchronous motor as control object, and designs a robust high order sliding mode controller with differentiator, through the state feedback linearization to decouple the system. The simulation results show that, despite the existence of parameter uncertainties and external disturbances, the system still has a better dynamic performance and robustness, which is due to higher order sliding mode control converge within limited time. Comparing with the traditional sliding mode control, high order sliding mode control eliminates the chattering phenomenon. And the better test results prove the feasibility of the

The parameters and state estimation of permanent magnet synchronous motor has been more concerned in motor control. As the motor itself is a typical nonlinear, multivariable system with strong coupling, there are a lot algorithms to improve the motor control performance in recent years. Earlier off-line estimation of the static dynamic system can not satisfy the control requirements; the use of extended Kalman filter (EKF) usually have a group of high order nonlinear equations, which is not conducive to the calculation (Yan, 2006), and its stability is also a local stable; In least squares procedure, the matrix forgotten

error of angular position does not exceed 0.1 rad even though the load perturbation.

*time (second)*

figure, direct axis current *di* is very near reference 0 *dref i* .

surface within limited time.

0.5 1 1.5 2 2.5


Fig. 6. Tracking error of motor with the load torque disturbance.

*error (rad)*

theory.

**2.2.2 States estimation** 

*load torque (Nm)*

Fig. 4. Four quadrant run and quadrature/direct axis currents.

*s 2*

Fig. 5. Sliding mode variables <sup>2</sup> *s* , <sup>2</sup> *s* and <sup>2</sup> *s* .

*2*

The figure 4 is the speed and d-q axis current of synchronous motor. The motor in the fourquadrant operation, with acceleration, deceleration, has good dynamic performance. In this figure, direct axis current *di* is very near reference 0 *dref i* .

The figure 5 shows that sliding mode variable 2*s* converge into origin in three dimensional surface within limited time.

Fig. 6. Tracking error of motor with the load torque disturbance.

The Figure 6 shows the controller is strong robustness versus the load torque variations. The error of angular position does not exceed 0.1 rad even though the load perturbation.

To sum up, this section takes the multiple-input multiple-output nonlinear permanent magnet synchronous motor as control object, and designs a robust high order sliding mode controller with differentiator, through the state feedback linearization to decouple the system. The simulation results show that, despite the existence of parameter uncertainties and external disturbances, the system still has a better dynamic performance and robustness, which is due to higher order sliding mode control converge within limited time. Comparing with the traditional sliding mode control, high order sliding mode control eliminates the chattering phenomenon. And the better test results prove the feasibility of the theory.

### **2.2.2 States estimation**

244 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -100

*id iq*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*time (second)*


<sup>2</sup>*<sup>s</sup>* <sup>2</sup>*s*


*s* .


0

*s 2 '*

1

2 -10

Fig. 5. Sliding mode variables <sup>2</sup> *s* , <sup>2</sup> *s* and <sup>2</sup>


0

*s*

2 *s*

*''*

*2*

5 10

15



*s 2* -0.02

0

0.02



Fig. 4. Four quadrant run and quadrature/direct axis currents.

*Current (A)*

0

*speed (rad/s)*

50

100

The parameters and state estimation of permanent magnet synchronous motor has been more concerned in motor control. As the motor itself is a typical nonlinear, multivariable system with strong coupling, there are a lot algorithms to improve the motor control performance in recent years. Earlier off-line estimation of the static dynamic system can not satisfy the control requirements; the use of extended Kalman filter (EKF) usually have a group of high order nonlinear equations, which is not conducive to the calculation (Yan, 2006), and its stability is also a local stable; In least squares procedure, the matrix forgotten factor (Poznyak et al, 1999; Poznyak, 1999) is used to solve non-static parameter identification; as a result of sliding mode control with strong robustness and global convergence, In recent years, sliding mode observer (Floret-Pontet, 2001; Koshkouei, 2002) has been used for dynamic system state and parameter estimation, but the observer feedback gain is usually not easy to choose.

With the development of nonlinear theory, in order to enhance the performance of permanent magnet synchronous motor, many advanced control strategies have been proposed and used in motor control, which requires the state of motor can be measured, such as mechanical angular position, rotational speed, the electrical current and so on. Hence mechanical, electromagnetic or photoelectric sensor are needed, as well known to all, the sensors have many other shortcomings such as drift, friction, high costs, as well as electromagnetic interference caused by additional conductors. Therefore, the control system should be as possible as release the use of sensors to ensure the reliability and stability, which requires the system observer to precisely estimate the value of the state.

The high order sliding mode control is widely used in last decade, which take high order derivetives of sliding mode variables to substitute original discrete control, so that the chattering disappears in the high order differentiation. This section uses a high order sliding mode observer with differentiator algorithm to estimates the value of state variables. In this case, it removes the speed and current sensors of motor, and a better control precision and accurate state estimation are obtained.

In this section, the mathematical model of PMSM is the same with above section (33). In order to make control effectiveness more smooth, the relative order is raised artificially. Considering control input *u* as a new input, original sliding mode variable (35) and (37) are transformed into

$$\begin{aligned} \ddot{s}\_1 &= A\_1 \text{"+} B\_{11} \text{"\'} \dot{u}\_1\\ s\_2^{(4)} &= A\_2 \text{"+} B\_{21} \text{"\'} \dot{u}\_1 + B\_{22} \text{"\'} \dot{u}\_2 \end{aligned} \tag{51}$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 247

Next, it still is high order sliding mode controller design that make the sliding mode variables converge into origin within limited time in the sliding mode surface. In another

{ | ( , ) ( , ) ( , ) ( , ) 0}

<sup>1</sup> <sup>2</sup> [, ]*<sup>T</sup> s ss* , 1 2 *u uu* [,] , this control object is equivalent to stable of following multi-

'' *s A Bu* (53)

*r*

*u w Lhx B A w*

After decoupling, the relative degree of 1*s* equals 2, so 2 order sliding mode control law is

1 1 11 1 *w ss s*

01 01 01 01 01 1 01 1 11

For the motor's angular position control, 4 order sliding mode control law is used. In this

22 2 2 2 2 22

Similarly, the output 02 12 22 *zzz* , , of 3 order differentiator is used to estimated sliding mode

02 02 02 02 02 2 02 2 12

 

*zv v zs zsz zv v zv zv z zv v zv zv z*

12 12 12 12 12 02 12 02 22

22 22 22 22 22 12 22 12 32

, | | sgn( )

*w s s s s ss*

sgn{ 3[( ) ( ) | | ] sgn[ (| |

3/4

, | | sgn( ) ; , | | sgn( ) ;

2/3

1/2

*zv v zs zsz*

, | | sgn( )

<sup>2</sup>*s* effected by *u*1 and *u*2 , the output of this system are coupled. Here, input-

<sup>1</sup> ( ( )) ' [ ' ] ( )

1/2

1/2

(56)

6 4 3 1/12 4

1 is a positive constant. Now, we use the output of sliding mode differentiator 01 *z*

1

sgn( | | s gn( )) (55)

2 to make the system converge within limited

(57)

(58)

 (54)

(52)

2 2222

*S x X s xt s xt s xt s xt*

{ | ( , ) ( , ) 0}

word, it should satisfy following conditions.

input multi-output system in limited time.

and 11 *z* to estimate the value of 1*s* and 1*s* .

case, we only adjust a single parameter

  Let (4)

Due to (4)

adopted

Where,

time.

variables 222 *sss* , , .

1 11

*S x X s xt s xt*

output feedback linearization technology is used to decouple system.

1

*LL hx*

*g f*

11 11 11 01

*z zv*

*f r*

sgn( )

3 1/6 3/4 2 22 2

32 32 32 22

*z zv*

sgn( )


*s ss s*

Then, the coefficient matrix of original system *A*<sup>1</sup> , *A*<sup>2</sup> , *B*<sup>11</sup> , *B*21 and *B*22 become new matrixes 1 *A* ' , 2 *A* ' , 11 *B* ' , 21 *B* ' and 22 *B* ' . Where,

$$\begin{aligned} A\_1 &= k\_4 (k\_4 x\_3 + k\_5 x\_2 x\_4 + k\_6 x\_1) + k\_5 [(k\_1 x\_3 + k\_2) x\_4 + k\_3 x\_2] \mathbf{x}\_4 \\ &+ k\_5 x\_2 (k\_7 x\_2 + k\_8 x\_2 x\_3 + k\_9 x\_4 + k\_{10} u\_2) - \ddot{\mathbf{x}}\_{3\tau f} \\\\ A\_2 &= (k\_1 k\_5 x\_4^2 + k\_1 k\_7 x\_3 + k\_1 k\_8 x\_3^2 + k\_2 k\_7 + k\_2 k\_8 x\_3 + k\_3^2) [(k\_1 x\_3 + k\_2) x\_4 + k\_3 x\_2] \\ &+ (k\_1 k\_4 x\_4 + k\_1 k\_7 x\_2 + 2 k\_1 k\_8 x\_2 x\_3 + k\_1 k\_9 x\_4 + k\_2 k\_8 x\_2 + k\_1 k\_{10} u\_2 + k\_1 k\_3 x\_4) (k\_4 x\_3 \\ &+ k\_5 x\_2 x\_4 + k\_6 u\_1) + 2 k\_1 k\_5 x\_2 x\_4 + k\_1 k\_6 u\_1 + k\_1 k\_6 x\_3 + k\_2 k\_9 + k\_1 k\_3 x\_3 + (k\_1 k\_4 x\_3 \\ &+ k\_2 k\_3) (k\_7 x\_2 + k\_8 x\_2 x\_3 + k\_9 x\_4 + k\_{10} u\_2) - x\_{\text{ref}}^{40} \\ &+ k\_{11} "= k\_6 \ : \\ B\_{11} "= k\_7 k\_6 x\_4 \ : \\\ B\_{22} "= (k\_1 x\_3 + k\_2) k\_{10} \ : \end{aligned}$$

246 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

factor (Poznyak et al, 1999; Poznyak, 1999) is used to solve non-static parameter identification; as a result of sliding mode control with strong robustness and global convergence, In recent years, sliding mode observer (Floret-Pontet, 2001; Koshkouei, 2002) has been used for dynamic system state and parameter estimation, but the observer

With the development of nonlinear theory, in order to enhance the performance of permanent magnet synchronous motor, many advanced control strategies have been proposed and used in motor control, which requires the state of motor can be measured, such as mechanical angular position, rotational speed, the electrical current and so on. Hence mechanical, electromagnetic or photoelectric sensor are needed, as well known to all, the sensors have many other shortcomings such as drift, friction, high costs, as well as electromagnetic interference caused by additional conductors. Therefore, the control system should be as possible as release the use of sensors to ensure the reliability and stability,

The high order sliding mode control is widely used in last decade, which take high order derivetives of sliding mode variables to substitute original discrete control, so that the chattering disappears in the high order differentiation. This section uses a high order sliding mode observer with differentiator algorithm to estimates the value of state variables. In this case, it removes the speed and current sensors of motor, and a better control precision and

In this section, the mathematical model of PMSM is the same with above section (33). In order to make control effectiveness more smooth, the relative order is raised artificially. Considering control input *u* as a new input, original sliding mode variable (35) and (37) are

2 2 21 1 22 2

Then, the coefficient matrix of original system *A*<sup>1</sup> , *A*<sup>2</sup> , *B*<sup>11</sup> , *B*21 and *B*22 become new

1 4 43 524 61 5 13 2 4 32 4 5 2 7 2 8 2 3 9 4 10 2 3

1 4 4 1 7 2 1 8 2 3 1 9 4 2 8 2 1 10 2 1 3 4 4 3 52 4 61 152 4 161 193 29 133 143

(4)

*kkx kkx kkxx kkx kkx kk u kkx kx kxx ku kkxx kku kkx kk kkx kkx*

' ( ) [( ) ]

*A k kx kxx ku k kx k x kx x kx kx kxx kx k u x* 

2 15 4 17 3 183 27 283 3 13 2 4 32

*A kkx kkx kkx kk kkx k kx k x kx*

 

' ( )[( ) ] ( 2 )( ) 2 (

11 6 *B k* ' ;

21 1 6 4 *B kkx* ' ;

22 1 3 2 10 *B kx k k* '( ) ;

*s A Bu Bu*

'' '

(51)

1 1 11 1

*s A Bu*

() *ref*

22 2

2 3 7 2 8 2 3 9 4 10 2 3 )( ) *ref kk kx kxx kx k u x*

' '

(4)

which requires the system observer to precisely estimate the value of the state.

feedback gain is usually not easy to choose.

accurate state estimation are obtained.

matrixes 1 *A* ' , 2 *A* ' , 11 *B* ' , 21 *B* ' and 22 *B* ' . Where,

transformed into

Next, it still is high order sliding mode controller design that make the sliding mode variables converge into origin within limited time in the sliding mode surface. In another word, it should satisfy following conditions.

$$\begin{aligned} S\_1 &= \{ \mathbf{x} \in X \mid s\_1(\mathbf{x}, t) = \dot{s}\_1(\mathbf{x}, t) = 0 \} \\ S\_2 &= \{ \mathbf{x} \in X \mid s\_2(\mathbf{x}, t) = \dot{s}\_2(\mathbf{x}, t) = \dddot{s}\_2(\mathbf{x}, t) = \dddot{s}\_2(\mathbf{x}, t) = 0 \} \end{aligned} \tag{52}$$

Let (4) <sup>1</sup> <sup>2</sup> [, ]*<sup>T</sup> s ss* , 1 2 *u uu* [,] , this control object is equivalent to stable of following multiinput multi-output system in limited time.

$$\mathbf{s} = A' + B'\mathbf{u} \tag{53}$$

Due to (4) <sup>2</sup>*s* effected by *u*1 and *u*2 , the output of this system are coupled. Here, inputoutput feedback linearization technology is used to decouple system.

$$\mu = \frac{1}{L\_{\text{g}}L\_{f}^{r-1}h(\mathbf{x})} (w - L\_{f}^{r}h(\mathbf{x})) = B^{r-1} \cdot [-A^{\prime} + w] \tag{54}$$

After decoupling, the relative degree of 1*s* equals 2, so 2 order sliding mode control law is adopted

$$w\_1 = -a\_1 \operatorname{sgn}(\dot{s}\_1 + |s\_1|^{1/2} \operatorname{sgn}(s\_1))\tag{55}$$

Where, 1 is a positive constant. Now, we use the output of sliding mode differentiator 01 *z* and 11 *z* to estimate the value of 1*s* and 1*s* .

$$\begin{aligned} \dot{z}\_{01} &= v\_{01}, \quad v\_{01} = -\lambda\_{01} \left| z\_{01} - s\_1 \right|^{1/2} \operatorname{sgn}(z\_{01} - s\_1) + z\_{11} \\ \dot{z}\_{11} &= -\lambda\_{11} \operatorname{sgn}(z\_{11} - v\_{01}) \end{aligned} \tag{56}$$

For the motor's angular position control, 4 order sliding mode control law is used. In this case, we only adjust a single parameter 2 to make the system converge within limited time.

$$\begin{split} \mathbf{w}\_{2} &= -\alpha\_{2} \operatorname{sgn} \{ \ddot{\mathbf{s}}\_{2} + \Re \{ (\ddot{\mathbf{s}}\_{2})^{6} + (\dot{\mathbf{s}}\_{2})^{4} + |\mathbf{s}\_{2}|^{3} \}^{1/12} \operatorname{sgn} [\ddot{\mathbf{s}}\_{2} + (|\dot{\mathbf{s}}\_{2}|^{4} \\ &+ |\mathbf{s}\_{2}|^{3})^{1/6} \operatorname{sgn} (\dot{\mathbf{s}}\_{2} + 0.5 \left| \mathbf{s}\_{2} \right|^{3/4} \operatorname{sgn} (\mathbf{s}\_{2})) \} \end{split} \tag{57}$$

Similarly, the output 02 12 22 *zzz* , , of 3 order differentiator is used to estimated sliding mode variables 222 *sss* , , .

$$\begin{aligned} \dot{z}\_{02} &= v\_{02}, \quad v\_{02} = -\dot{\lambda}\_{02} \left| z\_{02} - s\_2 \right|^{3/4} \operatorname{sgn}(z\_{02} - s\_2) + z\_{12}; \\ \dot{z}\_{12} &= v\_{12}, \quad v\_{12} = -\dot{\lambda}\_{12} \left| z\_{12} - v\_{02} \right|^{2/3} \operatorname{sgn}(z\_{12} - v\_{02}) + z\_{22}; \\ \dot{z}\_{22} &= v\_{22}, \quad v\_{22} = -\dot{\lambda}\_{22} \left| z\_{22} - v\_{12} \right|^{1/2} \operatorname{sgn}(z\_{22} - v\_{12}) + z\_{32} \\ \dot{z}\_{32} &= -\dot{\lambda}\_{32} \operatorname{sgn}(z\_{32} - v\_{22}) \end{aligned} \tag{58}$$

Generally speaking, in the actual system not all the state variables is measurable. Sometimes, due to the limitation of condition, some state variables can not be measured. Therefore, it requires the controller can estimate state variables of system as possible as accurate.

PMSM only uses the position sensors, taking the use of high order sliding mode control techniques, so that its speed and the current state variable are estimated online. In this way, it avoids the use of other sensors, at the same time ensures the motor position tracking progress.

In the design of controller, we have obtained that

$$
\dot{\mathbf{s}}\_1 = k\_4 \mathbf{x}\_3 + k\_5 \mathbf{x}\_2 \mathbf{x}\_4 + k\_6 \boldsymbol{\mu}\_2 - \mathbf{x}\_{3ref}
$$

$$
\dot{\mathbf{s}}\_2 = \mathbf{x}\_2 - \dot{\mathbf{x}}\_{1ref}
$$

$$
\ddot{\mathbf{s}}\_2 = \ddot{\mathbf{x}}\_1 - \ddot{\mathbf{x}}\_{1ref} = (k\_1 \mathbf{x}\_3 + k\_2)\mathbf{x}\_4 + k\_3 \mathbf{x}\_2 - \ddot{\mathbf{x}}\_{1ref} \tag{59}
$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 249

a

real reference

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

time (second)

From the figure 7 above, it can be seen that the permanent magnet synchronous motor control system has good performance. This figure shows the permanent magnet synchronous motor can precisely track the given position. And the error between reference and the actual position feedback is shown in Figure 7 below. The maximal error does not

(a)

0 1 2 3 4 5 6 7 8 9 10

(b)

0 1 2 3 4 5 6 7 8 9 10

time (second)

The figure 8 above shows the motor angular speed by derivative of the motor's angular position. The figure 8 below shows the error between the estimation of the electrical angular

0


> -10 -5 0 5 10


Fig. 8. Speed estimation and error curve of PMSM.

error (rad/s)

speed.

speed (rad/s)

Fig. 7. Position tracking and error curve of PMSM.

error (rad)

exceed to 0.08 rad.

10

20

position (rad)

30

From the above equations, we can calculate the speed estimation of synchronous motor <sup>221</sup>*ref xsx* , so the current estimation expressed by

$$\begin{aligned} \tilde{\mathbf{x}}\_3 &= \frac{1}{k\_4} [\dot{\mathbf{s}}\_1 - k\_5 \tilde{\mathbf{x}}\_4 (\dot{\mathbf{s}}\_2 + \dot{\mathbf{x}}\_{1ref}) - k\_6 \boldsymbol{\mu}\_2] \\ \tilde{\mathbf{x}}\_4 &= \frac{\ddot{\mathbf{s}}\_2 - k\_3 (\dot{\mathbf{s}}\_2 + \dot{\mathbf{x}}\_{1ref}) + \ddot{\mathbf{x}}\_{1ref}}{k\_1 \tilde{\mathbf{x}}\_3 + k\_2} \end{aligned} \tag{60}$$

For calculating 3 *x* and 4 *x* , considering sliding mode variables 1*s* , 2*s* , 2 *s* and *u*2 as known value, adopt recursive algorithm to get

$$\begin{aligned} \tilde{\boldsymbol{x}}\_{3(j+1)} &= \frac{1}{k\_4} [\dot{\mathbf{s}}\_{1(j)} - k\_5 \tilde{\mathbf{x}}\_{4(j)} (\dot{\mathbf{s}}\_{2(j)} + \dot{\mathbf{x}}\_{1ref}) - k\_6 \boldsymbol{\mu}\_{2(j)}] \\ \tilde{\boldsymbol{x}}\_{4(j+1)} &= \frac{\ddot{\mathbf{s}}\_{2(j)} - k\_3 (\dot{\mathbf{s}}\_{2(j)} + \dot{\mathbf{x}}\_{1ref}) + \ddot{\mathbf{x}}\_{1ref}}{k\_1 \ddot{\mathbf{x}}\_{3(j)} + k\_2} \quad j = 1, 2, \dots \end{aligned} \tag{61}$$

Where, *j* is the *j* -th sample point of system, *j* 1 is the next sample point. Through the above recursive equation, current estimation 3 *x* and 4 *x* are obtained. Take these estimation into the control system so that save the sensors. Thereby, system become more simple and reliability.

In the simulation, we use the DutyMAX95-BSC060300 permanent magnetic synchronous motor. The parameters of motor are 3 *P* , *R* 3.3 , 0.027 *L H <sup>d</sup>* , 0.0034 *L H <sup>q</sup>* ,

0.341 *<sup>f</sup> Wb* , *B Nms* 0.0034 , <sup>2</sup> *J kg m* 0.0037 . A phase current of the maximum accepted value is 6.0*A* , the load torque maximum value is 6*N m* , and angular velocity is 3000*rpm* .

The parameter of controller are 1 5 and 2 50 ; the parameter of sliding mode differentiator are 01 2 , 11 1.5 , 02 25 , 12 25 , 22 33 and 32 500 .

Fig. 7. Position tracking and error curve of PMSM.

248 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Generally speaking, in the actual system not all the state variables is measurable. Sometimes, due to the limitation of condition, some state variables can not be measured. Therefore, it requires the controller can estimate state variables of system as possible as

PMSM only uses the position sensors, taking the use of high order sliding mode control techniques, so that its speed and the current state variable are estimated online. In this way, it avoids the use of other sensors, at the same time ensures the motor position tracking

1 43 524 62 3*ref s kx kxx ku x*

2 21*ref sxx*

2 1 1 13 2 4 32 1 ( ) *ref ref*

From the above equations, we can calculate the speed estimation of synchronous motor

3 1 54 2 1 62

*x s kx s x ku*

<sup>1</sup> [ ( )]

*ref*

*ref ref*

2 32 1 1

*kx k*

3( 1) 1( ) 5 4( ) 2( ) 1 6 2( )

*j j j j ref j*

*j j ref ref*

Where, *j* is the *j* -th sample point of system, *j* 1 is the next sample point. Through the above recursive equation, current estimation 3 *x* and 4 *x* are obtained. Take these estimation into the control system so that save the sensors. Thereby, system become more simple and

In the simulation, we use the DutyMAX95-BSC060300 permanent magnetic synchronous motor. The parameters of motor are 3 *P* , *R* 3.3 , 0.027 *L H <sup>d</sup>* , 0.0034 *L H <sup>q</sup>* ,

 *<sup>f</sup> Wb* , *B Nms* 0.0034 , <sup>2</sup> *J kg m* 0.0037 . A phase current of the maximum accepted value is 6.0*A* , the load torque maximum value is 6*N m* , and angular velocity is

 25 , 22 

50 ; the parameter of sliding mode

500 .

33 and 32

5 and 2

 25 , 12 

<sup>1</sup> [ ( )]

( ) 1,2,

2( ) 3 2( ) 1 1

*s ks x x x j kx k*

*x s kx s x ku*

 1.5 , 02 

1 3( ) 2

*j*

*s ks x x*

( )

13 2

4

For calculating 3 *x* and 4 *x* , considering sliding mode variables 1*s* , 2*s* , 2

*k*

4

4

*k*

4( 1)

*j*

*x*

*s x x kx k x kx x* (59)

(60)

(61)

*s* and *u*2 as known

accurate.

progress.

reliability.

3000*rpm* .

0.341

differentiator are 01

In the design of controller, we have obtained that

<sup>221</sup>*ref xsx* , so the current estimation expressed by

value, adopt recursive algorithm to get

The parameter of controller are 1

 2 , 11 

From the figure 7 above, it can be seen that the permanent magnet synchronous motor control system has good performance. This figure shows the permanent magnet synchronous motor can precisely track the given position. And the error between reference and the actual position feedback is shown in Figure 7 below. The maximal error does not exceed to 0.08 rad.

Fig. 8. Speed estimation and error curve of PMSM.

The figure 8 above shows the motor angular speed by derivative of the motor's angular position. The figure 8 below shows the error between the estimation of the electrical angular speed.

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 251

order sliding mode surface, which makes the lower sliding mode surface smooth. That is the

This subsection focuses on a state estimation of PMSM online. In the practical systems, not all the state variables are measurable, or because of objective reasons they are often not easy to measure. In this section, we just use the motor position sensor, through the high order sliding mode control with differentiator, to achieve the state variables of motor estimation online. The simulation results show that the PMSM control system has good dynamic performance, while the electrical angular speed and d-q axis current are

In high precious servo control, the disturbance load will impact servo control. Therefore, the estimation of the disturbance load is very necessary to reduce its influence. Usually in the actual system, the disturbance load torque is often random and uncertain. So, this requires the controller can estimate the value of state variables as accurately as possible. This section will use the arbitrary order sliding mode differentiator, to calculate the high order derivative of sliding mode variables online, so as to avoid the complexity of differential calculation. Then, through the expression of the unknown disturbance load torque, it is estimated. Take the estimation as system input, thereby enhancing the system performance. In the simulation, the position and current sensors of PMSM are used. Adopt high order sliding mode control, its disturbance torque is estimated online. Then, the unknown uncertain external disturbance torque can be entered as a known value so that improve the

In order to facilitate the description, the mathematical model of motor still use system state equation (31) in d-q axis coordinate. The meaning of the parameters remains unchanged.

> 3 4 22 [( ) ] *T p L L x x Bx Jx l dq f*

is the estimation of external torque. 2 *<sup>x</sup>* and 2 *<sup>x</sup>*

2 11 21 *ref*

Therefore, the above equation (63) can solve the estimation of angular speed and angular

221 221

Until now, if we can get the 1 and 2 order derivative of sliding mode variable 2*s* , the

*xsx xsx* 

*sxx xx*

*ref ref*

*ref ref*

(62)

(63)

(64)

can be solved. According to the

*s* in equation (64) can be

are the estimation of

Then, from the mathematical model of PMSM, the following solution can easily get

From the controller of previous section, the following expression can be obtained

221

*sxx*

reason why high order sliding mode control can eliminate the chattering.

estimated precisely.

**2.2.3 Torque disturbance identification**

motor position tracking accuracy.

angular speed and angular acceleration respectively.

estimation of state variable 2 *x* and its differentiation 2 *x*

estimated by the output of differentiator <sup>12</sup> *z* and <sup>22</sup> *z* .

principle of high order sliding mode differentiator, <sup>2</sup> *s* and <sup>2</sup>

In equation (62), *Tl*

acceleration.

Fig. 9. The direct/quadrature axis currents and their estimations.

For permanent magnet synchronous motor, its angular position, speed and current are system state values. The figure 9(a), (b), (c), (d) shows that the estimated value and actual current value of direct axis and quadrature axis respectively. The figure 9(e), (f) are the error between actual current value and the estimated value. In this figure, the error of direct axis current is between 1.0 10 ( ) <sup>4</sup> *A* , and the error of quadrature axis is between 1.0 10 ( ) <sup>3</sup> *A* .

Fig. 10. System sliding mode variables curve.

The figure 10 is the convergence curve of sliding mode variable and its high order derivatives. From the figure we can clearly see that the discrete control law acts on the high order sliding mode surface, which makes the lower sliding mode surface smooth. That is the reason why high order sliding mode control can eliminate the chattering.

This subsection focuses on a state estimation of PMSM online. In the practical systems, not all the state variables are measurable, or because of objective reasons they are often not easy to measure. In this section, we just use the motor position sensor, through the high order sliding mode control with differentiator, to achieve the state variables of motor estimation online. The simulation results show that the PMSM control system has good dynamic performance, while the electrical angular speed and d-q axis current are estimated precisely.

### **2.2.3 Torque disturbance identification**

250 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

iq-estimation (A)

iq (A)





> -30 -20 -10 0 10 20 30

s(3)

The figure 10 is the convergence curve of sliding mode variable and its high order derivatives. From the figure we can clearly see that the discrete control law acts on the high

s'

*s*

0 2 x 10-3

error (A)

For permanent magnet synchronous motor, its angular position, speed and current are system state values. The figure 9(a), (b), (c), (d) shows that the estimated value and actual current value of direct axis and quadrature axis respectively. The figure 9(e), (f) are the error between actual current value and the estimated value. In this figure, the error of direct axis current is between 1.0 10 ( ) <sup>4</sup> *A* , and the error of quadrature axis is between 1.0 10 ( ) <sup>3</sup> *A* .

0 5 10

(d)

0 5 10

(f)

0 5 10

time (second)

0 5 10

0 5 10

time (second)

(b)

0 5 10

(c)

0 5 10

(e)

0 5 10

time (second)

Fig. 9. The direct/quadrature axis currents and their estimations.

0 5 10

0 5 10

*ss*

time (second)

(a)





> -4 -3 -2 -1 0 1

Fig. 10. System sliding mode variables curve.

s

''

s

*s*

0 1 x 10-4

error (A)

id-estimation (A)

id (A)

In high precious servo control, the disturbance load will impact servo control. Therefore, the estimation of the disturbance load is very necessary to reduce its influence. Usually in the actual system, the disturbance load torque is often random and uncertain. So, this requires the controller can estimate the value of state variables as accurately as possible. This section will use the arbitrary order sliding mode differentiator, to calculate the high order derivative of sliding mode variables online, so as to avoid the complexity of differential calculation. Then, through the expression of the unknown disturbance load torque, it is estimated. Take the estimation as system input, thereby enhancing the system performance. In the simulation, the position and current sensors of PMSM are used. Adopt high order sliding mode control, its disturbance torque is estimated online. Then, the unknown uncertain external disturbance torque can be entered as a known value so that improve the motor position tracking accuracy.

In order to facilitate the description, the mathematical model of motor still use system state equation (31) in d-q axis coordinate. The meaning of the parameters remains unchanged. Then, from the mathematical model of PMSM, the following solution can easily get

$$
\tilde{T}\_l = p[(L\_d - L\_q)\mathbf{x}\_3 + \boldsymbol{\nu}\_f]\mathbf{x}\_4 - B\tilde{\mathbf{x}}\_2 - I\tilde{\dot{\mathbf{x}}}\_2 \tag{62}
$$

In equation (62), *Tl* is the estimation of external torque. 2 *<sup>x</sup>* and 2 *<sup>x</sup>* are the estimation of angular speed and angular acceleration respectively.

From the controller of previous section, the following expression can be obtained

$$\begin{aligned} \dot{\mathbf{s}}\_2 &= \mathbf{x}\_2 - \dot{\mathbf{x}}\_{1ref} \\ \ddot{\mathbf{s}}\_2 &= \ddot{\mathbf{x}}\_1 - \ddot{\mathbf{x}}\_{1ref} = \dot{\mathbf{x}}\_2 - \ddot{\mathbf{x}}\_{1ref} \end{aligned} \tag{63}$$

Therefore, the above equation (63) can solve the estimation of angular speed and angular acceleration.

$$\begin{aligned} \tilde{\mathbf{x}}\_2 &= \dot{\mathbf{s}}\_2 + \dot{\mathbf{x}}\_{1ref} \\ \tilde{\dot{\mathbf{x}}}\_2 &= \ddot{\mathbf{s}}\_2 + \ddot{\mathbf{x}}\_{1ref} \end{aligned} \tag{64}$$

Until now, if we can get the 1 and 2 order derivative of sliding mode variable 2*s* , the estimation of state variable 2 *x* and its differentiation 2 *x* can be solved. According to the principle of high order sliding mode differentiator, <sup>2</sup> *s* and <sup>2</sup> *s* in equation (64) can be estimated by the output of differentiator <sup>12</sup> *z* and <sup>22</sup> *z* .

$$\begin{aligned} \dot{z}\_{02} &= \nu\_0 \quad \nu\_0 = -130 \left| z\_{02} - z\_2 \right|^{2/3} \text{sgn}(z\_{02} - z\_2) + z\_{12} \\ \dot{z}\_{12} &= \nu\_1 \quad \nu\_1 = -150 \left| z\_{12} - \nu\_0 \right|^{1/2} \text{sgn}(z\_{12} - \nu\_0) + z\_{22} \\ \dot{z}\_{22} &= -500 \left| z\_{22} - \nu\_1 \right| \end{aligned} \tag{65}$$

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 253

The dSPACE is a equipment of control exploitation and test system based on MATLAB/Simulink that is from Germany. It implements seamless link with the MATLAB/Simulink completely. It can complete the control algorithm design, test and implementation, overcoming the shortage of the traditional control system, for example, the difficult to achieve the complex algorithm and the long development cycle. It has

Taking DS1005PPC control board as the core, with DS2001AD acquisition board, DS2002/2003 multi-channel AD acquisition board, CP4002 Multi-I/O board, DS2102DA output board, DS3002 incremental encoder interface board, we constitute a standard component hardware parts of dSPACE DS1005 system, which is used in this experiment. After the completion of the experimental platform, the development steps of control system

1. MATLAB/Simulink modeling and off-line simulation. Take use of MATLAB/Simulink to establish a mathematical model for the simulation object, and design control

2. Input/output interface (I/O) experimental model. In the MATLAB/Simulink environment, we need to retain module that is downloaded to the dSPACE. Select the real-time control required for I/O modules from the RTI library. Replace the original connection relationship with the hardware interface, and configure I/O parameters. In some special cases, we also need to set up hardware and software interrupt priority

programs. At the same time, complete the system off-line simulation.

Fig. 12. Position tracking curve with load torque disturbance.

advantages of high speed, ease to use and user-friendly.

for PMSM based on the dSPACE include the following points:

**2.4 Experimental results and analysis** 

levels.

Fig. 11. External disturbance load torque and its estimation of motor. Available into the equation (62).

$$\tilde{T}\_l = p\mathbf{j}\{(L\_d - L\_q)\mathbf{x}\_3 + \boldsymbol{\nu}\_f\}\mathbf{x}\_4 - B(z\_{12} + \dot{\mathbf{x}}\_{1ref}) - \mathbf{j}(z\_{22} + \ddot{\mathbf{x}}\_{1ref})\tag{66}$$

Through calculation online, the estimation of disturbance load is obtained. Take the estimated value into the control system so that the uncertain disturbance load become the determine input. In this case, the system performance is improved effectively.

From the figure 11 we can see that, taking use of high order sliding mode with differentiator, disturbance load torque get a better estimation. Disturbance torque is estimated online successfully so that it is no longer unknown uncertainties factor. It also improves the performance of the system. The maximum torque value is 2*N m* in the figure. The sliding mode variable converges into origin at the 0.25s.

The figure 12 shows that the actual angular position track reference of PMSM with the disturbance load. From the figure we can see that the maximum error of the angular position is not more than 0.12 rad. The system gets a better control performance.

Fig. 12. Position tracking curve with load torque disturbance.

### **2.4 Experimental results and analysis**

252 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

02 0 0 02 2 02 2 12

 

*z zs zsz z z zz*

Fig. 11. External disturbance load torque and its estimation of motor. Available into the

determine input. In this case, the system performance is improved effectively.

position is not more than 0.12 rad. The system gets a better control performance.

The sliding mode variable converges into origin at the 0.25s.

3 4 12 1 22 1 [( ) ] ( ) ( ) *T p L L x x Bz x Jz x l d q f*

Through calculation online, the estimation of disturbance load is obtained. Take the estimated value into the control system so that the uncertain disturbance load become the

From the figure 11 we can see that, taking use of high order sliding mode with differentiator, disturbance load torque get a better estimation. Disturbance torque is estimated online successfully so that it is no longer unknown uncertainties factor. It also improves the performance of the system. The maximum torque value is 2*N m* in the figure.

The figure 12 shows that the actual angular position track reference of PMSM with the disturbance load. From the figure we can see that the maximum error of the angular

*ref ref* (66)

, 130| | sgn( ) , 150| | sgn( )

12 1 1 12 0 12 0 22

 

22 22 1

*z z*

 

 

equation (62).

500| |

2/3

1/2

  (65)

The dSPACE is a equipment of control exploitation and test system based on MATLAB/Simulink that is from Germany. It implements seamless link with the MATLAB/Simulink completely. It can complete the control algorithm design, test and implementation, overcoming the shortage of the traditional control system, for example, the difficult to achieve the complex algorithm and the long development cycle. It has advantages of high speed, ease to use and user-friendly.

Taking DS1005PPC control board as the core, with DS2001AD acquisition board, DS2002/2003 multi-channel AD acquisition board, CP4002 Multi-I/O board, DS2102DA output board, DS3002 incremental encoder interface board, we constitute a standard component hardware parts of dSPACE DS1005 system, which is used in this experiment.

After the completion of the experimental platform, the development steps of control system for PMSM based on the dSPACE include the following points:


Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 255

Step 1: Off-line simulation. According to the principle of high order sliding mode control and differentiator, combining with chapter 5 of the application for permanent magnet synchronous motor, the theoretical simulation is researched in the MATLAB/Simulink firstly. In detail, set the sampling frequency and differential equation solution, and save the

Step 2: After the control algorithm verification, remove the inverter model and motor model replaced by the physical prototypes of actual system. And then complete all of the system interface, including the A /D, D/A, I/O, PWM and other interfaces of the dSPACE.

Step 3: Check all connections are correct. After that, start the dSPACE. Compile and download files real-time (RTI) in the environment of MATLAB/Simulink. At this moment, algorithm program code is downloaded to the DSP core program area of dSPACE controller; Step 4: Start the dSPACE/ControlDesk. Create an experimental file .prj in the interface, and design the required .lay layer file. Observe compiler-generated variable file .sdf in order to

Step 5: After the completion of the above, check the status of external devices is good or not. Finally, start bus power, while start system operation in dSPACE/ControlDesk interface.

The controlled object in experiments uses non-salient pole permanent magnet synchronous

Name Value Unit

Resistance *R* 3.052

Inductance 8.4 *Ld mH*

Rating Power *P* 1.0 *kW*

Torque *T* 3.3 *N m*

Pole-pairs 4 *Np* ---

Voltage *U* 300 *V*

Speed *nN* 3000 *r* /min

Rotary inertia *J* 0.00026 <sup>2</sup> *kg m*

Host-computer control surface adopts visual man-machine surface the dSPACE/ControDesk to realize data acquisition and display. Figure 14 is pictorial diagram based on the dSPACE control system. The system consists of inverter, isolation circuit,

Table 2. Parameters of PMSM in experiment d.

detection circuit, power circuit and etc.

Afterwards, compile on-line to generate. cof configuration file;

facilitate observe the real-time dynamic performance of the system;

motor of Delta's ASMT series, whose main parameters are as follows table 2:

iii. Experimental procession:

.mdl model file;

iv. Controlled device:


The figure 13 is a control system in MATLAB/Simulink environment with the dSPACE/RTI module.

Fig. 13. MATLAB/Simulink environment based on dSPACE/RTI control system.



Table 1. The list of experimental equipments.

iii. Experimental procession:

254 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

3. The dSPACE/RTW provides tools to automatically generate code and download. Since MATLAB and dSPACE with seamless connectivity features, a simple operation can complete real-time C code generation, compile, link and download for the target system. In other word, model is downloaded into target board DS1005PPC as running program. 4. The dSPACE integrated experiment and debugging. The dSPACE provides real-time ControlDesk software as well, which changes the parameters and real-time control. The figure 13 is a control system in MATLAB/Simulink environment with the dSPACE/RTI

Fig. 13. MATLAB/Simulink environment based on dSPACE/RTI control system.

2. Test system using high sliding mode control whether it can release chattering

Name Type Unit Amount dSPACE controller DS1005 Dais 1 DC regulated power WYK-303B2 Dais 1 Slide-wire rheostat BX8D-3/7 Dais 3

Universal meter LINI-T/UT58A Dais 1

Industrial computer ADLINK Dais 1

S-100-24 Dais 1

<sup>4</sup>Dais 1

1. Validate feasibility of high order sliding mode control in PMSM;

Ondoscope Tektronix/TDS202

3. Test system using high sliding mode control whether it has robustness.

module.

i. Content and intention:

Switch regulated

Table 1. The list of experimental equipments.

power

phenomenon;

ii. Equipments:

Step 1: Off-line simulation. According to the principle of high order sliding mode control and differentiator, combining with chapter 5 of the application for permanent magnet synchronous motor, the theoretical simulation is researched in the MATLAB/Simulink firstly. In detail, set the sampling frequency and differential equation solution, and save the .mdl model file;

Step 2: After the control algorithm verification, remove the inverter model and motor model replaced by the physical prototypes of actual system. And then complete all of the system interface, including the A /D, D/A, I/O, PWM and other interfaces of the dSPACE. Afterwards, compile on-line to generate. cof configuration file;

Step 3: Check all connections are correct. After that, start the dSPACE. Compile and download files real-time (RTI) in the environment of MATLAB/Simulink. At this moment, algorithm program code is downloaded to the DSP core program area of dSPACE controller; Step 4: Start the dSPACE/ControlDesk. Create an experimental file .prj in the interface, and design the required .lay layer file. Observe compiler-generated variable file .sdf in order to facilitate observe the real-time dynamic performance of the system;

Step 5: After the completion of the above, check the status of external devices is good or not. Finally, start bus power, while start system operation in dSPACE/ControlDesk interface. iv. Controlled device:

The controlled object in experiments uses non-salient pole permanent magnet synchronous motor of Delta's ASMT series, whose main parameters are as follows table 2:


Table 2. Parameters of PMSM in experiment d.

Host-computer control surface adopts visual man-machine surface the dSPACE/ControDesk to realize data acquisition and display. Figure 14 is pictorial diagram based on the dSPACE control system. The system consists of inverter, isolation circuit, detection circuit, power circuit and etc.

Fig. 14. Control system hardware circuit with dSPACE interface.

### v. Waveform:

Because the research of nonlinear system high order sliding mode control theory still is in primary stage, it is face with much challenge. For example, it strictly requires all of the system functions are smooth, and norm-bounded. Otherwise, there is the higher derivative of reference value in control law. In our experiment, 60V DC regulated power is supplied. Experiment is tested under the conditions above. The main test results are following.

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 257

High order sliding mode control law has high derivative of reference signal, so the reference signal must be smooth and continues enough function. For testing speed dynamic response of PMSM in the experiment, the reference signal is set as Fig. 15. Actual measurement of

By comparing Fig. 15 and Fig 16, nonlinear PMSM holds good dynamic tracking character

 Fig. 17. Steady speed of clockwise/ Anti-clockwize displayed in dSPACE/ControlDesk.

> -101.5 -101 -100.5 -100 -99.5 -99 -98.5 -98

? /? ?

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

<sup>2</sup> 2.002 2.004 2.006 2.008 2.01 2.012 2.014 2.016 2.018 2.02 -102

? ? /?

? ? /?

Fig. 18. Speed curve of PMSM in MATLAB/Simulink using traditional sliding mode control. Fig. 17 is steady speed clockwise/anti-clockwise curve of PMSM. It is can be seen that the

The Fig. 18 is outline simulation speed waveform of PMSM using traditional sliding mode control. It displays anti-clockwise speed waveform of PMSM. And the Fig. 19 is outline simulation speed curve of PMSM using high order sliding mode. After partial amplification, comparing with Fig. 18, high order sliding mode control is provided with the ability of avoidance chattering. But, its algorithm is more complicated than tradition. The adjusting

speed dynamic response is shown in Fig. 16.

with high order sliding mode control.


PMSM also takes on good steady performance.



? /? ?

time is longer, too.




0

Fig. 15. Speed reference curve of PMSM.

Fig. 16 Dynamic speed feedback curve.

256 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Because the research of nonlinear system high order sliding mode control theory still is in primary stage, it is face with much challenge. For example, it strictly requires all of the system functions are smooth, and norm-bounded. Otherwise, there is the higher derivative of reference value in control law. In our experiment, 60V DC regulated power is supplied.

Experiment is tested under the conditions above. The main test results are following.

Fig. 14. Control system hardware circuit with dSPACE interface.

v. Waveform:

Fig. 15. Speed reference curve of PMSM.

Fig. 16 Dynamic speed feedback curve.

High order sliding mode control law has high derivative of reference signal, so the reference signal must be smooth and continues enough function. For testing speed dynamic response of PMSM in the experiment, the reference signal is set as Fig. 15. Actual measurement of speed dynamic response is shown in Fig. 16.

By comparing Fig. 15 and Fig 16, nonlinear PMSM holds good dynamic tracking character with high order sliding mode control.

Fig. 17. Steady speed of clockwise/ Anti-clockwize displayed in dSPACE/ControlDesk.

Fig. 18. Speed curve of PMSM in MATLAB/Simulink using traditional sliding mode control.

Fig. 17 is steady speed clockwise/anti-clockwise curve of PMSM. It is can be seen that the PMSM also takes on good steady performance.

The Fig. 18 is outline simulation speed waveform of PMSM using traditional sliding mode control. It displays anti-clockwise speed waveform of PMSM. And the Fig. 19 is outline simulation speed curve of PMSM using high order sliding mode. After partial amplification, comparing with Fig. 18, high order sliding mode control is provided with the ability of avoidance chattering. But, its algorithm is more complicated than tradition. The adjusting time is longer, too.

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 259

Fig. 21. Speed curve of 0.5 *N m* disturbance load in dSPACE/ControlDesk.

The PMSM is a typical complex system because of elevated temperature, saturation, time delay and a good many elements. These reasons lead to the synchronous motor is nonlinear, variation parameter, close coupled system. For the sake of testing high order sliding control, which is insensitive to the parameter uncertainness and disturbance, experiment is injected about 0.5 *N m* external load disturbance at the 0.01 second. The speed actual measurement waveform is shown in Fig. 21. From this figure, speed curve is smooth without flutter. The

Fig. 19. Speed curve of PMSM in MATLAB/Simulink using high order sliding mode control.

To validate experiment intention 2 and 3, show high order sliding mode control with free chattering and robustness, the experiment designs traditional sliding mode controller, too. Simulation and actual measurement are recorded in order to compare with high order sliding mode.

Fig. 20. Speed curve in dSPACE/ControlDesk using traditional/high order sliding mode control.

Due to the traditional sliding mode control uses discontinues control law acting on sliding mode manifold surface, chattering problem is caused. The great of the coefficient in sliding mode control law, the faster of convergence, when the system enter into sliding mode, chattering phenomenon is more obvious. The left figure of Fig. 20 is actual measurement speed curve of PMSM, which adopts conventional sliding mode control. The control law is *uK s <sup>i</sup>* sgn , in current loop 1 *K* 5 , in speed loop 2 *K* 8.2 . From the comparison of Fig. 20, we can obtain a conclusion that chattering is released in high order sliding mode.

Hereto, both of simulation and experiment results prove that high order sliding mode control can reduce the chattering phenomenon which exists in conventional. Following experiment will test the robustness of high order sliding mode control.

258 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

**2 2.002 2.004 2.006 2.008 2.01 2.012 2.014 2.016 2.018 2.02**

? ? /?

Fig. 19. Speed curve of PMSM in MATLAB/Simulink using high order sliding mode control. To validate experiment intention 2 and 3, show high order sliding mode control with free chattering and robustness, the experiment designs traditional sliding mode controller, too. Simulation and actual measurement are recorded in order to compare with high order

 Fig. 20. Speed curve in dSPACE/ControlDesk using traditional/high order sliding mode

20, we can obtain a conclusion that chattering is released in high order sliding mode.

experiment will test the robustness of high order sliding mode control.

Due to the traditional sliding mode control uses discontinues control law acting on sliding mode manifold surface, chattering problem is caused. The great of the coefficient in sliding mode control law, the faster of convergence, when the system enter into sliding mode, chattering phenomenon is more obvious. The left figure of Fig. 20 is actual measurement speed curve of PMSM, which adopts conventional sliding mode control. The control law is *uK s <sup>i</sup>* sgn , in current loop 1 *K* 5 , in speed loop 2 *K* 8.2 . From the comparison of Fig.

Hereto, both of simulation and experiment results prove that high order sliding mode control can reduce the chattering phenomenon which exists in conventional. Following

**-100.05 -100 -99.95 -99.9**




? /? ?

sliding mode.

control.




0

Fig. 21. Speed curve of 0.5 *N m* disturbance load in dSPACE/ControlDesk.

The PMSM is a typical complex system because of elevated temperature, saturation, time delay and a good many elements. These reasons lead to the synchronous motor is nonlinear, variation parameter, close coupled system. For the sake of testing high order sliding control, which is insensitive to the parameter uncertainness and disturbance, experiment is injected about 0.5 *N m* external load disturbance at the 0.01 second. The speed actual measurement waveform is shown in Fig. 21. From this figure, speed curve is smooth without flutter. The

Robust High Order Sliding Mode Control of Permanent Magnet Synchronous Motors 261

Levant A. (1993). Sliding order and sliding accuracy in sliding mode control. *International* 

Levant A. (1998). Robust exact differentiation via sliding mode technique. *Automatica*,

Levant A., Pridor A., Ben-Asher J.Z., Gitizadeh R. & Yaesh I. (2000). 2-sliding mode

Levant A. (2003). Higher-order sliding modes, differentiation and output feedback

Bartolini G.; Ferrara A. & Usai E. (1997). Application of a sub-optimal discontinuous control

Bartolini G., Ferrara A. & Giacomini. (1999). A robust control design for a class of uncertain

Bartolini G., Ferrara A. & Punta E. (2000). Multi-input second order sliding mode hybrid

Fridman L. & Levant A. (2002). High order sliding mode. In: *Sliding Mode Control in* 

Utkin V.I. (1992). *Sliding modes in control and optimization*. Springer-Verlag, ISBN: 3540535

Glumineau A., Hamy M., Lanier C. & Moog C. (1993). Robust control of a brushless servo

Ziribi M., Ramirez Sira H. & Ngai A. (2001). Static and dynamic sliding mode control

Caravani P. & Gennaro S. D. (1998). Robust control of synchronous motor with non-linear

Yan Y., Zhu J., Guo Y., & Lu H. (2006). Modeling and simulation of direct torque controlled

Poznyak A. S. & Medel J. (1999). Matrix forgetting with adaptation. *International Journal of* 

Poznyak, A. S. (1999) Matrix forgetting factor. *International Journal of Systems Science*, vol.30,

Floret-Pontet F. & Lamnabhi-Lagarrigue F. (2001). Parameter identification methodology

*Systems Science*, vol.30, No.(8), pp. 865–878, ISSN :0020-7721.

implementation in aircraft pitch control. *Journal of Guidance Control and Dynamics*,

control. *International Journal of Control*, Vol.76, No.9, pp.924-941, ISSN: 0020-

algorithm for uncertain second order systems. *International Journal of Robust and* 

nonlinear systems featuring a second order sliding mode. *International Journal of* 

control of constrained manipulators. *Dynamics and Control*, Vol. 10, pp.277-296,

*Engineering,* W. Perruquetti and J.P. Barbot, editors, pp.53-102. Marcel Dekker,

motor via sliding mode. *International Journal of Control*, vol.58, pp.979–990, ISSN:

schemes for a permanent magnet stepper motor. *International Journal of Control*,

and parameter uncertainties. *Automatica*, vol.34, pp.445-450, ISSN: 0005-

pmsm drive system incorporating structural and saturation saliencies. *Industry Applications Conference*. *41st IAS Annual Meeting.*, vol.1, pp. 76–83, Oct.

using sliding mode observers. *International Journal of Control*, vol.74, pp.1743-1753,

*Journal of Control*, Vol. 58, No.6, pp.1247-1263, ISSN: 0020-7179

*Nonlinear Control*, Vol. 7, No.4, pp.299-319, ISSN: 1049-8923

*Control*, Vol. 72, No.4, pp.321-331, ISSN: 0020-7179

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7179

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ISBN: 0824706714, New York

vol.74, No.2, pp.103-117, ISSN: 0020-7179.

No.2, pp.165-174, ISSN :0020-7721.

experiment result illuminates high order sliding mode control reserves robustness of conventional sliding mode.

Experimentally verified, high order sliding mode control provides an effective method to improve accuracy and robustness further for nonlinear systems

### **3. Conclusion**

This chapter applies the research of nonlinear control and high order sliding mode control theory in PMSM control, and achieves robust control for a PMSM in spit of the internal parameter uncertainties and unknown external disturbance load torque. The simulation results show good performance; in addition, the estimation online of system state variables is also one of the hot issues in the control field. In this chapter, a new design based on high order sliding mode with differentiator for PMSM, access to the state variable estimation; Besides, unknown uncertain load impacts the performance of motor control. In order to improve system performance, this chapter also achieves external disturbance load estimation online. It makes sure the load can be accurately estimated.

This chapter described dSPACE physics experiment control platform the build and development process in detail. Through the dSPACE real-time control platform, the nonlinear high order sliding mode control theory research is applied to the control of permanent magnet synchronous motor. The experimental results and simulation results are consistently indicate that synchronous motor has better dynamic performance and steady accuracy, proves the feasibility of this technology in practical application systems; It is also verified by high order sliding mode control technique that preserves the robustness of traditional sliding mode control. The high order sliding mode essentially eliminates the chattering caused by discrete control law. From another point of view, the simulation and physical experiment provide a certain reference value for the nonlinear systems high order sliding mode control further application.

### **4. Acknowledgment**

I would like to express my gratitude to all those who helped me during the writing. A special acknowledgement should be shown to Professor Salah Laghrouche, from whose lectures I benefited greatly. I am particularly indebted to Prof. Liu Weiguo, who gave me kind encouragement and useful instructions all through my writing. Finally I wish to extend my thanks to my university. This chapter is supported by the Basic Research Foundation of Northwestern Polytechnical University.

### **5. References**


260 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

experiment result illuminates high order sliding mode control reserves robustness of

Experimentally verified, high order sliding mode control provides an effective method to

This chapter applies the research of nonlinear control and high order sliding mode control theory in PMSM control, and achieves robust control for a PMSM in spit of the internal parameter uncertainties and unknown external disturbance load torque. The simulation results show good performance; in addition, the estimation online of system state variables is also one of the hot issues in the control field. In this chapter, a new design based on high order sliding mode with differentiator for PMSM, access to the state variable estimation; Besides, unknown uncertain load impacts the performance of motor control. In order to improve system performance, this chapter also achieves external disturbance load

This chapter described dSPACE physics experiment control platform the build and development process in detail. Through the dSPACE real-time control platform, the nonlinear high order sliding mode control theory research is applied to the control of permanent magnet synchronous motor. The experimental results and simulation results are consistently indicate that synchronous motor has better dynamic performance and steady accuracy, proves the feasibility of this technology in practical application systems; It is also verified by high order sliding mode control technique that preserves the robustness of traditional sliding mode control. The high order sliding mode essentially eliminates the chattering caused by discrete control law. From another point of view, the simulation and physical experiment provide a certain reference value for the nonlinear systems high order

I would like to express my gratitude to all those who helped me during the writing. A special acknowledgement should be shown to Professor Salah Laghrouche, from whose lectures I benefited greatly. I am particularly indebted to Prof. Liu Weiguo, who gave me kind encouragement and useful instructions all through my writing. Finally I wish to extend my thanks to my university. This chapter is supported by the Basic Research Foundation of

Emelyanov, S.V.; Korovin, S.K. & Levantovskiy, A. (1990). New class of second order sliding

Emelyanov, S.V.; Korovin, S.K., & Levant, A. (1996). High-order sliding modes in control

systems. *Computational mathematics and modeling*, Vol.7, No.3, pp.294-318, ISSN:

algorithm. *Mathematical modeling*, Vol.2, No.3, pp. 85-100

improve accuracy and robustness further for nonlinear systems

estimation online. It makes sure the load can be accurately estimated.

sliding mode control further application.

Northwestern Polytechnical University.

**4. Acknowledgment** 

**5. References** 

1046-283X

conventional sliding mode.

**3. Conclusion** 


Koshkouei A. & Zinober A. (2002). Sliding mode observers for a class of nonlinear systems. *Proceedings of American Control Conference*, vol. 8, pp. 2106 – 2111, Anchorage, Alaska, USA, 2002.

**14** 

*Algeria* 

**Sliding Controller** 

*1University of Mascara,* 

**of Switched Reluctance Motor** 

Ahmed Tahour1 and Abdel Ghani Aissaoui2

*2IRECOM Laboratory, University of Sidi Bel Abbes,* 

Switched reluctance motors (SRMs) can be applied in many industrial applications due to their cost advantages and ruggedness. The switched reluctance motor is simple to construct. It is not only features a salient pole stator with concentrated coils, which allows earlier winding and shorter end turns than other types of motors, but also features a salient pole rotor, which has no conductors or magnets and is thus the simplest of all electric machine rotors. Simplicity makes the SRM inexpensive and reliable, and together with its high speed capacity and high torque to inertia ratio, makes it a superior choice in

The dynamics of these systems are highly nonlinear and their models inevitable contain parametric uncertainties and unmodeled dynamics. The application of non linear robust control techniques is a necessity for successful operation electrical system. The industrial applications necessitate speed/position variators having high dynamic performances, a good precision in permanent regime, a high capacity of overload and robustness to the different perturbations. Thus, the recourse to robust control algorithms is desirable in

Variable structure control with sliding mode, is one of the effective non linear robust control approaches. Sliding Mode Control (SMC) has attracted considerable attention because it provides a systematic approach to the problem of maintaining stability. It has been studied extensively to tackle problems of the nonlinear dynamic control systems. The sliding mode control can offer many good properties such as good performance against unmodelled dynamics systems, insensitivity to parameter variation, external disturbance rejection and

Sliding mode control has long proved its interests. Among them, relative simplicity of design, control of independent motion (as long as sliding conditions are maintained), invariance to process dynamics characteristics and external perturbations, wide variety of operational modes such as regulation, trajectory control [1], model following [2] and

However, the motor is highly nonlinear and operates in saturation to maximize the output torque. Moreover, the motor torque is a nonlinear function of current and rotor position. This highly coupled nonlinear and complex structure of the SRM make the design of the

**1. Introduction** 

different applications.

fast dynamic [5, 9].

observation [3].

controller difficult [4].

stabilization and in tracking trajectories [1, 2].

## **Sliding Controller of Switched Reluctance Motor**

Ahmed Tahour1 and Abdel Ghani Aissaoui2 *1University of Mascara, 2IRECOM Laboratory, University of Sidi Bel Abbes, Algeria* 

### **1. Introduction**

262 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Koshkouei A. & Zinober A. (2002). Sliding mode observers for a class of nonlinear systems.

Alaska, USA, 2002.

*Proceedings of American Control Conference*, vol. 8, pp. 2106 – 2111, Anchorage,

Switched reluctance motors (SRMs) can be applied in many industrial applications due to their cost advantages and ruggedness. The switched reluctance motor is simple to construct. It is not only features a salient pole stator with concentrated coils, which allows earlier winding and shorter end turns than other types of motors, but also features a salient pole rotor, which has no conductors or magnets and is thus the simplest of all electric machine rotors. Simplicity makes the SRM inexpensive and reliable, and together with its high speed capacity and high torque to inertia ratio, makes it a superior choice in different applications.

The dynamics of these systems are highly nonlinear and their models inevitable contain parametric uncertainties and unmodeled dynamics. The application of non linear robust control techniques is a necessity for successful operation electrical system. The industrial applications necessitate speed/position variators having high dynamic performances, a good precision in permanent regime, a high capacity of overload and robustness to the different perturbations. Thus, the recourse to robust control algorithms is desirable in stabilization and in tracking trajectories [1, 2].

Variable structure control with sliding mode, is one of the effective non linear robust control approaches. Sliding Mode Control (SMC) has attracted considerable attention because it provides a systematic approach to the problem of maintaining stability. It has been studied extensively to tackle problems of the nonlinear dynamic control systems. The sliding mode control can offer many good properties such as good performance against unmodelled dynamics systems, insensitivity to parameter variation, external disturbance rejection and fast dynamic [5, 9].

Sliding mode control has long proved its interests. Among them, relative simplicity of design, control of independent motion (as long as sliding conditions are maintained), invariance to process dynamics characteristics and external perturbations, wide variety of operational modes such as regulation, trajectory control [1], model following [2] and observation [3].

However, the motor is highly nonlinear and operates in saturation to maximize the output torque. Moreover, the motor torque is a nonlinear function of current and rotor position. This highly coupled nonlinear and complex structure of the SRM make the design of the controller difficult [4].

Sliding Controller of Switched Reluctance Motor 265

The switched reluctance motor has a simple construction, but the solution of its mathematical models is relatively difficult due to its dominant non-linear behaviour. The flux linkage is a function of two variables, the current I and the rotor position (angle *θ*). The instantaneous voltage across the terminals of a phase of an SR motor winding is related

*j j*

effects, the flux linked in an SRM phase varies as a function of rotor position

*i i di V RI <sup>j</sup> i dt* 

 

*V RI*

(, ) *<sup>j</sup>*

(, ) (, ) 1,....3 *j j*

, the instantaneous inductance, and term

(2)

 

While excluding saturation and mutual inductance effects, the flux in each phase is given by

(, ) () *jj j* 

 *i Li* 

(1)

*t* 

is the

(3)

 

and the

*i*

Because of the double salience construction of the SR motor and the magnetic saturation

*t*

Fig. 2. Control of SRM.

**2.2 Machine equation** 

With *j* 1,....3

In which:

where *<sup>j</sup>*

*i* 

the linear equation

instantaneous back e.m.f.

to the flux linked in the winding by Faraday's law as:

phase current. Equation (1) can be expanded as

*j j*

*<sup>t</sup>*

is defined as *L i* ( ,)

Section 2, investigates a case study of sliding mode control. In a more general study, the third section develops sliding mode controllers for switched reluctance motor drive; the proposed controller is described, and used to control the speed of the switched reluctance motor. Simulation results are given to show the effectiveness of this controller. Conclusions are summarized in the last section.

### **2. SRM model**

### **2.1 Description of the system**

In a switched reluctance machine, only the stator presents windings, while the rotor is made of steel laminations without conductors or permanent magnets. This very simple structure reduces greatly its cost. Motivated by this mechanical simplicity together with the recent advances in the power electronics components, much research has being developed in the last decade. The SRM, when compared with the AC and DC machines, shows two main advantages:


The switched reluctance machine motion is produced because of the variable reluctance in the air gap between the rotor and the stator. When a stator winding is energized, producing a single magnetic field, reluctance torque is produced by the tendency of the rotor to move to its minimum reluctance position [5].

A cross-sectional view is presented in figure 1.

Fig. 1. Switched reluctance motor.

The schematic diagram of the speed control system under study is shown in figure 2. The power circuit consists with the *H*-bridge asymmetric type converter whose output is connected to the stator of the switched reluctance machine. Each phase has two IGBTS and two diodes. The parameters of the switched reluctance motor are given in the Appendix [5, 6].

The SMC inputs are obtained by manipulating the speed reference and feedback, while the SMC output is integrated to produce the current reference.

264 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Section 2, investigates a case study of sliding mode control. In a more general study, the third section develops sliding mode controllers for switched reluctance motor drive; the proposed controller is described, and used to control the speed of the switched reluctance motor. Simulation results are given to show the effectiveness of this controller. Conclusions

In a switched reluctance machine, only the stator presents windings, while the rotor is made of steel laminations without conductors or permanent magnets. This very simple structure reduces greatly its cost. Motivated by this mechanical simplicity together with the recent advances in the power electronics components, much research has being developed in the last decade. The SRM, when compared with the AC and DC machines, shows two main



The switched reluctance machine motion is produced because of the variable reluctance in the air gap between the rotor and the stator. When a stator winding is energized, producing a single magnetic field, reluctance torque is produced by the tendency of the rotor to move

The schematic diagram of the speed control system under study is shown in figure 2. The power circuit consists with the *H*-bridge asymmetric type converter whose output is connected to the stator of the switched reluctance machine. Each phase has two IGBTS and two diodes.

The SMC inputs are obtained by manipulating the speed reference and feedback, while the

The parameters of the switched reluctance motor are given in the Appendix [5, 6].

SMC output is integrated to produce the current reference.

magnetically, and electrically from the other machine phases;

are summarized in the last section.

**2.1 Description of the system** 

or magnets on the rotor;

to its minimum reluctance position [5].

Fig. 1. Switched reluctance motor.

A cross-sectional view is presented in figure 1.

**2. SRM model** 

advantages:

Fig. 2. Control of SRM.

#### **2.2 Machine equation**

The switched reluctance motor has a simple construction, but the solution of its mathematical models is relatively difficult due to its dominant non-linear behaviour. The flux linkage is a function of two variables, the current I and the rotor position (angle *θ*).

The instantaneous voltage across the terminals of a phase of an SR motor winding is related to the flux linked in the winding by Faraday's law as:

$$V\_j = RI\_j + \frac{\partial \Psi\_j(i, \theta)}{\partial t} \tag{1}$$

With *j* 1,....3

Because of the double salience construction of the SR motor and the magnetic saturation effects, the flux linked in an SRM phase varies as a function of rotor position and the phase current. Equation (1) can be expanded as

$$dV\_j = RI\_j + \frac{\partial \Psi\_j(i, \theta)}{\partial \mathbf{i}} \frac{d\mathbf{i}}{dt} + \frac{\partial \Psi\_j(i, \theta)}{\partial \theta} \alpha \mathbf{o} \qquad j = 1, \dots, 3 \tag{2}$$

In which:

$$
\rho = \frac{\partial \theta}{\partial t}
$$

where *<sup>j</sup> i* is defined as *L i* ( ,) , the instantaneous inductance, and term *t* is the instantaneous back e.m.f.

While excluding saturation and mutual inductance effects, the flux in each phase is given by the linear equation

$$\mathbf{^j\Psi}\_{j}(\theta,\mathbf{i}\_{j}) = \mathbf{L}(\theta)\mathbf{i}\_{j} \tag{3}$$

It can be written as

$$V\_j = RI\_j + L(\theta) \frac{\partial \mathbf{i}}{\partial t} + \mathbf{i} \frac{\partial L(\theta)}{\partial \theta} \alpha \qquad \mathbf{j} = \mathbf{1}, \dots \mathbf{3} \tag{4}$$

Sliding Controller of Switched Reluctance Motor 267

With the bounded function *f*(x) 0 0 *f* ( ) *x f f cons t* tan and the control as a relay function

have different signs if 0 0

<sup>2</sup> ( ) ( ) ( )( ( ) *eq c dc c <sup>L</sup> P t I t VIt* 

> ( ) () *dc e c <sup>V</sup> T It*

Where *Kt* is a proportional torque constant and ( ) *<sup>t</sup> I t* is the equivalent dc-link current

The electromagnetic dynamic model of a switched reluctance motor and loads can be

*t J*

 ( ) *T T e l f*

then electromagnetic torque can be further simplified as

*r t*( ) is the reference input and u is given by:

*<sup>r</sup> u f <sup>t</sup>* 

(10)

(11)

( ) *T KI t e tt* (12)

(13)

.

or 0 0 *u u sign e u cons t* ( ) tan

*t*

0 0

(figure(3)) of the tracking error ( ) *<sup>x</sup> e rt*

*u*

The values of *e* and 0 () () *e r <sup>f</sup> x u sign e t t*

The equivalent total phase power becomes [9, 10]

The electromagnetic torque over the switching period is then

Fig. 3. Relay control.

If () ( ) () *c tt*

*dc It K It V* 

providing electromagnetic torque.

expressed as follows [11,12, 13]:

**3.2 Sliding mode controller** 

0 0

*u if e*

*u if e* 

The total energy associated with the three phases ( 3) *n* is given by

$$\mathcal{W}\_{total} = \frac{1}{2} \sum\_{j=1}^{3} L(\theta + (n - j - 1)\theta\_s) I\_j^{(2)}$$

with

$$\theta\_s = 2\pi (\frac{1}{N\_r} - \frac{1}{N\_s}) \tag{5}$$

Each phase inductance displaced by an angle *s* .

The average torque can be written as the superposition of the torque of the individual motor phases:

$$T\_e = \sum\_{plus \approx = 1}^{n} T\_{phase} \tag{6}$$

and the motor total torque by

$$T\_e = \frac{\partial \mathcal{W}\_{total}}{d\theta} = \frac{1}{2} \sum\_{j=1}^{3} \frac{\partial L(\theta + (n-j-1)\theta\_s)}{\partial \theta} I\_j^2 \tag{7}$$

The mechanical equations are

$$J\frac{\partial \phi \alpha}{\partial t} = T\_e - T\_l - f \alpha \tag{8}$$

Where *V* - the terminal voltage, *I* - the phase current, *R* - the phase winding resistance, the flux linked by the winding, *J* - the moment of inertia, *f* - the friction coefficient, *L*( ) - the instantaneous inductance, *Nr* number of rotor poles, *Ns* number of stator poles, *Tl* is the torque load and *Te* is the total torque.

### **3. SRM sliding mode speed controller**

### **3.1 Sliding mode principle**

Sliding modes is phenomenon may appear in a dynamic system governed by ordinary differential equations with discontinuous right-hand sides. It may happen that the control as a function of the system state switches at high frequency, this motion is called sliding mode. It may be enforced in the simplest tracking relay system with the state variable x(t) [7, 8]:

$$\frac{\partial \mathbf{x}}{\partial t} = f(\mathbf{x}) + \mathbf{u} \tag{9}$$

266 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

( ) ( ) 1,....3 *j j i L V RI L i <sup>j</sup> <sup>t</sup>*

<sup>1</sup> ( ( 1) ) <sup>2</sup> *total <sup>s</sup> <sup>j</sup> <sup>j</sup>*

1 1 2( ) *<sup>s</sup> N N r s*

*s* . The average torque can be written as the superposition of the torque of the individual motor

> *n e phase phase T T*

> > 3

2

*d*

*j <sup>W</sup> L nj T I*

1

*e l J TTf <sup>t</sup>*

Where *V* - the terminal voltage, *I* - the phase current, *R* - the phase winding resistance, the flux linked by the winding, *J* - the moment of inertia, *f* - the friction coefficient, *L*( )

instantaneous inductance, *Nr* number of rotor poles, *Ns* number of stator poles, *Tl* is the

Sliding modes is phenomenon may appear in a dynamic system governed by ordinary differential equations with discontinuous right-hand sides. It may happen that the control as a function of the system state switches at high frequency, this motion is called sliding mode. It may be enforced in the simplest tracking relay system with the state variable x(t) [7, 8]:

( ) *<sup>x</sup> <sup>f</sup> x u*

*t* 

*total <sup>s</sup> <sup>e</sup> <sup>j</sup>*

1

1 ( ( 1) )

 

 

1

 

Each phase inductance displaced by an angle

and the motor total torque by

The mechanical equations are

torque load and *Te* is the total torque.

**3.1 Sliding mode principle** 

**3. SRM sliding mode speed controller** 

*W L nj* 

The total energy associated with the three phases ( 3) *n* is given by

 

3 2

 *I*

(4)

(5)

(6)

2

(8)

(9)


 

(7)

It can be written as

with

phases:

With the bounded function *f*(x) 0 0 *f* ( ) *x f f cons t* tan and the control as a relay function (figure(3)) of the tracking error ( ) *<sup>x</sup> e rt t r t*( ) is the reference input and u is given by:

$$\mu = \begin{cases} \mu\_0 & \text{if } \ e > 0 \\ -\mu\_0 & \text{if } \ e < 0 \end{cases} \text{ or } \mu = \mu\_0 \text{sign}(e) \qquad \mu\_0 = \text{constant}$$

Fig. 3. Relay control.

The values of *e* and 0 () () *e r <sup>f</sup> x u sign e t t* have different signs if 0 0 *<sup>r</sup> u f <sup>t</sup>* .

### **3.2 Sliding mode controller**

The equivalent total phase power becomes [9, 10]

$$P\_{eq}(t) = I\_c^2(t) \langle \alpha \frac{\partial L(\theta)}{\partial \theta} = V\_{dc} I\_c(t) \tag{10}$$

The electromagnetic torque over the switching period is then

$$T\_c = (\frac{V\_{dc}}{\alpha})I\_c(t) \tag{11}$$

If () ( ) () *c tt dc It K It V* then electromagnetic torque can be further simplified as

$$T\_e = K\_t I\_t \text{(t)}\tag{12}$$

Where *Kt* is a proportional torque constant and ( ) *<sup>t</sup> I t* is the equivalent dc-link current providing electromagnetic torque.

The electromagnetic dynamic model of a switched reluctance motor and loads can be expressed as follows [11,12, 13]:

$$\frac{\partial \rho o}{\partial t} = \frac{\{T\_e - T\_l - fo\}}{\mathcal{J}} \tag{13}$$

From (11) and (12), (13) can be obtained:

$$\frac{\partial \alpha o}{\partial t} = \frac{(K\_t I\_t(t) - T\_l - f \, o)}{J} \tag{14}$$

Sliding Controller of Switched Reluctance Motor 269

1 1 1 1

*a*

2

*<sup>x</sup> if S <sup>t</sup> <sup>b</sup>*

a and b are proportional and derivative gain constant respectively, and 121 2

The speed regulation of the SRM, despite its mechanical simplicity, is not simple to achieve. In the previous sections, we saw the importance that the values of the commutation angles have to torque oscillations. A linear controller as the PI regulator presents good results for the SRM speed control [8]. However, the controller will be only valid for a given operating point. Therefore, some authors have investigated recently non-linear controllers based on the sliding mode [11,12,16] applied to SRM speed control. In this section, we discuss and illustrate the advantages and drawbacks of the SRM speed control by using a PI regulator

To show the sliding mode controller performances we have simulated the system described in figure 2. The simulation of the starting mode without load is done. The simulation is realized using the SIMULINK software in MATLAB environment. Figure 5 shows the

The saturating function in the PI block diagram is necessary since during the transient, if the current demanded is high and if the speed reference is also high, then the f.e.m. produced will prevent the current to grow. Hence, the maximum current value of the block *saturation* 

2

*if Sx*

*if Sx*

*<sup>x</sup> if S <sup>t</sup>*

0 0

1

1

0

(20)

 *and* are

0

 , , 

Where:

real constants.

Fig. 4. A prescribed sliding line in phase plane.

**4. Simulation result** 

and a sliding mode controller.

has been fixed in 25A.

performances of the sliding mode controller.

Speed control can be implemented by a sliding-mode variable structure controller, but a discontinuous torque control signal would cause chattering of the speed response. In order to enable smooth torque control and reduce the chattering problem ( ) *<sup>t</sup> I t* must be smoothed according to (11). The phase variable state representation of Fig. 4 can be used to develop the required control scheme. It can be simplified as:

$$
\begin{bmatrix}
\frac{\partial \mathbf{x}\_1}{\partial t} \\
\frac{\partial \mathbf{x}\_2}{\partial t}
\end{bmatrix} = \begin{bmatrix}
0 & -\frac{R}{L(\theta)}
\end{bmatrix} \cdot \begin{bmatrix}
\mathbf{x}\_1 \\
\mathbf{x}\_2
\end{bmatrix} + \begin{bmatrix}
0 \\
\frac{1}{L(\theta)}
\end{bmatrix} \mathcal{U} + \begin{bmatrix}
\mathbf{1} \\
0
\end{bmatrix} \cdot \begin{pmatrix}
\mathbf{1} \\
0
\end{pmatrix} \tag{15}
$$

Where 1 *ref x* , *ref* is the demand rotor speed, 2 *ref x II* , and *U* is a control signal which is used to control the current error, irrespective of drive system parameter variations. The sliding line in the phase plane diagram [Fig. 4] can be described as follows:

$$S = \alpha \rho - \alpha\_{r\circ} \tag{16}$$

from the equation (13) and (15) , we can be obtains

$$\frac{\partial \mathbf{S}}{\partial t} = \frac{\mathbf{K}\_t}{\mathbf{J}} I - \frac{f}{\mathbf{J}} \alpha \mathbf{o} - \frac{\mathbf{T}\_l}{\mathbf{J}} - \frac{\partial \alpha\_{ref}}{\partial t} \tag{17}$$

the current of control is given by

$$I\_c = I\_c^{eq} + I\_c^{n}$$

With

$$I\_c^{eq} = \frac{1}{K\_t} (f \frac{\partial o\_{ref}}{\partial t} + fo + T\_l) \tag{18}$$

sgn( ( )) *<sup>n</sup> c w IK S* 

To satisfy the existence condition of the sliding-mode speed controller, the following must be satisfied:

$$\lim\_{S \to 0} S \frac{dS}{dt} \ll = 0 \tag{19}$$

The controller can be designed as follows:

$$\mathbf{U} = a\mathbf{x}\_1 + b\frac{\partial \mathbf{x}\_1}{\partial t}$$

Where:

268 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

( () ) *KI t T f tt l*

(14)

*l*

*ref* (16)

(18)

(19)

(15)

*t J*

Speed control can be implemented by a sliding-mode variable structure controller, but a discontinuous torque control signal would cause chattering of the speed response. In order to enable smooth torque control and reduce the chattering problem ( ) *<sup>t</sup> I t* must be smoothed according to (11). The phase variable state representation of Fig. 4 can be used to develop

1

*<sup>t</sup> J J <sup>x</sup> <sup>T</sup> <sup>U</sup> x x <sup>R</sup> <sup>J</sup> <sup>L</sup>*

which is used to control the current error, irrespective of drive system parameter variations.

*ref t l <sup>S</sup> K T <sup>f</sup> <sup>I</sup> tJ J J t*

*eq n cc c II I*

<sup>1</sup> ( ) *eq ref c l*

> sgn( ( )) *<sup>n</sup> c w IK S*

To satisfy the existence condition of the sliding-mode speed controller, the following must

0 lim 0

1 *<sup>x</sup> U ax b*

*S dS <sup>S</sup> dt* 1

*t* 

*I J fT K t* 

 

*t*

0

*ref* is the demand rotor speed, 2 *ref x II* , and *U* is a control signal

(17)

<sup>1</sup> . <sup>1</sup> . .( ) <sup>0</sup> <sup>0</sup> ( ) ( )

2 2

*x f K*

*t L*

*t*

The sliding line in the phase plane diagram [Fig. 4] can be described as follows:

*S* 

From (11) and (12), (13) can be obtained:

the required control scheme. It can be simplified as:

1

from the equation (13) and (15) , we can be obtains

the current of control is given by

The controller can be designed as follows:

Where 1 *ref x* , 

With

be satisfied:

$$a = \begin{cases} a\_1 & \text{if } S \ge 0 \\ \beta\_1 & \text{if } S \ge 1 \end{cases}$$

$$b = \begin{cases} a\_2 & \text{if } S \frac{\partial \chi\_1}{\partial t} > 0 \\ \beta\_2 & \text{if } S \frac{\partial \chi\_1}{\partial t} < 0 \end{cases} \tag{20}$$

a and b are proportional and derivative gain constant respectively, and 121 2 , , *and* are real constants.

Fig. 4. A prescribed sliding line in phase plane.

### **4. Simulation result**

The speed regulation of the SRM, despite its mechanical simplicity, is not simple to achieve. In the previous sections, we saw the importance that the values of the commutation angles have to torque oscillations. A linear controller as the PI regulator presents good results for the SRM speed control [8]. However, the controller will be only valid for a given operating point. Therefore, some authors have investigated recently non-linear controllers based on the sliding mode [11,12,16] applied to SRM speed control. In this section, we discuss and illustrate the advantages and drawbacks of the SRM speed control by using a PI regulator and a sliding mode controller.

To show the sliding mode controller performances we have simulated the system described in figure 2. The simulation of the starting mode without load is done. The simulation is realized using the SIMULINK software in MATLAB environment. Figure 5 shows the performances of the sliding mode controller.

The saturating function in the PI block diagram is necessary since during the transient, if the current demanded is high and if the speed reference is also high, then the f.e.m. produced will prevent the current to grow. Hence, the maximum current value of the block *saturation*  has been fixed in 25A.

Sliding Controller of Switched Reluctance Motor 271

wants to continue the speed regulation at 90 / *rd s* , it will be necessary to produce a braking torque when the speed error is negative. The controller by sliding mode gives good result compared to that of regulator PI. The increase speed is reduced during the application of a negative torque. However, when the negative torque load is applied at 0,5s *t* , the speed oscillations become significant. Previous results in Fig. 6 showed significant oscillations in

> *on* value.

96,3 /s *rd min* , as shown in Fig. 6. Now, if one

PI Sliding mode

Fig. 6. PI and sliding speed regulation with *Tl* 2Nm applied at 0,6s *t* .

controller are owed to a slower reaction of the current, as shown in figure 7.

Figure 7 shows the very good performances reached by the sliding mode controller. Indeed, one notes that the overshoot is less important in the case of the sliding regulator, with a best

For this test, the sliding controller proves to be well more robust because the speed curve is hardly of its reference. On the other hand, the speed signal evolution obtained with the PI controller deviates about 10% from its reference value (figure 7). The speed tracking is satisfactory, and the torque ripple is low. These results demonstrate the robustness of the drive under unpredictable load conditions. The decreasing speed oscillations with the PI

In order to test the robustness of the proposed control, we have studied the speed

The figure 8 shows the tests of the robustness: a) The robustness tests concerning the

variation of the resistances, b) the robustness tests in relation to inertia variations.

In our case, the minimum speed stays

the speed signal due to an initial bad choice of the

response time without increasing the overshoot.

performances. Two cases are considered:

**4.1 Robustness** 

1. Inertia variation,

2. Stator resistance variation.

Fig. 5.b shows the speed regulation for a reference of 120rad/s , with 0 *on* and 38 *off* . Values of PI parameters *Kp* and *Ki* have been optimized in order to have the best compromise between response time and overshoot. Fig. 5.b shows good results for the speed regulation with weak speed oscillations in the permanent regime. Using a PI controller, a good compromise for *Kp* and *Ki* parameters has been found in order to have weak speed oscillations. The values found were: 0,18 *Kp* and 2,85 *Ki* .However, if a load is applied, the speed oscillations will increase, as illustrated in Fig. 5.b. These results have been obtained with a load of *Tl* 1,5Nm applied at 0,6s *t* .

Fig. 5 shows the very good performance reached by the sliding mode controller. Indeed, one notes that the overshoot is less important in the case of the sliding mode regulator, with a best response time without increasing the overshoot. Follow, we show the robustness of the PI and the sliding mode controller for the same operating condition.

Fig. 5. PI and sliding speed regulation with *Tl* 1,5Nm applied at *t* 0,6s .

Follow, the speed regulation operates in a supplementary quadrant. That means we are going to do a speed regulation with a negative torque load but maintaining the speed reference. Fig. 6 shows the results obtained for a negative load of *Tl* 2Nm also applied at *t* 0,6s .

Fig. 6 shows that the speed regulation is not assured anymore after 0,6s *t* . From (8), the motor speed is given by

$$\alpha = \frac{\{T\_e - T\_l\}}{f} \tag{21}$$

Therefore, the minimum speed for a possible regulation without producing a braking torque, meaning that the regulator will have its reference current to zero, will be

$$
\alpha\_{\min} = \frac{-T\_l}{f} \tag{22}
$$

270 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

 *off* . Values of PI parameters *Kp* and *Ki* have been optimized in order to have the best compromise between response time and overshoot. Fig. 5.b shows good results for the speed regulation with weak speed oscillations in the permanent regime. Using a PI controller, a good compromise for *Kp* and *Ki* parameters has been found in order to have weak speed oscillations. The values found were: 0,18 *Kp* and 2,85 *Ki* .However, if a load is applied, the speed oscillations will increase, as illustrated in Fig. 5.b. These results

Fig. 5 shows the very good performance reached by the sliding mode controller. Indeed, one notes that the overshoot is less important in the case of the sliding mode regulator, with a best response time without increasing the overshoot. Follow, we show the robustness of the

*on* and

Fig. 5.b shows the speed regulation for a reference of 120rad/s , with 0

have been obtained with a load of *Tl* 1,5Nm applied at 0,6s *t* .

PI and the sliding mode controller for the same operating condition.

 PI Sliding mode Fig. 5. PI and sliding speed regulation with *Tl* 1,5Nm applied at *t* 0,6s .

Follow, the speed regulation operates in a supplementary quadrant. That means we are going to do a speed regulation with a negative torque load but maintaining the speed reference. Fig. 6 shows the results obtained for a negative load of *Tl* 2Nm also applied at

Fig. 6 shows that the speed regulation is not assured anymore after 0,6s *t* . From (8), the

( ) *T T e l f*

> *Tl f*

Therefore, the minimum speed for a possible regulation without producing a braking

(21)

*min* (22)

torque, meaning that the regulator will have its reference current to zero, will be

38

*t* 0,6s .

motor speed is given by

In our case, the minimum speed stays 96,3 /s *rd min* , as shown in Fig. 6. Now, if one wants to continue the speed regulation at 90 / *rd s* , it will be necessary to produce a braking torque when the speed error is negative. The controller by sliding mode gives good result compared to that of regulator PI. The increase speed is reduced during the application of a negative torque. However, when the negative torque load is applied at 0,5s *t* , the speed oscillations become significant. Previous results in Fig. 6 showed significant oscillations in the speed signal due to an initial bad choice of the *on* value.

Fig. 6. PI and sliding speed regulation with *Tl* 2Nm applied at 0,6s *t* .

Figure 7 shows the very good performances reached by the sliding mode controller. Indeed, one notes that the overshoot is less important in the case of the sliding regulator, with a best response time without increasing the overshoot.

For this test, the sliding controller proves to be well more robust because the speed curve is hardly of its reference. On the other hand, the speed signal evolution obtained with the PI controller deviates about 10% from its reference value (figure 7). The speed tracking is satisfactory, and the torque ripple is low. These results demonstrate the robustness of the drive under unpredictable load conditions. The decreasing speed oscillations with the PI controller are owed to a slower reaction of the current, as shown in figure 7.

### **4.1 Robustness**

In order to test the robustness of the proposed control, we have studied the speed performances. Two cases are considered:


The figure 8 shows the tests of the robustness: a) The robustness tests concerning the variation of the resistances, b) the robustness tests in relation to inertia variations.

Sliding Controller of Switched Reluctance Motor 273

Sliding controller

Fig. 7. Simulation results of speed control.

### PI controller

272 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

PI controller

Fig. 7. Simulation results of speed control.

Sliding Controller of Switched Reluctance Motor 275

The simulation results show that the proposed controller is superior to conventional controller in robustness and in tracking precision. The simulation study clearly indicates the superior performance of sliding control, because it is inherently adaptive in nature. It appears from the response properties that it has a high performance in presence of the plant parameters uncertain and load disturbances. It is used to control system with unknown model. The control of speed by SMC gives fast dynamic response without overshoot and zero steady-state error. The controller contains only two structures and the only way of changing them is by switching. A major drawback of this system is chattering, which is

Throughout this section the motor parameters used to verify the design principles are: Number of phase 3, Number of stator poles 6, Pole arc 30°, Number of rotor poles 4, Pole arc 30°, Maximum inductance 60mH, Minimum inductance 8mH, resistance 1,3Ω, Moment

V.Utkin, J.Guldner, J.Shi "Sliding mode control in electromechanical systems" Ed Taylor

V. I. Utkin, "Sliding mode control design principles and applications to electric drives", *IEEE* 

S. K. Panda and P. K. Dash, "Application of nonlinear control to switched reluctance motors:

W. Perruquetti, J.P.Barbot "Sliding Mode Control In Engineering" Ed Marcel Dekker, 2002 Ahmed TAHOUR, Hamza ABID2, Abdel Ghani AISSAOUI " Speed Control of Switched

F. Soares and P.J. "Costa Branco Simulation of a 6/4 Switched Reluctance Motor Based on

R. A. De Carlo, S. H. Zak and G. P. Matthews, "Variable structure control of nonlinear

J. Y. Hung, W. Gao, and J. C. Hung, "Variable structure control : a survey", *IEEE Trans.* 

D. G. Taylor, "Nonlinear control of electric machines: An overview", *IEEE Control Systems* 

J.J.E. Slotine, J.K. Hedrick and E.A. Misawa, "On sliding observers for nonlinear systems",

E. Y. Y. Ho and P. C. Sen, "Control dynamics of speed drive systems using sliding mode

A feedback linearization approach," *Proc.Inst. Elect. Eng.,* vol. 143, pt. B, no. 5, pp.

Reluctance Motor Using Fuzzy Sliding Mode" Advances in Electrical and

Matlab/Simulink Environment "aerospace and electronic system. IEEE

multivariable systems: a tutorial", Proceedings IEEE*,* Vol. 76, No. 3, March 1988, pp.

Transactions of the ASME: Journal of Dynamic Systems Measurement and Control*,* 

controllers with integral compensation" IEEE Trans.Ind. Applicat*.*, vol. 27, pp. 883–

caused by a fast switching of the controller structure

of inertia 0,0013Kg;m2, friction 0,0183Nm/s, Inverter voltage 150v.

*Trans.* Industrial Electronics*,* Vol. 40, No. 1, February 1993

Computer Engineering Volume 8 (15), Number 1 (29), 2008.

*Industrial Electronics,* Vol.40, No. 1, February 1993, pp. 2-22.

*magazine,* Vol. 14, No. 6, December 1994, pp. 41-51.

transactions, Vol 37 pp989-1009, July 2001.

**6. Appendix** 

**7. References** 

and Francis 1999.

371–379,1996.

212-232.

Vol. 109, pp. 245-252, 1987.

892, Sept./Oct. 1991.

Fig. 8. Test of robustness.

Figure 8-b shows the parameter variation does not allocate performances of proposed control. The speed response is insensitive to parameter variations of the machine, without overshoot and without static error. The other performances are maintained.

For the robustness of control, a decrease or increase of the moment of inertia, the resistances doesn't have any effects on the performances of the technique used (figure 8.a and 8.b). An increase of the moment of inertia gives best performances, but it presents a slow dynamic response (figure 8.b). The controller suggested gives good performances although the parameters are unknown.

### **5. Conclusion**

This chapter presents a new approach to robust speed control for switched reluctance motor. It develops a simple robust controller to deal with parameters uncertain and external disturbances and takes full account of system noise, digital implementation and integral control. The control strategy is based on SMC approaches.

The simulation results show that the proposed controller is superior to conventional controller in robustness and in tracking precision. The simulation study clearly indicates the superior performance of sliding control, because it is inherently adaptive in nature. It appears from the response properties that it has a high performance in presence of the plant parameters uncertain and load disturbances. It is used to control system with unknown model. The control of speed by SMC gives fast dynamic response without overshoot and zero steady-state error. The controller contains only two structures and the only way of changing them is by switching. A major drawback of this system is chattering, which is caused by a fast switching of the controller structure

### **6. Appendix**

274 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

a) Different values of resistance of stator

J 50%J 2J

0 0.2 0.4 0.6 0.8 1 1.2

t(s)

b) Different values of moment of inertia

Figure 8-b shows the parameter variation does not allocate performances of proposed control. The speed response is insensitive to parameter variations of the machine, without

For the robustness of control, a decrease or increase of the moment of inertia, the resistances doesn't have any effects on the performances of the technique used (figure 8.a and 8.b). An increase of the moment of inertia gives best performances, but it presents a slow dynamic response (figure 8.b). The controller suggested gives good performances although the

This chapter presents a new approach to robust speed control for switched reluctance motor. It develops a simple robust controller to deal with parameters uncertain and external disturbances and takes full account of system noise, digital implementation and integral

overshoot and without static error. The other performances are maintained.

control. The control strategy is based on SMC approaches.

0

20

40

60

speed (rd/s)

Fig. 8. Test of robustness.

parameters are unknown.

**5. Conclusion** 

80

100

120

Throughout this section the motor parameters used to verify the design principles are:

Number of phase 3, Number of stator poles 6, Pole arc 30°, Number of rotor poles 4, Pole arc 30°, Maximum inductance 60mH, Minimum inductance 8mH, resistance 1,3Ω, Moment of inertia 0,0013Kg;m2, friction 0,0183Nm/s, Inverter voltage 150v.

### **7. References**


B. K. Bose, "Sliding mode control of induction motor," in Proc. IEEEIAS Annu. Meeting, 1985, pp. 479–486.

**15** 

Birou M.T. Iulian

*Romania* 

**Robust Control of Sensorless AC Drives Based** 

In most of the modern drive systems with alternating current (AC) machines which require rotor speed control, the main task is to design and develop different controllers, able to achieve high dynamic performance and to maintain the system response within specified tolerances, for a wide range of speed and torque values, for parameter variations and for external perturbations like: total inertia moment, friction coefficient, etc. (Leonhard, 1985). Various concepts for controlled AC drives without speed sensor (sensorless control) have been developed in the past few years (Holtz, 2002; Rajashekara et al., 1996; Vas, 1998). Ongoing research has focused on providing sustained operation at high dynamic performance at very low speed, including zero speed and zero stator frequency (Akatsu & Kawamura, 2000; Holtz & Quan, 2002; Hurst et al., 1998; Lascu et al., 2005). In speed sensorless control, motor parameter sensitivity is an important and large discussed and analyzed problem (Akpolat et al., 2000; Toliyat et al., 2003). In many existing speed identification algorithms, the rotor speed is estimated based on the rotor flux observer. Therefore, these algorithms are, to a certain degree, machine parameter dependent. The solution proposed in this chapter is to apply robust control to sensorless AC drive systems. The designing procedure of the speed controllers can be very difficult, if a complex mathematical model of the plant (here of the AC machine) is used. But robust controllers keep the dynamic and stability performance of the controlled system even if structured or unstructured uncertainties appear. That's why, robust speed controllers can be designed by using simplified models of the AC machines, and have to be used in a complex structure based on the field-oriented control (FOC) principle (Birou & Pavel, 2008). Thus, the requirements of a digital control application are: a flexible control structure, reduced hardware configuration and a good dynamic behavior of the controlled process. The last two aspects can be realized by finding a compromise between the reducing of the control cycle times and the increasing of controller complexity. For industrial applications the

Two different algorithms will be presented to estimate the rotor speed in this chapter, one based on the model reference adaptive system (MRAS) and the other on a full order observer (FOO). The speed identification algorithms, the designing procedure of the optimal H controller and the robust control of the sensorless driving system will be accomplished by simulated and experimental results. Based on the results obtained, advantages and

disadvantages of the proposed control structures will be discussed.

**1. Introduction** 

hardware costs are also important.

*Technical University of Cluj-Napcoa, Department of Electrical Drives and Robots* 

**on Adaptive Identification** 


## **Robust Control of Sensorless AC Drives Based on Adaptive Identification**

Birou M.T. Iulian

*Technical University of Cluj-Napcoa, Department of Electrical Drives and Robots Romania* 

### **1. Introduction**

276 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

B. K. Bose, "Sliding mode control of induction motor," in Proc. IEEEIAS Annu. Meeting,

Tzu-Shien Chuang and Charles Pollock "Robust Speed Control of a Switched Reluctance

Roy A. McCann, Mohammad S. Islam,*,* I.Husain*,"* Application of a Sliding-Mode Observer

Zheng Hongtao, Qiao Bin, Guo Zhijiang, and Jiang Jingping. (2001) "Modeling of Switched

Gobbi, K. Ramar and N.C. Sahoo."Fuzzy Iterative Technique for Torque Ripple

Gobbi. R and K. Ramar, "Optimization Techniques for a Hysteresis Current Controller to

Electronics, Vol. 44, No. 6, DECEMBER 1997 pp.800.808

Systems. Vol. 37, Issue. 9, July 2009. pp. 982 – 1004.

JANUARY/FEBRUARY 2001 pp.51.58

Vector Drive Using Variable Structure Approach" IEEE Transactions On Industrial

for Position and Speed Estimation in Switched Reluctance Motor Drives" IEEE Transactions On Industrial On Industry Applications, Vol. 37, NO. 1,

Reluctance Drive Based on Variable Structure Fuzzy-Neural Networks" Fifth International Conference on Electrical Machines and Systems, (ICEMS 2001), Vol. 2,

Minimization in Switched Reluctance Motors" Electric Power Components and

Minimize Torque Ripple in SR Motors", IET Electric Power Applications, 2009.

1985, pp. 479–486.

pp.1250-1253

Vol.3, Issues 5, pp 453-460

In most of the modern drive systems with alternating current (AC) machines which require rotor speed control, the main task is to design and develop different controllers, able to achieve high dynamic performance and to maintain the system response within specified tolerances, for a wide range of speed and torque values, for parameter variations and for external perturbations like: total inertia moment, friction coefficient, etc. (Leonhard, 1985). Various concepts for controlled AC drives without speed sensor (sensorless control) have been developed in the past few years (Holtz, 2002; Rajashekara et al., 1996; Vas, 1998). Ongoing research has focused on providing sustained operation at high dynamic performance at very low speed, including zero speed and zero stator frequency (Akatsu & Kawamura, 2000; Holtz & Quan, 2002; Hurst et al., 1998; Lascu et al., 2005). In speed sensorless control, motor parameter sensitivity is an important and large discussed and analyzed problem (Akpolat et al., 2000; Toliyat et al., 2003). In many existing speed identification algorithms, the rotor speed is estimated based on the rotor flux observer. Therefore, these algorithms are, to a certain degree, machine parameter dependent. The solution proposed in this chapter is to apply robust control to sensorless AC drive systems. The designing procedure of the speed controllers can be very difficult, if a complex mathematical model of the plant (here of the AC machine) is used. But robust controllers keep the dynamic and stability performance of the controlled system even if structured or unstructured uncertainties appear. That's why, robust speed controllers can be designed by using simplified models of the AC machines, and have to be used in a complex structure based on the field-oriented control (FOC) principle (Birou & Pavel, 2008). Thus, the requirements of a digital control application are: a flexible control structure, reduced hardware configuration and a good dynamic behavior of the controlled process. The last two aspects can be realized by finding a compromise between the reducing of the control cycle times and the increasing of controller complexity. For industrial applications the hardware costs are also important.

Two different algorithms will be presented to estimate the rotor speed in this chapter, one based on the model reference adaptive system (MRAS) and the other on a full order observer (FOO). The speed identification algorithms, the designing procedure of the optimal H controller and the robust control of the sensorless driving system will be accomplished by simulated and experimental results. Based on the results obtained, advantages and disadvantages of the proposed control structures will be discussed.

Robust Control of Sensorless AC Drives Based on Adaptive Identification 279

mechanical and electrical variables (speed, torque, currents, voltages, etc.). In our drive system, the simplified transfer function describes the linear model of the AC machine corresponding to a steady state working point and is presented in Fig. 1. A vector control strategy will be applied to control the variable speed electrical drive systems discussed in this chapter. For the proposed FOC of the AC machine, the rotor flux vector is considered to be the reference system. In this case the speed controller has as input the speed error Δ*nr* and computes the control variable as the active component of the stator current *i*\*Active=*isqλr*, as

\* nr nr

\*

*3 3 L L m z = mi = z i 2 2 L L*

*m m <sup>e</sup> p p <sup>s</sup> <sup>r</sup> <sup>r</sup> sq r r*

with *Lm* and *Lr* the magnetic (mutual) and rotor inductance respectively, *zp* the number of

*m = Js + B + m e r rr* 

where *J* is the total moment of inertia, *B* the friction coefficient, *s* the derivate symbol and ω*<sup>r</sup>*

60 *r r p n*

with ω the electrical angular velocity and *nr* the rotor speed. The simplified transfer function

If we apply to the speed control loop presented in Fig. 1 the module criteria and consider the

. *ms Fse t m* 

*z* 

*n K <sup>K</sup> Ps K i Ts Ts*

<sup>2</sup>

non compensable component described by the dead-time transfer function:

2

\* <sup>1</sup> 1 1 *r m sq r m m*

K1

me <sup>K</sup>

P(s) - Simplified model of an AC machine

mr


T s <sup>m</sup> 2 1

*r*

, (3)

. (4)

, (5)

, (2)

, (1)

Ftm(s)

Dead time

Fig. 1. Simplified speed close-loop control structure of an ac machine.

described by Equation 1.

nr

+


Using the torque producing expression:

pole pairs and the motion equation

the rotor angular velocity

 Speed controller

of the rotor-flux oriented AC machine can be written:

C(s)

i \* Active

### **2. Control of AC machines**

Electrical machines are the major and most efficient source to generate motion for a large number of applications in a wide range of power (from μW to several hundred of MW). Among all types of electromechanical converters, the AC machines are now, from fare, the most produced and used in variable speed applications, because of their high performance/cost ratio. If for low power applications (i.e. servo drives), generally permanent magnet synchronous machines (PM-SM) and for very high powers electrical excited synchronous machines (SM) are used, the largest number of applications use rotor cage induction machines (IM) because of their higher mechanical robustness and lower cost (Birou et al., 2010; Holtz, 2002; Kelemen & Imecs, 1991; Leonhard, 1985; Moreira et al., 1991; Wieser, 1998; Trzynadlowski, 1994). The main disadvantage of using in the past the IM as motion source in variable speed applications, namely the difficulty to precisely control speed and/or torque, is now compensate by using:


Depending on the dynamic performances, energy efficiency demands and final cost of the electrical drive system, following control strategies can be used:


In the designing procedure of the controllers, it is important to know the transfer function of the process. A transfer function which describes exactly the behavior of the AC machine is almost impossible to obtain, because of the nonlinearities of the mathematical model of the machine. Consequently a simplified transfer function of the process is used to design the speed controller. Then, the control law is introduced in the not simplified and nonlinear "original" control structure, in order to simulate and analyze the dynamic behavior of the 278 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Electrical machines are the major and most efficient source to generate motion for a large number of applications in a wide range of power (from μW to several hundred of MW). Among all types of electromechanical converters, the AC machines are now, from fare, the most produced and used in variable speed applications, because of their high performance/cost ratio. If for low power applications (i.e. servo drives), generally permanent magnet synchronous machines (PM-SM) and for very high powers electrical excited synchronous machines (SM) are used, the largest number of applications use rotor cage induction machines (IM) because of their higher mechanical robustness and lower cost (Birou et al., 2010; Holtz, 2002; Kelemen & Imecs, 1991; Leonhard, 1985; Moreira et al., 1991; Wieser, 1998; Trzynadlowski, 1994). The main disadvantage of using in the past the IM as motion source in variable speed applications, namely the difficulty to precisely control




Depending on the dynamic performances, energy efficiency demands and final cost of the




In the designing procedure of the controllers, it is important to know the transfer function of the process. A transfer function which describes exactly the behavior of the AC machine is almost impossible to obtain, because of the nonlinearities of the mathematical model of the machine. Consequently a simplified transfer function of the process is used to design the speed controller. Then, the control law is introduced in the not simplified and nonlinear "original" control structure, in order to simulate and analyze the dynamic behavior of the

**2. Control of AC machines** 

current);

inverter.

and control methods.

large number of applications;

energy efficiency in variable speed control;

speed and/or torque, is now compensate by using:

control (DTC) strategies of AC drive systems;

electrical drive system, following control strategies can be used:

mechanical and electrical variables (speed, torque, currents, voltages, etc.). In our drive system, the simplified transfer function describes the linear model of the AC machine corresponding to a steady state working point and is presented in Fig. 1. A vector control strategy will be applied to control the variable speed electrical drive systems discussed in this chapter. For the proposed FOC of the AC machine, the rotor flux vector is considered to be the reference system. In this case the speed controller has as input the speed error Δ*nr* and computes the control variable as the active component of the stator current *i*\*Active=*isqλr*, as described by Equation 1.

Fig. 1. Simplified speed close-loop control structure of an ac machine.

Using the torque producing expression:

$$\dot{\mathbf{q}}\_{\text{Jm}\_{\text{F}}} = \frac{3}{2} \boldsymbol{\Sigma}\_{\text{p}} \frac{\mathbf{L}\_{m}}{\mathbf{L}\_{r}} \boldsymbol{\mathfrak{T}} m \left( \dot{\mathbf{q}}\_{s} \boldsymbol{\underline{\mathbf{V}}}\_{r}^{\*} \right) = \frac{3}{2} \boldsymbol{\mathcal{Z}}\_{\text{p}} \frac{\mathbf{L}\_{m}}{\mathbf{L}\_{r}} \boldsymbol{\Psi}\_{r} \dot{\mathbf{q}}\_{s\mathbf{q},\boldsymbol{\lambda}\_{r}} \tag{1}$$

with *Lm* and *Lr* the magnetic (mutual) and rotor inductance respectively, *zp* the number of pole pairs and the motion equation

$$
\delta m\_c = \text{Jso}\_r + B\alpha\_r + m\_{r\text{ }\prime} \tag{2}
$$

where *J* is the total moment of inertia, *B* the friction coefficient, *s* the derivate symbol and ω*<sup>r</sup>* the rotor angular velocity

$$
\rho u\_r = \frac{\alpha}{z\_p} = n\_r \frac{2\pi}{60} \,\mathrm{\,\, \,}\tag{3}
$$

with ω the electrical angular velocity and *nr* the rotor speed. The simplified transfer function of the rotor-flux oriented AC machine can be written:

$$P(\mathbf{s}) = \frac{n\_r}{i\_{sq\lambda r}^\*} = K\_1 \frac{K\_2}{T\_m \mathbf{s} + \mathbf{1}} = \frac{K\_m}{T\_m \mathbf{s} + \mathbf{1}} \,. \tag{4}$$

If we apply to the speed control loop presented in Fig. 1 the module criteria and consider the non compensable component described by the dead-time transfer function:

$$F\_{t.m}\left(\mathbf{s}\right) = e^{-\tau\_m s} \,\prime \tag{5}$$

the speed controller will have a transfer function as follows:

$$\mathbf{C}\_{Pl}\left(\mathbf{s}\right) = \mathbf{K}\_P \cdot \left(\mathbf{1} + \frac{\mathbf{1}}{T\_i \mathbf{s}}\right). \tag{6}$$

Robust Control of Sensorless AC Drives Based on Adaptive Identification 281

The proposed solution is based on a FOC structure with AC machine, using for speed estimation a model reference adaptive system (MRAS) algorithm and a full order observer

> Field Oriented Control

 r

Fig. 2. Block diagram of the speed sensorless vector control of the induction machine

In order to achieve sensorless control, the rotor speed estimation has to be indirectly derived based on the measured stator voltages and currents. Therefore, a mathematical model of the induction machine is needed. The model is described in the stationary (stator) reference frame. The block diagram of the MRAS speed identification is shown in Fig. 3. It contains a reference model, an adjustable model and an adaptive algorithm. Both models have as inputs the stator voltages and currents. The reference model outputs a performance index *p*

p

 p

Fig. 3. Model reference adaptive system algorithm (MRAS) for speed identification.

*m*

In order to estimate the rotor speed accurately, the performance index of the reference model has to be robust over the entire speed range and insensitive to the machine parameters. According to the equations of the induction machine, we can obtain the value of

> *r r s s s s*

, (9)

*<sup>d</sup> <sup>L</sup> di u Ri L dt L dt*

 

**3.1 Model reference adaptive system algorithm for speed identification** 

us

P(s)

MRAS Algorithm or Full Order Observer

is

. The difference between the two values is

to its real value.

Adaptive algorithm

(FOO) respectively, like presented in Fig. 2.

\*

and the adjustable model a performance index *p*

is

the rotor flux phasor based on stator equations:

used by the adaptive algorithm to converge the estimated speed

Reference model

us

Adjustable model

C(s) nr

 nr

It results a classical PI controller, having *Kp* - the proportional coefficient and *Ki*=*Kp*/*Ti* - the integrator coefficient. Considering the two first-order integrator type transfer functions of the direct loop having constant times of different ranges, with *<sup>m</sup> Tm* ,

$$K\_m e^{-(\tau\_m + T\_m)s} \equiv K\_m e^{-T\_m s} \, \tag{7}$$

the equivalent closed-loop transfer function of Fig.1 becomes:

$$H\_o \text{(s)} = \frac{K\_P K\_m (1 + T\_i s)}{T\_i s (T\_m s + 1) + K\_P K\_m (1 + T\_i s)}\tag{8}$$

### **3. Sensorless control of AC machines based on adaptive identification**

The common accepted definition of sensorless control for electrical drives means the need of speed and/or torque control of an electrical machine without using any mechanical speed or position measuring device placed on the rotor ax. Recently, sensorless control of AC drives is a prolific research area and many viable solutions have been proposed and implemented. It combines favorably the cost advantage with increased reliability due to the absence of the mechanical sensor and its communication cable. Speed sensorless AC drives are today well established in industrial applications where no persistent operation at lower speed occurs.

The main philosophy in sensorless control is to use the electrical machine itself as a "sensor" by offering the necessary information able to estimate its position or speed (Consoli et al., 2003; Lorenz 2010). Several techniques have been developed and published in application with both FOC and DTC of AC drives. A first category comprises signal injection techniques, based on spectral analysis which use either the natural (if it exists), or an artificial created, anisotropy of the magnetic field of the AC machine (Briz et al., 2004; Degner & Lorenz, 2000; Holtz 2006; Kim & Lorenz, 2004). By injecting appropriate voltage signals in the stator (mainly high frequency signals) and analyzing the obtained current or voltage harmonics, valuable information can be extracted to determine the rotor position. A second category comprises techniques which estimate the rotor position/speed starting from the real process (drive system) and from the mathematical model of the machine by using different identification algorithms, like:


280 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

<sup>1</sup> <sup>1</sup> *PI P*

It results a classical PI controller, having *Kp* - the proportional coefficient and *Ki*=*Kp*/*Ti* - the integrator coefficient. Considering the two first-order integrator type transfer functions of

> ( ) *m mT s Ts <sup>m</sup> Ke Ke m m*

(1 ) ( ) ( 1) (1 ) *Pm i <sup>o</sup>*

The common accepted definition of sensorless control for electrical drives means the need of speed and/or torque control of an electrical machine without using any mechanical speed or position measuring device placed on the rotor ax. Recently, sensorless control of AC drives is a prolific research area and many viable solutions have been proposed and implemented. It combines favorably the cost advantage with increased reliability due to the absence of the mechanical sensor and its communication cable. Speed sensorless AC drives are today well established in industrial applications where no persistent operation at lower speed occurs. The main philosophy in sensorless control is to use the electrical machine itself as a "sensor" by offering the necessary information able to estimate its position or speed (Consoli et al., 2003; Lorenz 2010). Several techniques have been developed and published in application with both FOC and DTC of AC drives. A first category comprises signal injection techniques, based on spectral analysis which use either the natural (if it exists), or an artificial created, anisotropy of the magnetic field of the AC machine (Briz et al., 2004; Degner & Lorenz, 2000; Holtz 2006; Kim & Lorenz, 2004). By injecting appropriate voltage signals in the stator (mainly high frequency signals) and analyzing the obtained current or voltage harmonics, valuable information can be extracted to determine the rotor position. A second category comprises techniques which estimate the rotor position/speed starting from the real process (drive system) and from the mathematical model of the machine by





compensation of nonlinearities and disturbances (Holtz & Quan, 2002);

*i m Pm i*

*Ts T s K K Ts*

*CsK*

*K K Ts H s*

**3. Sensorless control of AC machines based on adaptive identification** 

the direct loop having constant times of different ranges, with *<sup>m</sup> Tm* ,

*i*

*T s* 

. (6)

, (7)

(8)

the speed controller will have a transfer function as follows:

the equivalent closed-loop transfer function of Fig.1 becomes:

using different identification algorithms, like:

Jansen et al., 1994);

Pucci, 2005; Landau, 1979; Lascu et al., 2005);

networks and genetic algorithms (Zadeh, 1996) .

The proposed solution is based on a FOC structure with AC machine, using for speed estimation a model reference adaptive system (MRAS) algorithm and a full order observer (FOO) respectively, like presented in Fig. 2.

Fig. 2. Block diagram of the speed sensorless vector control of the induction machine

### **3.1 Model reference adaptive system algorithm for speed identification**

In order to achieve sensorless control, the rotor speed estimation has to be indirectly derived based on the measured stator voltages and currents. Therefore, a mathematical model of the induction machine is needed. The model is described in the stationary (stator) reference frame. The block diagram of the MRAS speed identification is shown in Fig. 3. It contains a reference model, an adjustable model and an adaptive algorithm. Both models have as inputs the stator voltages and currents. The reference model outputs a performance index *p*

and the adjustable model a performance index *p* . The difference between the two values is used by the adaptive algorithm to converge the estimated speed to its real value.

Fig. 3. Model reference adaptive system algorithm (MRAS) for speed identification.

In order to estimate the rotor speed accurately, the performance index of the reference model has to be robust over the entire speed range and insensitive to the machine parameters. According to the equations of the induction machine, we can obtain the value of the rotor flux phasor based on stator equations:

$$\frac{d\underline{\Psi\_r}}{dt} = \frac{L\_r}{L\_m} \left( \underline{u}\_s - R\_s \underline{i}\_s - L\_\sigma \frac{d\underline{i}\_s}{dt} \right) \tag{9}$$

where *Lσ* is the equivalent inductance and *Rs* is the stator resistance. The same rotor flux phasor based on the rotor equations:

$$\frac{d\underline{\Psi\_r}}{dt} = -\frac{1}{\tau\_r}\underline{\Psi\_r} + j\alpha\underline{\Psi\_r} + \frac{L\_m}{\tau\_r}\underline{i}\_{s'} \tag{10}$$

where τ*r*=*Lr*/*Rr* is the rotor time constant. Considering the electromotive induced voltage (back EMF) being:

$$
\underline{\underline{\varepsilon}} = \frac{L\_m}{L\_r} \frac{d\underline{\underline{\Psi}}\_r}{dt} \tag{11}
$$

Robust Control of Sensorless AC Drives Based on Adaptive Identification 283

22 2 ^ 1 1 - - *<sup>m</sup> sq sq sq sd m sd m sd sd sq md mq mq md*

*md mq sd r r*

*mq md sq r r*

*ii i*

^

 

 

, (19)

. (20)

*p p* (21)

, (22)

to the actual stator current *is* and

is minimized in some sense. This

(18)

rotor

*<sup>L</sup> di di L di di L di di p ii i i i i L dt dt L dt dt L dt dt*

^ *md* 1 1

^ *mq* 1 1

Equation 17 is used for the reference model and Equation 18 for the adjustable model. The

speed in order to converge the performance index of the adjustable model to the performance index of the reference model (converge the error to zero). In designing the adaptive mechanism of the presented MRAS structure, it is necessary to ensure the stability of the control system and the convergence of the estimated speed to the real one. Based on the hyper-stability theory (Landau, 1979), following adaptive mechanism is used in order to

> *K K dt p i*

where, *Kp* and *Ki* are the gain parameters of the adaptive algorithm, limited only by noise considerations and having for our control structure the particular values *Kp*=3 and *Ki*=10. The MRAS algorithm presented above can also be used for on-line identification of some parameter of the induction machine, namely the stator resistance, the equivalent inductance

The speed estimation strategy with full order observer (FOO) is based on the fundamental excitation variables as information source, like presented in Fig. 4. The rotor speed estimator

such that the error *i*s- *is*

will be done by using a full-order observer for the estimated stator current, rotor flux and

^ <sup>1</sup> *s r <sup>s</sup> s r <sup>s</sup> <sup>r</sup> s s*

*d i <sup>R</sup> L u R R jLi j ki i dt <sup>L</sup>*

*m*

 

, (23)

^ ^

 

*di ii i*

*r r r r r*

having the two formal magnetizing current components described by:

*dt*

*di*

*dt*

 

is the input for the adaptive algorithm, see Fig. 3. This algorithm estimates the

^ 

**3.2 Speed and rotor flux estimator based on a full order observer** 

is based on comparing the stator current estimate value *is*

 

guarantee the system stability:

or the rotor time constant.

updating the estimated speed

rotor speed, described by equations:

 ^

error between the two performance indexes

and decoupling Equation 9 on the stationary (stator-fixed) reference frame *d*-*q*, we obtain:

$$
\sigma\_d = \mathfrak{u}\_{sd} - \mathcal{R}\_s \mathfrak{i}\_{sd} - \mathcal{L}\_\sigma \frac{d\mathfrak{i}\_{sd}}{dt} \, \, \, \, \tag{12}
$$

$$
\epsilon\_q = \mu\_{sq} - R\_s i\_{sq} - L\_\sigma \frac{d\dot{\mathbf{u}}\_{sq}}{dt} \,. \tag{13}
$$

Considering a formal magnetizing current

$$
\underline{i}\_m = \frac{1}{L\_m} \underline{\Psi\_r} \tag{14}
$$

and decupling Equation 10 on the fix reference frame *d*-*q*, we have:

$$\mathcal{e}\_{md} = -\frac{\mathcal{L}\_m}{L\_r} \left( \frac{1}{\tau\_r} \mathbf{i}\_{md} + \alpha \mathbf{i}\_{mq} - \frac{1}{\tau\_r} \mathbf{i}\_{sd} \right) \tag{15}$$

$$e\_{m\eta} = -\frac{L\_m}{L\_r} \left( \frac{1}{\tau\_r} \dot{\mathbf{i}}\_{m\eta} - \alpha \dot{\mathbf{i}}\_{m\mathbf{l}} - \frac{1}{\tau\_r} \dot{\mathbf{i}}\_{s\eta} \right). \tag{16}$$

The reference model is described based on Equations 12 and 13 and is parameter dependent, namely with the stator resistance *Rs* and the equivalent inductance *L*. In the reference model there are no integral operations, so the model can be used also for low speed estimation. To improve the robustness of the reference model, one of the two machine parameters can be avoided by choosing an optimal way to define the reference model performance index *p*. To eliminate the effect of the inductance *L*, Equations 12 and 13 are cross multiplied by the derivates of the two stator current components and we obtain:

$$p = \mu\_{sd} \frac{d\dot{\mathbf{i}}\_{sq}}{dt} - \mu\_{sq} \frac{d\dot{\mathbf{i}}\_{sd}}{dt} - R\_s \left(\dot{\mathbf{i}}\_{sd} \frac{d\dot{\mathbf{i}}\_{sq}}{dt} - \dot{\mathbf{i}}\_{sq} \frac{d\dot{\mathbf{i}}\_{sd}}{dt}\right). \tag{17}$$

Equation 17 describes the performance index of the reference model. To obtain the performance index of the adjustable model, same mathematical operations applied to Equations 15 and 16 give:

282 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

where *Lσ* is the equivalent inductance and *Rs* is the stator resistance. The same rotor flux

<sup>1</sup> *r m r r <sup>s</sup> r r*

 

*sd*

*sq*

*di*

, (10)

(11)

, (12)

. (13)

, (15)

. (16)

. (17)

*<sup>L</sup>* (14)

*<sup>d</sup> <sup>L</sup> <sup>j</sup> <sup>i</sup>*

where τ*r*=*Lr*/*Rr* is the rotor time constant. Considering the electromotive induced voltage

*m r r L d*

*L dt*

and decoupling Equation 9 on the stationary (stator-fixed) reference frame *d*-*q*, we obtain:

*e*

*d sd s sd di e u Ri L dt*

*q sq s sq*

*i*

2

2

*L*

*L*

and decupling Equation 10 on the fix reference frame *d*-*q*, we have:

*e u Ri L dt*

> 1 *m r m*

*<sup>m</sup>* 1 1 *md md mq sd r r r*

*sq sq sd sd sd sq s sd sq di di di di pu u Ri i dt dt dt dt* 

Equation 17 describes the performance index of the reference model. To obtain the performance index of the adjustable model, same mathematical operations applied to

 

 

 

*<sup>L</sup> e ii i*

*<sup>m</sup>* 1 1 *mq mq md sq r r r*

The reference model is described based on Equations 12 and 13 and is parameter dependent, namely with the stator resistance *Rs* and the equivalent inductance *L*. In the reference model there are no integral operations, so the model can be used also for low speed estimation. To improve the robustness of the reference model, one of the two machine parameters can be avoided by choosing an optimal way to define the reference model performance index *p*. To eliminate the effect of the inductance *L*, Equations 12 and 13 are cross multiplied by the derivates of the two stator current components and we

*<sup>L</sup> e ii i*

*dt*

phasor based on the rotor equations:

Considering a formal magnetizing current

(back EMF) being:

obtain:

Equations 15 and 16 give:

$$\hat{\mathbf{p}} = \frac{\mathbf{L}\_m}{\mathbf{L}\_r} \frac{\mathbf{1}}{\tau\_r} \left( \mathbf{i}\_{sd} \frac{d\mathbf{i}\_{sq}}{dt} - \mathbf{i}\_{sq} \frac{d\mathbf{i}\_{sd}}{dt} \right) \cdot \frac{\mathbf{L}\_m}{\mathbf{L}\_r} \frac{\mathbf{1}}{\tau\_r} \left( \mathbf{i}\_{md} \frac{d\mathbf{i}\_{sq}}{dt} - \mathbf{i}\_{mq} \frac{d\mathbf{i}\_{sd}}{dt} \right) \cdot \frac{\mathbf{L}\_m}{\mathbf{L}\_r}^2 \alpha \left( \mathbf{i}\_{mq} \frac{d\mathbf{i}\_{sq}}{dt} - \mathbf{i}\_{md} \frac{d\mathbf{i}\_{sd}}{dt} \right) \tag{18}$$

having the two formal magnetizing current components described by:

$$\frac{d\mathbf{i}\_{md}}{dt} = -\frac{1}{\tau\_r}\mathbf{i}\_{md} - \stackrel{\frown}{\operatorname{co}}\mathbf{i}\_{mq} + \frac{1}{\tau\_r}\mathbf{i}\_{sd} \tag{19}$$

$$\frac{d\mathbf{i}\_{mq}}{dt} = -\frac{1}{\tau\_r}\mathbf{i}\_{mq} + \stackrel{\frown}{\operatorname{co}}\mathbf{i}\_{md} + \frac{1}{\tau\_r}\mathbf{i}\_{sq} \,. \tag{20}$$

Equation 17 is used for the reference model and Equation 18 for the adjustable model. The error between the two performance indexes

$$
\hat{\omega} = \stackrel{\frown}{p-\stackrel{\frown}{p}}\tag{21}
$$

is the input for the adaptive algorithm, see Fig. 3. This algorithm estimates the rotor speed in order to converge the performance index of the adjustable model to the performance index of the reference model (converge the error to zero). In designing the adaptive mechanism of the presented MRAS structure, it is necessary to ensure the stability of the control system and the convergence of the estimated speed to the real one. Based on the hyper-stability theory (Landau, 1979), following adaptive mechanism is used in order to guarantee the system stability:

$$
\stackrel{\frown}{\alpha}o = K\_p \mathfrak{x} + K\_i \int \mathfrak{z} dt \,\tag{22}
$$

where, *Kp* and *Ki* are the gain parameters of the adaptive algorithm, limited only by noise considerations and having for our control structure the particular values *Kp*=3 and *Ki*=10. The MRAS algorithm presented above can also be used for on-line identification of some parameter of the induction machine, namely the stator resistance, the equivalent inductance or the rotor time constant.

### **3.2 Speed and rotor flux estimator based on a full order observer**

The speed estimation strategy with full order observer (FOO) is based on the fundamental excitation variables as information source, like presented in Fig. 4. The rotor speed estimator is based on comparing the stator current estimate value *is* to the actual stator current *is* and updating the estimated speed such that the error *i*s- *is* is minimized in some sense. This will be done by using a full-order observer for the estimated stator current, rotor flux and rotor speed, described by equations:

$$L\_{\sigma} \frac{d\hat{\underline{\boldsymbol{i}}}\_{s}}{dt} = \underline{\boldsymbol{\mu}}\_{s} - \left(\boldsymbol{R}\_{s} + \boldsymbol{R}\_{r} + j\rho \boldsymbol{o}\_{k}\boldsymbol{L}\_{\sigma}\right)\hat{\underline{\boldsymbol{i}}}\_{s} + \left(\frac{\boldsymbol{R}\_{r}}{\boldsymbol{L}\_{m}} - j\rho \boldsymbol{o}^{\prime}\right)\underline{\hat{\underline{\boldsymbol{\Psi}}}}\_{r} + k\_{1}\left(\underline{\boldsymbol{i}}\_{s} - \underline{\boldsymbol{i}}\_{s}\right)\boldsymbol{\prime}\tag{23}$$

$$\frac{d\stackrel{\frown}{d\Psi}\_{r}}{dt} = R\_{r}\stackrel{\frown}{\underline{i}}\_{s} - \left[\frac{R\_{r}}{L\_{m}} + j\left(\boldsymbol{\alpha}\_{\lambda} - \boldsymbol{\alpha}\right)\right] \stackrel{\frown}{\underline{\Psi}}\_{r} + k\_{2}\left(\dot{\underline{i}}\_{s} - \dot{\underline{i}}\_{s}\right),\tag{24}$$

$$\frac{d\stackrel{\frown}{d\,o}}{dt} = \mathfrak{Re}\left\{k\_3 \left(\dot{\underline{\mathbf{i}}}\_s - \dot{\underline{\mathbf{i}}}\_s\right)\right\},\tag{25}$$

Robust Control of Sensorless AC Drives Based on Adaptive Identification 285

The simulated speed of this process, the estimated speed based on the MRAS algorithm and the speed error estimation are presented in Fig. 6. Problems that may occur by derivation of

0

500

1000

speed [rpm.]

Fig. 6. Rotor speed, estimated speed based on the MRAS algorithm and speed error for a

Simulated results confirm that the advantage of using the MRAS algorithm, for the control of a sensorless driving system, is a high dynamic performance at speed and torque steps. The error of the estimated speed in the starting process is relative big, because of the sensitivity of the control structure with speed estimators to parameter variations (rotor time constant) and external perturbations (moment of inertia, friction coefficient). By modifying the gain parameters from Equation 22 we can avoid this, but we disturb the control performance parameters (overshooting, stationary error). So, increasing robustness of the sensorless control is needed. The proposed solution is to apply the robust control theory to the AC drive, by designing an optimal H controller, to ensure the stability and robustness

A control system is considered to be robust if it is insensitive to internal process parameter variations or external perturbations (McFarlene & Glover, 1990; Safonov, 1980). In driving


The main goal of a robust controller is to compensate the effects introduced by the variations of the sensitive elements described above to the dynamic process of the controlled

The designing process of a robust controller may follow different methods, applying various robust control system synthesis techniques (Ball & Helton, 1993; Chiang & Safonov, 1992; Doyle et al., 1989; Morari & Zafiriou, 1990; Zames, 1996). The main methods are based on geometric-analytical, frequency domain or steady-state approaches like: Hardy space based,

1500

2000

<sup>0</sup> 0.5 <sup>1</sup> 1.5 -500

estimated speed

time [sec.]

speed error

the measured stator currents can be avoided using specific digital algorithms.

<sup>0</sup> 0.5 <sup>1</sup> 1.5 -500

time [sec.]

starting process at rated speed with rated torque.

performances of the driving system.

**4. Robust control of AC drives** 

(especially by robot arms);

system


systems with AC machines, the most sensible elements are:

0

500

1000

rotor speed [rpm.]

1500

2000

where is the speed of the reference frame and *k1, k2* and *k3* are the gain parameters of the algorithm, calculated from the Ricatti equation.

Fig. 4. Full order observer (FOO), for speed and rotor flux estimation

The speed estimator must converge significantly faster than the mechanical speed control loop in order to ensure good tracking. So, the dynamics of the speed estimator can be neglected as seen from the much slower flux and speed dynamics and thus it can be considered only that value of estimated speed and stator current which have converged to quasi steady-state values. The control structure based on the MRAS algorithm presented in 3.1 will be implemented on a driving system, composed of an induction machine with the following main catalog values:


Simulated results of the MRAS algorithm are presented in Fig. 5, where the reference model performance index *p*, the adjustable model performance index *p* (Equations 17 and 18), and error ε (Equation 21) are for a starting process to the rated speed with rated load torque.

Fig. 5. The performance index of reference model *p*, adjustable model *p* and error index .

284 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

*<sup>d</sup> <sup>R</sup> Ri j ki i*

^ ^

where is the speed of the reference frame and *k1, k2* and *k3* are the gain parameters of the

is \_ +

The speed estimator must converge significantly faster than the mechanical speed control loop in order to ensure good tracking. So, the dynamics of the speed estimator can be neglected as seen from the much slower flux and speed dynamics and thus it can be considered only that value of estimated speed and stator current which have converged to quasi steady-state values. The control structure based on the MRAS algorithm presented in 3.1 will be implemented on a driving system, composed of an induction machine with the

Simulated results of the MRAS algorithm are presented in Fig. 5, where the reference model

0 1 .5 <sup>0</sup>

e rro r o f th e p e rfo rm a n c e in d e xe s

<sup>0</sup> 0 .5 <sup>1</sup> 1 .5 -1 0 0

Fig. 5. The performance index of reference model *p*, adjustable model *p*

tim e [s e c .]

a d ju s ta b le m o d e l p e rfo rm a n c e in d e x

0 1 .5 <sup>0</sup>

re fe re n c e m o d e l p e rfo rm a n c e in d e x

error ε (Equation 21) are for a starting process to the rated speed with rated load torque.

*<sup>d</sup> ek i i*

 is

 

*m*

*dt* 

Stator current estimator

Fig. 4. Full order observer (FOO), for speed and rotor flux estimation

performance index *p*, the adjustable model performance index *p*

*dt L*

algorithm, calculated from the Ricatti equation.

following main catalog values:

100 200 300

100 200 300



us is

^ ^^ <sup>2</sup> *r r <sup>r</sup> <sup>s</sup> <sup>r</sup> s s*

3 *s s*

, (24)

Speed and flux observer

, (25)

 <sup>r</sup>

(Equations 17 and 18), and

and error index .

The simulated speed of this process, the estimated speed based on the MRAS algorithm and the speed error estimation are presented in Fig. 6. Problems that may occur by derivation of the measured stator currents can be avoided using specific digital algorithms.

Fig. 6. Rotor speed, estimated speed based on the MRAS algorithm and speed error for a starting process at rated speed with rated torque.

Simulated results confirm that the advantage of using the MRAS algorithm, for the control of a sensorless driving system, is a high dynamic performance at speed and torque steps. The error of the estimated speed in the starting process is relative big, because of the sensitivity of the control structure with speed estimators to parameter variations (rotor time constant) and external perturbations (moment of inertia, friction coefficient). By modifying the gain parameters from Equation 22 we can avoid this, but we disturb the control performance parameters (overshooting, stationary error). So, increasing robustness of the sensorless control is needed. The proposed solution is to apply the robust control theory to the AC drive, by designing an optimal H controller, to ensure the stability and robustness performances of the driving system.

### **4. Robust control of AC drives**

A control system is considered to be robust if it is insensitive to internal process parameter variations or external perturbations (McFarlene & Glover, 1990; Safonov, 1980). In driving systems with AC machines, the most sensible elements are:


The main goal of a robust controller is to compensate the effects introduced by the variations of the sensitive elements described above to the dynamic process of the controlled system

The designing process of a robust controller may follow different methods, applying various robust control system synthesis techniques (Ball & Helton, 1993; Chiang & Safonov, 1992; Doyle et al., 1989; Morari & Zafiriou, 1990; Zames, 1996). The main methods are based on geometric-analytical, frequency domain or steady-state approaches like: Hardy space based,

Robust Control of Sensorless AC Drives Based on Adaptive Identification 287

A(s)

Real plant P(s)

+

. (28)

. (29)

+

(G) the

(30)

. (31)

C(s) PN(s) I+M(s)

Fig. 7. Diagram of the controlled process using the additive and multiplicative uncertainty.

The extended H control theory is used to design a robust speed-control solution for AC driving systems. It has to satisfy the robust stability characteristics of the control structure as well as the dynamic performances of the driving system. Considering a Laplace transform

*Cmxn* of a multivariable system with *n* inputs and *m* outputs and

*G Gj* sup [ ( )] 

For a single input-single output system the H norm of a transfer function can be defined as:

*G Gj* max ( ) 

The H optimal control designing problem in the particular case of applying the small gain problem is to form an augmented plant of the process, *P*(*s*) like in Fig. 8, with the weighting functions *W*1(*s*), *W*2(*s*), *W*3(*s*) applied to the signals: error *e*, command *u* and output *y* respectively, so that the weighted (*y*11*, y*12, *y*13) and not weighted *y*2 system outputs can be

11 1 1 1 2 1 1 1 2

*y We W u P x Wu WP u*

1 1

0 () ( ) 0 () ()

*N N*

 

 ( ) *N N*

2 3

*W sP s*

*W s*

( )

*N N*

*N*

() () ()

*W s W sP s*

*I Ps*

+ e u y

c

matrix *G(s)*

defined as:

\_

**4.2 Optimal H controller design for the driving system** 

greatest singular value of matrix *G*, the H norm of *G(s)* can be defined as:

12 2 2 2 13 3 3 2 2 12

*y Wu Wu y Wy WPu y e u Pu*

*P s*

and the real, perturbed process *P*(*s*) can be described as:

optimal H2 and H techniques (Green & Limebeer, 1995; Ionescu et al., 1998; Kwakernaak, 1993; Mita et al., 1998), linear quadratic optimal LQG (Kucera, 1993), LQG/LTR (Doyle et al., 1989) or LQR (Chiang & Safonov, 1992) techniques and square root based synthesis techniques (Apkarian & Morris, 1993). An optimal H technique will be used in this chapter for the robust control of the driving system because of the relative simple designing process of the controller and the high robust performances obtained (Bryson, 1996). The designing procedure for the optimal controller will start by describing the driving system with AC machine in steady state equations as a linear multivariable system.

### **4.1 Mathematical model of the real driving system with AC machine**

Fig. 1 presents the closed loop control system of an AC drive. The controlled process is described by a simplified model *P*(*s*) of the AC machine (valuable for both the induction and the synchronous machine). The robust control problem is to design an optimal speed controller *C*(*s*) able to satisfy the robust-stability and robust-performance criteria of the controlled system. The difference between the real physical system (in our case the AC driving system) and his mathematical model, difference defined as mathematical uncertainty may have several causes, namely:


The mathematical uncertainties can be structural uncertainties, based on parameter variations of the dynamic process like rotor time constant and nonstructural uncertainties, frequency dependent like magnetic saturation or external perturbations (moment of inertia, load torque). The study of system robustness based on the H control theory is based on describing the model uncertainty as transfer function (matrix) different from the nominal one. The most used methods to describe them are like additive uncertainties, multiplicative uncertainties or a superposition of both uncertainties, like presented in Fig. 7. Using them, the real process can be written based on the modeled one (nominal plant) as:

$$P(\mathbf{s}) = P\_N(\mathbf{s}) + \Delta\_A(\mathbf{s})\tag{26}$$

in the case of additive uncertainties, and :

$$P(\mathbf{s}) = (I + \Delta\_M(\mathbf{s}))P\_N(\mathbf{s})\tag{27}$$

in the case of multiplicative uncertainties, where *PN*(*s*) is the nominal (rated) plant, *P*(*s*) is the real plant (perturbed process), A(*s*) is the additive uncertainty and M(*s*) is the multiplicative uncertainty. Analyzing the dynamic behavior of the process in frequency domain by using additive or multiplicative uncertainties, the model will describe better the real system in stationary frame or at lower frequencies and the uncertainties will increase at higher frequencies.

286 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

optimal H2 and H techniques (Green & Limebeer, 1995; Ionescu et al., 1998; Kwakernaak, 1993; Mita et al., 1998), linear quadratic optimal LQG (Kucera, 1993), LQG/LTR (Doyle et al., 1989) or LQR (Chiang & Safonov, 1992) techniques and square root based synthesis techniques (Apkarian & Morris, 1993). An optimal H technique will be used in this chapter for the robust control of the driving system because of the relative simple designing process of the controller and the high robust performances obtained (Bryson, 1996). The designing procedure for the optimal controller will start by describing the driving system with AC

Fig. 1 presents the closed loop control system of an AC drive. The controlled process is described by a simplified model *P*(*s*) of the AC machine (valuable for both the induction and the synchronous machine). The robust control problem is to design an optimal speed controller *C*(*s*) able to satisfy the robust-stability and robust-performance criteria of the controlled system. The difference between the real physical system (in our case the AC driving system) and his mathematical model, difference defined as mathematical




The mathematical uncertainties can be structural uncertainties, based on parameter variations of the dynamic process like rotor time constant and nonstructural uncertainties, frequency dependent like magnetic saturation or external perturbations (moment of inertia, load torque). The study of system robustness based on the H control theory is based on describing the model uncertainty as transfer function (matrix) different from the nominal one. The most used methods to describe them are like additive uncertainties, multiplicative uncertainties or a superposition of both uncertainties, like presented in Fig. 7. Using them,

in the case of multiplicative uncertainties, where *PN*(*s*) is the nominal (rated) plant, *P*(*s*) is the real plant (perturbed process), A(*s*) is the additive uncertainty and M(*s*) is the multiplicative uncertainty. Analyzing the dynamic behavior of the process in frequency domain by using additive or multiplicative uncertainties, the model will describe better the real system in stationary frame or at lower frequencies and the uncertainties will increase at

*Ps P s s* () () () *N A* (26)

*Ps I s P s* ( ) ( ( )) ( ) *M N* (27)

machine in steady state equations as a linear multivariable system.

describes the process only around this working point;


the real process can be written based on the modeled one (nominal plant) as:

high number of variables and parameter involved;

uncertainty may have several causes, namely:

in the case of additive uncertainties, and :

modeled;

higher frequencies.

**4.1 Mathematical model of the real driving system with AC machine** 

Fig. 7. Diagram of the controlled process using the additive and multiplicative uncertainty.

### **4.2 Optimal H controller design for the driving system**

The extended H control theory is used to design a robust speed-control solution for AC driving systems. It has to satisfy the robust stability characteristics of the control structure as well as the dynamic performances of the driving system. Considering a Laplace transform matrix *G(s)Cmxn* of a multivariable system with *n* inputs and *m* outputs and (G) the greatest singular value of matrix *G*, the H norm of *G(s)* can be defined as:

$$\left\|\mathbf{G}\right\|\_{\alpha} \stackrel{\Lambda}{=} \sup\_{\boldsymbol{\alpha} \in \mathfrak{M}} \overline{\sigma}[\mathbf{G}(j\boldsymbol{\alpha})]\,. \tag{28}$$

For a single input-single output system the H norm of a transfer function can be defined as:

$$\left\|\mathbf{G}\right\|\_{\infty} = \max\_{o} \left\|\mathbf{G}(jo)\right\|\,. \tag{29}$$

The H optimal control designing problem in the particular case of applying the small gain problem is to form an augmented plant of the process, *P*(*s*) like in Fig. 8, with the weighting functions *W*1(*s*), *W*2(*s*), *W*3(*s*) applied to the signals: error *e*, command *u* and output *y* respectively, so that the weighted (*y*11*, y*12, *y*13) and not weighted *y*2 system outputs can be defined as:

$$\begin{cases} y\_{11} = \mathcal{W}\_1 e = \mathcal{W}\_1 (\mu\_1 - P\_N x\_2) = \mathcal{W}\_1 \mu\_1 - \mathcal{W}\_1 P\_N \mu\_2 \\ y\_{12} = \mathcal{W}\_2 \mu = \mathcal{W}\_2 \mu\_2 \\ y\_{13} = \mathcal{W}\_3 y = \mathcal{W}\_3 P\_N \mu\_2 \\ y\_2 = e = \mu\_1 - P\_N \mu\_2 \end{cases} \tag{30}$$

and the real, perturbed process *P*(*s*) can be described as:

$$P(\mathbf{s}) = \begin{bmatrix} \mathcal{W}\_1(\mathbf{s}) & -\mathcal{W}\_1(\mathbf{s})P\_N(\mathbf{s}) \\ 0 & \mathcal{W}\_2(\mathbf{s}) \\ 0 & \mathcal{W}\_3(\mathbf{s})P\_N(\mathbf{s}) \\ I & -P\_N(\mathbf{s}) \end{bmatrix}. \tag{31}$$

Fig. 8. Structure of speed control system with weighted process.

The second step of the robust control designing problem is to find an optimal stabilizing H controller having the structure presented in Fig. 9. The optimal stabilizing H controller is described by the control law:

$$
\mu\_2(\mathbf{s}) = \mathbf{C}(\mathbf{s}) y\_2(\mathbf{s}) \tag{32}
$$

Robust Control of Sensorless AC Drives Based on Adaptive Identification 289

Considering the robust stability and robust performance criteria, the weighting functions for the optimal H controller are chosen and then the iterative computing process continues, until the norm condition is full fit. The performance design specifications of the speed control loop with the H controller are imposed in frequency domain (Ionescu et al., 1998;



According to them, following weighting functions have been considered to describe the

<sup>1</sup> 1 (3 1) ( ) ( ) <sup>100</sup>

*<sup>s</sup> W s*

1 1

1 3

*W s Ws s*

Respecting condition imposed by Equation 34 the corresponding H speed controller is:

2 3 2

*s s H s sss*

S(s)

Fig. 10. Weighting functions *W1-1*(*s*), *W3-1*(*s*) and sensitivity functions *S*(*s*) and *T*(*s*).

frequency (rad./sec.)

2327 22211 16495 ( ) 822951 548632 91442

The inverse weighting functions *W1-1*(*s*) and *W3-1*(*s*) and the sensitivity functions *S*(*s*) and *T*(*s*) are presented in Fig. 10. From the diagram results the influence of the weighting function *W3-1*(*s*) to limit the peak value of *T*(*s*) function. The output of the speed controller,

<sup>1</sup> <sup>150</sup> ( ) ( ) <sup>145</sup>

where represents the actual step value. The iterative process continues, until the graphic representation in Bode diagram of cost function Ty1u1 reach its maximum value in the

2

1 1 0,9999 *Ty u* . (37)

. (38)

W1 -1

> W3 -1

T(s)

, (36)

perturbed AC drive system with variable moment of inertia and friction coefficient:

1

i.e. the active current component, was limited.

[dB]

*W s*

3

proximity of 0 dB axis. In our case, for =39,75 we obtain the infinite norm

Morari & Zafiriou, 1990):

band of 100 rad/sec.

least 100 times to approximate 0.3333 rad/sec).

so that the infinity norm of the cost function *Ty1u1*, defined as:

$$T\_{y1u1}(\mathbf{s}) = \begin{bmatrix} \mathcal{W}\_1(\mathbf{s})\mathcal{S}(\mathbf{s}) \\ \mathcal{W}\_2(\mathbf{s})\mathcal{R}(\mathbf{s}) \\ \mathcal{W}\_3(\mathbf{s})T(\mathbf{s}) \end{bmatrix} \tag{33}$$

is minimized and is less then one (Doyle et al., 1989; Kwakernaak, 1993):

<sup>1</sup> <sup>1</sup> <sup>1</sup> *Ty <sup>u</sup>* , (34)

where

$$\begin{cases} S(s) = \left[ I + P(s)C(s) \right]^{-1} \\ R(s) = C(s) \cdot \left[ I + P(s)C(s) \right]^{-1} = C(s) \cdot S(s) \quad . \end{cases} \tag{35}$$

$$T(s) = P(s)C(s) \cdot \left[ I + P(s)C(s) \right]^{-1} = I - S(s)$$

288 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

u2

P(s)

K (s)

 U (s) Robust controller C(s)

The second step of the robust control designing problem is to find an optimal stabilizing H controller having the structure presented in Fig. 9. The optimal stabilizing H controller is

1

*W sSs*

*W sT s* 

() () ()

()()

() ()

1

1

3

1

( ) ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) [ ( ) ( )] ( )

*Rs Cs I PsCs Cs Ss T s PsCs I PsCs I Ss*

11 2

*T s W sRs*

*y u*

is minimized and is less then one (Doyle et al., 1989; Kwakernaak, 1993):

( ) [ ( ) ( )]

*Ss I PsCs*

<sup>u</sup> PN(s)<sup>y</sup>

y1 y2 W1(s)

y11

y12

y13

W2(s)

W3(s)

2 2 *u s Csy s* () () () (32)

<sup>1</sup> <sup>1</sup> <sup>1</sup> *Ty <sup>u</sup>* , (34)

(33)

. (35)

u1=c +

Fig. 9. Robust H controller

described by the control law:

where

 \_ e

y2

C(s) Controller

> u1 u2

Fig. 8. Structure of speed control system with weighted process.

so that the infinity norm of the cost function *Ty1u1*, defined as:

Weighted process P(s)

Considering the robust stability and robust performance criteria, the weighting functions for the optimal H controller are chosen and then the iterative computing process continues, until the norm condition is full fit. The performance design specifications of the speed control loop with the H controller are imposed in frequency domain (Ionescu et al., 1998; Morari & Zafiriou, 1990):


According to them, following weighting functions have been considered to describe the perturbed AC drive system with variable moment of inertia and friction coefficient:

$$\begin{cases} \frac{1}{\mathcal{W}\_1(s)} = \mathcal{W}\_1^{-1}(s) = \frac{1}{\mathcal{Y}} \cdot \frac{\left(3s+1\right)^2}{100} \\ \frac{1}{\mathcal{W}\_3(s)} = \mathcal{W}\_3^{-1}(s) = \frac{150}{s+145} \end{cases} \tag{36}$$

where represents the actual step value. The iterative process continues, until the graphic representation in Bode diagram of cost function Ty1u1 reach its maximum value in the proximity of 0 dB axis. In our case, for =39,75 we obtain the infinite norm

$$\left\| T\_{y^1u1} \right\|\_{\otimes} = 0, \text{\(9999\)}.\tag{37}$$

Respecting condition imposed by Equation 34 the corresponding H speed controller is:

$$H\_{\rm av}(\mathbf{s}) = \frac{2327\mathbf{s}^2 + 22211\mathbf{s} + 16495}{\mathbf{s}^3 + 822951\mathbf{s}^2 + 548632\mathbf{s} + 91442} \,\mathrm{.}\tag{38}$$

The inverse weighting functions *W1-1*(*s*) and *W3-1*(*s*) and the sensitivity functions *S*(*s*) and *T*(*s*) are presented in Fig. 10. From the diagram results the influence of the weighting function *W3-1*(*s*) to limit the peak value of *T*(*s*) function. The output of the speed controller, i.e. the active current component, was limited.

Fig. 10. Weighting functions *W1-1*(*s*), *W3-1*(*s*) and sensitivity functions *S*(*s*) and *T*(*s*).

The logarithmic Bode diagram and the Nyquist diagram of the direct-loop transfer function of the weighted process are presented in Fig. 11. According to them we establish the following stability parameters: crossover band 153,7 rad./sec., stability margins: gain margin = 130,3 dB, phase margin = 86,8°. For the same performance and robust stability specifications, a great number of weighting functions described by Equation 36 can be chosen, so the solution of designing an optimal H controller is not unique (Chiang & Safonov, 1992; Zames, 1996). To analyze if the speed control structure with the H controller presented in Equation 38 is robust stable, we apply the stability theorem for a perturbation in the drive system, namely a highest variation of total inertia moment from Jmot to 10Jmot and of the friction coefficient Bmot to 100\*Bmot. The condition

$$\left\| \overline{\Delta\_M}(\mathbf{s}) T(\mathbf{s}) \right\|\_{\infty} < 1 \tag{39}$$

Robust Control of Sensorless AC Drives Based on Adaptive Identification 291

Knowing the expression of the complementary sensibility function T(s), the condition of

Concluding, the control system with H controller remains robust stable for a variation of 10 times of the total inertial moment of the system, related to the catalogue one. Fig. 12 shows the direct-loop transfer function Hd(s) family curves for the PI controller and for the optimal H, controller, at variations of the inertial moment of the synchronous motor, starting at Jmot value, from 3, 5, 7 and 10 times of this value. As we can see from the presented graphs, at the variations of J, though the PI controller doesn't go in instability, thus it is more sensitive at the parameter variations than the H controller. This shows a better robustness of the H

Considering that the largest variation of the friction coefficient of the mechanical system is 100\*Bmot catalogue value. A greater friction coefficient practically means a smaller value of the time constant as like of the amplification factor of the fixed part. Considering the above mentioned, the transfer functions of the nominal process and of the disturbed processes are:

( ) <sup>1</sup>

*<sup>m</sup> <sup>N</sup> m m m*

*<sup>K</sup> P s T s*

 

uncertainty, in the case of 100\**Bmot* friction coefficient, can be written as:

*s*

10-4 10-2 <sup>100</sup> <sup>102</sup> <sup>104</sup> -100

at different inertia moment values.

10\*Jmot

frequency (rad./sec.)

Jmot


0

50

100

150

 Hd [dB] *<sup>K</sup> P s*

*m*

PI

/100 ( ) ( /100) 1

Using Equation 27 and the time constant of the nominal process, the maximal multiplicative

99 99 ( ) 100 1,232 100 *<sup>M</sup>*

Fig. 12. Bode diagram of direct-loop transfer functions for PI (left) and H controller (right),

*T s s*

Hd [dB]


0

50

100

*T s*

*<sup>M</sup>*( ) ( ) 0,9359 *sTs* . (42)

. (43)

. (44)

H

Jmot

10\*Jmot

10-4 10-2 <sup>100</sup> <sup>102</sup> <sup>104</sup> -100

frequency (rad./sec.)

robust stability can be determined, and is:

**4.2.2 Stability analyze for a variation from Bmot to 100Bmot**

optimal regulator.

must be tested, where *M*( )*s* represents the greatest multiplicative uncertainty for the nominal plant.

#### **4.2.1 Stability analyze for a variation from Jmot to 10Jmot**

Considering the calculus way of the transfer function of the process, a ten times growing of the inertial moment, practically means a ten time growing of the time constant of the fixed part. The transfer functions of the nominal process and of the disturbed process are:

$$\begin{cases} P\_N(\mathbf{s}) = \frac{K\_m}{T\_m \mathbf{s} + \mathbf{1}}\\ P(\mathbf{s}) = \frac{K\_m}{10T\_m \mathbf{s} + \mathbf{1}} \end{cases} \tag{40}$$

Using Equation 27 and considering Tm=1,232 sec., the maximum multiplicative uncertainty in the case of ten times growing the inertial moment J, can be modeled:

$$\overline{\Delta\_M}(s) = \frac{-9T\_m s}{10T\_m s + 1} = \frac{-11,097 s}{12,33s + 1} \tag{41}$$

Fig. 11. Direct-loop transfer function of the weighted process in Bode and Nyquist diagram.

290 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

The logarithmic Bode diagram and the Nyquist diagram of the direct-loop transfer function of the weighted process are presented in Fig. 11. According to them we establish the following stability parameters: crossover band 153,7 rad./sec., stability margins: gain margin = 130,3 dB, phase margin = 86,8°. For the same performance and robust stability specifications, a great number of weighting functions described by Equation 36 can be chosen, so the solution of designing an optimal H controller is not unique (Chiang & Safonov, 1992; Zames, 1996). To analyze if the speed control structure with the H controller presented in Equation 38 is robust stable, we apply the stability theorem for a perturbation in the drive system, namely a highest variation of total inertia moment from Jmot to 10Jmot

must be tested, where *M*( )*s* represents the greatest multiplicative uncertainty for the

Considering the calculus way of the transfer function of the process, a ten times growing of the inertial moment, practically means a ten time growing of the time constant of the fixed

( ) <sup>1</sup>

*<sup>m</sup> <sup>N</sup> m m m*

*<sup>K</sup> P s T s*

*<sup>K</sup> P s*

*m*

Fig. 11. Direct-loop transfer function of the weighted process in Bode and Nyquist diagram.

( ) 10 1

Using Equation 27 and considering Tm=1,232 sec., the maximum multiplicative uncertainty

<sup>9</sup> 11,097 ( ) 10 1 12,33 1

*Ts s*

*T s s*

*T s*

part. The transfer functions of the nominal process and of the disturbed process are:

 

in the case of ten times growing the inertial moment J, can be modeled:

*s*

frequency (rad/sec)

phase margin = 86,8° at B = 153,7 rad/sec

gain gain margin = 130,3 dB

[dB]

phase

*<sup>m</sup> <sup>M</sup>*

() () 1 *sTs <sup>M</sup>* (39)

, (40)

(41)

and of the friction coefficient Bmot to 100\*Bmot. The condition

**4.2.1 Stability analyze for a variation from Jmot to 10Jmot**

nominal plant.

Knowing the expression of the complementary sensibility function T(s), the condition of robust stability can be determined, and is:

$$\left\| \overline{\Delta\_M}(\mathbf{s}) T(\mathbf{s}) \right\|\_{\mathfrak{D}} = 0, \mathfrak{P} \mathfrak{B} \mathbf{\tilde{S}} \mathbf{\tilde{s}} \,. \tag{42}$$

Concluding, the control system with H controller remains robust stable for a variation of 10 times of the total inertial moment of the system, related to the catalogue one. Fig. 12 shows the direct-loop transfer function Hd(s) family curves for the PI controller and for the optimal H, controller, at variations of the inertial moment of the synchronous motor, starting at Jmot value, from 3, 5, 7 and 10 times of this value. As we can see from the presented graphs, at the variations of J, though the PI controller doesn't go in instability, thus it is more sensitive at the parameter variations than the H controller. This shows a better robustness of the H optimal regulator.

### **4.2.2 Stability analyze for a variation from Bmot to 100Bmot**

Considering that the largest variation of the friction coefficient of the mechanical system is 100\*Bmot catalogue value. A greater friction coefficient practically means a smaller value of the time constant as like of the amplification factor of the fixed part. Considering the above mentioned, the transfer functions of the nominal process and of the disturbed processes are:

$$\begin{cases} P\_N(s) = \frac{K\_m}{T\_m s + 1} \\ P(s) = \frac{K\_m \;/\; 100 \; s}{(T\_m \;/\; 100)s + 1} \end{cases} \tag{43}$$

Using Equation 27 and the time constant of the nominal process, the maximal multiplicative uncertainty, in the case of 100\**Bmot* friction coefficient, can be written as:

$$\overline{\Delta\_M}(s) = \frac{-99}{T\_m s + 100} = \frac{-99}{1,232s + 100} \,\text{.}\tag{44}$$

Fig. 12. Bode diagram of direct-loop transfer functions for PI (left) and H controller (right), at different inertia moment values.

The robust stability condition will be expressed as:

$$\left\| \overline{\Delta\_M}(\mathbf{s}) T(\mathbf{s}) \right\|\_{\infty} = 0, 9999 \,\text{J} \,\text{.}\tag{45}$$

Robust Control of Sensorless AC Drives Based on Adaptive Identification 293

The power converter is a 1kW voltage source converter, composed by a diode rectifier and a PWM inverter. The PM-SM is a three phase machine, having following catalog

Fig. 14 presents the simulated results for a speed control of the synchronous machine with the designed PI, H2, and H controller, having all parameters at the nominal value, for a starting process to the rated rotor speed with no load torque (left diagram) and with rated load torque (right diagram). In Fig. 15 the speed response is for a perturbed plant with total friction coefficient Btot=50Bmot for the same imposed speed step without (left) and with nominal load torque (right). The speed response for a similar simulation of a perturbed plant with total moment of inertia variation Jtot=11Jmot is presented in Fig. 16 for a no load

For the nominal plant the dynamic performances at a speed step are similar for all three controllers. It is normal to be so, because the controllers have been designed using a simplified model of the machine, working in the steady-state nominal point. The advantage of using optimal H robust controller is evident in the presented simulations when the nominal plant is perturbed, by changing the load torque, the total moment of inertia or the

Fig. 17 presents the speed response for a speed control of the synchronous machine with the designed PI, H2, and H controllers for a starting process at rated speed with no load followed at 0.3 sec. by a rated load step (1.7 Nm). Fig. 18 details the imposed torque step *mr* and the obtained electromagnetic torque *me* and dynamic torque *mj*. A starting process at rated speed, followed by a stopping process for the PM-SM without load and with rated load is presented in Fig. 19. A simulation of the starting process with low speed step is also

In conclusion, we consider that the H optimal robust controller ensures good dynamical performances and stability for a domain of variation large enough of the parameters that can be modified in the process. In applications, where electromechanical parameter variations or load perturbations appear (such as robot control), performant drive systems

with AC machines can be considered, by using robust speed (or position) controllers.





presented in Fig. 20 for all three types of designed controllers.

start (left) and a rated load start (right).

friction coefficient.


values:

We can find that the control system with the H controller remains robust stable for a 100 times variation of the friction coefficient.

### **5. Simulated and experimental results**

Some results of the sensorless control of AC machines based on adaptive identification are presented in main paragraph 3. Because of its sensitivity to parameter variations and external perturbations, a robust solution for the speed controller was proposed and discussed in paragraph 4. Simulated results of an AC drive system controlled by a robust optimal H controller (chapter 5.1) will demonstrate the advantage of using robust control in applications with permanent variation of system parameter or external perturbations, (i.e. robot arms, traction systems, etc.). Finally, some experimental results of a robust control system, with sensorless AC drive, based on adaptive identification, will be presented.

### **5.1 Simulated results of robust controlled driving system with permanent magnet synchronous machine**

A variable speed, FOC structure, of the PM-SM with cancelled longitudinal reaction, fed by a PWM voltage source inverter is presented in Fig.13. The stator current vector split into components leads to the characteristic loops of a FOC system. Rotor position information is obtained from the encoder and rotor speed is computed. The control system was simulated in Matlab/Simulink. The speed controller was designed and simulated using more optimizations criteria, to be compared with. The designed optimal robust H controller was implemented (see Equation 33), a PI controller (see Equation 6) and an optimal H2 controller based on the same performance design specifications like the optimal robust H controller.

Fig. 13. Variable speed VC structure of PM-SM with different types of speed controller.

292 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

We can find that the control system with the H controller remains robust stable for a 100

Some results of the sensorless control of AC machines based on adaptive identification are presented in main paragraph 3. Because of its sensitivity to parameter variations and external perturbations, a robust solution for the speed controller was proposed and discussed in paragraph 4. Simulated results of an AC drive system controlled by a robust optimal H controller (chapter 5.1) will demonstrate the advantage of using robust control in applications with permanent variation of system parameter or external perturbations, (i.e. robot arms, traction systems, etc.). Finally, some experimental results of a robust control system, with sensorless AC drive, based on adaptive identification, will be presented.

**5.1 Simulated results of robust controlled driving system with permanent magnet** 

A variable speed, FOC structure, of the PM-SM with cancelled longitudinal reaction, fed by a PWM voltage source inverter is presented in Fig.13. The stator current vector split into components leads to the characteristic loops of a FOC system. Rotor position information is obtained from the encoder and rotor speed is computed. The control system was simulated in Matlab/Simulink. The speed controller was designed and simulated using more optimizations criteria, to be compared with. The designed optimal robust H controller was implemented (see Equation 33), a PI controller (see Equation 6) and an optimal H2 controller based on the same performance design specifications like the optimal robust H controller.

> r - rotor position

is \*

is

*<sup>M</sup>*( ) ( ) 0,9999 *sTs* . (45)

PWM inverter

Encoder

PM-SM

Rectifier

The robust stability condition will be expressed as:

times variation of the friction coefficient.

**synchronous machine** 

nr \*

nr

isq\*

DIGITAL CONTROL UNIT

\* =0

field oriented control

Fig. 13. Variable speed VC structure of PM-SM with different types of speed controller.

longitudinal reaction isd

speed controller

> speed computation

**5. Simulated and experimental results** 

The power converter is a 1kW voltage source converter, composed by a diode rectifier and a PWM inverter. The PM-SM is a three phase machine, having following catalog values:


Fig. 14 presents the simulated results for a speed control of the synchronous machine with the designed PI, H2, and H controller, having all parameters at the nominal value, for a starting process to the rated rotor speed with no load torque (left diagram) and with rated load torque (right diagram). In Fig. 15 the speed response is for a perturbed plant with total friction coefficient Btot=50Bmot for the same imposed speed step without (left) and with nominal load torque (right). The speed response for a similar simulation of a perturbed plant with total moment of inertia variation Jtot=11Jmot is presented in Fig. 16 for a no load start (left) and a rated load start (right).

For the nominal plant the dynamic performances at a speed step are similar for all three controllers. It is normal to be so, because the controllers have been designed using a simplified model of the machine, working in the steady-state nominal point. The advantage of using optimal H robust controller is evident in the presented simulations when the nominal plant is perturbed, by changing the load torque, the total moment of inertia or the friction coefficient.

Fig. 17 presents the speed response for a speed control of the synchronous machine with the designed PI, H2, and H controllers for a starting process at rated speed with no load followed at 0.3 sec. by a rated load step (1.7 Nm). Fig. 18 details the imposed torque step *mr* and the obtained electromagnetic torque *me* and dynamic torque *mj*. A starting process at rated speed, followed by a stopping process for the PM-SM without load and with rated load is presented in Fig. 19. A simulation of the starting process with low speed step is also presented in Fig. 20 for all three types of designed controllers.

In conclusion, we consider that the H optimal robust controller ensures good dynamical performances and stability for a domain of variation large enough of the parameters that can be modified in the process. In applications, where electromechanical parameter variations or load perturbations appear (such as robot control), performant drive systems with AC machines can be considered, by using robust speed (or position) controllers.

Robust Control of Sensorless AC Drives Based on Adaptive Identification 295

time (sec)

PI

nr (rev/min)

Fig. 19. Starting and fast stopping process with PI, H2 and H controllers for a nominal plant

H2 <sup>H</sup>

PI

time (sec)

Fig. 20. Speed response at low speed step (300 rev/min) with PI, H2 and H controllers.

m e

m r

m j

H

time (sec)

H2

H2

H

PI

time (sec)

nr (rev/min)

time (sec)

with no load (left) and with rated load (right).

H2, PI

H

nr (rev/min)

H

H2

PI

Fig. 17. Speed response for speed and torque steps Fig. 18. Torque diagrams.

Fig. 14. Starting process with PI, H2 and H controller for a nominal plant without load torque (left) and with rated torque (right).

Fig. 15. Starting process with PI, H2 and H controller for a perturbed plant (moment of inertia Jtot=11Jmot ) without load torque (left) and with rated torque (right).

Fig. 16. Starting process with PI, H2 and H controller for a perturbed plant (friction coefficient B=50Bmot) without load torque (left) and with rated torque (right).

294 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Fig. 14. Starting process with PI, H2 and H controller for a nominal plant without load

Fig. 15. Starting process with PI, H2 and H controller for a perturbed plant (moment of

Fig. 16. Starting process with PI, H2 and H controller for a perturbed plant (friction coefficient B=50Bmot) without load torque (left) and with rated torque (right).

inertia Jtot=11Jmot ) without load torque (left) and with rated torque (right).

H2

nr (rev/min)

nr (rev/min)

H

nr (rev/min)

PI

H PI

time (sec)

H PI

time (sec)

H2

time (sec)

H2

H2

time (sec)

time (sec)

H

PI

nr (rev/min)

time (sec)

H2

torque (left) and with rated torque (right).

H PI

PI <sup>H</sup>

H2

nr (rev/min)

nr (rev/min)

Fig. 17. Speed response for speed and torque steps Fig. 18. Torque diagrams.

Fig. 19. Starting and fast stopping process with PI, H2 and H controllers for a nominal plant with no load (left) and with rated load (right).

Fig. 20. Speed response at low speed step (300 rev/min) with PI, H2 and H controllers.

Robust Control of Sensorless AC Drives Based on Adaptive Identification 297

The three control loops (presented also in Fig. 13), namely: speed control loop, implemented with a robust controller; minor loop, necessary for the field-orientation based on the rotor flux and the stator current control loop of the PWM generator are software implemented. In order to obtain the best approach speed and accuracy a fixed point, fractional arithmetic was used. The program modules are: *main*( ) the main loop realizing the hardware initialization and the communication functions with the main processor and the other dedicated processors; *tint*( ) interrupt handling routine executing the current control loop and minor control loop and *int\_4*( ) interrupt handling routine executing the PWM current control loop. The TMS320C5x assembler language was used to achieve an effective software implementation for the *main*  routine and ANSI-C language was used for the control routines *tint* and *int\_4*. Library functions (in C or assembly language) are used for handling the incremental encoder, the A/D converter, the PWM control signal generator and the communication through the serial interface with the screen terminal. The interrupts are generated by programmable counters. So *tint* interrupt is generated by processor internal programmable counter and has a 100s

period. *Int\_4* interrupt is generated by the PWM module and has a 10s period.

algorithm used to determine the rotor speed.

The dynamic evolution of rotor speed, estimated speed based on the MRAS algorithm and speed error for a robust controlled (using the optimal H controller) starting process of the sensorless driving system with induction machine at rated speed step with no load torque is presented in Fig. 22. It can be observed, that the maximum speed error in the dynamic process (starting process) is less then 9 rpm, that means a relative error (speed error/real speed) of around 0,6%, which is a considerable better value then the results obtained for the same sensorless drive system, without a robust speed controller. The relative sped error of the sensorless robust control, after the transitory dynamic process is ended and the drive system works at stabilized speed, is less then 0.05%, which means a really performant estimation (less then 0,7 rpm.). The differences between the two sensorless control structures, the one without a robust control (speed control results presented in Fig.6) and the one with robust control (speed control results presented in Fig. 22), can be explained by the increased robustness imposed by the optimal H controller, to a process (the AC drive system) which is sensitive to parameter variations (rotor time constant) and external perturbations (moment of inertia, friction coefficient), because of the adaptive estimation

Fig. 22. Speed, estimated speed and speed error for a robust controlled starting process.

### **5.2 Experimental results of robust controlled sensorless driving system with induction machine and adaptive speed estimation**

The control unit is based on the fixed-point TMS320C50 Digital Motor Control Board developed by Texas Instruments. The complete schematic of the control architecture is presented in Fig. 21. It is composed of:


Fig. 21. Hardware structure of the robust controlled sensorless driving system with induction machine and adaptive speed estimation.

296 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

The control unit is based on the fixed-point TMS320C50 Digital Motor Control Board developed by Texas Instruments. The complete schematic of the control architecture is

a high speed TDM type serial bus for interconnection with the master processor and

the control board (target hardware) consisting in a microcomputer, based on the



 a debugging computer interfaced to the target hardware through a serial RS232 driver and an emulation and debugging module XDS510. This computer will be used only in

TMS320C5x

 PWM (FPGA)

POWER CONVERTER

is \*

is

us

θr

INDUCTION MACHINE

 MUX A/D

 ENCODER INTERFACE

EPROM

JTAG

UART

TDM

SERIAL BUS

Fig. 21. Hardware structure of the robust controlled sensorless driving system with

**5.2 Experimental results of robust controlled sensorless driving system with** 

TMS320C5x signal processor and the following dedicated modules:

generates the three output signals for the power converter;

**induction machine and adaptive speed estimation** 

presented in Fig. 21. It is composed of:

with the other dedicated processors.



the software developing and debugging phase.

XDS510

HOST

COMPUTER

(PC)

RS232

induction machine and adaptive speed estimation.

The three control loops (presented also in Fig. 13), namely: speed control loop, implemented with a robust controller; minor loop, necessary for the field-orientation based on the rotor flux and the stator current control loop of the PWM generator are software implemented. In order to obtain the best approach speed and accuracy a fixed point, fractional arithmetic was used. The program modules are: *main*( ) the main loop realizing the hardware initialization and the communication functions with the main processor and the other dedicated processors; *tint*( ) interrupt handling routine executing the current control loop and minor control loop and *int\_4*( ) interrupt handling routine executing the PWM current control loop. The TMS320C5x assembler language was used to achieve an effective software implementation for the *main*  routine and ANSI-C language was used for the control routines *tint* and *int\_4*. Library functions (in C or assembly language) are used for handling the incremental encoder, the A/D converter, the PWM control signal generator and the communication through the serial interface with the screen terminal. The interrupts are generated by programmable counters. So *tint* interrupt is generated by processor internal programmable counter and has a 100s period. *Int\_4* interrupt is generated by the PWM module and has a 10s period.

The dynamic evolution of rotor speed, estimated speed based on the MRAS algorithm and speed error for a robust controlled (using the optimal H controller) starting process of the sensorless driving system with induction machine at rated speed step with no load torque is presented in Fig. 22. It can be observed, that the maximum speed error in the dynamic process (starting process) is less then 9 rpm, that means a relative error (speed error/real speed) of around 0,6%, which is a considerable better value then the results obtained for the same sensorless drive system, without a robust speed controller. The relative sped error of the sensorless robust control, after the transitory dynamic process is ended and the drive system works at stabilized speed, is less then 0.05%, which means a really performant estimation (less then 0,7 rpm.). The differences between the two sensorless control structures, the one without a robust control (speed control results presented in Fig.6) and the one with robust control (speed control results presented in Fig. 22), can be explained by the increased robustness imposed by the optimal H controller, to a process (the AC drive system) which is sensitive to parameter variations (rotor time constant) and external perturbations (moment of inertia, friction coefficient), because of the adaptive estimation algorithm used to determine the rotor speed.

Fig. 22. Speed, estimated speed and speed error for a robust controlled starting process.

Robust Control of Sensorless AC Drives Based on Adaptive Identification 299

0,31 sec. and the developed electromagnetic torque of the AC machine. The stator current was limited in the starting process to a 3,5 times value of the rated current (*Is,max*=17A). The estimated speed, the rotor speed and the speed error are presented in Fig. 26. The maximum relative speed error in the transitory process (around 1,4%), is significantly greater as for a rated speed step and the zero convergence time of the speed error is significantly longer. It is an expected result because in all presented FOC structures, by using a speed controller in the active control loop of the AC drive (Fig. 2 and Fig. 13), the rotor speed is directly controlled while the electromagnetic torque is indirectly controlled. In applications where a direct torque control is needed, the dynamic performances of torque response can be

Fig. 25. Performance indexes of MRAS algorithm (left diagram), imposed load torque step

Fig. 26. Speed evolution for a starting process and a rated load step at *t*=0.3 sec.

and developed electromagnetic torque (right diagram).

, and the error ε), the imposed load torque step at

the adjustable model performance index *p*

improved.

The speed and estimation speed error for the same control structure at a full loaded starting process (with nominal imposed torque) are presented in Fig. 23. The time constant of the transitory starting process is as expected a few times bigger than in the starting process with no load (Fig.22.) and the relative speed error is around 0,3%, that means less then at no-load start. That can be explained, due to the fact that the speed estimator itself is less sensitive to parameter variations when it works with nominal inputs. The dynamic evolution of the different currents of the induction machine for the same process are presented in Fig. 24, namely the stator current, the active (torque producing) and reactive (flux producing) components of the stator current and the estimated magnetizing current.

Fig. 23. Rotor speed and speed error at rated load starting process.

Fig. 24. Different currents of the IM at a starting process with rated load.

The dynamic behavior of the AC drive, if a rated load step is imposed to the system running at a stable rotor speed (rated speed), is tested. Fig. 25 presents the performance indexes of the MRAS based speed identification algorithm (the reference model performance index *p*, 298 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

The speed and estimation speed error for the same control structure at a full loaded starting process (with nominal imposed torque) are presented in Fig. 23. The time constant of the transitory starting process is as expected a few times bigger than in the starting process with no load (Fig.22.) and the relative speed error is around 0,3%, that means less then at no-load start. That can be explained, due to the fact that the speed estimator itself is less sensitive to parameter variations when it works with nominal inputs. The dynamic evolution of the different currents of the induction machine for the same process are presented in Fig. 24, namely the stator current, the active (torque producing) and reactive (flux producing)

components of the stator current and the estimated magnetizing current.

Fig. 23. Rotor speed and speed error at rated load starting process.

Fig. 24. Different currents of the IM at a starting process with rated load.

The dynamic behavior of the AC drive, if a rated load step is imposed to the system running at a stable rotor speed (rated speed), is tested. Fig. 25 presents the performance indexes of the MRAS based speed identification algorithm (the reference model performance index *p*, the adjustable model performance index *p* , and the error ε), the imposed load torque step at 0,31 sec. and the developed electromagnetic torque of the AC machine. The stator current was limited in the starting process to a 3,5 times value of the rated current (*Is,max*=17A). The estimated speed, the rotor speed and the speed error are presented in Fig. 26. The maximum relative speed error in the transitory process (around 1,4%), is significantly greater as for a rated speed step and the zero convergence time of the speed error is significantly longer. It is an expected result because in all presented FOC structures, by using a speed controller in the active control loop of the AC drive (Fig. 2 and Fig. 13), the rotor speed is directly controlled while the electromagnetic torque is indirectly controlled. In applications where a direct torque control is needed, the dynamic performances of torque response can be improved.

Fig. 25. Performance indexes of MRAS algorithm (left diagram), imposed load torque step and developed electromagnetic torque (right diagram).

Fig. 26. Speed evolution for a starting process and a rated load step at *t*=0.3 sec.

Robust Control of Sensorless AC Drives Based on Adaptive Identification 301

Birou, I. & Pavel, S. (2008). Sensorless AC Driving Systems Based on Adaptive Identification

Briz, F., Degner, M., Garcia, P. & Lorenz, R (2004). Comparison of Saliency–Based Sensorless

Caruana, C., Asher, G., Bradley, K. & Woolfson, M. (2003). Flux Positon Estimation in Cage

Cirrincione, M. & Pucci, M. (2005). An MRAS-Based Sensorless High-Performance Induction

Consoli, A., Scarcella, G., Testa, A. (2003). Using the Induction Motor as Flux Sensor: New

Doyle, J., Glover, K., Khargonekar, P. & Francis, B. (1989). State space solutions to standard

Green, M. & Limebeer, D. (1995) *Linear Robust Control*. Prentice-Hall, Englewood Cliffs, 1995 Hinkkanen, M. (2004) Analysis and Design of Full-Order Flux Observers for Sensorless

Holtz, J. & Quan, J. (2002). Sensorless Vector Control of Induction Motors at Very Low

Holtz J. (2006). Sensorless Control of Induction Machines – with or without Signal Injection?.

Hurst, K., Habetler, T., Griva, G. & Profumo, F. (1998). Zero-speed tacholess induction

Bryson, A. (1996). Optimal control. In *IEEE Control Systems*, Vol. 16, No.3, pp. 26-33

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90, No. 8, August 2002, pp. 1359-1368, ISSN: 0018-9219

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7-8 oct. 2010, pp. 42-47, ISSN 1842-4805

No.4, pp. 1107-1115, ISSN: 0093-9994

Massachusetts, 1992

831-847

1033-1040, ISSN: 0278-0046

Algorithms and Robust Control Strategies. In *Proceedings of the 9th International Conference DAS'2008,* May 22-24, Suceava, Romania, pp. 93-96, ISSN 1844-5038 Birou, I., Maier, V., Pavel, S. & Rusu, C. (2010). Control of AC Drives; a Balance Between

Dynamic Performance, Energy Efficiency and Cost Constrains. In: *Anals of University of Craiova, Electrical Engineering series,* No. 34, Vol. II, Craiova, Romania,

Control for AC Machines. In *IEEE Transactions on Industry Applications*, Vol.40,

Induction Machines Using Synchronous HF injection and Kalman Filtering. In *IEEE Transactions on Industry Applications*, Vol.39, No.5, pp. 1372-1378, ISSN: 0093-9994 Chiang, R. & Safonov, M. (1992). *Robust Control Toolbox. User's Guide*. The Math Works Inc.,

Motor Drive With a Predictive Adaptive Model. In *IEEE Transactions on Industrial* 

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The results confirm, that the robust control, using an optimal H speed controller, of the sensorless AC drives, based on adaptive identification with MRAS algorithm for speed estimation, assure:


but needs greater computing time for the control loop, that means more powerful real time computing systems (with high frequency digital signal processors) and increasing costs.

### **6. Conclusion**

Sensorless control of AC drive systems became in the last years a challenge for intelligent motion control. To estimate rotor speed or position, by using adaptive identification algorithms based on model reference, is a relative easy task and became therefore a valuable solution for applications where not high precision of speed/position estimation is needed. The main disadvantage of the MRAS identification algorithms is the relative high sensitivity to machine parameter variations (i.e. rotor time constant) and to external perturbations (i.e. moment of inertia, friction coefficient). Using the robust control theory seams to be a good solution to increase the speed/position estimation accuracy and to apply the same estimation method for a large number of motion control applications. The optimal H speed controller makes the AC drive system stable to a large scale of parameter variations and perturbations and provides high dynamic performance as well as accurate speed estimation to the sensorless control structure.

### **7. Acknowledgment**

The author would like to acknowledge the Alexander von Humboldt Foundation for their support in the research activity, which made possible an important part of the presented results.

### **8. References**


300 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

The results confirm, that the robust control, using an optimal H speed controller, of the sensorless AC drives, based on adaptive identification with MRAS algorithm for speed


but needs greater computing time for the control loop, that means more powerful real time computing systems (with high frequency digital signal processors) and increasing costs.

Sensorless control of AC drive systems became in the last years a challenge for intelligent motion control. To estimate rotor speed or position, by using adaptive identification algorithms based on model reference, is a relative easy task and became therefore a valuable solution for applications where not high precision of speed/position estimation is needed. The main disadvantage of the MRAS identification algorithms is the relative high sensitivity to machine parameter variations (i.e. rotor time constant) and to external perturbations (i.e. moment of inertia, friction coefficient). Using the robust control theory seams to be a good solution to increase the speed/position estimation accuracy and to apply the same estimation method for a large number of motion control applications. The optimal H speed controller makes the AC drive system stable to a large scale of parameter variations and perturbations and provides high dynamic performance as well as accurate speed estimation

The author would like to acknowledge the Alexander von Humboldt Foundation for their support in the research activity, which made possible an important part of the presented

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estimation, assure:

**6. Conclusion** 

to the sensorless control structure.

ISSN: 0093-9994

Vol.47, No.2, pp. 315-324, ISSN: 0278-0046

Vol. 29, No. 2, Great Britain, pp. 1-12, ISSN: 0005-1098

March 1-4, 2003, pp. 1819-1825

**7. Acknowledgment** 

results.

**8. References** 


Paolo Mercorelli

*Germany*

**16**

*Faculty of Automotive Engineering, Ostfalia University of Applied Sciences*

**Permanent Magnet Synchronous Motors** 

**A Robust Decoupling Estimator to Identify** 

**Electrical Parameters for Three-Phase** 

As the high field strength neodymium-iron-boron (NdFeB) magnets become commercially available and affordable, the sinusoidal back electromagnetic force (emf) permanent magnet synchronous motors (PMSMs) are receiving increasing attention due to their high speed, high power density and high efficiency. These characteristics are very favourable for high performance applications, e.g., robotics, aerospace, and electric ship propulsion systems Rahman et al. (1996), Ooshima et al. (2004). PMSMs as traction motors are common in electric or hybrid road vehicles, but not yet widely used for rail vehicles. Although the traction PMSM has many advantages, just a few prototypes of vehicles were built and tested. The following two new prototypes of rail vehicles with traction PMSMs, which were presented at the InnoTrans fair in Berlin 2008, were the Alstom AGV high speed train and the Skoda Transportation low floor tram 15T ForCity. The greatest advantage of the PMSM is its low volume in contrast to other types of motors, which makes a direct drive of wheels possible. However, the traction drive with PMSM must meet special requirements typical for overhead-line-fed vehicles. The drives and especially their control should be robust to a wide range of overhead line voltage tolerance (typically from −30% to +20% ), voltage surges and input filter oscillations. These features may cause problems during flux weakening operation, which must be used for several reasons. The typical reason is to obtain constant power operation in a wide speed range and to reach nominal power during low speed (commonly 1/3 of the maximum speed). In the case of common traction motors such as asynchronous or DC motors, it is possible to reach the constant power region using flux weakening. This is also possible for traction PMSM, however, a problem with high back emf arises. In the report by Dolecek (2009), the usage of a flux weakening control strategy for PMSM as a prediction control structure is shown to improve the dynamic performance of traditional feedback control strategies. This is obtained in terms, for instance, of overshoot and rising time. It is known that, an accurate knowledge of the model and its parameters is necessary for realizing an effective prediction control. To achieve desired system performance, advanced control systems are usually required to provide fast and accurate response, quick disturbance recovery and parameter variations insensitivity Rahman et al. (2003). Acquiring accurate models for systems under investigation is usually the fundamental part in advanced control system designs. For instance, proper implementation of flux weakening control requires the knowledge of synchronous machine parameters. The most

**1. Introduction**


## **A Robust Decoupling Estimator to Identify Electrical Parameters for Three-Phase Permanent Magnet Synchronous Motors**

Paolo Mercorelli *Faculty of Automotive Engineering, Ostfalia University of Applied Sciences Germany*

#### **1. Introduction**

302 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

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Lorenz, R., (2010). Core Sensor Integration Technologies for Advanced Electrical Drives and

Mita, T., Hirata, M., Murata, K. & Zhang, H. (1998). H Control Versus Disturbance-

Morari, M., Zafiriou, E. (1990). *Robust Process Control*, Prentice Hall, Englewood Cliffs, 1990 Moreira, J., Hung, K., Lipo, T. & Lorenz, R. (1991). A simple and robust adaptive controller

Rajashekara, K., Kawamura, A. & Matsuse, K. (1996). *Sensorless Control of AC Motor Drives.*

Safonov, M. (1980). *Stability and Robustness of Multivariable Feedback Systems*. MIT Press, 1980 Toliyat, H., Levi, E. & Raina, M. (2003). A Review of RFO Induction Motor Parameter

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Zadeh, L. (1996). The evolution of systems analysis and control: a personal perspective. In

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Vas, P. (1998). *Sensorless Vector and Direct Torque Control*, Oxford University Press, 1998 Wieser, R. (1998). Optimal rotor flux regulation for fast-accelerating induction machines in

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*Machine Drives*. Omikk Publisher, Budapest, 1991

*Meeting.* Vol. 2, pp. 977-984, ISBN: 0-7803-8486-5

Analysis and comparison of alternative methods. In *IEEE Transactions on Industrial* 

IPM Synchronous Machine Drives. In *Proceedings of the 39th IEEE-IAS Annual* 

*European Control Conference*, Groningen, Netherlands, March 1-4, 2003, pp. 334-337.

Sensorless Induction Machine Drives Without Signal Injection. In *IEEE Transactions* 

Power Converters-Keynote Speaker, In *Proc. of Int. Conference on Development and Application Systems DAS'2010,* May 27-29, 2010, Suceava, Romania, ISSN 1844-5020 McFarlane, D., Glover, K. (1990). Robust Controller Design Using Normalized Coprime

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the field-weakening region*. IEEE Transactions on Industry Applications*, Vol.34, No.5,

As the high field strength neodymium-iron-boron (NdFeB) magnets become commercially available and affordable, the sinusoidal back electromagnetic force (emf) permanent magnet synchronous motors (PMSMs) are receiving increasing attention due to their high speed, high power density and high efficiency. These characteristics are very favourable for high performance applications, e.g., robotics, aerospace, and electric ship propulsion systems Rahman et al. (1996), Ooshima et al. (2004). PMSMs as traction motors are common in electric or hybrid road vehicles, but not yet widely used for rail vehicles. Although the traction PMSM has many advantages, just a few prototypes of vehicles were built and tested. The following two new prototypes of rail vehicles with traction PMSMs, which were presented at the InnoTrans fair in Berlin 2008, were the Alstom AGV high speed train and the Skoda Transportation low floor tram 15T ForCity. The greatest advantage of the PMSM is its low volume in contrast to other types of motors, which makes a direct drive of wheels possible. However, the traction drive with PMSM must meet special requirements typical for overhead-line-fed vehicles. The drives and especially their control should be robust to a wide range of overhead line voltage tolerance (typically from −30% to +20% ), voltage surges and input filter oscillations. These features may cause problems during flux weakening operation, which must be used for several reasons. The typical reason is to obtain constant power operation in a wide speed range and to reach nominal power during low speed (commonly 1/3 of the maximum speed). In the case of common traction motors such as asynchronous or DC motors, it is possible to reach the constant power region using flux weakening. This is also possible for traction PMSM, however, a problem with high back emf arises. In the report by Dolecek (2009), the usage of a flux weakening control strategy for PMSM as a prediction control structure is shown to improve the dynamic performance of traditional feedback control strategies. This is obtained in terms, for instance, of overshoot and rising time. It is known that, an accurate knowledge of the model and its parameters is necessary for realizing an effective prediction control. To achieve desired system performance, advanced control systems are usually required to provide fast and accurate response, quick disturbance recovery and parameter variations insensitivity Rahman et al. (2003). Acquiring accurate models for systems under investigation is usually the fundamental part in advanced control system designs. For instance, proper implementation of flux weakening control requires the knowledge of synchronous machine parameters. The most

dq-model is considered in this work.

Fig. 1. Park's transformation for the motor

⎤ ⎦ = ⎡ ⎢ ⎣

⎡ ⎢ ⎣

<sup>−</sup> *Rs Lq* <sup>−</sup> *Ld Lq ωel*(*t*)

an isotropic motor with *Ld* � *Lq* = *Ldq*, it follows:

<sup>−</sup> *Rs Ldq* 2 sin(*ωelt*) 3

Parameters for Three-Phase Permanent Magnet Synchronous Motors

2 cos(*ωelt*) 3

1 3

2 cos(*ωelt*) 3

−2 sin(*ωelt*) 3

1 3

*Mm* <sup>=</sup> <sup>3</sup>

*ωel*(*t*)

� � *id*(*t*) *iq*(*t*) � + � 1 *Ldq* 0 0 <sup>1</sup> *Ldq* � �

*Mm* <sup>=</sup> <sup>3</sup> 2

*ωel*(*t*)

2 sin(*ωel*−2*π*/3) 3

<sup>305</sup> A Robust Decoupling Estimator to Identify Electrical

2 cos(*ωel*−2*π*/3) 3

1 3

2 cos(*ωel*−2*π*/3) 3

−2 sin(*ωel*−2*π*/3) 3

1 3

� � *id*(*t*) *iq*(*t*)

The dynamic model of the synchronous motor in d-q-coordinates can be represented as

� + � 1 *Ld* 0 0 <sup>1</sup> *Lq*

In (3) and (4), *id*(*t*), *iq*(*t*), *ud*(*t*) and *uq*(*t*) are the dq-components of the stator currents and voltages in synchronously rotating rotor reference frame, *ωel*(*t*) is the rotor electrical angular speed, the parameters *Rs*, *Ld*, *Lq*, Φ and *p* are the stator resistance, d-axis and q-axis inductance, the amplitude of the permanent magnet flux linkage, and *p* the number of couples of permanent magnets, respectively. At the end, *Mm* indicates the motor torque. Considering

2 sin(*ωel*+2*π*/3) 3

⎤ ⎥ ⎦

⎡ ⎣

⎤ ⎥ ⎦

� −

⎡ ⎣

*ia*(*t*) *ib*(*t*) *ic* (*t*)

� 0 Φ*ωel*(*t*)

⎤

*ua*(*t*) *ub*(*t*) *uc*(*t*) ⎤

⎦ , (1)

⎦ . (2)

, (3)

�

2 cos(*ωel*+2*π*/3) 3

1 3

2 cos(*ωel*+2*π*/3) 3

−2 sin(*ωel*+2*π*/3) 3

1 3

� � *ud*(*t*) *uq*(*t*)

<sup>2</sup> *<sup>p</sup>*{Φ*iq*(*t*)+(*Ld* <sup>−</sup> *Lq*)*id*(*t*)*iq*(*t*)}. (4)

*ud*(*t*) *uq*(*t*) � −

*p*Φ*iq*(*t*), (6)

� 0 Φ*ωel*(*t*) �

, (5)

*ud*(*t*) *uq*(*t*) *u*0(*t*)

⎡ ⎣

⎡ ⎣

� *did*(*t*) *dt diq*(*t*) *dt*

> � *did*(*t*) *dt diq*(*t*) *dt*

� = � <sup>−</sup> *Rs Ldq*

follows:

and

and

*id*(*t*) *iq*(*t*) *i*0(*t*)

> � = � <sup>−</sup> *Rs Ld Lq Ld ωel*(*t*)

⎤ ⎦ =

common parameters required for the implementation of such advanced control algorithms are the classical simplified model parameters: *Ld* - the direct axis self-inductance, *Lq* - the quadrature axis self-inductance, and Φ - the permanent magnet flux linkage. Prior knowledge of the previously mentioned parameters and the number of pole pairs p allows for the implementation of torque control through the use of current vector control. Techniques have been proposed for the parameters' identification of PMSM from different perspectives, such as offline Kilthau et al. (2002), Weisgerber et al. (1997) and online identification of PMSM electrical parameters Mobarakeh et al. (2001), Khaburi et al. (2003). These technique are based on the decoupled control of linear systems when the motor's mechanical dynamics are ignored. Using a decoupling control strategy, internal dynamics may be almost obscured, but it is useful to remember that there are no limitations in the controllability and observability of the system. In the report by Mercorelli et al. (2003) a decoupling technique is used to control a permanent magnets machine more efficiently in a sensorless way using an observer. The work described by Liu et al. (2008) investigates the possibility of using a numerical approach Particle Swarm Optimization (PSO) as a promising alternative. PSO approach uses a system with a known model structure but unknown parameters. The parameter identification problem can be treated as an optimisation problem, involving comparison of the system output with the model output. The discrepancy between the system and model outputs is minimised by optimisation based on a fitness function, which is defined as a measure of how well the model output fits the measured system output. This approach utilises numerical techniques for the optimisation, and it can incur in difficult non-convex optimisation problems because of the nonlinearity of the motor model. Despite limitation on the frequency range of identification, this paper proposes a dynamic observer based on an optimised decoupling technique to estimate *Ldq* and *Rs* parameters. The proposed optimisation technique, similar to that presented by Mercorelli (2009) applies a procedure based on minimum variance error to minimise the effects of non-exact cancelation due to the decoupling controller. In the meantime, the paper proposes a particular observer that identifies the permanent magnet flux using the estimated *Ldq* and *Rs* parameters. The whole structure of the observer is totally new. The limit of this observer for the estimation of the permanent magnet flux is given by the range of work frequency. In fact, examining the theoretical structure of the observer, these limits appear evident and are validated with simulated data, as the estimation becomes inaccurate for low and high velocity of the motor. Because of the coupled nonlinear system structure, a general expression of limits is not easy to find. The paper is organised in the following way: a sketch of the model of the synchronous motor and its behaviour are given in Section 2, Section 3 is devoted to deriving, proposing and discussing the dynamic estimator, and Section 4 shows the simulation results using real data for a three-phase PMSM.

### **2. Model and behavior of a synchronous motor**

To aid advanced controller design for PMSM, it is very important to obtain an appropriate model of the motor. A good model should not only be an accurate representation of system dynamics but should also facilitate the application of existing control techniques. Among a variety of models presented in the literature since the introduction of PMSM, the two-axis dq-model obtained using Park's transformation is the most widely used in variable speed PMSM drive control applications Rahman et al. (2003) and Khaburi et al. (2003). The Park's dq-transformation is a coordinate transformation that converts the three-phase stationary variables into variables in a rotating coordinate system. In dq-transformation, the rotating coordinate is defined relative to a stationary reference angle as illustrated in Fig. 1. The dq-model is considered in this work.

2 Will-be-set-by-IN-TECH

common parameters required for the implementation of such advanced control algorithms are the classical simplified model parameters: *Ld* - the direct axis self-inductance, *Lq* - the quadrature axis self-inductance, and Φ - the permanent magnet flux linkage. Prior knowledge of the previously mentioned parameters and the number of pole pairs p allows for the implementation of torque control through the use of current vector control. Techniques have been proposed for the parameters' identification of PMSM from different perspectives, such as offline Kilthau et al. (2002), Weisgerber et al. (1997) and online identification of PMSM electrical parameters Mobarakeh et al. (2001), Khaburi et al. (2003). These technique are based on the decoupled control of linear systems when the motor's mechanical dynamics are ignored. Using a decoupling control strategy, internal dynamics may be almost obscured, but it is useful to remember that there are no limitations in the controllability and observability of the system. In the report by Mercorelli et al. (2003) a decoupling technique is used to control a permanent magnets machine more efficiently in a sensorless way using an observer. The work described by Liu et al. (2008) investigates the possibility of using a numerical approach Particle Swarm Optimization (PSO) as a promising alternative. PSO approach uses a system with a known model structure but unknown parameters. The parameter identification problem can be treated as an optimisation problem, involving comparison of the system output with the model output. The discrepancy between the system and model outputs is minimised by optimisation based on a fitness function, which is defined as a measure of how well the model output fits the measured system output. This approach utilises numerical techniques for the optimisation, and it can incur in difficult non-convex optimisation problems because of the nonlinearity of the motor model. Despite limitation on the frequency range of identification, this paper proposes a dynamic observer based on an optimised decoupling technique to estimate *Ldq* and *Rs* parameters. The proposed optimisation technique, similar to that presented by Mercorelli (2009) applies a procedure based on minimum variance error to minimise the effects of non-exact cancelation due to the decoupling controller. In the meantime, the paper proposes a particular observer that identifies the permanent magnet flux using the estimated *Ldq* and *Rs* parameters. The whole structure of the observer is totally new. The limit of this observer for the estimation of the permanent magnet flux is given by the range of work frequency. In fact, examining the theoretical structure of the observer, these limits appear evident and are validated with simulated data, as the estimation becomes inaccurate for low and high velocity of the motor. Because of the coupled nonlinear system structure, a general expression of limits is not easy to find. The paper is organised in the following way: a sketch of the model of the synchronous motor and its behaviour are given in Section 2, Section 3 is devoted to deriving, proposing and discussing the dynamic estimator,

and Section 4 shows the simulation results using real data for a three-phase PMSM.

To aid advanced controller design for PMSM, it is very important to obtain an appropriate model of the motor. A good model should not only be an accurate representation of system dynamics but should also facilitate the application of existing control techniques. Among a variety of models presented in the literature since the introduction of PMSM, the two-axis dq-model obtained using Park's transformation is the most widely used in variable speed PMSM drive control applications Rahman et al. (2003) and Khaburi et al. (2003). The Park's dq-transformation is a coordinate transformation that converts the three-phase stationary variables into variables in a rotating coordinate system. In dq-transformation, the rotating coordinate is defined relative to a stationary reference angle as illustrated in Fig. 1. The

**2. Model and behavior of a synchronous motor**

Fig. 1. Park's transformation for the motor

$$
\begin{bmatrix} u\_d(t) \\ u\_q(t) \\ u\_0(t) \end{bmatrix} = \begin{bmatrix} \frac{2\sin(\omega\_d t)}{3} & \frac{2\sin(\omega\_{cl} - 2\pi/3)}{3} & \frac{2\sin(\omega\_{cl} + 2\pi/3)}{3} \\ \frac{2\cos(\omega\_{cl} t)}{3} & \frac{2\cos(\omega\_{cl} - 2\pi/3)}{3} & \frac{2\cos(\omega\_{cl} + 2\pi/3)}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{bmatrix} \begin{bmatrix} u\_a(t) \\ u\_b(t) \\ u\_c(t) \end{bmatrix},\tag{1}
$$

$$
\begin{bmatrix}
\dot{i}\_d(t) \\
\dot{i}\_q(t) \\
\dot{i}\_0(t)
\end{bmatrix} = \begin{bmatrix}
\frac{2\cos(\omega\_{el}t)}{3} & \frac{2\cos(\omega\_{el} - 2\pi/3)}{3} & \frac{2\cos(\omega\_{el} + 2\pi/3)}{3} \\
\frac{-2\sin(\omega\_{el}t)}{3} & \frac{-2\sin(\omega\_{el} - 2\pi/3)}{3} & \frac{-2\sin(\omega\_{el} + 2\pi/3)}{3} \\
\frac{1}{3} & \frac{1}{3} & \frac{1}{3}
\end{bmatrix} \begin{bmatrix}
\dot{i}\_d(t) \\
\dot{i}\_b(t) \\
\dot{i}\_c(t)
\end{bmatrix}.
\tag{2}
$$

The dynamic model of the synchronous motor in d-q-coordinates can be represented as follows:

$$
\begin{bmatrix}
\frac{d\dot{u}\_{\ell}(t)}{dt} \\
\frac{d\dot{l}\_{q}(t)}{dt}
\end{bmatrix} = \begin{bmatrix}
\end{bmatrix} \begin{bmatrix}
\dot{i}\_{d}(t) \\
\dot{i}\_{q}(t)
\end{bmatrix} + \begin{bmatrix}
\frac{1}{L\_{q}} & 0 \\
0 & \frac{1}{L\_{q}}
\end{bmatrix} \begin{bmatrix}
u\_{d}(t) \\
u\_{q}(t)
\end{bmatrix} - \begin{bmatrix}
0 \\
\Phi\omega\_{\ell l}(t)
\end{bmatrix},\tag{3}
$$

and

$$M\_m = \frac{3}{2} p \{ \Phi \dot{i}\_q(t) + (L\_d - L\_q) \dot{i}\_d(t) \dot{i}\_q(t) \}. \tag{4}$$

In (3) and (4), *id*(*t*), *iq*(*t*), *ud*(*t*) and *uq*(*t*) are the dq-components of the stator currents and voltages in synchronously rotating rotor reference frame, *ωel*(*t*) is the rotor electrical angular speed, the parameters *Rs*, *Ld*, *Lq*, Φ and *p* are the stator resistance, d-axis and q-axis inductance, the amplitude of the permanent magnet flux linkage, and *p* the number of couples of permanent magnets, respectively. At the end, *Mm* indicates the motor torque. Considering an isotropic motor with *Ld* � *Lq* = *Ldq*, it follows:

$$
\begin{bmatrix}
\frac{di\_d(t)}{dt} \\
\frac{di\_q(t)}{dt}
\end{bmatrix} = \begin{bmatrix}
\end{bmatrix} \begin{bmatrix}
i\_d(t) \\
i\_q(t)
\end{bmatrix} + \begin{bmatrix}
\frac{1}{L\_{dq}} & 0 \\
0 & \frac{1}{L\_{dq}}
\end{bmatrix} \begin{bmatrix}
u\_d(t) \\
u\_q(t)
\end{bmatrix} - \begin{bmatrix}
0 \\
\Phi\omega\_{el}(t)
\end{bmatrix},\tag{5}
$$

and

$$M\_m = \frac{3}{2} p \Phi i\_q(t),\tag{6}$$

with the following movement equation:

$$M\_m - M\_{\overline{w}} = J \frac{d\omega\_{\text{mec}}(t)}{dt} \,, \tag{7}$$

then the decoupling of the dynamics is obtained via the following relationship:

<sup>+</sup> im <sup>1</sup>

(10) and a suitable dynamics for sake of estimation. Condition (10) is guaranteed if

*dt* <sup>=</sup> <sup>−</sup> *Rs*

<sup>ˆ</sup>*id*(*t*) + *ud*(*t*) *Ldq*

*Ldq* 0 0 <sup>1</sup> *Ldq*

<sup>307</sup> A Robust Decoupling Estimator to Identify Electrical

where the parameters *F*11, *F*12, *F*21, and *F*<sup>22</sup> are to be calculated in order to guarantee condition

*Ldq*

**Proposition 1.** *Considering the disturbance n*(Δ(*Ldq* <sup>−</sup> *<sup>L</sup>*<sup>ˆ</sup> *dq*)) *of Eq. (12) as a white noise, then*

 = *σ* 

**Proof 1.** *If Eqs. (12) and (13) are discretised using Implicit Euler with a sampling frequency equal to*

*id*(*k* − 2) + *b*1*ud*(*k* − 1)+

*id*(*k* − 2) + *b*1*ud*(*k* − 1) + *b*2*ud*(*k* − 2)+

*where the coefficients a, b, c*1*, c*2*, are to be estimated, and n*(*k*) *is assumed as white noises. The next*

*eid* (*<sup>k</sup>* <sup>+</sup>1) = *<sup>a</sup>*<sup>1</sup> <sup>ˆ</sup>*id*(*k*) + *<sup>a</sup>*<sup>2</sup> <sup>ˆ</sup>*id*(*<sup>k</sup>* <sup>−</sup>1) + *<sup>b</sup>*1*ud*(*k*) + *<sup>b</sup>*2*ud*(*<sup>k</sup>* <sup>−</sup>1) + *<sup>n</sup>*(*<sup>k</sup>* <sup>+</sup>1) + *<sup>c</sup>*1*n*(*k*) + *<sup>c</sup>*2*n*(*<sup>k</sup>* <sup>−</sup>1).

*F*<sup>11</sup> *F*<sup>12</sup> *<sup>F</sup>*<sup>21</sup> *<sup>F</sup>*<sup>22</sup> <sup>0</sup>

*id*(*t*) + *ud*(*t*) *Ldq*

*id*(*t*) <sup>−</sup> <sup>ˆ</sup>

*ts Ldq*(<sup>1</sup> + *ts Rs*

*Ldq* )

*ts Ldq*(<sup>1</sup> + *ts Rs* *id*(*t*)

*Ldq* )

*b*2*ud*(*k* − 2) + *n*(*k*) + *c*1*un*(*k* − 1) + *c*2*un*(*k* − 2). (16)

1 

*F*<sup>12</sup> = −*ωel*(*t*)*Ldq*. (11)

<sup>⊆</sup> im 0 1 

, (12)

*is obtained by minimising the*

*ud*(*k*), (14)

*ud*(*k*) + *n*(*k*). (15)

*n*(*k*) + *c*1*n*(*k* − 1) + *c*2*n*(*k* − 2), (17)

(18)

<sup>+</sup> *<sup>n</sup>*(Δ(*ωel*(*t*)(*Ldq* <sup>−</sup> *<sup>L</sup>*<sup>ˆ</sup> *dq*))), (13)

, (10)

im <sup>−</sup> *Rs Ldq*

<sup>−</sup> *Rs Ldq*

*d*ˆ*id*(*t*)

*the current minimum variance error σ*

*ts, then it follows that:*

*id*(*k*) = ˆ

*eid* (*k*) = *<sup>a</sup>*<sup>1</sup> <sup>ˆ</sup>

*sample is:*

*id*(*k*) + *a*<sup>1</sup> ˆ

*id*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) + *<sup>a</sup>*<sup>2</sup> <sup>ˆ</sup>

*estimation error of the parameters Ldq and Rs.*

*ωel*(*t*)

Parameters for Three-Phase Permanent Magnet Synchronous Motors

*did*(*t*)

where *<sup>n</sup>*(Δ(*Ldq* <sup>−</sup> *<sup>L</sup>*<sup>ˆ</sup> *dq*)) is the disturbance due to the inexact cancelation.

 *eid* (*t*)

*id*(*k* − 1) (1 + *ts Rs Ldq* ) +

*It is possible to assume an ARMAX model for the system represented by (15) and thus*

*ωel*(*t*)

Because of the possible inexact decoupling, it follows that:

*Ldq*

<sup>ˆ</sup>*id*(*k*) = <sup>ˆ</sup>

<sup>ˆ</sup>*id*(*k*) = <sup>ˆ</sup>*id*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) (1 + *ts Rs Ldq* ) +

*id*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) + *<sup>a</sup>*<sup>2</sup> <sup>ˆ</sup>

*Letting eid* (*k*) = *id*(*k*) <sup>−</sup> <sup>ˆ</sup>*id*(*k*) *as mentioned above, it follows that:*

*dt* <sup>=</sup> <sup>−</sup> *Rs*

where *pωmech*(*t*) = *ωel*(*t*) and *Mw* is an unknown mechanical load.

### **3. Structure of the decoupling dynamic estimator**

The present estimator uses the measurements of input voltages, currents and angular velocity of the motor to estimate the "d-q" winding inductance, the rotor resistance and amplitude of the linkage flux. The structure of the estimator is described in Fig. 2. This diagram shows how the estimator works. In particular, after having decoupled the system described in (5), the stator resistance *Rs* and the inductance *Ldq* are estimated through a minimum error variance approach. The estimated values *R*ˆ*<sup>s</sup>* and *Ldq* are used for to estimate of the amplitude of the linkage flux (Φˆ ).

Fig. 2. Conceptual structure of the whole estimator

#### **3.1 Decoupling structure and minimum error variance algorithm**

To achieve a decoupled structure of the system described in Eq. (5), a matrix **F** is to be calculated such that,

$$(\mathbf{A} + \mathbf{B}\mathbf{F})\mathcal{V} \subseteq \mathcal{V}\_{\prime} \tag{8}$$

where **u**(*t*) = **Fx**(*t*) is a state feedback with **u**(*t*)=[*ud*(*t*), *uq*(*t*)]*<sup>T</sup>* and **x**(*t*)=[*id*(*t*), *iq*(*t*)]*T*,

$$\mathbf{A} = \begin{bmatrix} -\frac{\mathcal{R}\_{\boldsymbol{\epsilon}}}{\mathcal{L}\_{dq}} \ \boldsymbol{\omega}\_{\boldsymbol{\epsilon}\boldsymbol{l}}(t) \\ -\frac{\mathcal{R}\_{\boldsymbol{\epsilon}}}{\mathcal{L}\_{dq}} \ \boldsymbol{\omega}\_{\boldsymbol{\epsilon}\boldsymbol{l}}(t) \end{bmatrix}, \mathbf{B} = \begin{bmatrix} \frac{1}{\mathcal{L}\_{dq}} & 0 \\ 0 & \frac{1}{\mathcal{L}\_{dq}} \end{bmatrix} \tag{9}$$

and V = *im*([0, 1] *<sup>T</sup>*) of Eq. (8), according to Basile et al. (1992), is a controlled invariant subspace. More explicitly it follows:

$$\mathbf{F} = \begin{bmatrix} F\_{11} \ F\_{12} \\ F\_{21} \ F\_{22} \end{bmatrix} / \text{and} \begin{bmatrix} \boldsymbol{\mu}\_d(t) \\ \boldsymbol{\mu}\_q(t) \end{bmatrix} = \mathbf{F} \begin{bmatrix} \dot{\boldsymbol{\iota}}\_d(t) \\ \dot{\boldsymbol{\iota}}\_q(t) \end{bmatrix} / \text{$$

4 Will-be-set-by-IN-TECH

The present estimator uses the measurements of input voltages, currents and angular velocity of the motor to estimate the "d-q" winding inductance, the rotor resistance and amplitude of the linkage flux. The structure of the estimator is described in Fig. 2. This diagram shows how the estimator works. In particular, after having decoupled the system described in (5), the stator resistance *Rs* and the inductance *Ldq* are estimated through a minimum error variance approach. The estimated values *R*ˆ*<sup>s</sup>* and *Ldq* are used for to estimate of the amplitude of the

*dωmec*(*t*)

*dt* , (7)

*Mm* − *Mw* = *J*

where *pωmech*(*t*) = *ωel*(*t*) and *Mw* is an unknown mechanical load.

**3. Structure of the decoupling dynamic estimator**

Fig. 2. Conceptual structure of the whole estimator

**3.1 Decoupling structure and minimum error variance algorithm**

⎡ ⎣ <sup>−</sup> *Rs Ldq*

<sup>−</sup> *Rs Ldq*

> � , and � *ud*(*t*) *uq*(*t*)

**A** =

**F** = � *F*<sup>11</sup> *F*<sup>12</sup> *F*<sup>21</sup> *F*<sup>22</sup>

To achieve a decoupled structure of the system described in Eq. (5), a matrix **F** is to be

where **u**(*t*) = **Fx**(*t*) is a state feedback with **u**(*t*)=[*ud*(*t*), *uq*(*t*)]*<sup>T</sup>* and **x**(*t*)=[*id*(*t*), *iq*(*t*)]*T*,

⎤ ⎦ , **B** = ⎡ ⎣

*<sup>T</sup>*) of Eq. (8), according to Basile et al. (1992), is a controlled invariant

� = **F** � *id*(*t*) *iq*(*t*)

*ωel*(*t*)

*ωel*(*t*)

(**A** + **BF**)V⊆V, (8)

⎤

� ,

⎦ , (9)

with the following movement equation:

linkage flux (Φˆ ).

calculated such that,

and V = *im*([0, 1]

subspace. More explicitly it follows:

then the decoupling of the dynamics is obtained via the following relationship:

$$\operatorname{im} \left( \begin{bmatrix} -\frac{\mathcal{R}\_s}{\mathcal{L}\_{dq}} \ \omega\_{el}(t) \\ -\frac{\mathcal{R}\_s}{\mathcal{L}\_{dq}} \ \omega\_{el}(t) \end{bmatrix} \right) + \operatorname{im} \left( \begin{bmatrix} \frac{1}{\mathcal{L}\_{dq}} & 0 \\ 0 & \frac{1}{\mathcal{L}\_{dq}} \end{bmatrix} \begin{bmatrix} F\_{11} \ F\_{12} \\ F\_{21} \ F\_{22} \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) \subseteq \operatorname{im} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \Big/ \tag{10}$$

where the parameters *F*11, *F*12, *F*21, and *F*<sup>22</sup> are to be calculated in order to guarantee condition (10) and a suitable dynamics for sake of estimation. Condition (10) is guaranteed if

$$F\_{12} = -\omega\_{el}(t)L\_{dq}.\tag{11}$$

$$\frac{d\dot{u}\_d(t)}{dt} = -\frac{R\_s}{L\_{dq}}\dot{\iota}\_d(t) + \frac{\mu\_d(t)}{L\_{dq}}\prime \tag{12}$$

Because of the possible inexact decoupling, it follows that:

$$\frac{d\hat{t}\_d(t)}{dt} = -\frac{R\_s}{L\_{dq}}\hat{i}\_d(t) + \frac{\mu\_d(t)}{L\_{dq}} + n(\Delta(\omega\_{el}(t)(L\_{dq} - \hat{L}\_{dq}))),\tag{13}$$

where *<sup>n</sup>*(Δ(*Ldq* <sup>−</sup> *<sup>L</sup>*<sup>ˆ</sup> *dq*)) is the disturbance due to the inexact cancelation.

**Proposition 1.** *Considering the disturbance n*(Δ(*Ldq* <sup>−</sup> *<sup>L</sup>*<sup>ˆ</sup> *dq*)) *of Eq. (12) as a white noise, then the current minimum variance error σ eid* (*t*) = *σ id*(*t*) <sup>−</sup> <sup>ˆ</sup> *id*(*t*) *is obtained by minimising the estimation error of the parameters Ldq and Rs.*

**Proof 1.** *If Eqs. (12) and (13) are discretised using Implicit Euler with a sampling frequency equal to ts, then it follows that:*

$$\hat{I}\_d(k) = \frac{\hat{i}\_d(k-1)}{(1 + t\_s \frac{R\_s}{L\_{dq}})} + \frac{t\_s}{L\_{dq}(1 + t\_s \frac{R\_s}{L\_{dq}})} u\_d(k) \,\tag{14}$$

$$\hat{i}\_d(k) = \frac{\hat{i}\_d(k-1)}{(1 + t\_s \frac{R\_s}{L\_{dq}})} + \frac{t\_s}{L\_{dq}(1 + t\_s \frac{R\_s}{L\_{dq}})} u\_d(k) + n(k). \tag{15}$$

*It is possible to assume an ARMAX model for the system represented by (15) and thus*

$$\begin{aligned} \dot{u}\_d(k) &= \hat{i}\_d(k) + a\_1 \hat{i}\_d(k-1) + a\_2 \hat{i}\_d(k-2) + b\_1 u\_d(k-1) + \\ b\_2 u\_d(k-2) &+ n(k) + c\_{1l} n(k-1) + c\_{2l} n(k-2). \end{aligned} \tag{16}$$

*Letting eid* (*k*) = *id*(*k*) <sup>−</sup> <sup>ˆ</sup>*id*(*k*) *as mentioned above, it follows that:*

$$e\_{\hat{\mathbf{i}}\_{d}}(k) = a\_1 \hat{\mathbf{i}}\_d(k-1) + a\_2 \hat{\mathbf{i}}\_d(k-2) + b\_1 \boldsymbol{u}\_d(k-1) + b\_2 \boldsymbol{u}\_d(k-2) + \\\ \begin{aligned} \boldsymbol{u} &= \mathbf{k}\_1 \mathbf{u}(k-1) + \mathbf{c}\_1 \boldsymbol{u}(k-2) + \mathbf{c}\_2 \boldsymbol{u}(k-2), \quad \text{(17)} \end{aligned} \tag{17}$$

*where the coefficients a, b, c*1*, c*2*, are to be estimated, and n*(*k*) *is assumed as white noises. The next sample is:*

$$e\_{\hat{l}\_d}(k+1) = a\_1 \hat{l}\_d(k) + a\_2 \hat{l}\_d(k-1) + b\_1 u\_d(k) + b\_2 u\_d(k-1) + n(k+1) + c\_1 n(k) + c\_2 n(k-1). \tag{18}$$

*The prediction at time "k" is:*

$$\hat{\mathbf{c}}\_{\hat{i}\_d}(k+1/k) = a\_1 \hat{\mathbf{i}}\_d(k) + a\_2 \hat{\mathbf{i}}\_d(k-1) + b\_1 \mathbf{u}\_d(k) + b\_2 \mathbf{u}\_d(k-1) + c\_1 \mathbf{u}(k) + c\_2 \mathbf{u}(k-1). \tag{19}$$
  $Consider \text{ $i$ }\text{ $t$ }:$ 

*Considering that:*

$$J = E\{\mathcal{e}\_{\bar{l}\_d}^2(k+1/k)\} = E\{ [\hat{\mathfrak{e}}\_{\bar{l}\_d}(k+1/k) + n(k+1)]^2 \},$$

*and assuming that the noise is not correlated to the signal eid* (*k*)*, it follows:*

$$\begin{aligned} E\{\left[\mathbb{\left[\ell\_{\vec{l}\_d}(k+1/k) + n(k+1)\right]^2\right] } = E\{\left[\mathbb{\left[\ell\_{\vec{l}\_d}(k+1/k)\right]^2\right] + \\ E\{\left[n(k+1)\right]^2\} = E\{\left[\mathbb{\left[\ell\_{\vec{l}\_d}(k+1/k)\right]^2\right] + \sigma\_{n'}^2} \end{aligned} \tag{20}$$

*where σ<sup>n</sup> is defined as the variance of the white noises. The goal is to find* ˆ*id*(*k*) *such that:*

$$
\mathfrak{e}\_{l\_d}(k+1/k) = 0.\tag{21}
$$

**Remark 1.** *The approximation in equation (23) is equivalent to considering* �*c*2� << �*c*1�*. In other words, this means that a noise model of the first order is assumed. An indirect validation of this assumption is given by the results. In fact, the final measurements show in general good results with*

<sup>309</sup> A Robust Decoupling Estimator to Identify Electrical

**Remark 2.** *At the end, the recursive least-squares method gives an estimation of the parameters Ldq and Rs. These calculated parameters Ldq and Rs minimize the current minimum variance error*

If the electrical part of the system "q" and "d" axes is considered, then, assuming that *ωel*(*t*) �

*diq*(*t*)

*dt* <sup>=</sup> *<sup>d</sup>*Φ(*t*)

*dt* <sup>+</sup> *<sup>R</sup>*ˆ*sid*(*t*) *<sup>ω</sup>el*(*t*) <sup>+</sup>

where K is a function to be calculated. Eq. (33) represents the estimators of Φ. If the error functions are defined as the differences between the true and the observed values, then:

*dt* � << � *<sup>d</sup>*Φ<sup>ˆ</sup> (*t*)

*dt* <sup>+</sup> *<sup>R</sup>*ˆ*sid*(*t*) *<sup>ω</sup>el*(*t*) <sup>+</sup>

K can be chosen to make Eq. (39) exponentially stable. To guarantee exponential stability, K

K > 0.

*dt* + *Rsid*(*t*) + *Ldqωel*(*t*)*iq*(*t*) + *uq*(*t*)

*dt* <sup>−</sup> *<sup>d</sup>*Φ<sup>ˆ</sup> (*t*)

*dt* is negligible. Using equation (33), Eq. (35) becomes

*<sup>ω</sup>el*(*t*) . (32)

*L*ˆ *dqωel*(*t*)*iq*(*t*) + *uq*(*t*) *ωel*(*t*)

*dt* . (35)

*dt* �, (36)

. (37)

*L*ˆ *dqωel*(*t*)*iq*(*t*) + *uq*(*t*) *ωel*(*t*)

*dt* <sup>=</sup> <sup>K</sup>Φ<sup>ˆ</sup> (*t*) − KΦ(*t*), (38)

*dt* <sup>+</sup> <sup>K</sup>Φ(*t*) = 0. (39)

*<sup>e</sup>*Φ(*t*) = <sup>Φ</sup>(*t*) <sup>−</sup> <sup>Φ</sup><sup>ˆ</sup> (*t*), (34)

, (33)

0, *iq*(*t*) � 0, and *id*(*t*) � 0, the following equation can be considered:

Parameters for Three-Phase Permanent Magnet Synchronous Motors

*diq*(*t*)

*de*Φ(*t*)

� *d*Φ(*t*)

*<sup>L</sup>*<sup>ˆ</sup> *dq*

*de*Φ(*t*)

*de*Φ(*t*)

*diq*(*t*)

*the proposed method.*

*<sup>σ</sup>*(*eid* (*t*)) = *<sup>σ</sup>*(*id*(*t*) <sup>−</sup> <sup>ˆ</sup>

**3.2 The dynamic estimator of** Φ

*d*Φˆ (*t*)

and

*id*(*t*))*.*

<sup>Φ</sup>(*t*) = <sup>−</sup> *Ldq*

*dt* <sup>=</sup> −KΦ<sup>ˆ</sup> (*t*) − K *<sup>L</sup>*<sup>ˆ</sup> *dq*

Consider the following dynamic system:

If the following assumption is given:

then in Eq. (35), the term *<sup>d</sup>*Φ(*t*)

*de*Φ(*t*)

and considering (34), then

must be

*dt* <sup>=</sup> <sup>K</sup>Φ<sup>ˆ</sup> (*t*) + <sup>K</sup><sup>3</sup>

Because of Eq. (32), (37) can be written as follows:

*It is possible to write (17) as*

$$n(k) = e\_{i\underline{i}}(k) - a\_1 \hat{\mathbf{i}}\_d(k-1) - a\_2 \hat{\mathbf{i}}\_d(k-2) - b\_1 \boldsymbol{u}\_d(k-1) - \\\ 
$$b\_2 \boldsymbol{u}\_d(k-2) - c\_1 \boldsymbol{n}(k-1) - c\_2 \boldsymbol{n}(k-2). \tag{22}$$
$$

*Considering the effect of the noise on the system as follows:*

$$c\_1 n(k-1) + c\_2 n(k-2) \approx c\_1 n(k-1),\tag{23}$$

*and using the Z-transform, then:*

$$N(z) = \hat{I}\_d(z) - a\_1 z^{-1} \hat{I}\_d(z) - a\_2 z^{-2} \hat{I}\_d(z) - b\_1 z^{-1} \mathcal{U}\_d(z) - b\_2 z^{-2} \mathcal{U}\_d(z) - c\_1 z^{-1} N(z) \tag{24}$$

*and*

$$N(z) = \frac{(1 - a\_1 z^{-1} - a\_2 z^{-2})}{1 + c\_1 z^{-1}} \mathbf{\hat{I}}\_d(z) - \frac{(b\_1 z^{-1} + b\_2 z^{-2})}{1 + c\_1 z^{-1}} \mathbf{\mathcal{U}}\_d(z). \tag{25}$$

*Inserting Eq. (25) into Eq. (19) after its Z-transform, and considering the approximation stated in (23) and Eq. (21), the following expression is obtained:*

$$\hat{I}\_d(z) = -\frac{(a\_1 + c\_1 + b\_1 z^{-1})}{b\_1(1 + c\_1 z^{-1}) + b\_2(1 + c\_1 z^{-1})} \mathcal{U}\_d(z). \tag{26}$$

*Using the Z-transform for Eq. (14) it follows:*

$$\mathcal{U}\_d(z) = \frac{t\_s}{L\_{dq} + t\_s R\_s - L\_{dq} z^{-1}} \mathcal{U}\_d(z). \tag{27}$$

*Comparing (26) with (27), we are left with a straightforward diophantine equation to solve. The diophantine equation gives the relationship between the parameters* Θ = [*a*1, *b*1, *b*2, *c*1] *as follows:*

$$-b\_1 = 0\tag{28}$$

$$a\_1 + c\_1 = t\_s \tag{29}$$

$$b\_1 + b\_2 = L\_{dq} + t\_s R\_s \tag{30}$$

$$b\_1c\_1 + b\_2c\_1 = -L\_{dq}.\tag{31}$$

*Guessed initial values for parameters Ldq, Rs are given. This yields initial values for the parameters* Θ = [*a*1, *b*1, *b*2, *c*1]*. New values for the vector* Θ *are calculated using the recursive least squares method.*

**Remark 1.** *The approximation in equation (23) is equivalent to considering* �*c*2� << �*c*1�*. In other words, this means that a noise model of the first order is assumed. An indirect validation of this assumption is given by the results. In fact, the final measurements show in general good results with the proposed method.*

**Remark 2.** *At the end, the recursive least-squares method gives an estimation of the parameters Ldq and Rs. These calculated parameters Ldq and Rs minimize the current minimum variance error <sup>σ</sup>*(*eid* (*t*)) = *<sup>σ</sup>*(*id*(*t*) <sup>−</sup> <sup>ˆ</sup> *id*(*t*))*.*

### **3.2 The dynamic estimator of** Φ

If the electrical part of the system "q" and "d" axes is considered, then, assuming that *ωel*(*t*) � 0, *iq*(*t*) � 0, and *id*(*t*) � 0, the following equation can be considered:

$$\Phi(t) = -\frac{L\_{dq}\frac{di\_q(t)}{dt} + R\_s i\_d(t) + L\_{dq}\omega\_{el}(t)i\_q(t) + \mu\_q(t)}{\omega\_{el}(t)}.\tag{32}$$

Consider the following dynamic system:

$$\frac{d\Phi(t)}{dt} = -\mathcal{K}\dot{\Phi}(t) - \mathcal{K}\left(\frac{\hat{\mathcal{L}}\_{dq}\frac{d\dot{i}\_q(t)}{dt} + \hat{\mathcal{R}}\_{\dot{e}l}\dot{i}\_d(t)}{\omega\_{\ell l}(t)} + \frac{\hat{\mathcal{L}}\_{dq}\omega\_{\ell l}(t)\dot{i}\_q(t) + u\_q(t)}{\omega\_{\ell l}(t)}\right),\tag{33}$$

where K is a function to be calculated. Eq. (33) represents the estimators of Φ. If the error functions are defined as the differences between the true and the observed values, then:

$$e\_{\Phi}(t) = \Phi(t) - \hat{\Phi}(t),\tag{34}$$

and

6 Will-be-set-by-IN-TECH

*<sup>e</sup>*ˆ*id* (*<sup>k</sup>* <sup>+</sup> 1/*k*) = *<sup>a</sup>*<sup>1</sup> <sup>ˆ</sup>*id*(*k*) + *<sup>a</sup>*<sup>2</sup> <sup>ˆ</sup>*id*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) + *<sup>b</sup>*1*ud*(*k*) + *<sup>b</sup>*2*ud*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) + *<sup>c</sup>*1*n*(*k*) + *<sup>c</sup>*2*n*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>). (19)

*<sup>N</sup>*(*z*) = <sup>ˆ</sup>*Id*(*z*) <sup>−</sup> *<sup>a</sup>*1*z*−<sup>1</sup> <sup>ˆ</sup>*Id*(*z*) <sup>−</sup> *<sup>a</sup>*2*z*−<sup>2</sup> <sup>ˆ</sup>*Iu*(*z*) <sup>−</sup> *<sup>b</sup>*1*z*−<sup>1</sup>*Ud*(*z*) <sup>−</sup> *<sup>b</sup>*2*z*−<sup>2</sup>*Ud*(*z*) <sup>−</sup> *<sup>c</sup>*1*z*−1*N*(*z*) (24)

*Inserting Eq. (25) into Eq. (19) after its Z-transform, and considering the approximation stated in (23)*

*b*1(1 + *c*1*z*−1) + *b*2(1 + *c*1*z*−1)

*Comparing (26) with (27), we are left with a straightforward diophantine equation to solve. The diophantine equation gives the relationship between the parameters* Θ = [*a*1, *b*1, *b*2, *c*1] *as follows:*

*Guessed initial values for parameters Ldq, Rs are given. This yields initial values for the parameters* Θ = [*a*1, *b*1, *b*2, *c*1]*. New values for the vector* Θ *are calculated using the recursive least squares*

<sup>ˆ</sup>*Id*(*z*) = <sup>−</sup> (*a*<sup>1</sup> <sup>+</sup> *<sup>c</sup>*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*1*z*−1)

*Id*(*z*) = *ts*

<sup>1</sup> <sup>+</sup> *<sup>c</sup>*1*z*−<sup>1</sup> <sup>ˆ</sup>*Id*(*z*) <sup>−</sup> (*b*1*z*−<sup>1</sup> <sup>+</sup> *<sup>b</sup>*2*z*−2)

(*<sup>k</sup>* <sup>+</sup> 1/*k*)} <sup>=</sup> *<sup>E</sup>*{[*e*ˆ*id* (*<sup>k</sup>* <sup>+</sup> 1/*k*) + *<sup>n</sup>*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)]2},

*<sup>E</sup>*{[*n*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)]2} <sup>=</sup> *<sup>E</sup>*{[*e*ˆ*id* (*<sup>k</sup>* <sup>+</sup> 1/*k*)]2} <sup>+</sup> *<sup>σ</sup>*<sup>2</sup>

*e*ˆ*id* (*k* + 1/*k*) = 0. (21)

*Ldq* <sup>+</sup> *tsRs* <sup>−</sup> *Ldqz*−<sup>1</sup> *Ud*(*z*). (27)

− *b*<sup>1</sup> = 0 (28) *a*<sup>1</sup> + *c*<sup>1</sup> = *ts* (29) *b*<sup>1</sup> + *b*<sup>2</sup> = *Ldq* + *tsRs* (30) *b*1*c*<sup>1</sup> + *b*2*c*<sup>1</sup> = −*Ldq*. (31)

*c*1*n*(*k* − 1) + *c*2*n*(*k* − 2) ≈ *c*1*n*(*k* − 1), (23)

*b*2*ud*(*k* − 2) − *c*1*n*(*k* − 1) − *c*2*n*(*k* − 2). (22)

<sup>1</sup> <sup>+</sup> *<sup>c</sup>*1*z*−<sup>1</sup> *Ud*(*z*). (25)

*Ud*(*z*). (26)

*<sup>n</sup>*, (20)

*The prediction at time "k" is:*

*It is possible to write (17) as*

*and using the Z-transform, then:*

*and*

*method.*

*J* = *E*{*e* 2 *id*

*<sup>E</sup>*{[*e*ˆ*id* (*<sup>k</sup>* <sup>+</sup> 1/*k*) + *<sup>n</sup>*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)]2} <sup>=</sup> *<sup>E</sup>*{[*e*ˆ*id* (*<sup>k</sup>* <sup>+</sup> 1/*k*)]2}<sup>+</sup>

*<sup>n</sup>*(*k*) = *eid* (*k*) <sup>−</sup> *<sup>a</sup>*<sup>1</sup> <sup>ˆ</sup>*id*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> *<sup>a</sup>*<sup>2</sup> <sup>ˆ</sup>*id*(*<sup>k</sup>* <sup>−</sup> <sup>2</sup>) <sup>−</sup> *<sup>b</sup>*1*ud*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)<sup>−</sup>

*<sup>N</sup>*(*z*) = (<sup>1</sup> <sup>−</sup> *<sup>a</sup>*1*z*−<sup>1</sup> <sup>−</sup> *<sup>a</sup>*2*z*−2)

*Considering the effect of the noise on the system as follows:*

*and Eq. (21), the following expression is obtained:*

*Using the Z-transform for Eq. (14) it follows:*

*and assuming that the noise is not correlated to the signal eid* (*k*)*, it follows:*

*where σ<sup>n</sup> is defined as the variance of the white noises. The goal is to find* ˆ*id*(*k*) *such that:*

*Considering that:*

$$\frac{d\epsilon\_{\Phi}(t)}{dt} = \frac{d\Phi(t)}{dt} - \frac{d\hat{\Phi}(t)}{dt}.\tag{35}$$

If the following assumption is given:

$$\|\frac{d\Phi(t)}{dt}\| << \|\frac{d\hat{\Phi}(t)}{dt}\|\_{\prime} \tag{36}$$

then in Eq. (35), the term *<sup>d</sup>*Φ(*t*) *dt* is negligible. Using equation (33), Eq. (35) becomes

$$\frac{d\epsilon\_{\Phi}(t)}{dt} = \mathcal{K}\hat{\Phi}(t) + \mathcal{K}\_{3} \left( \frac{\hat{\mathcal{L}}\_{dq}\frac{d\hat{\imath}\_{q}(t)}{dt} + \hat{\mathcal{R}}\_{s}\dot{\imath}\_{d}(t)}{\omega\_{\ell l}(t)} + \frac{\hat{\mathcal{L}}\_{dq}\omega\_{\ell l}(t)\dot{\imath}\_{q}(t) + \mu\_{q}(t)}{\omega\_{\ell l}(t)} \right). \tag{37}$$

Because of Eq. (32), (37) can be written as follows:

$$\frac{d\varepsilon\_{\Phi(t)}}{dt} = \mathcal{K}\dot{\Phi}(t) - \mathcal{K}\Phi(t),\tag{38}$$

and considering (34), then

$$\frac{d e\_{\Phi}(t)}{dt} + \mathcal{K}\Phi(t) = 0.\tag{39}$$

K can be chosen to make Eq. (39) exponentially stable. To guarantee exponential stability, K must be

K > 0.

To guarantee � *<sup>d</sup>*Φ(*t*) *dt* � << � *<sup>d</sup>*Φ<sup>ˆ</sup> (*t*) *dt* �, then K >> 0. The observer defined in (33) suffers from the presence of the derivative of the measured current. In fact, if measurement noise is present in the measured current, then undesirable spikes are generated by the differentiation. The proposed algorithm must cancel the contribution from the measured current derivative. This is possible by correcting the observed velocity with a function of the measured current, using a supplementary variable defined as

$$
\eta(t) = \Phi(t) + \mathcal{N}(i\_q(t)),
\tag{40}
$$

Using the implicit Euler method, the following velocity observer structure is obtained:

1 + *ts*

<sup>311</sup> A Robust Decoupling Estimator to Identify Electrical

**Remark 3.** *Assumption (36) states that the dynamics of the approximating observer should be faster than the dynamics of the physical system. This assumption is typical for the design of observers.*

**Remark 4.** *The estimator of Eq. (50) presents the following limitations: for low velocity of the motor* (*ωmec*.(*t*) << *ωmecn* (*t*))*, where ωmecn* (*t*) *represents the nominal velocity of the motor), the estimation of* Φ *becomes inaccurate. Because ωel*(*t*) *divides the state variable η, the observer described by (50) becomes hyperdynamic. Critical phases of the estimation are the starting and ending of the movement. Another critical phase is represented by high velocity regime. In fact, it has been proven through simulations that if ωmec*(*t*) >> *ωmecn* (*t*)*, then the observer described by (50) becomes hypodynamic. According to the simulation results, within some range of frequency, this hypo-dynamicity can be*

**Remark 5.** *The Implicit Euler method guarantees the finite time convergence of the observer for any choice of kapp. Nevertheless, any other method can demonstrate the validity of the presented results.*

Simulations have been performed using a special stand with a 58-kW traction PMSM. The stand consists of a PMSM, a tram wheel and a continuous rail. The PMSM is a prototype for low floor trams. The PMSM parameters are: nominal power 58 kW, nominal torque 852 Nm, nominal speed 650 rpm, nominal phase current 122 A and number of poles 44. The model parameters are: *R* = 0.08723 Ohm, *Ldq* = *Ld* = *Lq* = 0.8 mH, Φ = 0.167 Wb. Surface mounted NdBFe magnets are used in PMSM. The advantage of these magnets is their inductance, which is as great as 1.2 T, but theirs disadvantage is corrosion. The PMSM was designed to meet B curve requirements. The stand was loaded by an asynchronous motor. The engine has a nominal power 55 kW, a nominal voltage 380 V and nominal speed 589 rpm. Figures 3, 4, and 5 show the estimation of *Rs* stator resistance, *Ldq* inductance, and Φ magnet flux, respectively. These simulation results are obtained using values of *kapp* equal to 2 and 20 respectively. From these results, in particular from flux estimation, an improvement, passing from values of *kapp* = 2 *kapp* = 20, is visible. From these figures, the effect of the limit of the procedure discussed in remark 4 is visible at the beginning of the estimation. Figure 6 shows the angular velocity of the motor. In the present simulations, *t* = 0 corresponds to *ωel*(*t*) = 0.

<sup>+</sup> *kappωel*(*k*)*iq*(*k*) + *tsR*<sup>ˆ</sup> *skappid*(*k*)

*kappωel*(*k*) *L*ˆ *dq*

*L*ˆ *dq*

<sup>Φ</sup><sup>ˆ</sup> (*k*) = *<sup>η</sup>*(*k*) <sup>−</sup> *kappiq*(*k*), (50)

*iq*(*k*)+

*ts kapp L*ˆ *dq*

> *kappωel*(*k*) *L*ˆ *dq*

*uq*(*k*),

1 + *ts*

*appωel*(*k*)*iq*(*k*) *L*ˆ *dq*

Parameters for Three-Phase Permanent Magnet Synchronous Motors

*<sup>η</sup>*(*k*) = *<sup>η</sup>*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) 1 + *ts*

*kappωel*(*k*) *L*ˆ *dq*

where *ts* is the sampling period.

*compensated by a suitable choice of kapp.*

**4. Simulation results**

*Implicit Euler method is a straightforward one.*

+ *ts k*2

where N (*iq*(*t*)) is the function to be designed. Consider

$$\frac{d\eta(t)}{dt} = \frac{d\Phi(t)}{dt} + \frac{d\mathcal{N}(i\_q(t))}{dt} \tag{41}$$

and let

$$\frac{d\mathcal{N}(i\_q(t))}{dt} = \frac{d\mathcal{N}(i\_q)}{di\_q(t)} \frac{di\_q(t)}{dt} = \frac{\mathcal{K}\ddot{L}\_{dq}}{\omega\_{el}(t)} \frac{di\_q(t)}{dt}.\tag{42}$$

The purpose of (42) is to cancel the differential contribution from (33). In fact, (40) and (41) yield, respectively,

$$
\Phi(t) = \eta(t) - \mathcal{N}(i\_q(t)) \quad \text{and} \tag{43}
$$

$$\frac{d\Phi(t)}{dt} = \frac{d\eta(t)}{dt} - \frac{dN(i\_q(t))}{dt}.\tag{44}$$

Substituting (42) in (44) results in

$$\frac{d\hat{\Phi}(t)}{dt} = \frac{d\eta(t)}{dt} - \frac{\mathcal{K}\hat{L}\_{dq}}{\omega\_{el}(t)} \frac{d\dot{\imath}\_q(t)}{dt}.\tag{45}$$

Inserting Eq. (45) into Eq. (33), the following expression is obtained1:

$$\frac{d\eta(t)}{dt} - \frac{\mathcal{K}\hat{L}\_{dq}}{\omega\_{\ell l}(t)}\frac{d\mathbf{i}\_q(t)}{dt} = -\mathcal{K}\Phi(t) - \mathcal{K}\left(\frac{\hat{L}\_{dq}\frac{d\mathbf{i}\_q(t)}{dt} + \hat{\mathcal{R}}\_s\mathbf{i}\_d(t)}{\omega\_{\ell l}(t)} + \frac{\hat{L}\_{dq}\omega\_{\ell l}(t)\mathbf{i}\_q(t) + \boldsymbol{u}\_q(t)}{\omega\_{\ell l}(t)}\right), \tag{46}$$

then

$$\frac{d\eta(t)}{dt} = -\mathcal{K}\dot{\Phi}(t) - \mathcal{K}\frac{\left(\hat{\mathcal{R}}\_s i\_d(t) + \hat{L}\_{d\eta}\omega\_{el}(t)i\_q(t) + u\_q(t)\right)}{\omega\_{el}(t)}.\tag{47}$$

Letting N (*iq*(*t*)) = *kappiq*(*t*), where a parameter has been indicated with *kapp*, then from (42) ⇒ K <sup>=</sup> *kappωel*(*t*) *L*ˆ *dq* , and Eq. (43) becomes:

$$
\hat{\Phi}(t) = \eta(t) - k\_{app} i\_q(t). \tag{48}
$$

Finally, substituting (48) into (47) results in the following equation:

$$\frac{d\eta(t)}{dt} = -\frac{k\_{app}\omega\_{el}(t)}{\hat{L}\_{dq}} \left(\eta(t) - k\_{app}i\_{q}(t)\right) + \frac{k\_{app}}{\hat{L}\_{dq}} \left(\hat{R}\_{s}i\_{d}(t) + \hat{L}\_{dq}\omega\_{el}(t)i\_{q}(t) + u\_{q}(t)\right),$$

$$\hat{\Phi}(t) = \eta(t) - k\_{app}i\_{q}(t). \tag{49}$$

<sup>1</sup> Expression (33) works under the assumption (36): fast observer dynamics.

Using the implicit Euler method, the following velocity observer structure is obtained:

$$\eta(k) = \frac{\eta(k-1)}{1 + t\_s \frac{k\_{app}\omega\_d(k)}{L\_{dq}}} + \frac{t\_s \frac{k\_{app}^2 \omega\_d(k)i\_q(k)}{L\_{dq}} + k\_{app}\omega\_{el}(k)i\_q(k) + \frac{t\_s \dot{R}\_s k\_{app}i\_d(k)}{L\_{dq}}}{1 + t\_s \frac{k\_{app}\omega\_d(k)}{L\_{dq}}} i\_q(k) + \frac{t\_s \dot{R}\_s k\_{app} \omega\_d(k)}{1 + t\_s \frac{k\_{app}\omega\_d(k)}{L\_{dq}}} u\_q(k),$$
 
$$\Phi(k) = \eta(k) - k\_{app}i\_q(k),\tag{50}$$

where *ts* is the sampling period.

8 Will-be-set-by-IN-TECH

presence of the derivative of the measured current. In fact, if measurement noise is present in the measured current, then undesirable spikes are generated by the differentiation. The proposed algorithm must cancel the contribution from the measured current derivative. This is possible by correcting the observed velocity with a function of the measured current, using

*dt* <sup>+</sup>

The purpose of (42) is to cancel the differential contribution from (33). In fact, (40) and (41)

*diq*(*t*)

*d*N (*iq*(*t*))

*dt* <sup>=</sup> <sup>K</sup>*L*<sup>ˆ</sup> *dq ωel*(*t*)

*dt* <sup>−</sup> *<sup>d</sup>*<sup>N</sup> (*iq*(*t*))

*dt* <sup>−</sup> <sup>K</sup>*L*<sup>ˆ</sup> *dq ωel*(*t*)

*diq*(*t*)

Letting N (*iq*(*t*)) = *kappiq*(*t*), where a parameter has been indicated with *kapp*, then from (42)

 + *kapp L*ˆ *dq* 

*dt* <sup>+</sup> *<sup>R</sup>*ˆ*sid*(*t*) *<sup>ω</sup>el*(*t*) <sup>+</sup>

*R*ˆ*sid*(*t*) + *L*ˆ *dqωel*(*t*)*iq*(*t*) + *uq*(*t*)

<sup>Φ</sup><sup>ˆ</sup> (*t*) = *<sup>η</sup>*(*t*) <sup>−</sup> *kappiq*(*t*). (48)

<sup>Φ</sup><sup>ˆ</sup> (*t*) = *<sup>η</sup>*(*t*) <sup>−</sup> *kappiq*(*t*). (49)

*R*ˆ*sid*(*t*) + *L*ˆ *dqωel*(*t*)*iq*(*t*) + *uq*(*t*)

*dt* �, then K >> 0. The observer defined in (33) suffers from the

*<sup>η</sup>*(*t*) = <sup>Φ</sup><sup>ˆ</sup> (*t*) + <sup>N</sup> (*iq*(*t*)), (40)

*diq*(*t*)

<sup>Φ</sup><sup>ˆ</sup> (*t*) = *<sup>η</sup>*(*t*) − N (*iq*(*t*)) and (43)

*diq*(*t*)

*dt* (41)

*dt* . (44)

*dt* . (45)

*L*ˆ *dqωel*(*t*)*iq*(*t*) + *uq*(*t*) *ωel*(*t*)

*<sup>ω</sup>el*(*t*) . (47)

 , (46)

 ,

*dt* . (42)

To guarantee � *<sup>d</sup>*Φ(*t*)

Consider

and let

yield, respectively,

*dη*(*t*)

then

*dt* <sup>−</sup> <sup>K</sup>*L*<sup>ˆ</sup> *dq ωel*(*t*)

⇒ K <sup>=</sup> *kappωel*(*t*) *L*ˆ *dq*

*dη*(*t*)

*dt* <sup>=</sup> <sup>−</sup> *kappωel*(*t*)

*L*ˆ *dq*

Substituting (42) in (44) results in

*diq*(*t*)

*dη*(*t*)

*dt* � << � *<sup>d</sup>*Φ<sup>ˆ</sup> (*t*)

where N (*iq*(*t*)) is the function to be designed.

*dη*(*t*)

*d*Φˆ (*t*)

*d*Φˆ (*t*)

Inserting Eq. (45) into Eq. (33), the following expression is obtained1:

*dt* <sup>=</sup> −KΦ<sup>ˆ</sup> (*t*) − K

*dt* <sup>=</sup> −KΦ<sup>ˆ</sup> (*t*) − K

Finally, substituting (48) into (47) results in the following equation:

*η*(*t*) − *kappiq*(*t*)

<sup>1</sup> Expression (33) works under the assumption (36): fast observer dynamics.

, and Eq. (43) becomes:

*dt* <sup>=</sup> *<sup>d</sup>*<sup>N</sup> (*iq*)

*d*N (*iq*(*t*))

*dt* <sup>=</sup> *<sup>d</sup>*Φ(*t*)

*diq*(*t*)

*dt* <sup>=</sup> *<sup>d</sup>η*(*t*)

*dt* <sup>=</sup> *<sup>d</sup>η*(*t*)

*<sup>L</sup>*<sup>ˆ</sup> *dq*

a supplementary variable defined as

**Remark 3.** *Assumption (36) states that the dynamics of the approximating observer should be faster than the dynamics of the physical system. This assumption is typical for the design of observers.*

**Remark 4.** *The estimator of Eq. (50) presents the following limitations: for low velocity of the motor* (*ωmec*.(*t*) << *ωmecn* (*t*))*, where ωmecn* (*t*) *represents the nominal velocity of the motor), the estimation of* Φ *becomes inaccurate. Because ωel*(*t*) *divides the state variable η, the observer described by (50) becomes hyperdynamic. Critical phases of the estimation are the starting and ending of the movement. Another critical phase is represented by high velocity regime. In fact, it has been proven through simulations that if ωmec*(*t*) >> *ωmecn* (*t*)*, then the observer described by (50) becomes hypodynamic. According to the simulation results, within some range of frequency, this hypo-dynamicity can be compensated by a suitable choice of kapp.*

**Remark 5.** *The Implicit Euler method guarantees the finite time convergence of the observer for any choice of kapp. Nevertheless, any other method can demonstrate the validity of the presented results. Implicit Euler method is a straightforward one.*

### **4. Simulation results**

Simulations have been performed using a special stand with a 58-kW traction PMSM. The stand consists of a PMSM, a tram wheel and a continuous rail. The PMSM is a prototype for low floor trams. The PMSM parameters are: nominal power 58 kW, nominal torque 852 Nm, nominal speed 650 rpm, nominal phase current 122 A and number of poles 44. The model parameters are: *R* = 0.08723 Ohm, *Ldq* = *Ld* = *Lq* = 0.8 mH, Φ = 0.167 Wb. Surface mounted NdBFe magnets are used in PMSM. The advantage of these magnets is their inductance, which is as great as 1.2 T, but theirs disadvantage is corrosion. The PMSM was designed to meet B curve requirements. The stand was loaded by an asynchronous motor. The engine has a nominal power 55 kW, a nominal voltage 380 V and nominal speed 589 rpm. Figures 3, 4, and 5 show the estimation of *Rs* stator resistance, *Ldq* inductance, and Φ magnet flux, respectively. These simulation results are obtained using values of *kapp* equal to 2 and 20 respectively. From these results, in particular from flux estimation, an improvement, passing from values of *kapp* = 2 *kapp* = 20, is visible. From these figures, the effect of the limit of the procedure discussed in remark 4 is visible at the beginning of the estimation. Figure 6 shows the angular velocity of the motor. In the present simulations, *t* = 0 corresponds to *ωel*(*t*) = 0.

0 0.2 0.4 0.6 0.8 1

<sup>313</sup> A Robust Decoupling Estimator to Identify Electrical

Estimated values of Lqd inductance Real value of Lqd inductance

Estimated values of Lqd inductance Real value of Lqd inductance

Time (sec.)

0 0.2 0.4 0.6 0.8 1

Time (sec.)

Fig. 4. Estimated and real values of *Ldq* inductance for *kapp* = 2 (on the top) and *kapp* = 20

0

(on the bottom)

Ldq inductance (H)

2

4

Ldq inductance (H)

6

8

<sup>10</sup> x 10−4

Parameters for Three-Phase Permanent Magnet Synchronous Motors

x 10−4

x 10−4

Fig. 3. Estimated and real values of *Rs* stator resistance for *kapp* = 2 (on the top) and *kapp* = 20 (on the bottom)

10 Will-be-set-by-IN-TECH

0 0.2 0.4 0.6 0.8 1

Real value of the stator resistance Estimated values of the stator resistance

Time (sec.)

Estimated values of the stator resistance Real value of the stator resistance

0 0.2 0.4 0.6 0.8 1

Time (sec.)

Fig. 3. Estimated and real values of *Rs* stator resistance for *kapp* = 2 (on the top) and

0

0

0.05

0.1

0.15

Stator resistance (Ohm)

*kapp* = 20 (on the bottom)

0.2

0.25

0.05

0.1

0.15

Stator resistance (Ohm)

0.2

0.25

x 10 −3

x 10−3

Fig. 4. Estimated and real values of *Ldq* inductance for *kapp* = 2 (on the top) and *kapp* = 20 (on the bottom)

**5. Conclusions and future work**

**6. References**

general test of the present algorithm using a real motor.

Parameters for Three-Phase Permanent Magnet Synchronous Motors

Jersey-USA, Prentice Hall, 1992.

pages 1334-1345, December 2009.

vol. 39, no. 2, 2003, pages 408-416.

pages 569-575.

This paper considers a dynamic estimator for fully automated parameters identification for three-phase synchronous motors. The technique uses a decoupling procedure optimised by a minimum variance error to estimate the inductance and resistance of the motor. Moreover, a dynamic estimator is shown to identify the amplitude of the linkage flux using the estimated inductance and resistance. It is generally applicable and could also be used for the estimation of mechanical load and other types of electrical motors, as well as for dynamic systems with similar nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method in identification of PMSM model parameters and discusses the limits of the found theoretical and the simulation results. Future work includes the estimation of a mechanical load and the

<sup>315</sup> A Robust Decoupling Estimator to Identify Electrical

G. Basile and G. Marro. "Controlled and conditioned invariants in linear system theory", New

R. Dolecek, J. Novak, and O. Cerny. "Traction Permanent Magnet Synchronous Motor Torque Control with Flux Weakening", Radioengineering, Vol. 18, No. 4, December 2009 D.A. Khaburi and M. Shahnazari. "Parameters Identification of Permanent Magnet

Li Liu, Wenxin Liu and David A. Cartes. "Permanent Magnet Synchronous Motor

P. Mercorelli, K. Lehmann and S. Liu. "On Robustness Properties in Permanent Magnet

Symposium on Robust Control Design, 25th-27th June 2003, Milan (Italy). B.N. Mobarakeh, F. Meibody-Tabar, and F. M. Sargos. "On-line identification of PMSM

M. Ooshima, A. Chiba, A. Rahman and T. Fukao. "An improved control method of

M.A. Rahman and P. Zhou. "Analysis of brushless permanent magnet synchronous motors", IEEE Trans. Industrial Electronics, vol. 43, no. 2, 1996, pages 256-267. M.A. Rahman, D.M. Vilathgamuwa, M.N. Uddin and T. King-Jet. "Nonlinear control of

IEEE-IAS Annu. Meeting, vol. 1, Chicago, IL, 2001, pages 266-273.

of Computational Intelligence Research Vol.4, No.2 (2008), pages 211-218 P. Mercorelli. "Robust Feedback Linearization Using an Adaptive PD Regulator for a

machine", in Proc. IEEE IAS Annu. Meeting, 2002, pages 2289-2295.

Synchronous Machine in Vector Control", in Proc. of the 10th European Conference on Power Electronics and Applications, EPE 2003, Toulouse, France, 2-4 Sep. 2003. A. Kilthau, and J. Pacas. "Appropriate models for the controls of the synchronous reluctance

Parameter Identification using Particle Swarm Optimization", International Journal

Sensorless Control of a Throttle Valve", Mechatronics, a Journal of IFAC, Elsevier Science publishing. DOI: 10.1016/j.mechatronics.2009.08.008, Volume 19, Issue 8,

Machine Control Using Decoupling Controller", in Proc. of the 4th IFAC International

electrical parameters based on decoupling control", in Proc. of the Conf. Rec.

buried-type IPM bearingless motors considering magnetic saturation and magnetic pull variation", IEEE Transactions on Energy Conversion, vol. 19, no. 3, Sept. 2004,

interior permanent magnet synchronous motor", IEEE Trans. Industry Applications,

Fig. 5. Estimated and real values of the permanent magnet flux linkage for *kapp* = 2 (on the top) and *kapp* = 20 (on the bottom)

Fig. 6. Angular velocity

### **5. Conclusions and future work**

This paper considers a dynamic estimator for fully automated parameters identification for three-phase synchronous motors. The technique uses a decoupling procedure optimised by a minimum variance error to estimate the inductance and resistance of the motor. Moreover, a dynamic estimator is shown to identify the amplitude of the linkage flux using the estimated inductance and resistance. It is generally applicable and could also be used for the estimation of mechanical load and other types of electrical motors, as well as for dynamic systems with similar nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method in identification of PMSM model parameters and discusses the limits of the found theoretical and the simulation results. Future work includes the estimation of a mechanical load and the general test of the present algorithm using a real motor.

### **6. References**

12 Will-be-set-by-IN-TECH

0 0.5 1 1.5 2

Estimated values of the permanent magnet flux linkage Real values of the permanent magnet flux linkage

Time (sec.)

Estimated values of the permanent magnet flux linkage Real value of the permanent magnet flux linkage

0 0.5 1 1.5 2

Time (sec.)

Fig. 5. Estimated and real values of the permanent magnet flux linkage for *kapp* = 2 (on the

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Time (sec.)

−30 −25 −20 −15 −10 −5 0 5

0

Fig. 6. Angular velocity

Angular velocity (rad./sec.)

2

4

Permanent magnet flux linkage (Vs)

top) and *kapp* = 20 (on the bottom)

6

8

10

Permanent magnet flux linkage (Vs)

x 10 −3

x 10−3


S. Weisgerber, A. Proca, and A. Keyhani. "Estimation of permanent magnet motor parameters", in Proc. of the IEEE Ind. Appl. Soc. Annu. Meeting, New Orleans, LA, Oct. 1997, pp. 29U34. ˝

Carlos Olalla1, Abdelali El Aroudi1, Ramon Leyva1 and Isabelle Queinnec2

This chapter proposes a systematic approach for the synthesis of robust controllers for dc-dc converters. The approach is based on the Linear Matrix Inequalities (LMIs) framework and the associated optimization algorithms. The aim of this approach is to allow the designer to describe the uncertainty of the converter and to deal with the requirements of the application

The aforementioned dc-dc converters (see Figure 1) are devices that deliver a dc output voltage, with different properties from those in the input voltage (Erickson & Maksimovic, 1999). They are usually employed to adapt energy sources to the load requirements (or vice versa). These devices present several challenges regarding their robust control. First, the converter must maintain a tight regulation or tracking of the output. Moreover, the controller design is focused on maximizing the bandwidth of the closed-loop response in order to reject the usual disturbances that appear in these systems. Finally, the response of the converter must satisfy desirable transient characteristics, as for example, the shortest possible output settling time or the minimum overshoot. Besides of these common requirements, the converter can be affected by uncertainty in its components or by input or output disturbances

> DC-DC Converter

*ub*(*t*)

*Vg*(*t*) *Vo* (*t*) Load

Control Input

**1. Introduction**

beforehand.

that may appear.

+ \_

Fig. 1. General scheme of a dc-dc converter.

Input Voltage <sup>2</sup>*CNRS ; LAAS ; 7 avenue du colonel Roche, F-31077 Toulouse Cedex 4*

**LMI Robust Control of PWM Converters:** 

*Université de Toulouse ; UPS, INSA, INP, ISAE, UT1, UTM ;*

**An Output-Feedback Approach** 

+

Output Voltage

\_

<sup>1</sup>*Universitat Rovira i Virgili, 43007 Tarragona*

*LAAS ; F-31077 Toulouse Cedex 4*

<sup>1</sup>*Spain* <sup>2</sup>*France*

**17**

## **LMI Robust Control of PWM Converters: An Output-Feedback Approach**

Carlos Olalla1, Abdelali El Aroudi1, Ramon Leyva1 and Isabelle Queinnec2 <sup>1</sup>*Universitat Rovira i Virgili, 43007 Tarragona* <sup>2</sup>*CNRS ; LAAS ; 7 avenue du colonel Roche, F-31077 Toulouse Cedex 4 Université de Toulouse ; UPS, INSA, INP, ISAE, UT1, UTM ; LAAS ; F-31077 Toulouse Cedex 4* <sup>1</sup>*Spain* <sup>2</sup>*France*

### **1. Introduction**

14 Will-be-set-by-IN-TECH

316 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

S. Weisgerber, A. Proca, and A. Keyhani. "Estimation of permanent magnet motor

Oct. 1997, pp. 29U34.

˝

parameters", in Proc. of the IEEE Ind. Appl. Soc. Annu. Meeting, New Orleans, LA,

This chapter proposes a systematic approach for the synthesis of robust controllers for dc-dc converters. The approach is based on the Linear Matrix Inequalities (LMIs) framework and the associated optimization algorithms. The aim of this approach is to allow the designer to describe the uncertainty of the converter and to deal with the requirements of the application beforehand.

The aforementioned dc-dc converters (see Figure 1) are devices that deliver a dc output voltage, with different properties from those in the input voltage (Erickson & Maksimovic, 1999). They are usually employed to adapt energy sources to the load requirements (or vice versa). These devices present several challenges regarding their robust control. First, the converter must maintain a tight regulation or tracking of the output. Moreover, the controller design is focused on maximizing the bandwidth of the closed-loop response in order to reject the usual disturbances that appear in these systems. Finally, the response of the converter must satisfy desirable transient characteristics, as for example, the shortest possible output settling time or the minimum overshoot. Besides of these common requirements, the converter can be affected by uncertainty in its components or by input or output disturbances that may appear.

Fig. 1. General scheme of a dc-dc converter.

*vg*(*t*) *vo*(*t*)

*rL*

*ub*(*t*)

**Notation**

partitioned matrices.

*iL*(*t*)

*L*

*vc*(*t*)

(a) Buck converter with stray resistances.

**2. Modeling of uncertain dc-dc converters**

discussed and a simple example is shown.

**2.1 Model of the buck converter**

considered in the models.

+

*C*

− −

Fig. 3. Schematic of the buck and the buck-boost converter.

*rC* +

*iload*(*t*)

LMI Robust Control of PWM Converters: An Output-Feedback Approach 319

following strategies. First, the classic dynamic output-feedback control problem is treated. This approach can be carried out with a change of variables as in (Scherer et al., 1997). However, with such an approach the uncertainty must be modeled with elaborated models, as for example, weighting functions Wallis & Tymerski (2000). Therefore, the chapter also proposes the synthesis of output-feedback controllers based on the static case. Both the static output-feedback and a parametrization (Peaucelle & Arzelier, 2001b) to deal with dynamic output-feedback are considered. The advantages and drawbacks of the three approaches

For symmetric matrices *<sup>A</sup>* and *<sup>B</sup>*, *<sup>A</sup>* <sup>&</sup>gt; *<sup>B</sup>* means that *<sup>A</sup>* <sup>−</sup> *<sup>B</sup>* is positive definite. **<sup>A</sup>** denotes that

denotes the convex hull defined by *<sup>N</sup>* vertices *vj* <sup>∈</sup> **<sup>R</sup>***n*. The identity matrix of order *<sup>n</sup>* is noted as <sup>1</sup>*<sup>n</sup>* and the null *<sup>n</sup>* <sup>×</sup> *<sup>m</sup>* matrix is noted as <sup>00</sup> *<sup>n</sup>*,*m*. The symbol � denotes symmetric blocks in

This subsection shows the state-space averaged models of the buck and the buck-boost converters of Figures 3(a) and 3(b). The models are assumed to operate in Continuous Conduction Mode (CCM), i.e. the inductor current is always larger than zero. Besides of the averaged models, this section also introduces a model of the sampling effect caused by the PWM. Finally, at the end of the section, the uncertainty modeling of dc-dc converters is

The first model that is introduced considers a buck converter, which is characterized by linear averaged control-to-output dynamics. As stated in (Olalla et al., 2010b), the transfer functions of dc-dc converters can strongly depend on the stray resistances of the converter. Since the chapter considers different output-feedback synthesis approaches, these stray resistances are

Figure 3(a) shows the circuit diagram of a dc-dc buck converter where *v*o(*t*) is the output voltage, *v*g(*t*) is the line voltage and *i*load(*t*) is the load disturbance. The output voltage must be kept at a given reference *V*ref. The converter load is modeled as a linear resistor *R*. The stray resistances of the switch during the on and the off position are combined with the resistance

*vg*(*t*) *vo*(*t*) *vc*(*t*) *iL*(*t*)

*rL*

*L C*

(b) Buck-boost converter with stray resistances.

+

*rC* +

− −

*ub*(*t*)

*iload*(*t*)

*R*

*vj*, *j* = 1, . . . , *N*

*R*

shown in the chapter will be discussed and the results will be compared.

the matrix *<sup>A</sup>* is an unknown variable. *<sup>A</sup>*� denotes the transpose of *<sup>A</sup>*. Co

(a) Block diagram of a state-feedback system with controller **K** and error integration. (b) Block diagram of an output-feedback system with controller *G*(*s*).

Fig. 2. State-feedback and output-feedback block diagrams.

Nevertheless, most of the modeling approaches in the literature disregard these uncertainties. Moreover, due to the switching nature of the system, pulse-width modulation (PWM) is commonly used in the industry applications, while the models that are usually employed disregard that part of the dynamics (i.e. the high frequency dynamics) and other inherent nonlinearities, such as saturations and bilinear terms.

The chapter proposes a systematic approach to deal with these challenges, using the concepts of LMI control (Ben-Tal et al., 2009; Bernussou, 1996; Boyd et al., 1994; El Ghaoui & Niculescu, 2000; Pyatnitskii & Skorodinskii, 1982). Linear matrix inequalities have become an important topic in the field of Automatic Control due to the following facts. First of all, LMIs can be solved numerically by efficient computer algorithms (Gahinet et al., 1995; Löfberg, 2004; Sturm, 1999). Secondly, more and more methods have been developed to describe control problems in terms of LMI constraints. Finally, these methods are able to include descriptions of the uncertainty.

Some of the previous literature on LMI control of dc-dc converters are (Montagner et al., 2005; Olalla et al., 2009a; 2010a). In these papers, the uncertainty of the converter is taken into account and the control synthesis deals with different operating points. Nevertheless, they do not consider the stability of the system trajectories when the system changes from one operating point to another, nor they include other nonlinearities such as saturations. The versatility of LMI control has allowed to deal with some of these nonlinearities (Olalla et al., 2009b), (Olalla et al., 2011).

These approaches share the same feedback scheme, which is based upon state-feedback with error integration (Figure 2(a)). The main advantage of this approach is that the synthesis optimization problem can be posed as a convex semidefinite programming and that the implementation of the controller is simple. On the other hand, state-feedback requires sensing of the state variables, which may not be easily measurable or may require estimation in some cases. In practice, most of the designs that can be found in the power electronics literature employ output-feedback approaches since they usually rely on frequency-based concepts which are well-known by electrical engineers. This is the reason why this chapter focuses on LMI-based synthesis methods which may be applicable to the output-feedback scheme (Figure 2(b)), with the aim to derive robust controllers for dc-dc converters.

In order to introduce such synthesis methods, the chapter is organized as follows. The first section deals with modeling of dc-dc converters, the averaging method, the sampling effect of the pulse-width modulator and the uncertainty. Section II reviews some of the results of previous works on LMI synthesis for state-feedback approaches. Section III puts forward the problem of output-feedback and some of the strategies that can be employed to pose such problem in terms of semidefinite programming. Concretely, Section III proposes the

(a) Buck converter with stray resistances. (b) Buck-boost converter with stray resistances.

Fig. 3. Schematic of the buck and the buck-boost converter.

following strategies. First, the classic dynamic output-feedback control problem is treated. This approach can be carried out with a change of variables as in (Scherer et al., 1997). However, with such an approach the uncertainty must be modeled with elaborated models, as for example, weighting functions Wallis & Tymerski (2000). Therefore, the chapter also proposes the synthesis of output-feedback controllers based on the static case. Both the static output-feedback and a parametrization (Peaucelle & Arzelier, 2001b) to deal with dynamic output-feedback are considered. The advantages and drawbacks of the three approaches shown in the chapter will be discussed and the results will be compared.

### **Notation**

2 Will-be-set-by-IN-TECH

Nevertheless, most of the modeling approaches in the literature disregard these uncertainties. Moreover, due to the switching nature of the system, pulse-width modulation (PWM) is commonly used in the industry applications, while the models that are usually employed disregard that part of the dynamics (i.e. the high frequency dynamics) and other inherent

The chapter proposes a systematic approach to deal with these challenges, using the concepts of LMI control (Ben-Tal et al., 2009; Bernussou, 1996; Boyd et al., 1994; El Ghaoui & Niculescu, 2000; Pyatnitskii & Skorodinskii, 1982). Linear matrix inequalities have become an important topic in the field of Automatic Control due to the following facts. First of all, LMIs can be solved numerically by efficient computer algorithms (Gahinet et al., 1995; Löfberg, 2004; Sturm, 1999). Secondly, more and more methods have been developed to describe control problems in terms of LMI constraints. Finally, these methods are able to include descriptions

Some of the previous literature on LMI control of dc-dc converters are (Montagner et al., 2005; Olalla et al., 2009a; 2010a). In these papers, the uncertainty of the converter is taken into account and the control synthesis deals with different operating points. Nevertheless, they do not consider the stability of the system trajectories when the system changes from one operating point to another, nor they include other nonlinearities such as saturations. The versatility of LMI control has allowed to deal with some of these nonlinearities (Olalla et al.,

These approaches share the same feedback scheme, which is based upon state-feedback with error integration (Figure 2(a)). The main advantage of this approach is that the synthesis optimization problem can be posed as a convex semidefinite programming and that the implementation of the controller is simple. On the other hand, state-feedback requires sensing of the state variables, which may not be easily measurable or may require estimation in some cases. In practice, most of the designs that can be found in the power electronics literature employ output-feedback approaches since they usually rely on frequency-based concepts which are well-known by electrical engineers. This is the reason why this chapter focuses on LMI-based synthesis methods which may be applicable to the output-feedback scheme

In order to introduce such synthesis methods, the chapter is organized as follows. The first section deals with modeling of dc-dc converters, the averaging method, the sampling effect of the pulse-width modulator and the uncertainty. Section II reviews some of the results of previous works on LMI synthesis for state-feedback approaches. Section III puts forward the problem of output-feedback and some of the strategies that can be employed to pose such problem in terms of semidefinite programming. Concretely, Section III proposes the

(Figure 2(b)), with the aim to derive robust controllers for dc-dc converters.

*V*ref +


with controller *G*(*s*).

*d*(*t*)

(b) Block diagram of an output-feedback system

*G*(*s*) DC-DC

*y*(*t*)

*d*(*t*)

*<sup>x</sup>*int(*t*) *<sup>V</sup>* ref <sup>+</sup>

(a) Block diagram of a state-feedback system

Fig. 2. State-feedback and output-feedback block diagrams.

nonlinearities, such as saturations and bilinear terms.

with controller **K** and error integration.


of the uncertainty.

2009b), (Olalla et al., 2011).

**K** DC-DC

*x*(*t*) *y*(*t*)

For symmetric matrices *<sup>A</sup>* and *<sup>B</sup>*, *<sup>A</sup>* <sup>&</sup>gt; *<sup>B</sup>* means that *<sup>A</sup>* <sup>−</sup> *<sup>B</sup>* is positive definite. **<sup>A</sup>** denotes that the matrix *<sup>A</sup>* is an unknown variable. *<sup>A</sup>*� denotes the transpose of *<sup>A</sup>*. Co *vj*, *j* = 1, . . . , *N* denotes the convex hull defined by *<sup>N</sup>* vertices *vj* <sup>∈</sup> **<sup>R</sup>***n*. The identity matrix of order *<sup>n</sup>* is noted as <sup>1</sup>*<sup>n</sup>* and the null *<sup>n</sup>* <sup>×</sup> *<sup>m</sup>* matrix is noted as <sup>00</sup> *<sup>n</sup>*,*m*. The symbol � denotes symmetric blocks in partitioned matrices.

### **2. Modeling of uncertain dc-dc converters**

This subsection shows the state-space averaged models of the buck and the buck-boost converters of Figures 3(a) and 3(b). The models are assumed to operate in Continuous Conduction Mode (CCM), i.e. the inductor current is always larger than zero. Besides of the averaged models, this section also introduces a model of the sampling effect caused by the PWM. Finally, at the end of the section, the uncertainty modeling of dc-dc converters is discussed and a simple example is shown.

### **2.1 Model of the buck converter**

The first model that is introduced considers a buck converter, which is characterized by linear averaged control-to-output dynamics. As stated in (Olalla et al., 2010b), the transfer functions of dc-dc converters can strongly depend on the stray resistances of the converter. Since the chapter considers different output-feedback synthesis approaches, these stray resistances are considered in the models.

Figure 3(a) shows the circuit diagram of a dc-dc buck converter where *v*o(*t*) is the output voltage, *v*g(*t*) is the line voltage and *i*load(*t*) is the load disturbance. The output voltage must be kept at a given reference *V*ref. The converter load is modeled as a linear resistor *R*. The stray resistances of the switch during the on and the off position are combined with the resistance of the inductor and noted as:

$$\begin{aligned} r\_{on} &= r\_{d\_{on}} + r\_L \\ r\_{off} &= r\_{d\_{off}} + r\_L \end{aligned} \tag{1}$$

where in a general case *Cy* <sup>∈</sup> **<sup>R</sup>***q*×*n*, *<sup>E</sup>*yw <sup>∈</sup> **<sup>R</sup>***q*×*m*. Considering the load voltage *<sup>v</sup>*o(*t*) as the

LMI Robust Control of PWM Converters: An Output-Feedback Approach 321

�

In the buck-boost converter, matrices *A*on and *A*off are not equal, and therefore, the averaged model contains bilinear terms concerning the control input, the states and the disturbance inputs. According to those nonlinear terms, the linearized transfer function depends on the operating point, hence making the control subsystem design more difficult. In order to derive accurate transfer functions of the buck-boost converter for output-feedback approaches, the

For the buck-boost converter, the averaged model in the form of (3) contains bilinear terms,

(*R* + *rC*)*L*

⎤ ⎥ <sup>⎦</sup> , *Bw* <sup>=</sup>

(*R* + *rC*)*C*

*DVg CD*�2(*R* + *rC*)

*<sup>L</sup>* <sup>−</sup> *RrC* (*R* + *rC*)*L*

> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

0 0

.

, *Eyw* =

�

<sup>0</sup> <sup>−</sup> *RrC R* + *rC*

+ *Ax*˜(*t*) + *B*w*w*˜(*t*) + *B*<sup>u</sup> ˜*d*(*t*) + *B*n*<sup>x</sup> x*˜(*t*) ˜*d*(*t*) + *B*n*<sup>w</sup> w*˜(*t*) ˜*d*(*t*) (7)

⎡ ⎢ ⎣

*D*� *DR*<sup>2</sup> *<sup>D</sup>*�*R*(*D*�*<sup>R</sup>* <sup>+</sup> *rC*)+(*<sup>R</sup>* <sup>+</sup> *rC*)*req* �

> ⎤ ⎦ ,

, *B*n*<sup>x</sup>* =

⎡ ⎢ ⎣

*roff* . The dimensions of the system matrices are defined as

*RrC*

*R* (*<sup>R</sup>* <sup>+</sup> *rC*)*<sup>C</sup>* <sup>0</sup>

+ *Cyx*˜(*t*) + *E*yw*w*˜(*t*) + *C*yu*X* ˜*d*(*t*) + *C*yu*x*˜(*t*) ˜*d*(*t*) (9)

*<sup>L</sup>*(*<sup>R</sup>* <sup>+</sup> *rC*) <sup>−</sup> *<sup>R</sup>*

(*R* + *rC*)*L*

⎤ ⎥ ⎦ . (8)

*D* 1

*<sup>L</sup>* <sup>−</sup>*D*� *RrC*

<sup>0</sup> <sup>−</sup> *<sup>R</sup>*

(*R* + *rC*)*L*

⎤ ⎥ ⎥ ⎦ , ⎤ ⎥ ⎦ ,

(*R* + *rC*)*C*

�

. (6)

*R R* + *rC*

only output, these matrices are written as:

**2.2 Model of the buck-boost converter**

stray resistances are also taken into account.

*AX* + *B*w*W*�

*<sup>L</sup>* <sup>−</sup> *<sup>D</sup>*� *RrC*

<sup>−</sup>*D*� *<sup>R</sup>*

⎡ ⎢ ⎢ ⎣ *Vg L* � 1 +

*VgD*�

−*VgD*�

*<sup>A</sup>*, *<sup>B</sup>*n*<sup>x</sup>* , *<sup>B</sup>*n*<sup>w</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*n*, *<sup>B</sup>*<sup>u</sup> <sup>∈</sup> **<sup>R</sup>***n*×*m*, *<sup>B</sup>*<sup>w</sup> <sup>∈</sup> **<sup>R</sup>***n*×*<sup>l</sup>*

(*<sup>R</sup>* <sup>+</sup> *rC*)*<sup>L</sup> <sup>D</sup>*� *<sup>R</sup>*

*DrC <sup>D</sup>*�2(*<sup>R</sup>* <sup>+</sup> *rC*) <sup>+</sup>

*B*n*<sup>w</sup>* =

*DR*(*R* + *rC*) (−*D*�*R*(−*D*�*R* − *rC*)+(*R* + *rC*)*req*)*D*�*R*

*DR*(*R* + *rC*)

The averaged output *v*o(*t*) of the buck-boost converter can be written as:

−*D*�*R*(−*D*�*R* − *rC*)+(*R* + *rC*)*req*

*CyX* + *E*yw*W*�

⎡ ⎣ 1

(*<sup>R</sup>* <sup>+</sup> *rC*)*<sup>C</sup>* <sup>−</sup> <sup>1</sup>

and can be expressed as follows:

*dx*˜(*t*) *dt* <sup>=</sup>�

*A* =

⎡ ⎢ ⎣ −*req*

*Bu* =

where:

*X* =

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

being *req* = *Dron* + *D*�

*Y* + *y*˜(*t*) = �

*Cy* =

� *RrC R* + *rC*

The measurable states of the converter are noted as *xa*(*t*). Note that the time dependence of the variables may be omitted to simplify the notation.

The binary signal *u*b(*t*), which turns on and off the switches, is genereated by means of a Pulse Width Modulation (PWM) subcircuit, working at a constant frequency 1/*T*s. The switching period *T*<sup>s</sup> is equal to the sum of *t*on and *t*off. For a unit-amplitude sawtooth PWM, the duty-cycle *d*(*t*) = *t*on/(*t*on + *t*off) is the control input of the converter.

As shown in (Erickson & Maksimovic, 1999) and (Leyva et al., 2006), considering that the state-space matrices of the converter are [ *A*on, *B*on ] during *t*on and [ *A*off, *B*off ] during *t*off, the general state-space averaged model of a dc-dc converter can be written as:

$$\begin{aligned} \dot{\tilde{x}}(t) &= \left(A\_{\text{off}} + (A\_{\text{on}} - A\_{\text{off}})I\right)X + \left(B\_{\text{off}} + (B\_{\text{on}} - B\_{\text{off}})D\right) \begin{bmatrix} 1 \\ 0 \end{bmatrix} \mathcal{W} \\ &+ \left(A\_{\text{off}} + (A\_{\text{on}} - A\_{\text{off}})I\right)\tilde{x}(t) + \left(B\_{\text{off}} + (B\_{\text{on}} - B\_{\text{off}})D\right) \begin{bmatrix} 0 \\ 1 \end{bmatrix} \dot{w}(t) \\ &+ \left((A\_{\text{on}} - A\_{\text{off}})X + (B\_{\text{on}} - B\_{\text{off}})\begin{bmatrix} 1 \\ 0 \end{bmatrix} \mathcal{W}\right)\tilde{d}(t) \\ &+ \left((A\_{\text{on}} - A\_{\text{off}})\tilde{x}(t) + (B\_{\text{on}} - B\_{\text{off}})\begin{bmatrix} 0 \\ 1 \end{bmatrix} \tilde{w}(t)\right)\tilde{d}(t), \end{aligned} \tag{2}$$

where the equilibrium (noted with capital letters) and the incremental vectors (noted with tildes) are as follows. *<sup>X</sup>* and *<sup>x</sup>*˜ <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* correspond to the state vectors, *<sup>D</sup>* and ˜*<sup>d</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>m</sup>* are the control inputs, while *<sup>W</sup>* and *<sup>w</sup>*˜ <sup>∈</sup> **<sup>R</sup>***<sup>l</sup>* stand for the disturbance inputs.

In the buck converter, *A*on = *A*off, and the averaged model (2) can be rewritten as:

$$\frac{d\tilde{x}(t)}{dt} = \left(AX + B\_W W\right) + A\tilde{x}(t) + B\_W \tilde{w}(t) + B\_\mathbf{u}\tilde{d}(t) + B\_{\mathbf{h}\_\mathbf{u}}\tilde{w}(t)\tilde{d}(t) \tag{3}$$

where:

$$A = \begin{bmatrix} -\frac{r\_{\Omega}}{L} - \frac{Rr\_{\subset}}{(R+r\_{\subset})L} - \frac{R}{(R+r\_{\subset})L} \\ \frac{R}{(R+r\_{\subset})\mathbb{C}} - \frac{1}{(R+r\_{\subset})\mathbb{C}} \end{bmatrix}, B\_{\text{W}} = \begin{bmatrix} \frac{D}{L} & \frac{Rr\_{\subset}}{(R+r\_{\subset})L} \\ 0 & -\frac{R}{(R+r\_{\subset})\mathbb{C}} \end{bmatrix},$$

$$B\_{\text{U}} = \begin{bmatrix} \frac{V\_{\mathcal{S}}}{L} \\ 0 \end{bmatrix}, B\_{\text{R}\_{\text{U}}} = \begin{bmatrix} \frac{1}{L} \\ 0 \end{bmatrix},\tag{4}$$

$$\mathbf{X} = \begin{bmatrix} \frac{V\_{\mathcal{S}} \mathbf{D}}{1 + r\_{eq}/R} \\\\ \frac{\mathbf{R}}{1 + r\_{eq}/R} \end{bmatrix}, \mathbf{W} = \begin{bmatrix} V\_{\mathcal{S}} \\ \mathbf{0} \end{bmatrix}, \tilde{\mathbf{x}}(t) = \begin{bmatrix} \tilde{i}\_{L}(t) \\ \tilde{v}\_{0}(t) \end{bmatrix}, \tilde{w}(t) = \begin{bmatrix} \tilde{v}\_{\mathcal{S}}(t) \\ \dot{i}\_{\text{load}}(t) \end{bmatrix}, \mathbf{w}$$

being *req* = *Dron* + *D*� *roff* and *D*� = 1 − *D*. The dimensions of the system matrices are defined as *<sup>A</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*n*, *<sup>B</sup>*u, *<sup>B</sup>*n*<sup>w</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*m*, *<sup>B</sup>*<sup>w</sup> <sup>∈</sup> **<sup>R</sup>***n*×*<sup>l</sup>* .

Similarly, the averaged outputs of the buck converter can be written as:

$$\mathcal{Y} + \mathfrak{F}(t) = \left(\mathbb{C}\_{\mathcal{Y}} X + E\_{\mathcal{Y}\mathcal{W}} \mathcal{W}\right) + \mathbb{C}\_{\mathcal{Y}} \mathfrak{x}(t) + E\_{\mathcal{Y}\mathcal{W}} \mathfrak{w}(t) \tag{5}$$

where in a general case *Cy* <sup>∈</sup> **<sup>R</sup>***q*×*n*, *<sup>E</sup>*yw <sup>∈</sup> **<sup>R</sup>***q*×*m*. Considering the load voltage *<sup>v</sup>*o(*t*) as the only output, these matrices are written as:

$$\mathcal{C}\_{y} = \left[\frac{R r\_{\mathcal{C}}}{R + r\_{\mathcal{C}}} \, \frac{R}{R + r\_{\mathcal{C}}}\right], \\ E\_{yw} = \left[0 - \frac{R r\_{\mathcal{C}}}{R + r\_{\mathcal{C}}}\right]. \tag{6}$$

#### **2.2 Model of the buck-boost converter**

In the buck-boost converter, matrices *A*on and *A*off are not equal, and therefore, the averaged model contains bilinear terms concerning the control input, the states and the disturbance inputs. According to those nonlinear terms, the linearized transfer function depends on the operating point, hence making the control subsystem design more difficult. In order to derive accurate transfer functions of the buck-boost converter for output-feedback approaches, the stray resistances are also taken into account.

For the buck-boost converter, the averaged model in the form of (3) contains bilinear terms, and can be expressed as follows:

$$\frac{d\tilde{\mathbf{x}}(t)}{dt} = \left(AX + B\_{\mathbf{w}}W\right) + A\tilde{\mathbf{x}}(t) + B\_{\mathbf{w}}\tilde{w}(t) + B\_{\mathbf{u}}\tilde{d}(t) + B\_{\mathbf{n}\_{x}}\tilde{\mathbf{x}}(t)\tilde{d}(t) + B\_{\mathbf{n}\_{y}}\tilde{w}(t)\tilde{d}(t) \tag{7}$$

where:

4 Will-be-set-by-IN-TECH

*ron* = *rdon* + *rL*

The measurable states of the converter are noted as *xa*(*t*). Note that the time dependence of

The binary signal *u*b(*t*), which turns on and off the switches, is genereated by means of a Pulse Width Modulation (PWM) subcircuit, working at a constant frequency 1/*T*s. The switching period *T*<sup>s</sup> is equal to the sum of *t*on and *t*off. For a unit-amplitude sawtooth PWM,

As shown in (Erickson & Maksimovic, 1999) and (Leyva et al., 2006), considering that the state-space matrices of the converter are [ *A*on, *B*on ] during *t*on and [ *A*off, *B*off ] during *t*off,

*x*˜(*t*) + �

where the equilibrium (noted with capital letters) and the incremental vectors (noted with tildes) are as follows. *<sup>X</sup>* and *<sup>x</sup>*˜ <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* correspond to the state vectors, *<sup>D</sup>* and ˜*<sup>d</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>m</sup>* are the

(*R* + *rC*)*L*

⎤ ⎥ <sup>⎦</sup> , *<sup>B</sup>*<sup>w</sup> <sup>=</sup>

> � 1 *L* 0

� ˜*iL*(*t*) *v*˜o(*t*)

*roff* and *D*� = 1 − *D*. The dimensions of the system matrices are defined

�

, *w*˜(*t*) =

� ,

(*R* + *rC*)*C*

⎤

⎦ , *B*n*<sup>w</sup>* =

, *x*˜(*t*) =

.

*CyX* + *E*yw*W*�

� 1 0 � *W*� ˜*d*(*t*)

> � 0 1 � *w*˜(*t*)

*B*off + (*B*on − *B*off)*D*

*X* + �

the duty-cycle *d*(*t*) = *t*on/(*t*on + *t*off) is the control input of the converter.

the general state-space averaged model of a dc-dc converter can be written as:

*roff* <sup>=</sup> *rdof f* <sup>+</sup> *rL* (1)

� � 1 0 � *W*

1 � *w*˜(*t*)

(2)

(4)

*<sup>B</sup>*off + (*B*on <sup>−</sup> *<sup>B</sup>*off)*D*� � <sup>0</sup>

� ˜*d*(*t*),

+ *Ax*˜(*t*) + *B*w*w*˜(*t*) + *B*<sup>u</sup> ˜*d*(*t*) + *B*n*<sup>w</sup> w*˜(*t*) ˜*d*(*t*) (3)

⎡ ⎢ ⎣ *D L*

*RrC* (*R* + *rC*)*L* ⎤ ⎥ ⎦ ,

(*R* + *rC*)*C*

� *<sup>v</sup>*˜*g*(*t*) ˜ *<sup>i</sup>*load(*t*)

+ *Cyx*˜(*t*) + *E*yw*w*˜(*t*) (5)

� ,

<sup>0</sup> <sup>−</sup> *<sup>R</sup>*

of the inductor and noted as:

˙*x*˜(*t*) =�

*dx*˜(*t*) *dt* <sup>=</sup>�

> ⎡ ⎢ ⎣

−*req*

*A* =

*X* =

being *req* = *Dron* + *D*�

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

*VgD* 1 + *req*/*R R VgD* 1 + *req*/*R*

as *<sup>A</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*n*, *<sup>B</sup>*u, *<sup>B</sup>*n*<sup>w</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*m*, *<sup>B</sup>*<sup>w</sup> <sup>∈</sup> **<sup>R</sup>***n*×*<sup>l</sup>*

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

*Y* + *y*˜(*t*) = �

where:

+ �

+ �

+ �

the variables may be omitted to simplify the notation.

*<sup>A</sup>*off + (*A*on <sup>−</sup> *<sup>A</sup>*off)*U*�

*<sup>A</sup>*off + (*A*on <sup>−</sup> *<sup>A</sup>*off)*U*�

(*A*on − *A*off)*X* + (*B*on − *B*off)

control inputs, while *<sup>W</sup>* and *<sup>w</sup>*˜ <sup>∈</sup> **<sup>R</sup>***<sup>l</sup>* stand for the disturbance inputs.

(*<sup>R</sup>* <sup>+</sup> *rC*)*<sup>L</sup>* <sup>−</sup> *<sup>R</sup>*

(*<sup>R</sup>* <sup>+</sup> *rC*)*<sup>C</sup>* <sup>−</sup> <sup>1</sup>

*B*<sup>u</sup> =

, *W* =

Similarly, the averaged outputs of the buck converter can be written as:

⎡ ⎣ *Vg L* 0

� *Vg* 0 �

*AX* + *B*w*W*�

*<sup>L</sup>* <sup>−</sup> *RrC*

*R*

(*A*on − *A*off)*x*˜(*t*)+(*B*on − *B*off)

In the buck converter, *A*on = *A*off, and the averaged model (2) can be rewritten as:

$$A = \begin{bmatrix} -\frac{r\_{eq}}{L} - D' \frac{Rr\_{\subset}}{(R+r\_{\subset})L} & D' \frac{R}{(R+r\_{\subset})L} \\ -D' \frac{R}{(R+r\_{\subset})\overline{C}} & -\frac{1}{(R+r\_{\subset})\overline{C}} \end{bmatrix}, \quad B\_{\text{uv}} = \begin{bmatrix} D\frac{1}{L} - D' \frac{Rr\_{\subset}}{(R+r\_{\subset})L} \\ 0 & -\frac{R}{(R+r\_{\subset})\overline{C}} \end{bmatrix},$$

$$B\_{\text{u}} = \begin{bmatrix} \frac{V\_{\mathcal{S}}}{L} \left(1 + \frac{Dr\_{\subset}}{D'^2 (R+r\_{\subset})} + \frac{D'DR^2}{D'R(D'R+r\_{\subset}) + (R+r\_{\subset})r\_{\neq}}\right) \\ \frac{DV\_{\mathcal{S}}}{CD'^2 (R+r\_{\subset})} \end{bmatrix},$$

$$B\_{\text{n}\_{\varphi}} = \begin{bmatrix} 1 & -\frac{Rr\_{\subset}}{(R+r\_{\subset})L} \\ 0 & 0 \end{bmatrix},\tag{8}$$

$$X = \begin{bmatrix} \frac{V\_{\mathcal{S}}D'DR(R+r\_{\mathcal{C}})}{(-D'R(-D'R-r\_{\mathcal{C}})+(R+r\_{\mathcal{C}})r\_{\mathcal{C}\eta})D'R} \\\\ \frac{-V\_{\mathcal{S}}D'DR(R+r\_{\mathcal{C}})}{-D'R(-D'R-r\_{\mathcal{C}})+(R+r\_{\mathcal{C}})r\_{\mathcal{C}\eta}} \end{bmatrix}, \begin{bmatrix} R r\_{\mathcal{C}} \\\\ \frac{L(R+r\_{\mathcal{C}})}{R} - \frac{R}{(R+r\_{\mathcal{C}})L} \end{bmatrix}.$$

being *req* = *Dron* + *D*� *roff* . The dimensions of the system matrices are defined as *<sup>A</sup>*, *<sup>B</sup>*n*<sup>x</sup>* , *<sup>B</sup>*n*<sup>w</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*n*, *<sup>B</sup>*<sup>u</sup> <sup>∈</sup> **<sup>R</sup>***n*×*m*, *<sup>B</sup>*<sup>w</sup> <sup>∈</sup> **<sup>R</sup>***n*×*<sup>l</sup>* .

The averaged output *v*o(*t*) of the buck-boost converter can be written as:

$$\mathcal{Y} + \mathfrak{F}(t) = \left(\mathsf{C}\_{\mathcal{Y}}X + \mathsf{E}\_{\mathsf{y}\mathcal{W}}\mathcal{W}\right) + \mathsf{C}\_{\mathcal{Y}}\mathfrak{x}(t) + \mathsf{E}\_{\mathsf{y}\mathcal{W}}\mathfrak{w}(t) + \mathsf{C}\_{\mathsf{y}\mathcal{U}}X\tilde{d}(t) + \mathsf{C}\_{\mathsf{y}\mathcal{U}}\mathfrak{x}(t)\tilde{d}(t) \tag{9}$$

The equivalent hold transfer function with the Padé approximation writes

operating conditions, aging effects, etc... are sources of uncertainty.

described as an interconnection of known LTI systems.

**Definition 2.1.** *A polytopic system is a linear time-varying system*

subsection is focused on uncertain state-space models.

**2.4 Modeling of uncertainty**

uncertainties and LMI control are:

model:

uncertain and/or time-varying coefficients.

*GZOH*(*s*) = <sup>1</sup>

which is a strictly proper transfer function whose representation in state-space form could be:

LMI Robust Control of PWM Converters: An Output-Feedback Approach 323

As stated in (Gahinet et al., 1995), the notion of system uncertainty is of major importance in the field of robust control theory. First of all, one of the key features of feedback is that it reduces the effects of uncertainty. However, when designing a control system, the model used to represent the behavior of the plant is often approximated. The difference between the approximated model and the true model is called model uncertainty. Also the changes due to

The two main approaches shown in (Gahinet et al., 1995) when dealing with system

• Uncertain state-space models, relevant for systems described by dynamical equations with

• Linear-fractional representation (LFR) of uncertainty, in which the uncertain system is

While LFR models have had a main role in modern robust control synthesis methods such as in *μ*-synthesis (Zhou et al., 1996), state-space models have been used in convex optimization approaches (Boyd et al., 1994). Since this chapter presents approaches that do not employ the concept of structured singular value on which the *μ*-synthesis method is based, the following

If some of the physical parameters are approximated or unknown, or if there exists nonlinear or non-modeled dynamic effects, then the system can be described by an uncertain state-space

> *x*˙ = *Ax* + *Bu y* = *Cx* + *Du*

where the state-space matrices *A*, *B*, *C*, *D* depend on uncertain and/or time-varying parameters or vary in some bounded sets of the space of matrices. One of the state-space

> *x*˙ = *A*(*t*)*x* + *B*(*t*)*u y* = *C*(*t*)*x* + *D*(*t*)*u*

representations of relevance in LMI control problems is the class of polytopic models:

˙*x*˜*p*(*t*) = <sup>−</sup>(2/*Ts*)*x*˜*p*(*t*)+(2/*Ts*) ˜

where *x*˜*p*(*t*) is the state variable of the *GZOH*(*s*), ˜*d*(*t*) is its input and ˜*d*2(*t*) is its output.

1 + *s Ts* 2

*d*(*t*) ˜*d*2(*t*) = *<sup>x</sup>*˜*p*(*t*) (15)

(14)

(16)

(17)

Fig. 4. Waveforms of the PWM process.

where:

$$\mathbf{C}\_{y} = \left[ -D' \frac{R r\_{\text{C}}}{R + r\_{\text{C}}} \frac{R}{R + r\_{\text{C}}} \right], \mathbf{C}\_{yu} = \left[ \frac{R r\_{\text{C}}}{R + r\_{\text{C}}} \ 0 \right], E\_{yw} = \left[ 0 - \frac{R r\_{\text{C}}}{R + r\_{\text{C}}} \right]. \tag{10}$$

These models are employed in Section 3 to derive robust controllers for the buck and the buck-boost converters.

#### **2.3 Delay model for the PWM actuator**

The models presented above do not take into account the sampling effect of the modulation (Brown & Middlebrook, 1981; Erickson & Maksimovic, 1999) (see Figure 4).

Usually, the sampling effect is not considered, and only the linear gain of the modulator is taken into account. In a voltage-mode modulator, the duty-cycle input is usually constrained between zero and the amplitude of the sawtooth signal *VM*, and therefore the linear gain of this modulator is 1/*VM* (Erickson & Maksimovic, 1999). For simplicity the amplitude *VM* can be considered equal to one, such that the linear model shown previously is valid for a duty-cycle input *d* ∈ [0, 1].

However, the sampling effect can be taken into account in order to limit the control-loop bandwidth in the automatic control synthesis algorithms. Such an effect can be incorporated to the power stage model as a sampling at the switching frequency 1/*Ts* and a zero-order hold block, assuming that the switch is fired once every switching cycle *Ts* (Maksimovic, 2000). The equivalent transfer function for this sampling model is then:

$$G\_{ZOH}(s) = \frac{1 - e^{-sT\_s}}{sT\_s} \tag{11}$$

The exponential factor *e*−*sTs* can be approximated by a Padé function:

$$e^{-sT\_i} \approx \frac{\sum\_{k=0}^{n} -1^k c\_k T\_s s^k}{\sum\_{k=0}^{n} c\_k T\_s s^k},$$
 
$$c\_k = \frac{(2n - k)! n!}{2n! k! (n - k)!}, \qquad k = 0, 1, \cdots, n. \tag{12}$$

Taking the first order approximation *n* = 1 we obtain

$$e^{-sT\_s} \approx \frac{1 - (T\_s/2)\text{s}}{1 + (T\_s/2)\text{s}}\tag{13}$$

The equivalent hold transfer function with the Padé approximation writes

$$G\_{ZOH}(s) = \frac{1}{1 + s\frac{T\_s}{2}}\tag{14}$$

which is a strictly proper transfer function whose representation in state-space form could be:

$$\begin{cases} \dot{\mathfrak{x}}\_p(t) = -(2/T\_s)\mathfrak{x}\_p(t) + (2/T\_s)\check{d}(t) \\\\ \check{d}\_2(t) = \mathfrak{x}\_p(t) \end{cases} \tag{15}$$

where *x*˜*p*(*t*) is the state variable of the *GZOH*(*s*), ˜*d*(*t*) is its input and ˜*d*2(*t*) is its output.

#### **2.4 Modeling of uncertainty**

6 Will-be-set-by-IN-TECH

*T*s

*t*

, *Eyw* =

<sup>0</sup> <sup>−</sup> *RrC R* + *rC*

, *k* = 0, 1, ·, *n*. (12)

<sup>1</sup> + (*Ts*/2)*<sup>s</sup>* (13)

. (10)

(11)

*t*

*v*s

*t*on *t*off

(Brown & Middlebrook, 1981; Erickson & Maksimovic, 1999) (see Figure 4).

equivalent transfer function for this sampling model is then:

*e*

Taking the first order approximation *n* = 1 we obtain

The exponential factor *e*−*sTs* can be approximated by a Padé function:

∑*n*

*ck* <sup>=</sup> (2*<sup>n</sup>* <sup>−</sup> *<sup>k</sup>*)!*n*! 2*n*!*k*!(*n* − *k*)!

*e*

<sup>−</sup>*sTs* <sup>≈</sup> <sup>∑</sup>*<sup>n</sup>*

, *Cyu* =

These models are employed in Section 3 to derive robust controllers for the buck and the

The models presented above do not take into account the sampling effect of the modulation

Usually, the sampling effect is not considered, and only the linear gain of the modulator is taken into account. In a voltage-mode modulator, the duty-cycle input is usually constrained between zero and the amplitude of the sawtooth signal *VM*, and therefore the linear gain of this modulator is 1/*VM* (Erickson & Maksimovic, 1999). For simplicity the amplitude *VM* can be considered equal to one, such that the linear model shown previously is valid for a

However, the sampling effect can be taken into account in order to limit the control-loop bandwidth in the automatic control synthesis algorithms. Such an effect can be incorporated to the power stage model as a sampling at the switching frequency 1/*Ts* and a zero-order hold block, assuming that the switch is fired once every switching cycle *Ts* (Maksimovic, 2000). The

*GZOH*(*s*) = <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*sTs*

*<sup>k</sup>*=<sup>0</sup> <sup>−</sup>1*kckTssk*

*<sup>k</sup>*=<sup>0</sup> *ckTss<sup>k</sup>* ,

<sup>−</sup>*sTs* <sup>≈</sup> <sup>1</sup> <sup>−</sup> (*Ts*/2)*<sup>s</sup>*

*sTs*

 *RrC R* + *rC* 0 

*R R* + *rC*

*VM*

*u*b

Fig. 4. Waveforms of the PWM process.

**2.3 Delay model for the PWM actuator**

<sup>−</sup>*D*� *RrC R* + *rC*

*Cy* = 

buck-boost converters.

duty-cycle input *d* ∈ [0, 1].

where:

*d*

As stated in (Gahinet et al., 1995), the notion of system uncertainty is of major importance in the field of robust control theory. First of all, one of the key features of feedback is that it reduces the effects of uncertainty. However, when designing a control system, the model used to represent the behavior of the plant is often approximated. The difference between the approximated model and the true model is called model uncertainty. Also the changes due to operating conditions, aging effects, etc... are sources of uncertainty.

The two main approaches shown in (Gahinet et al., 1995) when dealing with system uncertainties and LMI control are:


While LFR models have had a main role in modern robust control synthesis methods such as in *μ*-synthesis (Zhou et al., 1996), state-space models have been used in convex optimization approaches (Boyd et al., 1994). Since this chapter presents approaches that do not employ the concept of structured singular value on which the *μ*-synthesis method is based, the following subsection is focused on uncertain state-space models.

If some of the physical parameters are approximated or unknown, or if there exists nonlinear or non-modeled dynamic effects, then the system can be described by an uncertain state-space model:

$$\begin{cases} \dot{\mathfrak{x}} = A\mathfrak{x} + Bu \\\\ y = \mathbb{C}\mathfrak{x} + Du \end{cases} \tag{16}$$

where the state-space matrices *A*, *B*, *C*, *D* depend on uncertain and/or time-varying parameters or vary in some bounded sets of the space of matrices. One of the state-space representations of relevance in LMI control problems is the class of polytopic models:

**Definition 2.1.** *A polytopic system is a linear time-varying system*

$$\begin{cases} \dot{\mathbf{x}} = A(t)\mathbf{x} + B(t)u \\ y = C(t)\mathbf{x} + D(t)u \end{cases} \tag{17}$$

*in which the matrix G*(*t*) = � *A*(*t*) *B*(*t*) *C*(*t*) *D*(*t*) � *varies within a fixed polytope of matrices*

$$\mathcal{G}(t) \in \text{Co}\{\mathcal{G}\_1, \dots, \mathcal{G}\_N\} := \left\{ \sum\_{j=1}^N \delta\_j \mathcal{G}\_j : \delta\_j \ge 0, \sum\_{j=1}^N \delta\_j = 1 \right\} \tag{18}$$

*d*(*t*)

*<sup>x</sup>*int(*t*) *<sup>V</sup>* ref <sup>+</sup>

Fig. 5. Block diagram of a state-feedback system with controller **K** and error integration.

This state-space representation has been previously used to derive robust control synthesis methods for dc-dc converters, which generally result in a state-feedback law that stabilizes the system for a certain range of uncertainty: parameter-dependent approaches for the linear dynamics of the converters are presented in (Montagner et al., 2005) and (Torres-Pinzon & Leyva, 2009) while (Hu, 2011) introduces a representation of the nonlinear dynamics. Consistent experimental results with tight performances are presented in (Olalla et al., 2009a; 2010a; 2011). The small-signal stabilization of nonlinear dc-dc converters is considered in (Olalla et al., 2009a; 2010a), where the converter is ensured to be stable in a range of operating points, but its trajectory between those points is not ensured to be stable due to the disregard of the nonlinear dynamics. These nonlinearities are taken into account in (Olalla et al., 2011) where also a less conservative polytopic uncertainty model is introduced. The state-feedback formulation of the control problem is of interest since (i) it may deliver better performance than some output-feedback approaches, (ii) it can be posed as a convex optimization problem with no conservatism or iterations and (iii) it is very simple to implement. However, the main disadvantage of state-feedback is that the full state vector must be available for measure, which is not always true. Therefore, it may require additional components and sensors to obtain the state or to implement estimators of the unaccessible states. Robust output-feedback approaches are then an alternative to derive robust controllers

LMI Robust Control of PWM Converters: An Output-Feedback Approach 325

Robust control via output-feedback has been the subject of extensive research in the field of automatic control (de Oliveira & Geromel, 1997; Garcia et al., 2004; Peaucelle & Arzelier, 2001a;b; Scherer et al., 1997; Skogestad & Postlethwaite, 1996), but it has been hardly employed in dc-dc converters (Rodriguez et al., 1999). Power electronics engineers tend to use current-mode approaches (Erickson & Maksimovic, 1999) that employ an inner current loop before applying the output-feedback loop and, in that way, ease the control of the dc-dc converter. However, current-mode approaches require current sensing, as state-feedback control, and they suffer from noise, since in some cases, as in peak-current control, the current waveform must be sensed accurately. Therefore, a plain output-feedback approach can be of interest in certain cases in which a simple control is required and the sensing of all the states

The most simple control problem in terms of an LMI formulation is the one in which all the system states are measurable. The state-feedback problem considers the stabilization of (22)

Since the state-feedback approach does not allow to eliminate steady-state error, an additional

*x*(*t*) + *Bw*(*p*)*w*(*t*). (23)

*A*(*p*) + *B*u(*p*)**K**

with a simple controller *<sup>u</sup>* <sup>=</sup> **<sup>K</sup>***x*, where **<sup>K</sup>** <sup>∈</sup> <sup>R</sup>*m*×*n*, as follows

*x*˙(*t*) =


with known performances.

of the converter is not possible.

**3.1 State-feedback control**

**K** DC-DC

*x*(*t*)� *y*(*t*)

*where G*1,..., *GN are the vertices of the polytope.*

In other words, *G*(*t*) is a convex combination of the matrices *G*1,..., *GN*. Polytopics models are also called *linear differential inclusions* LDI in (Boyd et al., 1994).

### **2.4.1 Example: Buck converter polytopic model**

Consider the buck converter model introduced in subsection 2.1, with *w*˜(*t*) = 0. For simplicity, the stray resistances are disregarded. If we take *R* and *Vg* as uncertain parameters of the converter, the uncertain system is described as follows

$$\begin{cases} \frac{d\tilde{\mathbf{x}}(t)}{dt} = \left(\sum\_{j=1}^{N} A\_j \delta\_j\right) \tilde{\mathbf{x}}(t) + \left(\sum\_{j=1}^{N} B\_{\mathrm{uj}} \delta\_j\right) \tilde{d}(t) \\\ \tilde{y}(t) = \mathbf{C}\_{\mathbf{y}} \tilde{\mathbf{x}}(t) + E\_{\mathrm{yw}} \tilde{w}(t) \end{cases} \tag{19}$$

with *δ<sup>j</sup>* ≥ 0, *N* ∑ *j*=1 *δ<sup>j</sup>* = 1. The uncertain matrices *Aj* and *B*u*<sup>j</sup>* are

$$A\_{j} = \begin{bmatrix} 0 & -\frac{1}{L} \\ \frac{1}{C} & -\frac{1}{R\_{j}C} \end{bmatrix}, \quad B\_{\text{uj}} = \begin{bmatrix} V\_{\text{gj}} \\ \frac{L}{L} \\ 0 \end{bmatrix}, \tag{20}$$

where *Rj* = {*Rmin Rmax Rmin Rmax*}, and *Vgj* = {*Vgmin Vgmin Vgmax Vgmax* }, which represents a uncertain polytope of four vertices (2 power the number of uncertain parameters, that are *Rj* and *Vgj* in this example).

#### **3. Robust control of dc-dc converters**

Consider a general LTI model with states *x*(*t*), controlled outputs *y*(*t*) and performance outputs *z*(*t*):

$$\Sigma: \begin{cases} \dot{x}(t) = Ax(t) + B\_{\text{\textquotedblleft}}w(t) + B\_{\text{\textquotedblleft}}u(t) \\ y(t) = \mathbb{C}\_{y}x(t) + E\_{y\text{\textquotedblleft}}w(t) + E\_{y\text{\textquotedblright}}u(t) \\ z(t) = \mathbb{C}\_{z}x(t) + E\_{z\text{\textquotedblleft}}w(t) + E\_{z\text{\textquotedblright}}u(t) \end{cases} . \tag{21}$$

It is possible to assume that some elements involved in the system matrices are uncertain or time-varying. For the sake of simplicity, the performance and measurable outputs are discarded, hence these uncertain elements are concentrated in matrices *A*, *Bw* and *Bu* and they are grouped in a vector *p*. Thus, matrices *A*, *Bw* and *Bu* depend on such uncertainty vector, and we can express (21) as function of these parameters:

$$
\dot{\mathbf{x}}(t) = A(p)\mathbf{x}(t) + B\_{\mathbf{w}}(p)\mathbf{w}(t) + B\_{\mathbf{u}}(p)\mathbf{u}(t) \,. \tag{22}
$$

8 Will-be-set-by-IN-TECH

⎧ ⎨ ⎩

In other words, *G*(*t*) is a convex combination of the matrices *G*1,..., *GN*. Polytopics models

Consider the buck converter model introduced in subsection 2.1, with *w*˜(*t*) = 0. For simplicity, the stray resistances are disregarded. If we take *R* and *Vg* as uncertain parameters of the

*x*˜(*t*) +

� *<sup>N</sup>* ∑ *j*=1 *B*u*jδ<sup>j</sup>* � ˜*d*(*t*)

> ⎡ ⎣ *Vgj L* 0

*x*˙(*t*) = *A*(*p*)*x*(*t*) + *Bw*(*p*)*w*(*t*) + *B*u(*p*)*u*(*t*). (22)

⎤

⎦ , (20)

. (21)

*N* ∑ *j*=1

*varies within a fixed polytope of matrices*

*N* ∑ *j*=1

*δ<sup>j</sup>* = 1

⎫ ⎬ ⎭

(18)

(19)

*δjGj* : *δ<sup>j</sup>* ≥ 0,

*in which the matrix G*(*t*) =

with *δ<sup>j</sup>* ≥ 0,

outputs *z*(*t*):

*N* ∑ *j*=1

and *Vgj* in this example).

**3. Robust control of dc-dc converters**

Σ :

�

*where G*1,..., *GN are the vertices of the polytope.*

**2.4.1 Example: Buck converter polytopic model**

⎧ ⎪⎨

⎪⎩

converter, the uncertain system is described as follows

*dx*˜(*t*) *dt* <sup>=</sup>

*Aj* =

⎧ ⎨ ⎩

vector, and we can express (21) as function of these parameters:

*A*(*t*) *B*(*t*) *C*(*t*) *D*(*t*)

*G*(*t*) ∈ Co{*G*1,..., *GN*} :=

are also called *linear differential inclusions* LDI in (Boyd et al., 1994).

� *<sup>N</sup>* ∑ *j*=1 *Ajδ<sup>j</sup>* �

*y*˜(*t*) = *Cyx*˜(*t*) + *E*yw*w*˜(*t*)

*δ<sup>j</sup>* = 1. The uncertain matrices *Aj* and *B*u*<sup>j</sup>* are

<sup>0</sup> <sup>−</sup> <sup>1</sup> *<sup>L</sup>* <sup>1</sup> *<sup>C</sup>* <sup>−</sup> <sup>1</sup> *RjC* ⎤ ⎥ <sup>⎦</sup> , *<sup>B</sup>*u*<sup>j</sup>* <sup>=</sup>

where *Rj* = {*Rmin Rmax Rmin Rmax*}, and *Vgj* = {*Vgmin Vgmin Vgmax Vgmax* }, which represents a uncertain polytope of four vertices (2 power the number of uncertain parameters, that are *Rj*

Consider a general LTI model with states *x*(*t*), controlled outputs *y*(*t*) and performance

It is possible to assume that some elements involved in the system matrices are uncertain or time-varying. For the sake of simplicity, the performance and measurable outputs are discarded, hence these uncertain elements are concentrated in matrices *A*, *Bw* and *Bu* and they are grouped in a vector *p*. Thus, matrices *A*, *Bw* and *Bu* depend on such uncertainty

*x*˙(*t*) = *Ax*(*t*) + *Bww*(*t*) + *B*u*u*(*t*) *y*(*t*) = *Cyx*(*t*) + *Eyww*(*t*) + *Eyuu*(*t*) *z*(*t*) = *Czx*(*t*) + *Ezww*(*t*) + *Ezuu*(*t*)

⎡ ⎢ ⎣ �

Fig. 5. Block diagram of a state-feedback system with controller **K** and error integration.

This state-space representation has been previously used to derive robust control synthesis methods for dc-dc converters, which generally result in a state-feedback law that stabilizes the system for a certain range of uncertainty: parameter-dependent approaches for the linear dynamics of the converters are presented in (Montagner et al., 2005) and (Torres-Pinzon & Leyva, 2009) while (Hu, 2011) introduces a representation of the nonlinear dynamics. Consistent experimental results with tight performances are presented in (Olalla et al., 2009a; 2010a; 2011). The small-signal stabilization of nonlinear dc-dc converters is considered in (Olalla et al., 2009a; 2010a), where the converter is ensured to be stable in a range of operating points, but its trajectory between those points is not ensured to be stable due to the disregard of the nonlinear dynamics. These nonlinearities are taken into account in (Olalla et al., 2011) where also a less conservative polytopic uncertainty model is introduced. The state-feedback formulation of the control problem is of interest since (i) it may deliver better performance than some output-feedback approaches, (ii) it can be posed as a convex optimization problem with no conservatism or iterations and (iii) it is very simple to implement. However, the main disadvantage of state-feedback is that the full state vector must be available for measure, which is not always true. Therefore, it may require additional components and sensors to obtain the state or to implement estimators of the unaccessible states. Robust output-feedback approaches are then an alternative to derive robust controllers with known performances.

Robust control via output-feedback has been the subject of extensive research in the field of automatic control (de Oliveira & Geromel, 1997; Garcia et al., 2004; Peaucelle & Arzelier, 2001a;b; Scherer et al., 1997; Skogestad & Postlethwaite, 1996), but it has been hardly employed in dc-dc converters (Rodriguez et al., 1999). Power electronics engineers tend to use current-mode approaches (Erickson & Maksimovic, 1999) that employ an inner current loop before applying the output-feedback loop and, in that way, ease the control of the dc-dc converter. However, current-mode approaches require current sensing, as state-feedback control, and they suffer from noise, since in some cases, as in peak-current control, the current waveform must be sensed accurately. Therefore, a plain output-feedback approach can be of interest in certain cases in which a simple control is required and the sensing of all the states of the converter is not possible.

#### **3.1 State-feedback control**

The most simple control problem in terms of an LMI formulation is the one in which all the system states are measurable. The state-feedback problem considers the stabilization of (22) with a simple controller *<sup>u</sup>* <sup>=</sup> **<sup>K</sup>***x*, where **<sup>K</sup>** <sup>∈</sup> <sup>R</sup>*m*×*n*, as follows

$$\dot{\mathbf{x}}(t) = \left(A(p) + B\_{\mathbf{u}}(p)\mathbf{K}\right)\mathbf{x}(t) + B\_{\mathbf{w}}(p)\mathbf{w}(t) \,. \tag{23}$$

Since the state-feedback approach does not allow to eliminate steady-state error, an additional

integral state can be introduced for the regulated output of the system, as shown in Figure 5. Once the augmented system has been rewritten in the form of (23), the following result, adapted from (Bernussou et al., 1989), points out a synthesis method to obtain a state-feedback controller that stabilizes quadratically the closed-loop system.

**Theorem 3.1.** *The system* (23) *is stabilizable by state-feedback u* = **K***x if and only if there exist a symmetric matrix* **<sup>W</sup>** <sup>∈</sup> **<sup>R</sup>***n*×*<sup>n</sup> and a matrix* **<sup>Y</sup>** <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup> such that*

$$\begin{cases} \mathbf{W} > \mathbf{0} \\ A\mathbf{W} + \mathbf{W}A' + B\_{\boldsymbol{\mu}}\mathbf{Y} + \mathbf{Y}'B\_{\boldsymbol{\mu}}' < \mathbf{0} \end{cases} \tag{24}$$

*x*(*t*) =

⎡ ⎣

Uncertainty:

*to the equilibrium.*

*iL*(*t*) *vo*(*t*) *x*int(*t*)

⎤

⎦*. Considering*

*A* =

Table 1. Buck-boost: converter parameters

*as p* = [*R D*� *δ*<sup>1</sup> *δ*2]*, where:*

*both functions are strictly decreasing.*

*modulator model can not be measured.*

Sampling effect:

⎡ ⎣

<sup>−</sup> *<sup>D</sup>*� *<sup>C</sup>* <sup>−</sup> <sup>1</sup> *RC* 0 0 10

*AX* + *BwW* +

⎤

⎦ , *Bw* =

*the linear dynamics of the buck-boost converter are then written as:*

0 *<sup>D</sup>*� *<sup>L</sup>* 0 � 0 1 0 �

LMI Robust Control of PWM Converters: An Output-Feedback Approach 327

⎡ ⎣ *D <sup>L</sup>* 0 <sup>0</sup> <sup>−</sup> <sup>1</sup> *C* 0 0

*Polytopic uncertainty (19) is introduced in the model of the converter to cope with the variations of D and R. The parameters of this example take the values shown in Table 1. Note that the transient performance requirements are only fulfilled when the trajectory starts from an equilibrium point. Consequently, the variations of D and R must be slow enough to allow the system states to return*

> Parameter Value *R* ∈ [10, 50] Ω *Vg* 12 *V D* [0, 0.7] *C* 200 *μ* F *L* 100 *μ* H *Ts* 5 *μ* s

*As in (Olalla et al., 2009a), additional variables are introduced, in order to remove the non affine dependence of the system matrices on the uncertain terms. The uncertainty parameter vector is defined*

*R* ∈ [*Rmin*, *Rmax*],

2

*Note that the uncertain model is inside a polytopic domain formed by N* = 24 *vertices. Also note that the multiplication between δ*<sup>2</sup> *and* 1/*R in the second row of Bu does not imply a new variable because*

*In this example the sampling effect has not been included in the converter, as the state variables of the*

*min*, *D*�

*max*, 1/*D*�

*max*, *Dmax*/*D*�

*max*],

*min*],

2 *min*].

*D*� ∈ [*D*�

*δ*<sup>1</sup> ∈ [1/*D*�

*<sup>δ</sup>*<sup>2</sup> <sup>∈</sup> [*Dmin*/*D*�

*Vref* = 0, (27)

⎤ ⎥

<sup>⎦</sup> . (29)

(30)

˙*x*˜(*t*) =*Ax*˜(*t*) + *Bww*˜(*t*) + *Buu*˜(*t*) (28)

⎡ ⎢ ⎣

*Vg D*�*L DVg D*�2*RC* 0

⎤

⎦ , *Bu* =

*then, the state-feedback is given by* **K** = **YW**−1*.*

**Proof.** *The proof uses a quadratic Lyapunov function V*(*x*) = *x*� **P***x,* **P** = **P**� > 0*, whose time-derivative along the trajectories of the closed-loop system x*˙ = (*A* + *Bu*)**K***x must be definite negative (Boyd et al., 1994). It follows that the following condition*

$$A^\prime \mathbf{P} + \mathbf{P}A + \mathbf{K}^\prime B\_\mu^\prime \mathbf{P} + \mathbf{P}B\_\mu \mathbf{K} < \mathbf{0} \tag{25}$$

*has to be satisfied. Finally, considering the left and right-hand multiplication of the previous condition by* **W** = **P**−1*, and the substitution of* **KW** = **Y***, LMI condition* (24) *follows.*

A single Lyapunov function can be used to guarantee the stability of an uncertain system. The following theorem yields the state-feedback condition in the case of a polytopic representation.

**Theorem 3.2.** *The uncertain system defined by a convex polytope* Co {*G*1,..., *GN*} *is quadratically stabilizable by state-feedback u* = **K***x if and only if there exist a symmetric positive definite matrix* **W** *and a matrix* **Y** *such that*

$$A\_j \mathbf{W} + \mathbf{W} A\_j' + B\_{uj} \mathbf{Y} + \mathbf{Y}' B\_{uj}' < \mathbf{0} \quad \forall j = 1, \ldots, N,\tag{26}$$

*then* **K** = **YW**−<sup>1</sup> *is a state-feedback matrix.*

The proof of this theorem is given in (Bernussou et al., 1989). It is worth to point out that there exist more recent works which have been concerned with the stability of polytopic uncertain systems considering in particular multiple Lyapunov functions instead of a single one (Apkarian et al., 2001; Bernussou & Oustaloup, 2002; Peaucelle & Arzelier, 2001c), in order to reduce the conservatism of the quadratic approach.

#### **Example 1.** *Buck-Boost Converter*

*In this example, an uncertain polytopic model of the buck-boost converter is presented and a robust state-feedback controller is derived.*

*Consider the buck-boost converter model introduced in Section 2.2. Since for state-feedback control, the capacitor voltage is considered measurable, the stray resistance rC is neglected. Also the stray resistances of the inductor and the semiconductor devices are disregarded. In order to obtain zero steady-state error between the voltage reference Vref and the output voltage vo* (*t*)*, the model is augmented with an additional state variable x*int(*t*)*, which stands for the integral of the output voltage error, i.e. x*int(*t*) = <sup>−</sup> *Vref* − *v*o(*t*) *dt. The state vector of the new model is then written as*

$$\mathbf{x}(t) = \begin{bmatrix} i\_L(t) \\ v\_o(t) \\ \mathbf{x}\_{\mathrm{int}}(t) \end{bmatrix}. \text{ Considering}$$

$$AX + B\_{\overline{w}}W + \begin{bmatrix} 0 \\ 1 \end{bmatrix} V\_{\text{ref}} = 0,\tag{27}$$

*the linear dynamics of the buck-boost converter are then written as:*

$$
\dot{\mathfrak{X}}(t) = A\mathfrak{X}(t) + B\_{\mathfrak{w}}\mathfrak{w}(t) + B\_{\mathfrak{u}}\mathfrak{u}(t) \tag{28}
$$

$$A = \begin{bmatrix} 0 & \frac{D'}{L} & 0\\ -\frac{D'}{C} & -\frac{1}{RC} & 0\\ 0 & 1 & 0 \end{bmatrix},\ B\_{\overline{w}} = \begin{bmatrix} \frac{D}{L} & 0\\ 0 & -\frac{1}{C}\\ 0 & 0 \end{bmatrix},\ B\_{\overline{u}} = \begin{bmatrix} \frac{V\_{\overline{s}}}{DV\_{\overline{s}}}\\ \frac{DV\_{\overline{s}}}{D^{2}RC}\\ 0 \end{bmatrix}.\tag{29}$$

Uncertainty:

10 Will-be-set-by-IN-TECH

integral state can be introduced for the regulated output of the system, as shown in Figure 5. Once the augmented system has been rewritten in the form of (23), the following result, adapted from (Bernussou et al., 1989), points out a synthesis method to obtain a state-feedback

**Theorem 3.1.** *The system* (23) *is stabilizable by state-feedback u* = **K***x if and only if there exist a*

*time-derivative along the trajectories of the closed-loop system x*˙ = (*A* + *Bu*)**K***x must be definite*

*B*�

*has to be satisfied. Finally, considering the left and right-hand multiplication of the previous condition*

A single Lyapunov function can be used to guarantee the stability of an uncertain system. The following theorem yields the state-feedback condition in the case of a polytopic

**Theorem 3.2.** *The uncertain system defined by a convex polytope* Co {*G*1,..., *GN*} *is quadratically stabilizable by state-feedback u* = **K***x if and only if there exist a symmetric positive definite matrix* **W**

*B*�

The proof of this theorem is given in (Bernussou et al., 1989). It is worth to point out that there exist more recent works which have been concerned with the stability of polytopic uncertain systems considering in particular multiple Lyapunov functions instead of a single one (Apkarian et al., 2001; Bernussou & Oustaloup, 2002; Peaucelle & Arzelier, 2001c), in

*In this example, an uncertain polytopic model of the buck-boost converter is presented and a robust*

*Consider the buck-boost converter model introduced in Section 2.2. Since for state-feedback control, the capacitor voltage is considered measurable, the stray resistance rC is neglected. Also the stray resistances of the inductor and the semiconductor devices are disregarded. In order to obtain zero steady-state error between the voltage reference Vref and the output voltage vo* (*t*)*, the model is augmented with an additional state variable x*int(*t*)*, which stands for the integral of the output voltage*

*B*�

*<sup>u</sup>* <sup>&</sup>lt; <sup>00</sup> (24)

*<sup>u</sup>***<sup>P</sup>** + **<sup>P</sup>***Bu***<sup>K</sup>** < <sup>00</sup> (25)

*uj* <sup>&</sup>lt; <sup>00</sup> <sup>∀</sup>*<sup>j</sup>* <sup>=</sup> 1, . . . , *<sup>N</sup>*, (26)

*dt. The state vector of the new model is then written as*

**P***x,* **P** = **P**� > 0*, whose*

*A***W** + **W***A*� + *Bu***Y** + **Y**�

controller that stabilizes quadratically the closed-loop system.

*symmetric matrix* **<sup>W</sup>** <sup>∈</sup> **<sup>R</sup>***n*×*<sup>n</sup> and a matrix* **<sup>Y</sup>** <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup> such that*

*then, the state-feedback is given by* **K** = **YW**−1*.*

representation.

*and a matrix* **Y** *such that*

**W** > 00

**Proof.** *The proof uses a quadratic Lyapunov function V*(*x*) = *x*�

**P** + **P***A* + **K**�

*<sup>j</sup>* + *Buj***Y** + **Y**�

*by* **W** = **P**−1*, and the substitution of* **KW** = **Y***, LMI condition* (24) *follows.*

*negative (Boyd et al., 1994). It follows that the following condition*

*A*�

*Aj***W** + **W***A*�

order to reduce the conservatism of the quadratic approach.

*Vref* − *v*o(*t*)

*then* **K** = **YW**−<sup>1</sup> *is a state-feedback matrix.*

**Example 1.** *Buck-Boost Converter*

*state-feedback controller is derived.*

*error, i.e. x*int(*t*) = <sup>−</sup>

*Polytopic uncertainty (19) is introduced in the model of the converter to cope with the variations of D and R. The parameters of this example take the values shown in Table 1. Note that the transient performance requirements are only fulfilled when the trajectory starts from an equilibrium point. Consequently, the variations of D and R must be slow enough to allow the system states to return to the equilibrium.*


Table 1. Buck-boost: converter parameters

*As in (Olalla et al., 2009a), additional variables are introduced, in order to remove the non affine dependence of the system matrices on the uncertain terms. The uncertainty parameter vector is defined as p* = [*R D*� *δ*<sup>1</sup> *δ*2]*, where:*

$$\begin{aligned} \mathcal{R} & \in [\mathcal{R}\_{\min}, \mathcal{R}\_{\max}]\_{\prime} \\ D^{\prime} & \in [D\_{\min}^{\prime}, D\_{\max}^{\prime}]\_{\prime} \end{aligned} \tag{30}$$

$$\delta\_1 \in [1/D\_{\max}^{\prime}, 1/D\_{\min}^{\prime}]\_{\prime} \tag{30}$$

$$\delta\_2 \in [D\_{\min} / D\_{\max}^{\prime}, D\_{\max} / D\_{\min}^{\prime}].$$

*Note that the uncertain model is inside a polytopic domain formed by N* = 24 *vertices. Also note that the multiplication between δ*<sup>2</sup> *and* 1/*R in the second row of Bu does not imply a new variable because both functions are strictly decreasing.*

#### Sampling effect:

*In this example the sampling effect has not been included in the converter, as the state variables of the modulator model can not be measured.*

(a) Bode plot of closed loop transfer function from reference to output

(b) Pole location map of the closed loop system

*d*(*t*)

*the simulation results of the controller over the sixteen vertices of the set of matrices. Since the nonlinear terms are disregarded, the robust stability and performance of the converter is guaranteed while the converter remains in the considered operating points, assuming that the change between these operation points is sufficiently slow. In order to account for the neglected dynamics, see (Hu, 2011; Olalla et al.,*

LMI Robust Control of PWM Converters: An Output-Feedback Approach 329

Figure 7 shows the general diagram of an output-feedback control system. Depending on the structure of the controller *G*(*s*), two main approaches can be differentiated for the synthesis

Given the system Σ as described in (21), for the buck and the buck-boost converter *Eyu* = 0 can be considered. Then, in the case of a dynamic controller of order *k* with the following

� *x*˙*<sup>c</sup>* = **Ac***xc* + **Bc***y*

� *<sup>x</sup>*˙*cl* <sup>=</sup> <sup>A</sup>*xcl* <sup>+</sup> <sup>B</sup>*<sup>w</sup> z* = C*xc* + D*w*

*A* + *Bu***Dc***Cy Bu***Cc** *Bw* + *Bu***Dc***Eyw* **Bc***Cy* **Ac Bc***Eyw Cz* + *Ezu***Dc***Cy Ezu***Cc** *Ezw* + *Ezu***Dc***Dyw*

� *<sup>x</sup>*˙ <sup>=</sup> <sup>A</sup>*<sup>x</sup>* <sup>+</sup> <sup>B</sup>*<sup>w</sup>*

� (*A* + *Bu***K***Cy*) (*Bw* + *Bu***K***Eyw*) (*Cz* + *Ezu***K***Cy*) (*Ezw* + *Ezu***K***Eyw*)

*V*ref +

*2009b; 2011).*

structure

where

where

**3.2 Output-feedback control**

the closed loop system has the form

� A B C D

In the case of a static controller, **<sup>K</sup>** <sup>∈</sup> <sup>R</sup>*m*×*<sup>q</sup>*

� = ⎡ ⎣

and the closed loop system has the following structure

� =

� A B C E


Fig. 7. Block diagram of an output-feedback system with controller *G*(*s*).

of output-feedback controllers: static and dynamic controllers.

Σ*<sup>K</sup>* :

*T*Σ*<sup>K</sup>* :

*TK* :

*G*(*s*) DC-DC

*y*(*t*)

*<sup>u</sup>* <sup>=</sup> **Cc***xc* <sup>+</sup> **Dc***<sup>y</sup>* , (32)

*u* = **K***y* = **K**(*Cyx* + *Eyww*) (35)

*<sup>z</sup>* <sup>=</sup> <sup>C</sup>*<sup>x</sup>* <sup>+</sup> <sup>E</sup>*<sup>w</sup>* (36)

�

⎤

⎦ (34)

(33)

(37)

(c) Bode plot of closed loop transfer function from output disturbance to output. (d) Output voltage transient response to a step reference.

Fig. 6. Simulation results of Example 1, with controller **K**1, for the sixteen vertices of the uncertainty set.

Performance Specifications:

*Following the synthesis method shown in (Olalla et al., 2010a), the objective function to be minimized is the H*<sup>∞</sup> *norm of the transfer function between the output disturbance* ˜*iload*(*t*) *and the output voltage v*˜*o*(*t*)*.*

*In order to assure robust transient performances, the closed loop poles are constrained in an LMI region* S(*α*,*r*, *θ*)*, where the desired minimum damping ratio is set to θ* = <sup>√</sup> 1 2 *, the required maximum damped frequency is r* = <sup>1</sup> 10 2*π Ts , and the minimum decay rate, for a settling time lower than* 20 *ms, is set to α* = 200*.*

Results:

*The robust control synthesis algorithm yields a controller* **K**1*:*

$$\mathbf{K}\_1 = \begin{bmatrix} -0.31 & -0.25 & 194.70 \end{bmatrix} \tag{31}$$

*that ensures an H*∞ *norm from output disturbance to output voltage of 3.80 (11.6 dB). Figure 6 shows*

Fig. 7. Block diagram of an output-feedback system with controller *G*(*s*).

*the simulation results of the controller over the sixteen vertices of the set of matrices. Since the nonlinear terms are disregarded, the robust stability and performance of the converter is guaranteed while the converter remains in the considered operating points, assuming that the change between these operation points is sufficiently slow. In order to account for the neglected dynamics, see (Hu, 2011; Olalla et al., 2009b; 2011).*

#### **3.2 Output-feedback control**

Figure 7 shows the general diagram of an output-feedback control system. Depending on the structure of the controller *G*(*s*), two main approaches can be differentiated for the synthesis of output-feedback controllers: static and dynamic controllers.

Given the system Σ as described in (21), for the buck and the buck-boost converter *Eyu* = 0 can be considered. Then, in the case of a dynamic controller of order *k* with the following structure

$$\Sigma\_K \quad \colon \quad \begin{cases} \dot{\mathfrak{x}}\_{\mathcal{C}} = \mathbf{A}\_{\mathsf{C}} \mathfrak{x}\_{\mathcal{C}} + \mathbf{B}\_{\mathsf{C}} y \\ u = \mathbf{C}\_{\mathsf{C}} \mathfrak{x}\_{\mathcal{C}} + \mathbf{D}\_{\mathsf{C}} y \end{cases} \tag{32}$$

the closed loop system has the form

$$T\_{\Sigma\_{\mathcal{K}}} \quad : \quad \begin{cases} \dot{\mathfrak{x}}\_{cl} = \mathcal{A}\mathfrak{x}\_{cl} + \mathcal{B}w \\ z = \mathcal{C}\mathfrak{x}\_{\mathcal{C}} + \mathcal{D}w \end{cases} \tag{33}$$

where

12 Will-be-set-by-IN-TECH

−1.5

−1 −0.8 −0.6 −0.4 −0.2 0 0.2

reference.

Fig. 6. Simulation results of Example 1, with controller **K**1, for the sixteen vertices of the

*Following the synthesis method shown in (Olalla et al., 2010a), the objective function to be minimized is the H*<sup>∞</sup> *norm of the transfer function between the output disturbance* ˜*iload*(*t*) *and the output voltage*

*In order to assure robust transient performances, the closed loop poles are constrained in an LMI region*

*that ensures an H*∞ *norm from output disturbance to output voltage of 3.80 (11.6 dB). Figure 6 shows*

Amplitude

−1

−0.5

0.94

0.94

0.8

0.8

0.64

0.64

S(*α*,*r*, *θ*)

0

Imaginary Axis

0.5

1

1.5 x 10 <sup>5</sup>

−15 −10 −5 0

Real Axis

(b) Pole location map of the closed loop system

Step Response

<sup>0</sup> 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 <sup>−</sup>1.2

(d) Output voltage transient response to a step

1 2

**K**<sup>1</sup> = [−0.31 − 0.25 194.70] (31)

*Ts , and the minimum decay rate, for a settling time lower than* 20 *ms, is set to*

Time (sec)

*, the required maximum damped*

Pole−Zero Map

0.5 0.38 0.28 0.17

0.5 0.38 0.28 0.17 0.08

x 10 <sup>4</sup>

2e+004

0.08

1.5e+004

1e+004 5e+003

2e+004

1e+004 1.5e+004

5e+003

−150

−180 −90 0 90 180

uncertainty set.

*frequency is r* = <sup>1</sup>

output disturbance to output.

Performance Specifications:

10 2*π*

Phase (deg)

*v*˜*o*(*t*)*.*

*α* = 200*.* Results:

−30−20

Magnitude (dB)

reference to output

100 101 102 103 104 105

Frequency (Hz) (a) Bode plot of closed loop transfer function from

Bode Diagram

101 102 103 104

Frequency (Hz) (c) Bode plot of closed loop transfer function from

S(*α*,*r*, *θ*)*, where the desired minimum damping ratio is set to θ* = <sup>√</sup>

*The robust control synthesis algorithm yields a controller* **K**1*:*

Bode Diagram

−100

Magnitude (dB)

Phase (deg)

−50

0

$$
\begin{bmatrix}
\frac{\mathcal{A} \begin{bmatrix} \mathcal{B} \\ \mathcal{C} \end{bmatrix}}{\mathcal{C} \begin{bmatrix} \mathcal{D} \end{bmatrix}} = \begin{bmatrix}
A + B\_{\mathcal{U}} \mathbf{D}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} & B\_{\mathcal{U}} \mathbf{C}\_{\mathbf{c}} \\
\mathbf{B}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} & \mathbf{A}\_{\mathbf{c}} \\
\hline \mathbf{C}\_{\mathbf{c}} + E\_{zu} \mathbf{D}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} & E\_{zu} \mathbf{C}\_{\mathbf{c}} \end{bmatrix} \begin{bmatrix} \mathcal{B}\_{\mathbf{c}} E\_{\mathcal{Y}w} \\
\mathbf{B}\_{\mathbf{c}} E\_{\mathcal{Y}w} \\
\end{bmatrix} \tag{34}
$$

In the case of a static controller, **<sup>K</sup>** <sup>∈</sup> <sup>R</sup>*m*×*<sup>q</sup>*

$$
\mu = \mathbf{K}y = \mathbf{K}(\mathbb{C}\_{\mathcal{Y}}\mathfrak{x} + E\_{\mathcal{Y}}w) \tag{35}
$$

and the closed loop system has the following structure

$$T\_K \quad : \quad \begin{cases} \dot{\mathfrak{x}} = \mathcal{A}\mathfrak{x} + \mathcal{B}w \\ z = \mathcal{C}\mathfrak{x} + \mathcal{E}w \end{cases} \tag{36}$$

where

$$
\begin{bmatrix}
\frac{\mathcal{A} \begin{vmatrix} \mathcal{B} \end{vmatrix}}{\mathcal{C} \begin{vmatrix} \mathcal{E} \end{vmatrix}}
\end{bmatrix} = \begin{bmatrix}
\frac{(A + B\_{\mathcal{U}} \mathbf{K} \mathbf{C}\_{y})}{(\mathcal{C}\_{z} + E\_{z\mathcal{U}} \mathbf{K} \mathbf{C}\_{y}) \left| (E\_{zw} + E\_{z\mathcal{U}} \mathbf{K} E\_{yw}) \right. \end{bmatrix} \tag{37}
$$

For both problems the Lyapunov inequality is written, in a generic form:

$$\begin{cases} \mathbf{P} > \mathbf{0} \\ \mathcal{A}'\mathbf{P} + \mathbf{P}\mathcal{A} < \mathbf{0} \end{cases} \tag{38}$$

This restriction involves conservatism in the design, but it leads to a numerically tractable LMI problem, it produces controllers of reasonable order and it exploits all degrees of freedom in **P** (Scherer et al., 1997). Actually, if a single Lyapunov function **P** is considered, the following

LMI Robust Control of PWM Converters: An Output-Feedback Approach 331

Let *n* be the number of states of the plant, and let *k* be the order of the controller. Partition **P**

where **<sup>X</sup>** and **<sup>Y</sup>** are <sup>∈</sup> <sup>S</sup>*n*, and � is a symmetric positive definite matrix such that **PP**−<sup>1</sup> <sup>=</sup> <sup>1</sup>

� **X** 1 *M*� 00

**A**ˆ := *N***Ac***M*� + *N***Bc***Cy***X** + **Y***Bu***Cc***M*� + **Y**(*A* + *Bu***Dc***Cy*)**X**

where **<sup>A</sup>**<sup>ˆ</sup> , **<sup>B</sup>**<sup>ˆ</sup> , **<sup>C</sup>**<sup>ˆ</sup> have dimensions *<sup>n</sup>* <sup>×</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>×</sup> *<sup>m</sup>*, *<sup>q</sup>* <sup>×</sup> *<sup>n</sup>* respectively. If *<sup>M</sup>* and *<sup>N</sup>* have full row rank, and **A**ˆ , **B**ˆ , **C**ˆ , **D**ˆ , **X**, **Y** are given, the matrices **Ac**, **BcCc**, **Dc** can be computed. If *M* and *N*

�

�

*Cj***X** + *Ej***C**ˆ *Cj* + *Ej***D**ˆ *Cy*

<sup>1</sup>Π<sup>2</sup> =

which can be used in a congruence transformation to derive the LMI constraints. A detailed

Once the variables **A**ˆ , **B**ˆ , **C**ˆ , **D**ˆ , **X**, **Y** have been found, let us recover the original system by following this procedure. First we need to construct *M*, *N* and **P** that satisfy (45). *M* and

we assure **<sup>Y</sup>** <sup>&</sup>gt; <sup>00</sup> and **<sup>X</sup>** <sup>−</sup> **<sup>Y</sup>**−<sup>1</sup> <sup>&</sup>gt; <sup>00</sup> such that <sup>1</sup> <sup>−</sup> **YX** is nonsingular. Hence, *<sup>M</sup>* and *<sup>N</sup>* can

<sup>2</sup>B*<sup>j</sup>* =

, **P**−<sup>1</sup> =

, which leads to

�

� **X** *M M*� �

, Π<sup>2</sup> =

*A***X** + *B***C**ˆ *A* + *Bu***D***Cy* **A Y** ˆ *A* + **B**ˆ *Cy*

�

�

*Bj* + *Bu***D***Fj* **Y***Bj* + **B**ˆ *Fj*

> � **X** 1 1 **Y** �

�

� 1 **Y** 00 *N*� �

�

> 00 (48)

(44)

(45)

(46)

(47)

<sup>1</sup> can be found.

change of variable linearizes the control problem and makes it solvable with LMIs.

�

� **Y** *N N*� ∗

**P** =

**P**Π<sup>1</sup> = Π2, Π<sup>1</sup> =

� **X** *M*� � = � 1 00 �

**B**ˆ := *N***Bc** + **Y***Bu***Dc C**ˆ := **Cc***M*� + **Dc***Cy***X**

are square *n* = *k* and invertible, then **Ac**, **Bc**, **Cc**, **Dc** are unique.

<sup>1</sup>**P**AΠ<sup>1</sup> = Π�

Π�

The motivation for this change of variables lies in the following identities

<sup>1</sup>**P**B*<sup>j</sup>* = Π�

<sup>C</sup>*j*Π<sup>1</sup> <sup>=</sup> �

Π�

*<sup>N</sup>* should be chosen such that *NM*� <sup>=</sup> <sup>1</sup> <sup>−</sup> **YX**. With the following LMI:

<sup>2</sup>AΠ<sup>1</sup> =

<sup>1</sup>**P**Π<sup>1</sup> = Π�

� **X** 1 1 **Y** �

always be found. After that, Π<sup>1</sup> and Π<sup>2</sup> are also nonsingular, and **P** = Π2Π−<sup>1</sup>

and **P**−<sup>1</sup> as

holds.

From **PP**−<sup>1</sup> = 1 we infer **P**

The change of variables is as follows ⎧ ⎪⎪⎨

⎪⎪⎩

**D**ˆ := **Dc**

Π�

proof is given in (Scherer et al., 1997).

which depends non-linearly on **P** and the matrices of the controller (**K** in the static case or **Ac**, **Bc**, **Cc**, **Dc** in the dynamic case).

There exist several methods to linearize the output-feedback synthesis problem. For the dynamic output-feedback case, the results of reference (Scherer et al., 1997) are summarized. In the case of static output-feedback, the methods shown in (de Oliveira & Geromel, 1997) and (Peaucelle & Arzelier, 2001a) are employed.

### **3.2.1 Dynamic output-feedback**

The dynamic output-feedback synthesis method shown in (Scherer et al., 1997) employs the following transfer function parametrization defined from the exogenous input *w* = *wjRj* to the cost output *zj* = *Ljz* as follows:

$$T\_{\dot{j}}(s) = \frac{z\_{\dot{j}}(s)}{w\_{\dot{j}}(s)} := \begin{bmatrix} \frac{\mathcal{A}}{\mathcal{C}\_{\dot{j}}} \Big| \mathcal{B}\_{\dot{j}} \Big| \qquad \begin{matrix} \mathcal{B} \mathcal{R}\_{\dot{j}} \\ \hline L\_{\dot{j}} \mathcal{C} \Big| L\_{\dot{j}} \mathcal{D} \mathcal{R}\_{\dot{j}} \end{matrix} \Big| \qquad \begin{matrix} \begin{matrix} A + B\_{\mathcal{U}} \mathbf{D}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} & B\_{\mathcal{U}} \mathbf{C}\_{\mathbf{c}} \Big| B\_{\mathcal{J}} + B\_{\mathcal{U}} \mathbf{D}\_{\mathbf{c}} F\_{\hat{j}} \\ \hline \mathbf{B}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} & \mathbf{A}\_{\mathbf{c}} \end{matrix} \Big| \begin{matrix} B\_{\mathcal{J}} + B\_{\mathcal{U}} \mathbf{D}\_{\mathbf{c}} F\_{\hat{j}} \\ \hline \mathbf{C}\_{\mathcal{Y}} + E\_{\hat{j}} \mathbf{D}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} & E\_{\hat{j}} \mathbf{C}\_{\mathcal{C}} \Big| D\_{\mathcal{J}} + E\_{\hat{j}} \mathbf{D}\_{\mathbf{c}} F\_{\hat{j}} \end{matrix} \Big| \end{matrix} \tag{39}$$

where

$$B\_{\dot{j}} := B\_{\text{w}} R\_{\dot{j}}, \quad \mathbb{C}\_{\dot{j}} := L\_{\dot{j}} \mathbb{C}\_{\text{z}}, \quad D\_{\dot{j}} := L\_{\dot{j}} \mathbb{E}\_{\text{z} \mathbf{w}} R\_{\dot{j}}, \quad E\_{\dot{j}} := L\_{\dot{j}} \mathbb{E}\_{\text{z} \mathbf{w}}, \quad F\_{\dot{j}} := E\_{\text{y} \mathbf{w}} R\_{\dot{j}}.\tag{40}$$

To find a controller which stabilizes the closed-loop system, there must exist a quadratic Lyapunov function

$$V(\mathbf{x}\_{cl}) = \mathbf{x}\_{cl}' \mathbf{P} \mathbf{x}\_{cl\prime} \tag{41}$$

such that

$$\begin{cases} \mathbf{P} > \mathbf{0} \\ \mathcal{A}'\mathbf{P} + \mathbf{P}\mathcal{A} < \mathbf{0} \end{cases} \tag{42}$$

The LMI constraints are formulated for a transfer function *Tj*(*s*) = *LjT*(*s*)*Rj*, in terms of the state-space matrices A, B*j*, C*j*, D*j*. The goal is to synthesize an LTI controller Σ*<sup>K</sup>* that:


Generally, each transfer function *Tj* will satisfy each specification *Sj*, if there exists a Lyapunov matrix **P***<sup>j</sup>* > 0 that satisfies some LMI constraints in **P***j*. The control problem usually includes a number *i* of specifications. Therefore, the synthesis problem involves a set of matrix inequalities whose variables are:


Since this problem is nonlinear and hardly tractable numerically, the method shown in (Scherer et al., 1997) requires that all the specifications are satisfied with a single Lyapunov function, that is:

$$\mathbf{P}\_1 = \dots = \mathbf{P}\_l = \mathbf{P}.\tag{43}$$

This restriction involves conservatism in the design, but it leads to a numerically tractable LMI problem, it produces controllers of reasonable order and it exploits all degrees of freedom in **P** (Scherer et al., 1997). Actually, if a single Lyapunov function **P** is considered, the following change of variable linearizes the control problem and makes it solvable with LMIs.

Let *n* be the number of states of the plant, and let *k* be the order of the controller. Partition **P** and **P**−<sup>1</sup> as

$$\mathbf{P} = \begin{bmatrix} \mathbf{Y} \ N \\ N' \ \ast \end{bmatrix}, \quad \mathbf{P}^{-1} = \begin{bmatrix} \mathbf{X} \ M \\ M' \ \ast \end{bmatrix} \tag{44}$$

where **<sup>X</sup>** and **<sup>Y</sup>** are <sup>∈</sup> <sup>S</sup>*n*, and � is a symmetric positive definite matrix such that **PP**−<sup>1</sup> <sup>=</sup> <sup>1</sup> holds.

$$\text{From } \mathbf{P} \mathbf{P}^{-1} = \mathbf{1} \text{ we infer } \mathbf{P} \begin{pmatrix} \mathbf{X} \\ M' \end{pmatrix} = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix} \text{, which leads to }$$

$$\mathbf{PII}\_1 = \boldsymbol{\Pi}\_{2'} \quad \boldsymbol{\Pi}\_1 = \begin{bmatrix} \mathbf{X} & \mathbf{1} \\ M' \mathbf{ 0} \end{bmatrix} \quad \boldsymbol{\Pi}\_2 = \begin{bmatrix} \mathbf{1} & \mathbf{Y} \\ \mathbf{0} & N' \end{bmatrix} \tag{45}$$

The change of variables is as follows

14 Will-be-set-by-IN-TECH

which depends non-linearly on **P** and the matrices of the controller (**K** in the static case or **Ac**,

There exist several methods to linearize the output-feedback synthesis problem. For the dynamic output-feedback case, the results of reference (Scherer et al., 1997) are summarized. In the case of static output-feedback, the methods shown in (de Oliveira & Geromel, 1997) and

The dynamic output-feedback synthesis method shown in (Scherer et al., 1997) employs the following transfer function parametrization defined from the exogenous input *w* = *wjRj* to

> � = ⎡ ⎣

To find a controller which stabilizes the closed-loop system, there must exist a quadratic

The LMI constraints are formulated for a transfer function *Tj*(*s*) = *LjT*(*s*)*Rj*, in terms of the

• meets certain specifications (*H*2, *H*<sup>∞</sup> , pole placement,...) on a particular set of channels. Generally, each transfer function *Tj* will satisfy each specification *Sj*, if there exists a Lyapunov matrix **P***<sup>j</sup>* > 0 that satisfies some LMI constraints in **P***j*. The control problem usually includes a number *i* of specifications. Therefore, the synthesis problem involves a set of matrix

Since this problem is nonlinear and hardly tractable numerically, the method shown in (Scherer et al., 1997) requires that all the specifications are satisfied with a single Lyapunov

*V*(*xcl*) = *x*�

� **P** > 00 A�

state-space matrices A, B*j*, C*j*, D*j*. The goal is to synthesize an LTI controller Σ*<sup>K</sup>* that:

• additional auxiliary variables to minimize, for example, the norm cost *H*∞.

*Bj* := *BwRj*, *Cj* := *LjCz*, *Dj* := *LjEzwRj*, *Ej* := *LjEzu*, *Fj* := *EywRj*. (40)

**<sup>P</sup>** <sup>+</sup> **<sup>P</sup>**<sup>A</sup> <sup>&</sup>lt; <sup>00</sup> , (38)

*A* + *Bu***Dc***Cy Bu***Cc** *Bj* + *Bu***Dc***Fj* **Bc***Cy* **Ac Bc***Fj Cj* + *Ej***Dc***Cy Ej***Cc** *Dj* + *Ej***Dc***Fj*

*cl***P***xcl*, (41)

**<sup>P</sup>** <sup>+</sup> **<sup>P</sup>**<sup>A</sup> <sup>&</sup>lt; <sup>00</sup> (42)

**P**<sup>1</sup> = ... = **P***<sup>i</sup>* = **P**. (43)

⎤

⎦ (39)

For both problems the Lyapunov inequality is written, in a generic form:

**Bc**, **Cc**, **Dc** in the dynamic case).

**3.2.1 Dynamic output-feedback**

*zj*(*s*) *wj*(*s*) :<sup>=</sup>

*Tj*(*s*) =

Lyapunov function

where

such that

the cost output *zj* = *Ljz* as follows:

• internally stabilizes the system

inequalities whose variables are:

function, that is:

• the controller matrices **Ac**, **Bc**,**Cc**, **Dc**.

• the *i* Lyapunov matrices **P**1,..., **P***i*, one per specification.

(Peaucelle & Arzelier, 2001a) are employed.

� <sup>A</sup> <sup>B</sup>*<sup>j</sup>* C*j* D*j* � = � <sup>A</sup> <sup>B</sup>*Rj Lj*C *Lj*D*Rj*

� **P** > 00 A�

$$\begin{cases} \dot{\mathbf{A}} := N\mathbf{A}\_{\mathbf{c}}M' + N\mathbf{B}\_{\mathbf{c}}\mathbf{C}\_{\mathbf{y}}\mathbf{X} + \mathbf{Y}B\_{\mu}\mathbf{C}\_{\mathbf{c}}M' + \mathbf{Y}(A + B\_{\mu}\mathbf{D}\_{\mathbf{c}}\mathbf{C}\_{\mathbf{y}})\mathbf{X} \\ \dot{\mathbf{B}} := N\mathbf{B}\_{\mathbf{c}} + \mathbf{Y}B\_{\mu}\mathbf{D}\_{\mathbf{c}} \\ \dot{\mathbf{C}} := \mathbf{C}\_{\mathbf{c}}M' + \mathbf{D}\_{\mathbf{c}}\mathbf{C}\_{\mathbf{y}}\mathbf{X} \\ \dot{\mathbf{D}} := \mathbf{D}\_{\mathbf{c}} \end{cases} \tag{46}$$

where **<sup>A</sup>**<sup>ˆ</sup> , **<sup>B</sup>**<sup>ˆ</sup> , **<sup>C</sup>**<sup>ˆ</sup> have dimensions *<sup>n</sup>* <sup>×</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>×</sup> *<sup>m</sup>*, *<sup>q</sup>* <sup>×</sup> *<sup>n</sup>* respectively. If *<sup>M</sup>* and *<sup>N</sup>* have full row rank, and **A**ˆ , **B**ˆ , **C**ˆ , **D**ˆ , **X**, **Y** are given, the matrices **Ac**, **BcCc**, **Dc** can be computed. If *M* and *N* are square *n* = *k* and invertible, then **Ac**, **Bc**, **Cc**, **Dc** are unique.

The motivation for this change of variables lies in the following identities

$$\begin{aligned} \Pi\_1' \mathbf{P} \mathcal{A} \Pi\_1 &= \Pi\_2' \mathcal{A} \Pi\_1 = \begin{bmatrix} A \mathbf{X} + B \hat{\mathbf{C}} \ \mathbf{A} + B\_u \mathbf{D} \mathbf{C}\_y \\ \hat{\mathbf{A}} & \mathbf{Y} \mathbf{A} + \hat{\mathbf{B}} \mathbf{C}\_y \end{bmatrix} \\ \Pi\_1' \mathbf{P} \mathcal{B}\_j &= \Pi\_2' \mathcal{B}\_j = \begin{bmatrix} B\_j + B\_u \mathbf{D} \mathbf{f}\_j \\ \mathbf{Y} \mathbf{B}\_j + \hat{\mathbf{B}} \mathbf{f}\_j \end{bmatrix} \\ \mathcal{C}\_j \Pi\_1 &= \begin{bmatrix} \mathbf{C}\_j \mathbf{X} + E\_j \hat{\mathbf{C}} \ \mathbf{C}\_j + E\_j \hat{\mathbf{D}} \mathbf{C}\_y \end{bmatrix} \\ \Pi\_1' \mathbf{P} \Pi\_1 &= \Pi\_1' \Pi\_2 = \begin{bmatrix} \mathbf{X} \mathbf{1} \\ \mathbf{1} \ \mathbf{Y} \end{bmatrix} \end{aligned} \tag{47}$$

which can be used in a congruence transformation to derive the LMI constraints. A detailed proof is given in (Scherer et al., 1997).

Once the variables **A**ˆ , **B**ˆ , **C**ˆ , **D**ˆ , **X**, **Y** have been found, let us recover the original system by following this procedure. First we need to construct *M*, *N* and **P** that satisfy (45). *M* and *<sup>N</sup>* should be chosen such that *NM*� <sup>=</sup> <sup>1</sup> <sup>−</sup> **YX**. With the following LMI:

$$
\begin{bmatrix} \mathbf{X} \ \mathbf{1} \\ \mathbf{1} \ \mathbf{Y} \end{bmatrix} > \mathbf{0} \tag{48}
$$

we assure **<sup>Y</sup>** <sup>&</sup>gt; <sup>00</sup> and **<sup>X</sup>** <sup>−</sup> **<sup>Y</sup>**−<sup>1</sup> <sup>&</sup>gt; <sup>00</sup> such that <sup>1</sup> <sup>−</sup> **YX** is nonsingular. Hence, *<sup>M</sup>* and *<sup>N</sup>* can always be found. After that, Π<sup>1</sup> and Π<sup>2</sup> are also nonsingular, and **P** = Π2Π−<sup>1</sup> <sup>1</sup> can be found. Then **Dc**, **Cc**, **Bc** and **Ac** can be solved, in this order:

$$\begin{cases} \mathbf{D}\_{\mathbf{c}} := \mathbf{\hat{D}} \\ \mathbf{C}\_{\mathbf{c}} := \left( \hat{\mathbf{C}} - \mathbf{D}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} \mathbf{X} \right) M^{\prime - 1} \\ \mathbf{B}\_{\mathbf{c}} := N^{-1} \left( \hat{\mathbf{B}} - \mathbf{Y} \mathbf{B}\_{\mathcal{U}} \mathbf{D}\_{\mathbf{c}} \right) \\ \mathbf{A}\_{\mathbf{c}} := N^{-1} \left( \hat{\mathbf{A}} - N \mathbf{B}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} \mathbf{X} - \mathbf{Y} \mathbf{B}\_{\mathcal{U}} \mathbf{C}\_{\mathbf{c}} M^{\prime} - \mathbf{Y} \left( A + B\_{\mathcal{U}} \mathbf{D}\_{\mathbf{c}} \mathbf{C}\_{\mathcal{Y}} \right) \mathbf{X} \right) M^{\prime - 1} \end{cases} \tag{49}$$

*The algorithm yields a controller of the same order as the augmented plant, that is, the order of the*

LMI Robust Control of PWM Converters: An Output-Feedback Approach 333

*The objective of the design procedure is to minimize the H*∞ *norm of the disturbance to output transfer function. For such objective, a weighting function W*1(*s*) *for the error signal (i.e. for the sensitivity function) and a weighting function W*2(*s*) *for the output signal (i.e. for the complementary sensitivity function) are considered. Both weighting functions are depicted in Figure 8(b). In order to obtain small*

*Other performance requirements could have been included (pole placement, H*2*, ...) in the optimization problem, but they have not been used to maintain all the degrees of freedom in the research of the*

*The minimization algorithm yields the following controller transfer function (1 input, 1 output):*

**<sup>K</sup>**2(*s*) = <sup>−</sup>3.00 (*<sup>s</sup>* <sup>+</sup> *<sup>z</sup>*1)(*<sup>s</sup>* <sup>+</sup> *<sup>z</sup>*2)(*<sup>s</sup>* <sup>+</sup> *<sup>z</sup>*3,4)(*<sup>s</sup>* <sup>+</sup> *<sup>z</sup>*5,6)

*<sup>p</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>1.05 · <sup>10</sup>−<sup>2</sup> *<sup>z</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>4.39 · 102 <sup>+</sup> *<sup>j</sup>*6.04 · 102 *<sup>p</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>4.90 · 103 *<sup>z</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>4.39 · 102 <sup>−</sup> *<sup>j</sup>*6.04 · 102

*The maximum guaranteed gain peak from disturbance to output is γ* = 0.045 *(-26.93 dB). Figure 8 depicts the simulation results for the nominal frequency and time-domain response of the buck*

An alternative to the use of weighting functions and frequency dependent uncertainty models is to consider the static output-feedback case. Static output-feedback considers a gain **K** to set

The survey on output-feedback design methods (de Oliveira & Geromel, 1997) differentiates

1. Nonlinear programming methods. They work on the parametric space defined by **K** and **P** to find an optimal value of a cost variable, if any. The search is done by means of classical optimization methods as, for example, a gradient algorithm, primal or dual Levine-Athans' method, etc. The solution of the algorithm, which converges to a local optimum, strongly depends on an initial stabilizing gain, which must be found

2. Parametric optimization methods. These methods optimize the objective function for the parametric space defined by **P**, for some matrix **K**. The determination of the controller, if it exists, is decomposed in independent steps. These methods can be easily implemented

3. Convex programming methods. They solve a sufficient version of the Lyapunov inequality (38) obtained by the addition of constraints which lead to a convex feasibility set.

between several approaches to solve the synthesis of a static output gain as follows:

*<sup>p</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>6.66 · 104 *<sup>z</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>1.89 · 103 *<sup>p</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>1.36 · 109 *<sup>z</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>3.50 · 109

(*<sup>s</sup>* <sup>+</sup> *<sup>p</sup>*1)(*<sup>s</sup>* <sup>+</sup> *<sup>p</sup>*2)(*<sup>s</sup>* <sup>+</sup> *<sup>p</sup>*3,4)(*<sup>s</sup>* <sup>+</sup> *<sup>p</sup>*5,6) (50)

(51)

*original plant plus the order of the shaping filters.*

*steady-state error, W*1(*s*) *is very large at low frequencies.*

Performance Specifications:

*minimum H*∞ *norm.*

Results:

*where*

*converter.*

**3.2.2 Static output-feedback**

beforehand.

using LMI solvers.

up the feedback loop as *u* = **K***y*.

For a list of LMI constraints which respond to several specifications with this change of variables, it is recommended to read (Scherer et al., 1997).

### **Example 2.** *Buck Converter*

*In this example, the synthesis of an output-feedback controller for a buck converter is carried out. The objective of the synthesis algorithm is, again, to minimize the H*∞ *norm of the output disturbance to output voltage transfer function.*

*In this case the stray resistances of the converter are taken into account, since only the output signals are used. This design considers a unique output signal vo*(*t*) *to set-up a voltage-regulation operation.*

### Sampling effect:

*The sampling effect could be included in the converter model, in order to prevent the optimization algorithm to yield unrealistic results due to the switching action. However, in this case, a weighting function on the complementary sensitivity response can be used for this purpose.*

### Uncertainty:

*Polytopic uncertainty (19) can be introduced in the model of the converter to cope with the variations of the uncertain parameters, as the load or the input voltage. However, in the case of output-feedback the polytopic representation of uncertainty introduces nonlinear relationship between the variables of the inequalities. This problem is treated in (Courties, 1997; 1999) where a cross-decomposition algorithm is described to obtain a local optimum controller giving an initial feasible solution. The solution proposed in this example exploits the weighting transfer functions to obtain the expected sensitivity and complementary sensitivity responses. The parameters of this example take the values shown in Table 2. The synthesis algorithm closely follows the linearizing change of variables of (Scherer et al.,*


Table 2. Buck: converter parameters

*1997) and the methodology explained in chapters 5 and 6 of (Gahinet et al., 1995):*


*The algorithm yields a controller of the same order as the augmented plant, that is, the order of the original plant plus the order of the shaping filters.*

### Performance Specifications:

*The objective of the design procedure is to minimize the H*∞ *norm of the disturbance to output transfer function. For such objective, a weighting function W*1(*s*) *for the error signal (i.e. for the sensitivity function) and a weighting function W*2(*s*) *for the output signal (i.e. for the complementary sensitivity function) are considered. Both weighting functions are depicted in Figure 8(b). In order to obtain small steady-state error, W*1(*s*) *is very large at low frequencies.*

*Other performance requirements could have been included (pole placement, H*2*, ...) in the optimization problem, but they have not been used to maintain all the degrees of freedom in the research of the minimum H*∞ *norm.*

### Results:

16 Will-be-set-by-IN-TECH

Then **Dc**, **Cc**, **Bc** and **Ac** can be solved, in this order:

**<sup>C</sup>**<sup>ˆ</sup> <sup>−</sup> **Dc***Cy***<sup>X</sup>**

variables, it is recommended to read (Scherer et al., 1997).

� *M*�−<sup>1</sup>

�

*function on the complementary sensitivity response can be used for this purpose.*

*1997) and the methodology explained in chapters 5 and 6 of (Gahinet et al., 1995):*

*2. Then, the original plant is augmented with such filters to obtain a weighted plant.*

*1. First, the design specifications are expressed in terms of loop shapes and their corresponding shaping*

*3. Finally, the augmented plant is used in the optimization algorithm to derive a controller that meets*

**<sup>A</sup>**<sup>ˆ</sup> <sup>−</sup> *<sup>N</sup>***Bc***Cy***<sup>X</sup>** <sup>−</sup> **<sup>Y</sup>***Bu***Cc***M*� <sup>−</sup> **<sup>Y</sup>** �

For a list of LMI constraints which respond to several specifications with this change of

*In this example, the synthesis of an output-feedback controller for a buck converter is carried out. The objective of the synthesis algorithm is, again, to minimize the H*∞ *norm of the output disturbance to*

*In this case the stray resistances of the converter are taken into account, since only the output signals are used. This design considers a unique output signal vo*(*t*) *to set-up a voltage-regulation operation.*

*The sampling effect could be included in the converter model, in order to prevent the optimization algorithm to yield unrealistic results due to the switching action. However, in this case, a weighting*

*Polytopic uncertainty (19) can be introduced in the model of the converter to cope with the variations of the uncertain parameters, as the load or the input voltage. However, in the case of output-feedback the polytopic representation of uncertainty introduces nonlinear relationship between the variables of the inequalities. This problem is treated in (Courties, 1997; 1999) where a cross-decomposition algorithm is described to obtain a local optimum controller giving an initial feasible solution. The solution proposed in this example exploits the weighting transfer functions to obtain the expected sensitivity and complementary sensitivity responses. The parameters of this example take the values shown in Table 2. The synthesis algorithm closely follows the linearizing change of variables of (Scherer et al.,* Parameter Value *R* 1000 Ω *Vg* 55 *V C* 1000 *μ* F *L* 100 *μ* H *req* 150 *m*Ω *rC* 30 *m*Ω *Ts* 5 *μ* s

*A* + *Bu***Dc***Cy*

� **X** � *M*�−<sup>1</sup> (49)

**<sup>B</sup>**<sup>ˆ</sup> <sup>−</sup> **<sup>Y</sup>***Bu***Dc**

⎧ ⎪⎪⎨ **Dc** := **D**ˆ **Cc** := �

**Bc** := *N*−<sup>1</sup> �

**Ac** := *N*−<sup>1</sup> �

⎪⎪⎩

**Example 2.** *Buck Converter*

*output voltage transfer function.*

Table 2. Buck: converter parameters

Sampling effect:

Uncertainty:

*filters.*

*certain LMIs.*

*The minimization algorithm yields the following controller transfer function (1 input, 1 output):*

$$\mathbf{K}\_2(s) = -3.00 \frac{(s+z\_1)(s+z\_2)(s+z\_{3,4})(s+z\_{5,6})}{(s+p\_1)(s+p\_2)(s+p\_{3,4})(s+p\_{5,6})} \tag{50}$$

*where*

$$\begin{array}{l|l}p\_1 = -1.05 \cdot 10^{-2} & z\_1 = -4.39 \cdot 10^2 + j6.04 \cdot 10^2 \\ p\_2 = -4.90 \cdot 10^3 & z\_2 = -4.39 \cdot 10^2 - j6.04 \cdot 10^2 \\ p\_3 = -6.66 \cdot 10^4 & z\_3 = -1.89 \cdot 10^3 \\ p\_4 = -1.36 \cdot 10^9 & z\_4 = -3.50 \cdot 10^9 \\ \end{array} \tag{51}$$

*The maximum guaranteed gain peak from disturbance to output is γ* = 0.045 *(-26.93 dB). Figure 8 depicts the simulation results for the nominal frequency and time-domain response of the buck converter.*

#### **3.2.2 Static output-feedback**

An alternative to the use of weighting functions and frequency dependent uncertainty models is to consider the static output-feedback case. Static output-feedback considers a gain **K** to set up the feedback loop as *u* = **K***y*.

The survey on output-feedback design methods (de Oliveira & Geromel, 1997) differentiates between several approaches to solve the synthesis of a static output gain as follows:


*Applying the elimination lemma with*

 *A*�

*B*�

*we obtain*

*With* **G** =

**Algorithm 3.1.**

Uncertainty:

*are shown in Table 3.*

*and return to step 2.* **Example 3.** *Buck Converter*

*between the output and the reference.*

 **F***<sup>s</sup>* −**F** 

> *A*�

*With the following change of variables*

and *H*∞ (Peaucelle & Arzelier, 2001a;b).

*Q* = *A*�

**P** + **P***A* **P***B*<sup>2</sup>

<sup>2</sup>**P** 00

*, the previous inequality is written as*

 + *C*� *y***K**� 1 **F**� *<sup>s</sup>* −**F**� + **F***s* −**F** 

**K**�

process, a stabilizing gain that satisfies a given cost function can be obtained.

*3. Iterative step k second part. Solve the LMI* (52) *in which* **F** *and* **R** *are constant.*

*it is verified that* **K** = **F**−1**R** = **F**−1**FK***, and we obtain the result of (52).*

*1. Initialization. Step k* = 1*. Choose a stabilizing state-feedback gain. 2. Iterative step k first part. Solve the LMI* (52) *in which* **K***s is constant.*

*B*�

**P** + **P***A* **P***B*<sup>2</sup>

<sup>2</sup>**P** 00

**P** + **P***A* **P***B*<sup>2</sup>

LMI Robust Control of PWM Converters: An Output-Feedback Approach 335

**G**� + **G**

**<sup>K</sup>***Cy* <sup>−</sup><sup>1</sup>

**<sup>K</sup>***Cy* <sup>−</sup><sup>1</sup>

*<sup>s</sup>* = **<sup>F</sup>***s***F**<sup>−</sup>1, **<sup>R</sup>** = **FK**, (57)

*N* = **K***Cy* (54)

< 00 (55)

< 00 (56)

<sup>2</sup>**P** 00

This parametrization has been adapted to the minimization of the performance indexes *H*<sup>2</sup>

With the new introduced variables we can split the optimization problem into two linear steps; in the first step, **K***s* is kept constant, and the problem is solved for **P**, **R** and **F**. Then, in the next step, **R** and **F** are kept constant and the problem is solved for **P** and **K***s*. With this iterative

A key point of this approach is however related to the initialization step, for which and admissible stabilizing state-feedback has to be selected. On the other hand, this algorithm presents the advantage that the Lyapunov matrix **P** is set as a free variable in both steps.

*4. Final step. If the objective function satisfies the requirements, then stop* **K** = **F**−1**R***, else k = k + 1*

*In this example, the previous output-feedback parametrization is employed to derive a static controller for a buck converter, whose model was introduced in Section 2.2. The stray resistances of the model are taken into account in this case. Also, note that the augmented model contains an integrator of the error*

*The output-feedback parametrization allows to consider the polytopic uncertainty model shown in Section 2.4. In this case, the uncertain model is formed by N* = 22 *vertices. The values of the converter*

*B*�

 + *C*� *y***K**� 1 

(a) Bode plot of closed loop transfer function from reference to output. (b) Weighting functions for dynamic output-feedback synthesis.

(c) Bode plot of closed loop transfer function from output disturbance to output. (d) Output voltage transient response to a step reference.

Fig. 8. Simulation results of Example 2 with controller **K**2(*s*).

Frequency (Hz)

The proposed parametrization is based on the elimination lemma (Boyd et al., 1994) and the introduction of additional variables to obtain an iterative algorithm. It has been extracted from (Peaucelle & Arzelier, 2001a).

**Theorem 3.3.** *The Lyapunov inequality (38) can be rewritten as follows*

$$\begin{cases} \mathbf{P} > \mathbf{0} \\ \begin{bmatrix} A'\mathbf{P} + \mathbf{P}A \ \mathbf{P}B\_2 \\ B\_2'\mathbf{P} & \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{K}\_s' \\ \mathbf{1} \end{bmatrix} \begin{bmatrix} \mathbf{R}\mathbf{C}\_{\mathcal{Y}} - \mathbf{F} \end{bmatrix} + \begin{bmatrix} \mathbf{C}\_{\mathcal{Y}}'\mathbf{R}' \\ -\mathbf{F}' \end{bmatrix} \begin{bmatrix} \mathbf{K}\_s - \mathbf{1} \end{bmatrix} < \mathbf{0} \end{cases} \tag{52}$$

*where* **K***s is a state-feedback gain that stabilizes the system. At the optimum point, which depends on the objective function, the output-feedback controller is given by* **K** = **F**−1**R***.*

**Proof.** *The equation* (38) *can be written as the following product of matrices:*

$$
\mathbb{E}\left[\mathbf{1}\,\mathbf{C}\_{\mathcal{Y}}^{\prime}\mathbf{K}^{\prime}\right]\begin{bmatrix}A^{\prime}\mathbf{P}+\mathbf{P}A\,\mathbf{P}B\_{2}\\ \mathbf{B}\_{2}^{\prime}\mathbf{P}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{1}\\ \mathbf{K}\mathbf{C}\_{\mathcal{Y}}\end{bmatrix}<\mathbf{0}\tag{53}
$$

*Applying the elimination lemma with*

$$\mathbf{Q} = \begin{bmatrix} A'\mathbf{P} + \mathbf{P}A \mathbf{P}B\_2 \\ B\_2'\mathbf{P} & \mathbf{0} \end{bmatrix} \\ N = \mathbf{K}\mathbf{C}\_{\mathcal{Y}} \tag{54}$$

*we obtain*

18 Will-be-set-by-IN-TECH

Amplitude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

reference.

The proposed parametrization is based on the elimination lemma (Boyd et al., 1994) and the introduction of additional variables to obtain an iterative algorithm. It has been extracted

> **R***Cy* −**F** � + � *C*� *y***R**� −**F**�

*where* **K***s is a state-feedback gain that stabilizes the system. At the optimum point, which depends on*

**P** + **P***A* **P***B*<sup>2</sup>

� � 1 **K***Cy* �

<sup>2</sup>**P** 00

*B*�

Magnitude (dB)

<sup>10</sup>−4 <sup>10</sup>−2 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>2</sup> <sup>10</sup><sup>4</sup> −60

output-feedback synthesis.

(b) Weighting functions for dynamic

Frequency (Hz)

Step Response

Time (sec)

(d) Output voltage transient response to a step

� �

**<sup>K</sup>***<sup>s</sup>* <sup>−</sup><sup>1</sup> �

0 1 2 3 4 5 6

Bode Diagram

*W*2(*s*)

x 10<sup>−</sup><sup>3</sup>

<sup>&</sup>lt; <sup>00</sup> (52)

< 00 (53)

*W*1(*s*)

Bode Diagram

−20 −15 −10 −5 0

−90 −45 0 45

> −70 −60 −50 −40 −30 −20

Magnitude (dB)

Phase (deg)

reference to output.

Magnitude (dB)

Phase (deg)

Frequency (Hz)

Bode Diagram

Frequency (Hz)

Fig. 8. Simulation results of Example 2 with controller **K**2(*s*).

**Theorem 3.3.** *The Lyapunov inequality (38) can be rewritten as follows*

� + � **K**� *s* 1 � �

*the objective function, the output-feedback controller is given by* **K** = **F**−1**R***.* **Proof.** *The equation* (38) *can be written as the following product of matrices:*

**P** + **P***A* **P***B*<sup>2</sup>

<sup>2</sup>**P** 00

� 1 *C*� *y***K**� � � *A*�

(c) Bode plot of closed loop transfer function from

output disturbance to output.

from (Peaucelle & Arzelier, 2001a).

**P** > 00 � *A*�

*B*�

⎧ ⎨ ⎩

100 101 102 103 104 105

(a) Bode plot of closed loop transfer function from

100 101 102 103 104 105

$$
\begin{bmatrix}
\mathbf{A}^\prime \mathbf{P} + \mathbf{P} \mathbf{A} \mathbf{P} \mathbf{B}\_2 \\
\mathbf{B}\_2^\prime \mathbf{P} & \mathbf{0}
\end{bmatrix} + \begin{bmatrix}
\mathbf{C}\_y^\prime \mathbf{K}^\prime \\
\mathbf{1}
\end{bmatrix} \mathbf{G}^\prime + \mathbf{G} \begin{bmatrix}
\mathbf{K} \mathbf{C}\_y - \mathbf{1}
\end{bmatrix} < \mathbf{0} \tag{55}
$$

*With* **G** = **F***<sup>s</sup>* −**F** *, the previous inequality is written as*

$$
\begin{bmatrix} A'\mathbf{P} + \mathbf{P}A \ \mathbf{P}B\_2 \\ B\_2'\mathbf{P} & \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{C}\_y'\mathbf{K}' \\ \mathbf{1} \end{bmatrix} \begin{bmatrix} \mathbf{F}\_s' - \mathbf{F}' \end{bmatrix} + \begin{bmatrix} \mathbf{F}\_s \\ -\mathbf{F} \end{bmatrix} \begin{bmatrix} \mathbf{K}\mathbf{C}\_y - \mathbf{1} \end{bmatrix} < \mathbf{0} \tag{56}
$$

*With the following change of variables*

$$\mathbf{K}'\_{\mathbf{s}} = \mathbf{F}\_{\mathbf{s}} \mathbf{F}^{-1}, \qquad \mathbf{R} = \mathbf{F} \mathbf{K}, \tag{57}$$

*it is verified that* **K** = **F**−1**R** = **F**−1**FK***, and we obtain the result of (52).*

This parametrization has been adapted to the minimization of the performance indexes *H*<sup>2</sup> and *H*∞ (Peaucelle & Arzelier, 2001a;b).

With the new introduced variables we can split the optimization problem into two linear steps; in the first step, **K***s* is kept constant, and the problem is solved for **P**, **R** and **F**. Then, in the next step, **R** and **F** are kept constant and the problem is solved for **P** and **K***s*. With this iterative process, a stabilizing gain that satisfies a given cost function can be obtained.

A key point of this approach is however related to the initialization step, for which and admissible stabilizing state-feedback has to be selected. On the other hand, this algorithm presents the advantage that the Lyapunov matrix **P** is set as a free variable in both steps.

### **Algorithm 3.1.**


#### **Example 3.** *Buck Converter*

*In this example, the previous output-feedback parametrization is employed to derive a static controller for a buck converter, whose model was introduced in Section 2.2. The stray resistances of the model are taken into account in this case. Also, note that the augmented model contains an integrator of the error between the output and the reference.*

#### Uncertainty:

*The output-feedback parametrization allows to consider the polytopic uncertainty model shown in Section 2.4. In this case, the uncertain model is formed by N* = 22 *vertices. The values of the converter are shown in Table 3.*


Table 3. Buck: converter parameters

Sampling effect:

*Since only the output signal is used, the sampling effect is included in the converter model in order to obtain an accurate model up to half the switching frequency. This addition also prevents the optimization algorithms to yield unrealistic results due to the switching action.*

Performance Specifications:

*Following the synthesis method shown in (Olalla et al., 2010a), the objective function to be minimized is the H*<sup>∞</sup> *norm of the transfer function between the output disturbance* ˜ *iload*(*t*) *and the output voltage. As in the state-feedback case, the closed loop poles are constrained in an LMI region* S(*α*,*r*, *θ*)*, in order to assure robust transient performances. Again, the desired minimum damping ratio is set to θ* = √ 1 *,*

2 *the required maximum damped frequency is r* = <sup>1</sup> 10 2*π Ts , and the minimum decay rate, for a settling time lower than* 40 *ms, is set to α* = 100*.*

Results:

*The robust control synthesis algorithm yields a controller* **K**3*:*

$$\mathbf{K}\_3 = 4.472\tag{58}$$

−150

−270 −225 −180 −135 −90 −45 0

reference to output.

100 101 102 103 104 105

−50 −40 −30 −20 −10 0

output disturbance to output.

*yaug* =

 *xc y* 

(*<sup>s</sup>* <sup>+</sup> *<sup>p</sup>*) (62)

(60)

(61)

100 101 102 103 104 105

Frequency (Hz) (b) Bode plot of closed loop transfer function from

Bode Diagram

Magnitude (dB)

LMI Robust Control of PWM Converters: An Output-Feedback Approach 337

Phase (deg)

Step Response

<sup>0</sup> 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 <sup>0</sup>

Time (sec)

 *x*˙*c u* 

*A dynamic controller using the static output-feedback parametrization is derived in this subsection. As in the static case, the stray resistances and a pure integrator are considered. The system is augmented with an integrator and with the controller states to obtain a dynamic controller. For this case a simple*

(*s* + *z*)

(c) Output voltage transient response to a step

*uaug* =

**K** = **Ac Bc Cc Dc**

**K**(*s*) = *k*

Frequency (Hz) (a) Bode plot of closed loop transfer function from

> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

reference.

Fig. 9. Simulation results of Example 3 with controller **K**3.

 *x xc* 

*xaug* =

Amplitude

Bode Diagram

−100

Magnitude (dB)

Phase (deg)

*where*

*and*

**Example 4.** *Buck Converter*

*first-order controller is considered.*

−50

0

*that assures an H*∞ *norm from output disturbance to output voltage of 0.656 (-3.66 dB). The waveforms of a numerical simulation and corresponding Bode plots of the converter with controller* **K**<sup>3</sup> *over the four vertices of the set of matrices are depicted in Figure 9. It can be observed that the static output-feedback controller does not achieve the performance of the dynamic output-feedback controller* **K**2(*s*)*.*

### **3.2.3 Dynamic output-feedback with static parametrization**

The parametrization method shown in the previous subsection for the static case can be adapted to the synthesis of a dynamic output-feedback controller. The following theorem can be found in (Martenson, 1985; Nett et al., 1989).

**Theorem 3.4.** *The synthesis of a dynamic controller* Σ*<sup>K</sup> for the system* Σ *(for simplicity, in absence of perturbations (w* = 0*) ) can be expressed as the synthesis of a static controller for the augmented system* Σ*aug:*

$$\begin{array}{rcl} \Sigma\_{aug} &:& \begin{cases} \dot{\mathbf{x}}\_{aug} = \begin{bmatrix} A & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \mathbf{x}\_{aug} + \begin{bmatrix} \mathbf{0} & B \\ \mathbf{1} & \mathbf{0} \end{bmatrix} u\_{aug} \\\\ \mathbf{y}\_{aug} = \begin{bmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{C} & \mathbf{0} \end{bmatrix} \mathbf{x}\_{aug} + \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & D \end{bmatrix} u\_{aug} \end{array} \end{array} \tag{59}$$

(a) Bode plot of closed loop transfer function from reference to output. (b) Bode plot of closed loop transfer function from output disturbance to output.

(c) Output voltage transient response to a step reference.

Fig. 9. Simulation results of Example 3 with controller **K**3.

*where*

20 Will-be-set-by-IN-TECH

*Since only the output signal is used, the sampling effect is included in the converter model in order to obtain an accurate model up to half the switching frequency. This addition also prevents the*

*Following the synthesis method shown in (Olalla et al., 2010a), the objective function to be minimized*

*As in the state-feedback case, the closed loop poles are constrained in an LMI region* S(*α*,*r*, *θ*)*, in order to assure robust transient performances. Again, the desired minimum damping ratio is set to θ* = √

> 10 2*π*

*that assures an H*∞ *norm from output disturbance to output voltage of 0.656 (-3.66 dB). The waveforms of a numerical simulation and corresponding Bode plots of the converter with controller* **K**<sup>3</sup> *over the four vertices of the set of matrices are depicted in Figure 9. It can be observed that the static output-feedback*

The parametrization method shown in the previous subsection for the static case can be adapted to the synthesis of a dynamic output-feedback controller. The following theorem

**Theorem 3.4.** *The synthesis of a dynamic controller* Σ*<sup>K</sup> for the system* Σ *(for simplicity, in absence of perturbations (w* = 0*) ) can be expressed as the synthesis of a static controller for the augmented*

�

�

*xaug* +

*xaug* +

� 00 *B* 1 00

� 00 00 00 *D* � *uaug*

� *uaug*

� *A* 00 00 00

� 00 1 *C* 00

*controller does not achieve the performance of the dynamic output-feedback controller* **K**2(*s*)*.*

*iload*(*t*) *and the output voltage.*

*Ts , and the minimum decay rate, for a settling time*

**K**<sup>3</sup> = 4.472 (58)

1 2 *,*

(59)

*optimization algorithms to yield unrealistic results due to the switching action.*

*is the H*<sup>∞</sup> *norm of the transfer function between the output disturbance* ˜

*R* ∈ [10, 1000] Ω *Vg* ∈ [33, 55] *V C* 1000 *μ* F *L* 100 *μ* H *req* 150 mΩ *rc* 50 mΩ *Ts* 5 *μ* s

Parameter Value

Table 3. Buck: converter parameters

*the required maximum damped frequency is r* = <sup>1</sup>

*The robust control synthesis algorithm yields a controller* **K**3*:*

**3.2.3 Dynamic output-feedback with static parametrization**

⎧ ⎪⎪⎪⎪⎨

*x*˙*aug* =

*yaug* =

⎪⎪⎪⎪⎩

can be found in (Martenson, 1985; Nett et al., 1989).

Σ*aug* :

Performance Specifications:

*lower than* 40 *ms, is set to α* = 100*.*

Sampling effect:

Results:

*system* Σ*aug:*

$$\mathbf{x}\_{\text{aug}} = \begin{bmatrix} \mathbf{x} \\ \mathbf{x}\_{\text{c}} \end{bmatrix} \qquad \mathbf{u}\_{\text{aug}} = \begin{bmatrix} \dot{\mathbf{x}}\_{\text{c}} \\ \mathbf{u} \end{bmatrix} \qquad y\_{\text{aug}} = \begin{bmatrix} \mathbf{x}\_{\text{c}} \\ \mathbf{y} \end{bmatrix} \tag{60}$$

*and*

$$\mathbf{K} = \begin{bmatrix} \mathbf{A\_{c}} \ \mathbf{B\_{c}} \\ \mathbf{C\_{c}} \ \mathbf{D\_{c}} \end{bmatrix} \tag{61}$$

### **Example 4.** *Buck Converter*

*A dynamic controller using the static output-feedback parametrization is derived in this subsection. As in the static case, the stray resistances and a pure integrator are considered. The system is augmented with an integrator and with the controller states to obtain a dynamic controller. For this case a simple first-order controller is considered.*

$$\mathbf{K}(s) = k \frac{(s+z)}{(s+p)} \tag{62}$$

### Uncertainty:

*As in the previous subsection, the synthesis considers a polytopic model. A new LMI is introduced for every vertex of the uncertain model. The polytopic model used in this subsection considers the same uncertain load R and input voltage Vg, as shown in Table 4.*


Table 4. Buck: converter parameters

### Sampling effect:

*Again, the sampling effect is included in the converter model in order to obtain an accurate model up to half the switching frequency.*

#### Performance Specifications:

*As in the previous case, the objective is to minimize the H*∞ *norm of the transfer function between the disturbance and the output. Also, the closed loop poles are constrained in the same LMI region* S(*α*,*r*, *θ*)*, in order to assure robust transient performances.*

*The initialization step is constrained with the corresponding pole placement inequalities and its objective function tries to enforce the states of the dynamic controller, using a weight β on* **R** = **FK***.*

$$\begin{array}{l}\max f(\beta \mathbf{R}) \text{ subject to} \\ \mathbf{W} > \mathbf{0} \\ \mathbf{W}A^{\prime} + A\mathbf{W} + \mathbf{Y}^{\prime}B\_{\mu}^{\prime} + B\_{\text{ll}}\mathbf{Y} < \mathbf{0} \end{array} \tag{63}$$

−150

Phase (deg)

−60 −50 −40 −30 −20 −10 0

output disturbance to output.

LMIs, as *H*∞, *H*<sup>2</sup> or pole placement.

matrix variables or require an initial feasible result.

*state-feedback gain.*

**4. Conclusions**

Magnitude (dB)

Phase (deg)

reference to output.

10<sup>−</sup><sup>1</sup> 100 101 102 103 104 105

−100 −80 −60 −40 −20 0

Phase (deg)

0 0.2 0.4 0.6 0.8 1 1.2

reference.

*with the main drawback that the solution of the iterative algorithm may depend on the chosen initial*

In the chapter, it has been shown how a control formulation based on linear matrix inequalities can cope not only with academic optimization problems, but also with large real-life complex problems, since the numerical solution can be found by efficient computer algorithms. The synthesis (or analysis) of the control system can be made by solving an optimization problem, using the concepts of Lyapunov stability and positive definiteness. Besides, frequency-based and time domain performance requirements can also be posed in form of

LMI-based state-feedback synthesis methods have been already applied successfully in the field of power conversion, since they can be applied directly with no additional conservatism. However, output-feedback approaches require the linearization of the synthesis variables in order to be solvable with LMIs. Such linearization methods often impose changes in the

Amplitude

100 101 102 103 104 105

Frequency (Hz) (b) Bode plot of controller **K**4.

Step Response

<sup>0</sup> 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 <sup>−</sup>0.2

(d) Output voltage transient response to a step

Time (sec)

Bode Diagram

Magnitude (dB)

LMI Robust Control of PWM Converters: An Output-Feedback Approach 339

Frequency (Hz) (a) Bode plot of closed loop transfer function from

Bode Diagram

10<sup>−</sup><sup>1</sup> 100 101 102 103 104 105 106

Frequency (Hz) (c) Bode plot of closed loop transfer function from

Fig. 10. Simulation results of Example 4 with controller **K**4.

Bode Diagram

−100

Magnitude (dB)

−50

0

*where for this case of one input and one state of the dynamic controller can be considered <sup>β</sup>* <sup>=</sup> *<sup>β</sup>*<sup>1</sup> <sup>0</sup> 0 *β*<sup>2</sup> , *β*<sup>1</sup> > *β*2*. If an initial state-feedback is found then the iterative step minimizes the H*<sup>∞</sup> *norm γ.*

Results:

*For the present case the algorithm yields the following controller:*

$$\mathbf{K}\_{4} = \begin{bmatrix} \mathbf{A}\_{\mathbf{c}} \begin{vmatrix} \mathbf{B}\_{\mathbf{c}} \end{vmatrix} \\ \hline \mathbf{C}\_{\mathbf{c}} \begin{vmatrix} \mathbf{D}\_{\mathbf{c}} \end{vmatrix} \end{bmatrix} = \begin{bmatrix} -7475.4 \begin{vmatrix} -97.9 \\ 851.3 \end{vmatrix} \\ \hline \end{bmatrix} \tag{64}$$

*Note that the integrator is not included in the previous expression. The Bode plot of the controller with the integrator is depicted in Figure 10(b).*

*The guaranteed H*<sup>∞</sup> *norm with the controller* **K**<sup>4</sup> *achieves the value γ* = 0.571*. The simulation results are shown in Figure 10. It can be observed that the lag-lead compensation of the controller slightly changes the gain peak of the transfer function of interest, yielding slightly better output disturbance attenuation, but longer settling time with respect to the constant output feedback gain. Consequently, this method can be seen as an intermediate solution between dynamic and static output feedback,*

(a) Bode plot of closed loop transfer function from reference to output.

(b) Bode plot of controller **K**4.

(c) Bode plot of closed loop transfer function from output disturbance to output. (d) Output voltage transient response to a step reference.

Fig. 10. Simulation results of Example 4 with controller **K**4.

*with the main drawback that the solution of the iterative algorithm may depend on the chosen initial state-feedback gain.*

### **4. Conclusions**

22 Will-be-set-by-IN-TECH

*As in the previous subsection, the synthesis considers a polytopic model. A new LMI is introduced for every vertex of the uncertain model. The polytopic model used in this subsection considers the same*

> *R* ∈ [10, 1000] Ω *Vg* ∈ [33, 55] *V C* 1000 *μ* F *L* 100 *μ* H *req* 150 mΩ *rc* 50 mΩ *Ts* 5 *μ* s

*Again, the sampling effect is included in the converter model in order to obtain an accurate model up*

*As in the previous case, the objective is to minimize the H*∞ *norm of the transfer function between the disturbance and the output. Also, the closed loop poles are constrained in the same LMI region*

*The initialization step is constrained with the corresponding pole placement inequalities and its objective function tries to enforce the states of the dynamic controller, using a weight β on* **R** = **FK***.*

*B*�

, *β*<sup>1</sup> > *β*2*. If an initial state-feedback is found then the iterative step minimizes the H*<sup>∞</sup>

*where for this case of one input and one state of the dynamic controller can be considered <sup>β</sup>* <sup>=</sup> *<sup>β</sup>*<sup>1</sup> <sup>0</sup>

 = 

*Note that the integrator is not included in the previous expression. The Bode plot of the controller with*

*The guaranteed H*<sup>∞</sup> *norm with the controller* **K**<sup>4</sup> *achieves the value γ* = 0.571*. The simulation results are shown in Figure 10. It can be observed that the lag-lead compensation of the controller slightly changes the gain peak of the transfer function of interest, yielding slightly better output disturbance attenuation, but longer settling time with respect to the constant output feedback gain. Consequently, this method can be seen as an intermediate solution between dynamic and static output feedback,*

*<sup>u</sup>* + *Bu***<sup>Y</sup>** < <sup>00</sup>

−7475.4 −97.9 851.3 10.3

(63)

(64)

max *f*(*β***R**) subject to

**W***A*� + *A***W** + **Y**�

**W** > 00

*For the present case the algorithm yields the following controller:*

**K**<sup>4</sup> = **Ac Bc Cc Dc**

*the integrator is depicted in Figure 10(b).*

Parameter Value

*uncertain load R and input voltage Vg, as shown in Table 4.*

S(*α*,*r*, *θ*)*, in order to assure robust transient performances.*

Table 4. Buck: converter parameters

*to half the switching frequency.* Performance Specifications:

Uncertainty:

Sampling effect:

0 *β*<sup>2</sup> 

*norm γ.* Results:

> In the chapter, it has been shown how a control formulation based on linear matrix inequalities can cope not only with academic optimization problems, but also with large real-life complex problems, since the numerical solution can be found by efficient computer algorithms.

> The synthesis (or analysis) of the control system can be made by solving an optimization problem, using the concepts of Lyapunov stability and positive definiteness. Besides, frequency-based and time domain performance requirements can also be posed in form of LMIs, as *H*∞, *H*<sup>2</sup> or pole placement.

> LMI-based state-feedback synthesis methods have been already applied successfully in the field of power conversion, since they can be applied directly with no additional conservatism. However, output-feedback approaches require the linearization of the synthesis variables in order to be solvable with LMIs. Such linearization methods often impose changes in the matrix variables or require an initial feasible result.

The results presented in this chapter with LMI control of a buck and a buck-boost converter demonstrates the feasibility of this approach, but also shows some of the limitations.

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Olalla, C., Leyva, R., El Aroudi, A. & Queinnec, I. (2010a). LMI robust control design for boost

Olalla, C., Leyva, R., El Aroudi, A., Queinnec, I. & Tarbouriech, S. (2009b). Hinf control of

Olalla, C., Queinnec, I. & Leyva, R. (2010b). *Robust linear control of DC-DC converters: A practical*

Olalla, C., Queinnec, I., Leyva, R. & El Aroudi, A. (2011). Robust optimal control of bilinear

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*Industrial Electronics, IECON'09*, Porto.

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DC-DC converters with saturated inputs, *Proceedings of the IEEE Annual Conference on*

The buck-boost converter presents nonlinear dynamics; such control problem has been tackled with a state-feedback approach, so that the information about the converter states allows to consider the uncertainty coming from the disregarded dynamics, but also limits the achievable performance. Such limitation also allows to neglect the sampling effect of the modulation, since the effective feedback bandwidth of the control is well below the switching frequency.

On the other hand, the control of the buck converter, whose averaged dynamics are basically linear, has been dealt with an output-feedback realization. In this case, the closed-loop performance and its associated bandwidth can reach high frequencies, and it is advisable to take into account the sampling effect of the modulator. From the comparison between the three output-feedback approaches, the best results have been achieved with the *H*∞-based dynamic controller, but this approach also presents some limitations, as the impossibility to deal with easy-to-derive uncertainty models. Another drawback of this technique is that the choice of appropriate weighting functions must be made by trial and error and therefore this task requires good knowledge of the plant limits. Nevertheless, such limitations also appear in the dynamic output-feedback approach by static parametrization, since the results strongly depend on the initial feasible solution for the iterative algorithm.

Depending on the application of the dc-dc converter, it may be easier to implement a state-feedback controller or an output-feedback controller. For instance, the inductor current may not be accessible or the capacitor stray resistance cannot be assumed small enough.

Future research on LMI control of power converters could focus on the improvement of the output-feedback synthesis algorithms, which still require the tedious task of selecting weighting functions, even for the initial controller of the static feedback parametrization. Besides of the inherent limitations of output-feedback, the synthesis algorithms are still very conservative and do not lead to tight performances, when compared with state-feedback approaches.

### **5. References**


24 Will-be-set-by-IN-TECH

The results presented in this chapter with LMI control of a buck and a buck-boost converter

The buck-boost converter presents nonlinear dynamics; such control problem has been tackled with a state-feedback approach, so that the information about the converter states allows to consider the uncertainty coming from the disregarded dynamics, but also limits the achievable performance. Such limitation also allows to neglect the sampling effect of the modulation, since the effective feedback bandwidth of the control is well below the switching frequency. On the other hand, the control of the buck converter, whose averaged dynamics are basically linear, has been dealt with an output-feedback realization. In this case, the closed-loop performance and its associated bandwidth can reach high frequencies, and it is advisable to take into account the sampling effect of the modulator. From the comparison between the three output-feedback approaches, the best results have been achieved with the *H*∞-based dynamic controller, but this approach also presents some limitations, as the impossibility to deal with easy-to-derive uncertainty models. Another drawback of this technique is that the choice of appropriate weighting functions must be made by trial and error and therefore this task requires good knowledge of the plant limits. Nevertheless, such limitations also appear in the dynamic output-feedback approach by static parametrization, since the results strongly

Depending on the application of the dc-dc converter, it may be easier to implement a state-feedback controller or an output-feedback controller. For instance, the inductor current may not be accessible or the capacitor stray resistance cannot be assumed small enough. Future research on LMI control of power converters could focus on the improvement of the output-feedback synthesis algorithms, which still require the tedious task of selecting weighting functions, even for the initial controller of the static feedback parametrization. Besides of the inherent limitations of output-feedback, the synthesis algorithms are still very conservative and do not lead to tight performances, when compared with state-feedback

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**0**

**18**

*Italy*

**Analysis, Dimensioning and Robust Control**

**Currents Compensation in Electrical Mains**

*Center for Complex Automated Systems (CASY) Dept. of Electronics, Computer*

Harmonic pollution in the AC mains determines additional power losses and may cause malfunctioning or even damage to connected equipments. Distortion of the currents circulating on electric mains is mainly originated by non linear loads, as AC/DC uncontrolled rectifiers used for motor drives, that absorb undesired current harmonics. Therefore, local countermeasures have to be taken in order to keep the portion of grid affected by distortion as small as possible, hence preventing relevant power losses and "saving" other equipments,

Traditionally, passive filtering components have been adopted to cope with harmonic compensations, however they are affected by several drawbacks; they are very sensitive to network impedance variation and environmental conditions, moreover they need to be tuned on fixed frequencies. In order to overcome those limitations, in the last decades, thanks also to the fast growth in power electronics and control processor technologies, a remarkable research attempt has been devoted to the study of the so-called Active Power Filters (APFs), both from a theoretical and technological point of view (see Gyugyi & Strycula (1976), Akagi (1996), Singh & Al-Haddad (1999)). These devices are able to properly work in a wide range of operating conditions, providing better performance and overtaking intrinsic limitations of passive devices, they are far more insensitive to network impedance, they can be tuned onto different frequencies just varying some software parameters. Furthermore, the system reliability is improved, resonance phenomena are avoided and a diagnosis system can be implemented on the control processor to monitor the system variables and adopt some

In this chapter, the general issues related to analysis, dimensioning and control of a particular class of APFs, the so-called Shunt Active Filters (SAFs), are addressed; the main purpose of this kind of power system is to inject into mains a proper current, in order to cancel out, partially or totally, the power distortions generated by nonlinear loads. The SAFs considered in this work are based on a three-phases three-wires AC/DC boost converter topology (see

**1. Introduction**

connected to the rest of the grid.

recovery strategy in case of faulty conditions.

Fig. 1) connected in parallel to the distorting loads.

**of Shunt Active Filter for Harmonic**

Andrea Tilli, Lorenzo Marconi and Christian Conficoni

*Engineering and Systems (DEIS), University of Bologna,*

*Viale Risorgimento n.2, 40136 Bologna*


## **Analysis, Dimensioning and Robust Control of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains**

Andrea Tilli, Lorenzo Marconi and Christian Conficoni

*Center for Complex Automated Systems (CASY) Dept. of Electronics, Computer Engineering and Systems (DEIS), University of Bologna, Viale Risorgimento n.2, 40136 Bologna Italy*

### **1. Introduction**

26 Will-be-set-by-IN-TECH

342 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Peaucelle, D. & Arzelier, D. (2001a). An efficient numerical solution for *H*<sup>2</sup> static output feedback synthesis., *European Control Conference*, Porto, pp. 3800–3805. Peaucelle, D. & Arzelier, D. (2001b). An iterative method for mixed *H*2/*H*<sup>∞</sup> synthesis via static

Peaucelle, D. & Arzelier, D. (2001c). Robust performance analysis using LMI-based methods

Pyatnitskii, E. S. & Skorodinskii, V. I. (1982). Numerical methods of Lyapunov function

Rodriguez, H., Ortega, R. & Escobar, G. (1999). A robustly stable output feedback saturated

Scherer, C., Gahinet, P. & Chilali, M. (1997). Multiobjective output-feedback control via LMI

Skogestad, S. & Postlethwaite, I. (1996). *Multivariable Feedback Control: Analysis and Design*,

Sturm, J. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over

Torres-Pinzon, C. A. & Leyva, R. (2009). Fuzzy control in DC-DC converters: an LMI approach,

Wallis, G. F. & Tymerski, R. (2000). Generalized approach for *μ* synthesis of robust switching regulators, *IEEE Transactions on Aerospace and Electronic Systems* 36(2): 422–431. Zhou, K., Doyle, J. & Glover, K. (1996). *Robust and Optimal Control*, Prentice-Hall, New Jersey.

optimization, *IEEE Transactions on Automatic Control* 42(7): 896–911.

*Transactions on Automatic Control* 46(4): 624–630.

*Decision and Control, CDC'99.*, Vol. 3, pp. 2100–2105.

*Control Letters* 2(2): 130–135.

John Wiley and Sons, New York.

pp. 510–515.

http://fewcal.kub.nl/sturm.

output-feedback., *IEEE Conference on Decision and Control*, Las Vegas, pp. 3464–3469.

for real parametric uncertainty via parameter-dependent lyapunov functions, *IEEE*

construction and their application to the absolute stability problem, *Systems and*

controller for the boost dc-to-dc converter, *Proceedings of the IEEE Conference on*

symmetric cones, *Optimization Methods and Software* 11-12: 625–653. [Online]

*Proceedings of the IEEE Annual Conference on Industrial Electronics, IECON'09*, Porto,

Harmonic pollution in the AC mains determines additional power losses and may cause malfunctioning or even damage to connected equipments. Distortion of the currents circulating on electric mains is mainly originated by non linear loads, as AC/DC uncontrolled rectifiers used for motor drives, that absorb undesired current harmonics. Therefore, local countermeasures have to be taken in order to keep the portion of grid affected by distortion as small as possible, hence preventing relevant power losses and "saving" other equipments, connected to the rest of the grid.

Traditionally, passive filtering components have been adopted to cope with harmonic compensations, however they are affected by several drawbacks; they are very sensitive to network impedance variation and environmental conditions, moreover they need to be tuned on fixed frequencies. In order to overcome those limitations, in the last decades, thanks also to the fast growth in power electronics and control processor technologies, a remarkable research attempt has been devoted to the study of the so-called Active Power Filters (APFs), both from a theoretical and technological point of view (see Gyugyi & Strycula (1976), Akagi (1996), Singh & Al-Haddad (1999)). These devices are able to properly work in a wide range of operating conditions, providing better performance and overtaking intrinsic limitations of passive devices, they are far more insensitive to network impedance, they can be tuned onto different frequencies just varying some software parameters. Furthermore, the system reliability is improved, resonance phenomena are avoided and a diagnosis system can be implemented on the control processor to monitor the system variables and adopt some recovery strategy in case of faulty conditions.

In this chapter, the general issues related to analysis, dimensioning and control of a particular class of APFs, the so-called Shunt Active Filters (SAFs), are addressed; the main purpose of this kind of power system is to inject into mains a proper current, in order to cancel out, partially or totally, the power distortions generated by nonlinear loads. The SAFs considered in this work are based on a three-phases three-wires AC/DC boost converter topology (see Fig. 1) connected in parallel to the distorting loads.

The first step to properly design a SAF is the selection of suitable hardware components; as it will become clear in the next section, owing to the structural properties of the system, the sizing procedure cannot be considered apart from the canonical control aspects, hence a correct dimensioning algorithm (Ronchi & Tilli, 2002) is proposed to ensure feasibility of the desired control objectives and to minimize costs. In addition, according to such method, it is shown how a time-scale separation between different dynamics of SAF usually takes place "for free". This point is very useful for control design and stability analysis.

Once a correct hardware sizing has been carried out, the first control issues to deal with are: the current/power control algorithm and the load current analysis method adopted to define the filter current reference. Various solutions have been proposed in literature. As regards current/power harmonic tracking, in (Chandra et al., 2000) an hysteresis current control (Kazmierkowski & Malesani, 1998) is proposed, while in (Jeong & Woo, 1997) predictive current control is adopted. For what concerns the generation of the filter currents reference, beside Fast Fourier Transform Techniques, instantaneous power theory (Akagi et al., 1984), time domain correlation techniques (Van Harmelen & Enselin, 1993), notch filter theory (Rastogi et al., 1995) and other methods have been proposed. Solution based on state observer have been proposed, too, as in (Bhattacharya et al., 1995) and (Tilli et al., 2002).

However, what renders the SAF control problem challenging and different from other conventional tracking problems is the presence of peculiar and unstable internal dynamics, given by the voltage dynamics of the DC-link capacitor bank. This capacitor bank is the main energy storage element, which provides the voltage, modulated by the control, to steer the filter currents and, at the same time, is required to oscillate to exchange energy with the line and the load to compensate for current harmonics. Actually, this element needs to be carefully considered also in the previously-mentioned dimensioning stage; a correct capacitor sizing is crucial for control objective feasibility, whatever control technique is adopted. Moreover, it can be shown that, if perfect harmonic compensation is achieved, the DC-link voltage dynamics are unstable, due to the system parasitic resistances that lead to a slow discharge of the capacitor. Hence, a suitable stabilizing action for DC-link voltage dynamics needs to be provided. Since no additional circuit is used to feed the DC-link capacitor independently of the three-phase port used to inject currents, (see Fig. 1), the voltage stabilization would need to be integrated with the controller devoted to harmonic compensation (the AC/DC boost-based SAF is an underactuated system). This is a crucial point and it has to be tackled preserving harmonic compensation performances as far as possible.

Fig. 1. Shunt Active Filter scheme.

*theory* techniques (Khalil, 1996).

Section 5 through simulations.

Bipolar Transistors) and free-wheeling diodes.

**2. Shunt active filter model and control problem statement**

tracking and voltage stabilization control problems, using *averaging* and *singular perturbation*

<sup>345</sup> Analysis, Dimensioning and Robust Control

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

This chapter is organized as follows. In Section 2, the general framework is described, the SAF model is derived and the control objectives are formally defined. In Section 3, two methodological approaches are presented for the SAF components sizing. The first one is based on the knowledge of the load currents harmonic spectrum, the values selected for the hardware components are the minimums allowing the SAF to deal with the considered load distortion. Differently, the second approach is related to the maximum current of the AC/DC boost switching devices. In this case the selected components values are the minimums which enable the SAF to compensate for all possible loads giving distorted currents smaller or equal to the switches peak value. In Section 4 both the internal model-based power/current controller and the averaging voltage controller design are presented, stability analysis is carried out relying upon the time-scale separation imposed by the design algorithm; both the power and the voltage subsystem are proven to be asymptotically stable, then practical stability of the overall system is claimed exploiting general results on two time-scale averaged systems (Teel et al., 2003). The effectiveness of the proposed control solution is tested in

The scheme of the shunt active filter considered in this chapter is reported in Fig. 1, as mentioned in the introduction it is based on a three-phase three-wire AC/DC boost converter, where the main energy storage element is a DC-bus capacitor, while the inductances are exploited to steer the filter currents by means of the converter voltages. The switching devices of the three-leg bridge (also called "'inverter"') are usually realized by IGBTs (Insulated Gate

In this work a power/current controller, based on Internal Model Principle, (see Marconi et al. (2003), Marconi et al. (2004), Marconi et al. (2007)) is designed in order to cancel current harmonics, ensuring robustness with respect to SAF parameter uncertainties. By exploiting the internal model approach, the proposed solution also allows to merge and solve at the same time the two above-mentioned problems of current harmonics isolation and current reference tracking. As regards the robust stabilization of the DC-link voltage internal dynamics, a cascade control structure is proposed. An additional voltage controller, acting on the references of the power/current controller, is introduced. This controller is designed taking into account the structural voltage oscillations required for harmonic compensation and minimizing the impact on harmonic compensation. In particular, by exploiting a proper averaging (Sanders et al., 1991) of the capacitor voltage dynamics, the average value of the capacitor voltage is chosen as output variable to be controlled (Hanschke et al., 2006).

As far as the overall stability is concerned, the previously mentioned time-scale separation between portions of SAF dynamics can be effectively exploited to decouple power/current

Fig. 1. Shunt Active Filter scheme.

2 Will-be-set-by-IN-TECH

The first step to properly design a SAF is the selection of suitable hardware components; as it will become clear in the next section, owing to the structural properties of the system, the sizing procedure cannot be considered apart from the canonical control aspects, hence a correct dimensioning algorithm (Ronchi & Tilli, 2002) is proposed to ensure feasibility of the desired control objectives and to minimize costs. In addition, according to such method, it is shown how a time-scale separation between different dynamics of SAF usually takes place

Once a correct hardware sizing has been carried out, the first control issues to deal with are: the current/power control algorithm and the load current analysis method adopted to define the filter current reference. Various solutions have been proposed in literature. As regards current/power harmonic tracking, in (Chandra et al., 2000) an hysteresis current control (Kazmierkowski & Malesani, 1998) is proposed, while in (Jeong & Woo, 1997) predictive current control is adopted. For what concerns the generation of the filter currents reference, beside Fast Fourier Transform Techniques, instantaneous power theory (Akagi et al., 1984), time domain correlation techniques (Van Harmelen & Enselin, 1993), notch filter theory (Rastogi et al., 1995) and other methods have been proposed. Solution based on state observer

However, what renders the SAF control problem challenging and different from other conventional tracking problems is the presence of peculiar and unstable internal dynamics, given by the voltage dynamics of the DC-link capacitor bank. This capacitor bank is the main energy storage element, which provides the voltage, modulated by the control, to steer the filter currents and, at the same time, is required to oscillate to exchange energy with the line and the load to compensate for current harmonics. Actually, this element needs to be carefully considered also in the previously-mentioned dimensioning stage; a correct capacitor sizing is crucial for control objective feasibility, whatever control technique is adopted. Moreover, it can be shown that, if perfect harmonic compensation is achieved, the DC-link voltage dynamics are unstable, due to the system parasitic resistances that lead to a slow discharge of the capacitor. Hence, a suitable stabilizing action for DC-link voltage dynamics needs to be provided. Since no additional circuit is used to feed the DC-link capacitor independently of the three-phase port used to inject currents, (see Fig. 1), the voltage stabilization would need to be integrated with the controller devoted to harmonic compensation (the AC/DC boost-based SAF is an underactuated system). This is a crucial point and it has to be tackled

In this work a power/current controller, based on Internal Model Principle, (see Marconi et al. (2003), Marconi et al. (2004), Marconi et al. (2007)) is designed in order to cancel current harmonics, ensuring robustness with respect to SAF parameter uncertainties. By exploiting the internal model approach, the proposed solution also allows to merge and solve at the same time the two above-mentioned problems of current harmonics isolation and current reference tracking. As regards the robust stabilization of the DC-link voltage internal dynamics, a cascade control structure is proposed. An additional voltage controller, acting on the references of the power/current controller, is introduced. This controller is designed taking into account the structural voltage oscillations required for harmonic compensation and minimizing the impact on harmonic compensation. In particular, by exploiting a proper averaging (Sanders et al., 1991) of the capacitor voltage dynamics, the average value of the

capacitor voltage is chosen as output variable to be controlled (Hanschke et al., 2006).

As far as the overall stability is concerned, the previously mentioned time-scale separation between portions of SAF dynamics can be effectively exploited to decouple power/current

"for free". This point is very useful for control design and stability analysis.

have been proposed, too, as in (Bhattacharya et al., 1995) and (Tilli et al., 2002).

preserving harmonic compensation performances as far as possible.

tracking and voltage stabilization control problems, using *averaging* and *singular perturbation theory* techniques (Khalil, 1996).

This chapter is organized as follows. In Section 2, the general framework is described, the SAF model is derived and the control objectives are formally defined. In Section 3, two methodological approaches are presented for the SAF components sizing. The first one is based on the knowledge of the load currents harmonic spectrum, the values selected for the hardware components are the minimums allowing the SAF to deal with the considered load distortion. Differently, the second approach is related to the maximum current of the AC/DC boost switching devices. In this case the selected components values are the minimums which enable the SAF to compensate for all possible loads giving distorted currents smaller or equal to the switches peak value. In Section 4 both the internal model-based power/current controller and the averaging voltage controller design are presented, stability analysis is carried out relying upon the time-scale separation imposed by the design algorithm; both the power and the voltage subsystem are proven to be asymptotically stable, then practical stability of the overall system is claimed exploiting general results on two time-scale averaged systems (Teel et al., 2003). The effectiveness of the proposed control solution is tested in Section 5 through simulations.

### **2. Shunt active filter model and control problem statement**

The scheme of the shunt active filter considered in this chapter is reported in Fig. 1, as mentioned in the introduction it is based on a three-phase three-wire AC/DC boost converter, where the main energy storage element is a DC-bus capacitor, while the inductances are exploited to steer the filter currents by means of the converter voltages. The switching devices of the three-leg bridge (also called "'inverter"') are usually realized by IGBTs (Insulated Gate Bipolar Transistors) and free-wheeling diodes.

In this work the following notation is used to denote the SAF variables; *vmabc*=(*vma*, *vmb*, *vmc*)*<sup>T</sup>* is the mains voltage sinusoidal balanced and equilibrated tern, *im*=(*ima*, *imb*, *imc*)*<sup>T</sup>* are the mains currents, *il*=(*ila*, *ilb*, *ilc*)*<sup>T</sup>* are the load currents, while *i*=(*ia*, *ib*, *ic*)*<sup>T</sup>* are the filter currents. *L* indicates the value of the inductances, and *C* the DC-link bus capacitor value.

#### **2.1 Mathematical model**

Considering the inductors dynamics, the filter model can be expressed as

$$
\begin{bmatrix} v\_{ma}(t) \\ v\_{mb}(t) \\ v\_{mc}(t) \end{bmatrix} - L \frac{d}{dt} \begin{bmatrix} i\_a(t) \\ i\_b(t) \\ i\_c(t) \end{bmatrix} - R \begin{bmatrix} i\_a(t) \\ i\_b(t) \\ i\_c(t) \end{bmatrix} = \begin{bmatrix} u\_x(t) \\ u\_y(t) \\ u\_z(t) \end{bmatrix} v(t) - v\_{NK} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \tag{1}
$$

where *R* is the parasitic resistance related to the inductance *L* and to the cables, *vNK* is the voltage between the nodes *N* and *K* reported in Fig. 1, *v*(*t*) is the voltage on the DC-link capacitor, and *u*<sup>1</sup> = (*ux*, *uy*, *uz*)*<sup>T</sup>* is the switch command vector for the legs of the converter. Since a PWM (Pulse Width Modulation) strategy is assumed to control the inverter, the above-mentioned control inputs can be considered such that *u*1*<sup>i</sup>* ∈ [0, 1], *i* = *x*, *y*, *z*. According to the three-wire topology for any generic voltage/current vector *x* it holds

$$\sum\_{i=a,b,c} x\_i = 0 \tag{2}$$

*ux uy uz ua ub uc u<sup>α</sup> u<sup>β</sup>* 000 000 0 0 2/3 -1/3 -1/3 2/3 0 1/3 1/3 -2/3 1/3 1/√<sup>3</sup> -1/3 2/3 -1/3 -1/3 1/√<sup>3</sup> -2/3 1/3 1/3 -2/3 0 -1/3 -1/3 2/3 -1/3 -1/√<sup>3</sup> 1/3 -2/3 1/3 1/3 -1/√<sup>3</sup> 111 000 0 0

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

<sup>347</sup> Analysis, Dimensioning and Robust Control

Table 1. Control function table.

applying the following coordinates transformation

*iαβ*(*t*)=[*i<sup>α</sup> iβ*]

*uαβ*(*t*)=[*u<sup>α</sup> uβ*]

*αβTabc* <sup>=</sup> <sup>2</sup>

the SAF dynamics expressed in this *α* − *β* reference frame become

[*vm<sup>α</sup> vmβ*]

to adopt a further transformation from the two-phase current variables [*i<sup>α</sup> iβ*]

*dqTαβ* = *Vm*

*x* = [*xd xq*]

*diαβ dt* <sup>=</sup> <sup>−</sup> *<sup>R</sup>*

> *dv dt* <sup>=</sup> <sup>3</sup> <sup>2</sup>*<sup>C</sup> <sup>u</sup><sup>T</sup>*

real-virtual (imaginary) power variables defined as

where

*vmαβ* = [*vmαvmβ*]

3 <sup>1</sup> <sup>−</sup><sup>1</sup> <sup>2</sup> <sup>−</sup><sup>1</sup> 2

0 √3 2 − √3 2

*<sup>L</sup> <sup>I</sup>*2*iαβ*(*t*) <sup>−</sup> *<sup>v</sup>*(*t*)

*αβ*(*t*)*iαβ*(*t*)

tern can be expressed in the above-defined bi-dimensional reference frame as follows

according to the hypothesis of three-phase balanced sinusoidal line, the ideal main voltage

where *Vm* is the voltage amplitude and *ωm* the grid angular frequency. For what concerns the control vector *uabc*, in this reference frame the eight possible configurations of the switching network (reported in Tab. 1) can be mapped in the *α* − *β* plane, obtaining the vertexes and the origin of the feasibility space illustrated in Fig.2, while each point in the hexagon can be obtained as mean value in a PWM period. As it will become clear in the next sections, in order to simplify the control objectives definition and the controller design, it is very useful

*<sup>T</sup>* = *Vm*[*cos*(*ωm*) *sin*(*ωm*)]*<sup>T</sup>*

 *cos*(*ωmt*) *sin*(*ωmt*) −*sin*(*ωmt*) *cos*(*ωmt*)

*<sup>T</sup>* =*αβ Tabci*(*t*)

*<sup>T</sup>* =*αβ Tabcuabc*(*t*)

*<sup>T</sup>* =*αβ Tabcvm*

(7)

(8)

*<sup>T</sup>* to a two-phase

*<sup>L</sup> <sup>u</sup>αβ*(*t*) + <sup>1</sup>

*<sup>L</sup> vmαβ*

*<sup>T</sup>* =*dq <sup>T</sup>αβiαβ* (9)

 .

hence, from the sum of the scalar equations in (1) it follows that

$$v\_{NK} = \frac{u\_x(t) + u\_y(t) + u\_z(t)}{3} v(t) \tag{3}$$

defining

$$\boldsymbol{u}\_{abc} = \begin{bmatrix} \boldsymbol{u}\_{a}(t), \ \boldsymbol{u}\_{b}(t), \ \boldsymbol{u}\_{c}(t) \end{bmatrix}^{T} = \begin{bmatrix} \boldsymbol{u}\_{x}(t) \\ \boldsymbol{u}\_{y}(t) \\ \boldsymbol{u}\_{z}(t) \end{bmatrix} - \frac{\boldsymbol{u}\_{x}(t) + \boldsymbol{u}\_{y}(t) + \boldsymbol{u}\_{z}(t)}{3} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \tag{4}$$

it can be verified by direct computations that

$$[1\ 1\ 1]\mu\_{abc}(t) = 0\ \forall t \ge 0. \tag{5}$$

For what concerns the state equation relative to the capacitor voltage dynamics, it can be derived considering an ideal inverter and applying a power balance condition between the input and the output of the filter, then replacing (3) into (1), the complete filter model results

$$\begin{aligned} \frac{di}{dt} &= -\frac{R}{L}I\_3i(t) - \frac{v(t)}{L}u\_{abc}(t) + \frac{1}{L}v\_{mabc} \\ \frac{dv}{dt} &= \frac{1}{C}u\_{abc}^T(t)i(t) \end{aligned} \tag{6}$$

where the filter currents dynamics have been written in a more compact form with respect to (1), multiplying the current vector by the identity matrix of suitable dimension *I*3. Exploiting equations (3), (5), the system model can be reduced to the standard *two-phase planar representation* of a three-phase balanced systems (Krause et al., 1995), which can be obtained


Table 1. Control function table.

4 Will-be-set-by-IN-TECH

In this work the following notation is used to denote the SAF variables; *vmabc*=(*vma*, *vmb*, *vmc*)*<sup>T</sup>* is the mains voltage sinusoidal balanced and equilibrated tern, *im*=(*ima*, *imb*, *imc*)*<sup>T</sup>* are the mains currents, *il*=(*ila*, *ilb*, *ilc*)*<sup>T</sup>* are the load currents, while *i*=(*ia*, *ib*, *ic*)*<sup>T</sup>* are the filter currents.

> ⎡ ⎣ *ia*(*t*) *ib*(*t*) *ic*(*t*)

where *R* is the parasitic resistance related to the inductance *L* and to the cables, *vNK* is the voltage between the nodes *N* and *K* reported in Fig. 1, *v*(*t*) is the voltage on the DC-link capacitor, and *u*<sup>1</sup> = (*ux*, *uy*, *uz*)*<sup>T</sup>* is the switch command vector for the legs of the converter. Since a PWM (Pulse Width Modulation) strategy is assumed to control the inverter, the above-mentioned control inputs can be considered such that *u*1*<sup>i</sup>* ∈ [0, 1], *i* = *x*, *y*, *z*. According to the three-wire topology for any generic voltage/current vector *x* it holds

> ∑ *i*=*a*,*b*,*c*

*vNK* <sup>=</sup> *ux*(*t*) + *uy*(*t*) + *uz*(*t*)

⎡ ⎣

*ux*(*t*) *uy*(*t*) *uz*(*t*)

For what concerns the state equation relative to the capacitor voltage dynamics, it can be derived considering an ideal inverter and applying a power balance condition between the input and the output of the filter, then replacing (3) into (1), the complete filter model results

where the filter currents dynamics have been written in a more compact form with respect to (1), multiplying the current vector by the identity matrix of suitable dimension *I*3. Exploiting equations (3), (5), the system model can be reduced to the standard *two-phase planar representation* of a three-phase balanced systems (Krause et al., 1995), which can be obtained

*<sup>I</sup>*3*i*(*t*) <sup>−</sup> *<sup>v</sup>*(*t*)

*abc*(*t*)*i*(*t*)

⎤

*<sup>L</sup> uabc*(*t*) + <sup>1</sup>

⎤ ⎦ = ⎡ ⎣

*ux*(*t*) *uy*(*t*) *uz*(*t*)

⎤

⎦ *v*(*t*) − *vNK*

*xi* = 0 (2)

<sup>3</sup> *<sup>v</sup>*(*t*) (3)

⎡ ⎣ 1 1 1

⎤

⎦ (4)

(6)

<sup>⎦</sup> <sup>−</sup> *ux*(*t*) + *uy*(*t*) + *uz*(*t*) 3

[111]*uabc*(*t*) = 0 ∀*t* ≥ 0. (5)

*<sup>L</sup> vmabc*

⎡ ⎣ 1 1 1

⎤

⎦ (1)

*L* indicates the value of the inductances, and *C* the DC-link bus capacitor value.

Considering the inductors dynamics, the filter model can be expressed as

⎤ ⎦ − *R*

⎡ ⎣ *ia*(*t*) *ib*(*t*) *ic* (*t*)

hence, from the sum of the scalar equations in (1) it follows that

*uabc* = [*ua*(*t*), *ub*(*t*), *uc*(*t*)]*<sup>T</sup>* =

*di dt* <sup>=</sup> <sup>−</sup> *<sup>R</sup> L*

*dv dt* <sup>=</sup> <sup>1</sup> *CuT*

it can be verified by direct computations that

**2.1 Mathematical model**

defining

⎡ ⎣

*vma*(*t*) *vmb*(*t*) *vmc*(*t*)

⎤ <sup>⎦</sup> <sup>−</sup> *<sup>L</sup> <sup>d</sup> dt*

applying the following coordinates transformation

$$\begin{aligned} i\_{a\beta}(t) &= [i\_a \ i\_{\beta}]^T = {}^{a\beta} T\_{abc} i(t) \\ u\_{a\beta}(t) &= [u\_a \ u\_{\beta}]^T = {}^{a\beta} T\_{abc} u\_{abc}(t) \\ v\_{ma\beta} &= [v\_{ma} v\_{m\beta}]^T = {}^{a\beta} T\_{abc} v\_m \\ {}^{a\beta} T\_{abc} &= \frac{2}{3} \begin{bmatrix} 1 - \frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} \end{bmatrix} \end{aligned} \tag{7}$$

the SAF dynamics expressed in this *α* − *β* reference frame become

$$\begin{aligned} \frac{d\dot{u}\_{a\beta}}{dt} &= -\frac{R}{L}I\_2\dot{u}\_{a\beta}(t) - \frac{v(t)}{L}u\_{a\beta}(t) + \frac{1}{L}v\_{ma\beta} \\ \frac{dv}{dt} &= \frac{3}{2\mathsf{C}}u\_{a\beta}^T(t)\dot{u}\_{a\beta}(t) \end{aligned} \tag{8}$$

according to the hypothesis of three-phase balanced sinusoidal line, the ideal main voltage tern can be expressed in the above-defined bi-dimensional reference frame as follows

> [*vm<sup>α</sup> vmβ*] *<sup>T</sup>* = *Vm*[*cos*(*ωm*) *sin*(*ωm*)]*<sup>T</sup>*

where *Vm* is the voltage amplitude and *ωm* the grid angular frequency. For what concerns the control vector *uabc*, in this reference frame the eight possible configurations of the switching network (reported in Tab. 1) can be mapped in the *α* − *β* plane, obtaining the vertexes and the origin of the feasibility space illustrated in Fig.2, while each point in the hexagon can be obtained as mean value in a PWM period. As it will become clear in the next sections, in order to simplify the control objectives definition and the controller design, it is very useful to adopt a further transformation from the two-phase current variables [*i<sup>α</sup> iβ*] *<sup>T</sup>* to a two-phase real-virtual (imaginary) power variables defined as

$$\mathbf{x} = [\mathbf{x}\_d \ \mathbf{x}\_q]^T = {}^{dq}T\_{a\mathcal{B}}i\_{a\mathcal{B}} \tag{9}$$

where

$$^{dq}T\_{\alpha\beta} = V\_m \begin{bmatrix} \cos(\omega\_m t) & \sin(\omega\_m t) \\ -\sin(\omega\_m t) & \cos(\omega\_m t) \end{bmatrix}.$$

Fig. 2. Hexagon of feasible *uabc*.

In this so-called *synchronous* coordinate setting, aligned with the mains voltage vector, the model of the SAF is expressed as

$$\begin{aligned} \dot{\mathbf{x}} &= M(\mathbf{R}, L)\mathbf{x} - \frac{\upsilon}{L}\mathbf{u}\_{d\eta} + d\_0\\ \dot{\upsilon} &= \frac{\epsilon}{2}\mathbf{u}\_{d\eta}^T \mathbf{x} \end{aligned} \tag{10}$$

ensure the capability to steer the filter currents. On the other hand the ability of maintaining DC-link voltage into a suitable region is strictly related to the power exchanged with the mains, which in turn is affected by the current harmonics to be compensated for. The general control objective is then two-folds; one related to the tracking of current disturbances, the other concerns the voltage internal dynamics stabilization. In this paragraph a precise and feasible control problem is formally defined, recalling the considerations made above, and assuming that a suitable dimensioning, that will be deeply discussed in the next section, has

<sup>349</sup> Analysis, Dimensioning and Robust Control

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

Bearing in mind the power variables representation of a generic nonlinear load expressed in (12), it turns out that the only desired load component is *Xld*0, since it represents first-order harmonics aligned with the mains voltages, while the remaining part of the real component *xld* − *Xld*<sup>0</sup> is an oscillatory signal with null balance over a line period, and the imaginary component *xlq* represent a measure of the misalignment between mains ideal voltage and load currents (see Mohan et al. (1989)) and do not contribute to the power flow. In this respect, the terms *xld* − *Xld*0, *xlq* are undesired components which should be canceled by the injected filter currents, hence ideally the control problem can be formulated as a state tracking problem, for

a prefect tracking of this reference would ensure pure sinusoidal mains currents perfectly aligned with the mains voltages. However, this ideal objective is in contrast with the requirement to have a DC-link voltage bounded behavior. In order to formally motivate this claim, consider the steady state voltage dynamics in case perfect tracking of the power

the signal *�*Ψ(*x*∗(*t*)) which drives the integrator is periodic with period *T* = 1/ *fm*, and

*�L*(*M*(*R*, *L*)*x*∗)*Tx*<sup>∗</sup> which has negative mean value as long as parasitic resistance *R* or reference *x*∗ are not zero. By this, no matter the starting voltage value of the DC-link, the capacitor will be discharged and the voltage will drop, providing a loss of controllability of

To avoid this phenomenon, the reference must be revised, taking into account an additional power term, which should be drained from the line grid by the active filter, in order to compensate for its power losses. Following this motivation, and recalling that the unique useful component for the energy exchange is the real part of the power variables, the ideal

> 1/ *fm* 0

(*x*∗<sup>2</sup>

*<sup>d</sup>* (*τ*) + *<sup>x</sup>*∗<sup>2</sup>

*Xld*<sup>0</sup> <sup>−</sup> *xld* <sup>−</sup> *xlq<sup>T</sup>*

∗(*t*))

*<sup>T</sup> x*<sup>∗</sup> := *�*Ψ(*x*∗(*t*)) (14)

*<sup>q</sup>* (*τ*))*dτ* = 0 (16)

*<sup>ϕ</sup>*<sup>0</sup> <sup>=</sup> *<sup>x</sup>*<sup>∗</sup> + (*ϕ*<sup>0</sup> <sup>0</sup>)*<sup>T</sup>* (15)

<sup>∗</sup>)*Tx*∗, and the signal

(13)

been carried out.

the system.

system (10), of the following reference

*dv*<sup>2</sup>

reference signal (13) is modified as

in which *ϕ*<sup>0</sup> is a solution of the following equation

*Rϕ*<sup>2</sup>

*x*∗(*t*)=[*x*∗

reference *x*∗(*t*) is achieved, after some computations it results

*dt* <sup>=</sup> *�<sup>L</sup>* (*d*<sup>0</sup> <sup>+</sup> *<sup>M</sup>*(*R*, *<sup>L</sup>*)*x*∗(*t*) <sup>−</sup> *<sup>x</sup>*˙

it is composed by the sum of a zero mean value signal *�L*(*d*<sup>0</sup> − *x*˙

*x*∗

<sup>0</sup> − *Emdϕ*<sup>0</sup> + *R fm*

*<sup>d</sup> x*<sup>∗</sup> *q* ] *<sup>T</sup>* = 

where

$$d\_0 = \begin{bmatrix} \mathbf{E}\_{md}/L \\ 0 \end{bmatrix}, \ M(\mathbf{R}, L) = \begin{bmatrix} -\mathbf{R}/L & \omega\_m \\ -\omega\_m & -\mathbf{R}/L \end{bmatrix}, \ \mathbf{c} = \frac{\mathbf{3}}{\mathbf{C}\mathbf{E}\_{md}}, \ \mathbf{E}\_{md} = V\_m^2, \ u\_{d\boldsymbol{\eta}} = ^{d\boldsymbol{\eta}}T\_{a\boldsymbol{\xi}\boldsymbol{\theta}}\boldsymbol{u}\_{a\boldsymbol{\xi}} \tag{11}$$

it is further to notice that, since the filter currents, the mains voltage and the DC-link voltage are measurable, the full state (*x*,*v*) is available for feedback, moreover the actual control action *u* = [*ux uy uz*] can be determined from *uabc*, which in turn can be derived from *udq*.

As regards the load description, the same two-phase real-virtual power representation can be used, in particular following (Akagi et al., 1984), the load currents can be approximated as periodic signals given by the sum of a finite number N of harmonics, with frequencies multiple of *fm* = *ωm*/2*π*. Hence the load currents can be expressed in power variables as

$$\mathbf{x}\_{lj} = X\_{lj0} + \sum\_{n=1}^{N+1} X\_{ljn} \cos(n\omega\_m t + \psi\_{jn}), \quad j = d, q \tag{12}$$

where the harmonics amplitudes *Xld*0, *Xlq*0, *Xldn*, *Xlqn* and phases *ψdn*, *ψqn* are constants. Since the load currents and the mains voltages are measurable, also the variables (*xld*, *xlq*) will be considered known and available for control purpose.

#### **2.2 Problem statement and control objectives**

Roughly speaking the main control objective of the considered SAF is to steer the variables *xd*, *xq*, injecting power into the line to compensate for the load harmonics. However the ability of tracking current references relies upon the energy stored in the DC-link capacitor, which is the main power source of the filter, therefore another general objective is to keep the DC-link voltage confined in a suitable region, to avoid overcharge and, at the same time, to 6 Will-be-set-by-IN-TECH

c b

a

−0.8 −0.4 0 0.4 0.8

In this so-called *synchronous* coordinate setting, aligned with the mains voltage vector, the

it is further to notice that, since the filter currents, the mains voltage and the DC-link voltage are measurable, the full state (*x*,*v*) is available for feedback, moreover the actual control action

As regards the load description, the same two-phase real-virtual power representation can be used, in particular following (Akagi et al., 1984), the load currents can be approximated as periodic signals given by the sum of a finite number N of harmonics, with frequencies multiple of *fm* = *ωm*/2*π*. Hence the load currents can be expressed in power variables as

where the harmonics amplitudes *Xld*0, *Xlq*0, *Xldn*, *Xlqn* and phases *ψdn*, *ψqn* are constants. Since the load currents and the mains voltages are measurable, also the variables (*xld*, *xlq*) will be

Roughly speaking the main control objective of the considered SAF is to steer the variables *xd*, *xq*, injecting power into the line to compensate for the load harmonics. However the ability of tracking current references relies upon the energy stored in the DC-link capacitor, which is the main power source of the filter, therefore another general objective is to keep the DC-link voltage confined in a suitable region, to avoid overcharge and, at the same time, to

*u* = [*ux uy uz*] can be determined from *uabc*, which in turn can be derived from *udq*.

, *�* <sup>=</sup> <sup>3</sup>

*<sup>L</sup> udq* <sup>+</sup> *<sup>d</sup>*<sup>0</sup>

*CEmd*

, *Emd* = *<sup>V</sup>*<sup>2</sup>

*Xljncos*(*nωmt* + *ψjn*), *j* = *d*, *q* (12)

(10)

*<sup>m</sup>*, *udq* =*dq <sup>T</sup>αβuαβ* (11)

*<sup>x</sup>*˙ <sup>=</sup> *<sup>M</sup>*(*R*, *<sup>L</sup>*)*<sup>x</sup>* <sup>−</sup> *<sup>v</sup>*

*<sup>v</sup>*˙ <sup>=</sup> *�* 2 *uT dqx*

−*R*/*L ω<sup>m</sup>* −*ω<sup>m</sup>* −*R*/*L*

> *N*+1 ∑ *n*=1

*xlj* = *Xlj*<sup>0</sup> +

considered known and available for control purpose.

**2.2 Problem statement and control objectives**

−0.8

Fig. 2. Hexagon of feasible *uabc*.

model of the SAF is expressed as

, *M*(*R*, *L*) =

where

*d*<sup>0</sup> = *Emd*/*L* 0

−0.4

0

0.4

0.8

ensure the capability to steer the filter currents. On the other hand the ability of maintaining DC-link voltage into a suitable region is strictly related to the power exchanged with the mains, which in turn is affected by the current harmonics to be compensated for. The general control objective is then two-folds; one related to the tracking of current disturbances, the other concerns the voltage internal dynamics stabilization. In this paragraph a precise and feasible control problem is formally defined, recalling the considerations made above, and assuming that a suitable dimensioning, that will be deeply discussed in the next section, has been carried out.

Bearing in mind the power variables representation of a generic nonlinear load expressed in (12), it turns out that the only desired load component is *Xld*0, since it represents first-order harmonics aligned with the mains voltages, while the remaining part of the real component *xld* − *Xld*<sup>0</sup> is an oscillatory signal with null balance over a line period, and the imaginary component *xlq* represent a measure of the misalignment between mains ideal voltage and load currents (see Mohan et al. (1989)) and do not contribute to the power flow. In this respect, the terms *xld* − *Xld*0, *xlq* are undesired components which should be canceled by the injected filter currents, hence ideally the control problem can be formulated as a state tracking problem, for system (10), of the following reference

$$\mathbf{x}^\*(t) = \begin{bmatrix} \mathbf{x}\_d^\* \ \mathbf{x}\_q^\* \end{bmatrix}^T = \begin{bmatrix} \mathbf{X}\_{ld0} - \mathbf{x}\_{ld} \ \ -\mathbf{x}\_{lq} \end{bmatrix}^T \tag{13}$$

a prefect tracking of this reference would ensure pure sinusoidal mains currents perfectly aligned with the mains voltages. However, this ideal objective is in contrast with the requirement to have a DC-link voltage bounded behavior. In order to formally motivate this claim, consider the steady state voltage dynamics in case perfect tracking of the power reference *x*∗(*t*) is achieved, after some computations it results

$$\frac{dv^2}{dt} = \varepsilon L \left( d\_0 + M(\mathbf{R}, L) \mathbf{x}^\*(t) - \dot{\mathbf{x}}^\*(t) \right)^T \mathbf{x}^\* := \varepsilon \Psi(\mathbf{x}^\*(t)) \tag{14}$$

the signal *�*Ψ(*x*∗(*t*)) which drives the integrator is periodic with period *T* = 1/ *fm*, and it is composed by the sum of a zero mean value signal *�L*(*d*<sup>0</sup> − *x*˙ <sup>∗</sup>)*Tx*∗, and the signal *�L*(*M*(*R*, *L*)*x*∗)*Tx*<sup>∗</sup> which has negative mean value as long as parasitic resistance *R* or reference *x*∗ are not zero. By this, no matter the starting voltage value of the DC-link, the capacitor will be discharged and the voltage will drop, providing a loss of controllability of the system.

To avoid this phenomenon, the reference must be revised, taking into account an additional power term, which should be drained from the line grid by the active filter, in order to compensate for its power losses. Following this motivation, and recalling that the unique useful component for the energy exchange is the real part of the power variables, the ideal reference signal (13) is modified as

$$\mathbf{x}\_{\varphi\_0}^\* = \mathbf{x}^\* + (\varphi\_0 \ \mathbf{0})^T \tag{15}$$

in which *ϕ*<sup>0</sup> is a solution of the following equation

$$R\varphi\_0^2 - E\_{md}\varphi\_0 + Rf\_m \int\_0^{1/f\_m} (\mathbf{x}\_d^{\*2}(\tau) + \mathbf{x}\_q^{\*2}(\tau))d\tau = 0\tag{16}$$

this represents the power balancing condition which guarantees that the internal voltage dynamics in case of perfect tracking of the modified reference *x*∗ *<sup>ϕ</sup>*<sup>0</sup> is

$$\frac{dv^2(t)}{dt} = \varepsilon \Psi(\mathfrak{x}^\*\_{\varphi\_0}(t))\tag{17}$$

bound *vm*; hence, after a transient period, the controller can be switched on having the initial

<sup>351</sup> Analysis, Dimensioning and Robust Control

it has to be considered as an unknown variable, to be reconstructed by estimating the power

The aim of this section is to present a precise algorithm to properly select the SAF hardware components, two different design objectives are considered, the first is to select the minimal component values dependent on the level of current distortion imposed by the load, while the second is to find the minimum capacitor value necessary to compensate all the possible loads compatible with the maximum current rating of the inverter switching devices. Both the methods are control-oriented, that is they ensure the feasibility of control objectives stated in 2.2 and that control input saturation is avoided under nominal load and line voltage

The proposed design method is based on the model derived in Section 2, a further approximation is considered with respect to equation (6); the inductors are modeled as pure inductance, that is the parasitic resistance *R* is neglected, while ideal mains voltage tern and

The inductance value can be selected regardless the loads, hence this part of the design

The design criterion is based on the maximum current ripple Δ*IMpp* allowed for the filter currents; current ripple is a consequence of the PWM technique applied to obtain the reference

∗(*t*) + Δ*i*(*t*)

by simple computation it can be showed that the worst ripple case occurs when the desired

assuming that the DC-link voltage has constant value *V* in a PWM period, the peak to peak

*d*Δ*i*(*t*)

where the sampling period *Ts* and the PWM frequency *fPWM* are assumed already set before starting the sizing procedure. If the peak to peak ripple must be bounded by the desired

*abc*(*t*) + Δ*uabc*(*t*)

actual command vector *uabc*(*t*) and filter current *i*(*t*) are affected by a ripple component

*i*(*t*) = *i*

*uabc*(*t*) = *u*<sup>∗</sup>

*<sup>L</sup>*Δ*i*(*t*)

 *t*+*Ts*/2 *t*

substituting these expressions in the state equation (6) it turns out

Δ*Ipp* =

*abc*, it has to be bounded in order to limit high frequency distortion. The

*abc* is in the middle of a feasibility hexagon side (see Fig. 3). In this condition,

*dt dt* <sup>=</sup> *<sup>V</sup>*

*dt* <sup>=</sup> <sup>−</sup>Δ*uabc*(*t*)*v*(*t*) (23)

<sup>6</sup> *fPWML* (24)

(22)

*<sup>ϕ</sup>*<sup>0</sup> depends on parasitic resistance *R* through (16), hence

voltage value inside the admissible region as required by objective *B*.

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

losses by means of a proper elaboration of the DC-link voltage signal.

converter switches are considered as in the previous section.

procedure is the same for both the design objectives previously defined.

Finally it is further to remark that *x*∗

conditions.

**3.1 Inductance value selection**

command value *u*∗

command value *u*∗

current ripple is

**3. Shunt active filter sizing methodology**

and the right hand side *�*Ψ(*x*∗ *<sup>ϕ</sup>*<sup>0</sup> ) is periodic with period 1/ *fm* with zero mean value. A brief discussion is needed for the solutions of equation (16), it has two real positive solutions if the following condition is verified

$$E\_{md}^2 \ge 4\mathbb{R}^2 f\_m \int\_0^{1/f\_m} (\mathbf{x}\_d^{\*2}(\tau) + \mathbf{x}\_q^{\*2}(\tau)) d\tau \tag{18}$$

from a physical viewpoint relation (18) set an upper bound on the admissible undesired components which can be compensated and on the parasitic resistance *R*, however, as typically *Emd* >> *R*, this condition is not limitative at all. The two solutions of (16) under condition (18) are

$$\begin{split} \varphi\_0 &\approx \frac{R}{E\_{md}} f\_m \int\_0^{1/f\_m} (\mathbf{x}\_d^{\*2}(\tau) + \mathbf{x}\_q^{\*2}(\tau)) d\tau \approx 0 \\ \varphi\_0 &\approx \frac{E\_{md}}{R} \end{split} \tag{19}$$

the first solution, minimizing the power drained from the line grid to compensate the power losses, is the physically most plausible, because the power consumed by parasitic resistances in the filter is usually quite small, hence it will be considered throughout the chapter.

The control problem which will be faced in this work can now be precisely stated; the issue is to design the control vector *uabc* in a way such that the following objectives are fulfilled:

*A*) Given the reference signal *x*∗ *<sup>ϕ</sup>*<sup>0</sup> defined in (15), asymptotic tracking must be achieved, that is

$$\lim\_{t \to \infty} (\mathfrak{x}(t) - \mathfrak{x}^\*\_{\varphi\_0}) = 0; \tag{20}$$

*B*) Given a safe voltage range [*vm*, *vM*], with *vM* > *vm* > 0, and assuming *v*(*t*0) ∈ [*vm*, *vM*], it is required that

$$v(t) \in [v\_{m\prime}, v\_M], \forall t > t\_0;\tag{21}$$

it can be verified that the tracking of the modified power reference is potentially achievable keeping the voltage dynamics inside the safe region, only if the zero mean value oscillating component of *�*Ψ(*x*∗ *<sup>ϕ</sup>*<sup>0</sup> ) is properly bounded, this can be ensured by a suitable capacitor design. In the regulator design, saturation of the actual input *u*1, imposed by PWM strategy, will not be taken explicitly into account, also this approximation takes advantage of a correct sizing methodology; as it will become clear in Section 3, a suitable choice of the DC-link voltage lower bound *vm*, depending on the currents to be compensated for, has to be made to meet the constraint *u*<sup>1</sup> ∈ [0, 1], at least when the power tracking error is reasonably small.

A further consideration needs to be made on the requirement *v*(*t*0) ∈ [*vm*, *vM*]; according to the AC/DC boost converter theory (Mohan et al., 1989), the natural response of the system would lead the DC-link voltage at twice the line voltage peak value, due to the resonant behavior of the *LC* pair and the free-wheeling diodes of the switching bridge. If a proper design has been performed, this value is expected to be greater than the voltage range lower bound *vm*; hence, after a transient period, the controller can be switched on having the initial voltage value inside the admissible region as required by objective *B*.

Finally it is further to remark that *x*∗ *<sup>ϕ</sup>*<sup>0</sup> depends on parasitic resistance *R* through (16), hence it has to be considered as an unknown variable, to be reconstructed by estimating the power losses by means of a proper elaboration of the DC-link voltage signal.

### **3. Shunt active filter sizing methodology**

8 Will-be-set-by-IN-TECH

this represents the power balancing condition which guarantees that the internal voltage

*dt* <sup>=</sup> *�*Ψ(*x*<sup>∗</sup>

discussion is needed for the solutions of equation (16), it has two real positive solutions if the

from a physical viewpoint relation (18) set an upper bound on the admissible undesired components which can be compensated and on the parasitic resistance *R*, however, as typically *Emd* >> *R*, this condition is not limitative at all. The two solutions of (16) under

the first solution, minimizing the power drained from the line grid to compensate the power losses, is the physically most plausible, because the power consumed by parasitic resistances

The control problem which will be faced in this work can now be precisely stated; the issue is to design the control vector *uabc* in a way such that the following objectives are fulfilled:

(*x*(*t*) − *x*<sup>∗</sup>

*B*) Given a safe voltage range [*vm*, *vM*], with *vM* > *vm* > 0, and assuming *v*(*t*0) ∈ [*vm*, *vM*], it

it can be verified that the tracking of the modified power reference is potentially achievable keeping the voltage dynamics inside the safe region, only if the zero mean value oscillating

In the regulator design, saturation of the actual input *u*1, imposed by PWM strategy, will not be taken explicitly into account, also this approximation takes advantage of a correct sizing methodology; as it will become clear in Section 3, a suitable choice of the DC-link voltage lower bound *vm*, depending on the currents to be compensated for, has to be made to meet

A further consideration needs to be made on the requirement *v*(*t*0) ∈ [*vm*, *vM*]; according to the AC/DC boost converter theory (Mohan et al., 1989), the natural response of the system would lead the DC-link voltage at twice the line voltage peak value, due to the resonant behavior of the *LC* pair and the free-wheeling diodes of the switching bridge. If a proper design has been performed, this value is expected to be greater than the voltage range lower

the constraint *u*<sup>1</sup> ∈ [0, 1], at least when the power tracking error is reasonably small.

(*x*∗<sup>2</sup>

*<sup>d</sup>* (*τ*) + *<sup>x</sup>*∗<sup>2</sup>

*<sup>d</sup>* (*τ*) + *<sup>x</sup>*∗<sup>2</sup>

 1/ *fm* 0

 1/ *fm* <sup>0</sup> (*x*∗<sup>2</sup>

in the filter is usually quite small, hence it will be considered throughout the chapter.

lim *t*→∞ *<sup>ϕ</sup>*<sup>0</sup> is

*<sup>ϕ</sup>*<sup>0</sup> ) is periodic with period 1/ *fm* with zero mean value. A brief

*<sup>q</sup>* (*τ*))*dτ* ≈ 0

*<sup>ϕ</sup>*<sup>0</sup> defined in (15), asymptotic tracking must be achieved, that

*v*(*t*) ∈ [*vm*, *vM*], ∀*t* > *t*0; (21)

*<sup>ϕ</sup>*<sup>0</sup> ) is properly bounded, this can be ensured by a suitable capacitor design.

*<sup>ϕ</sup>*<sup>0</sup> ) = 0; (20)

*<sup>ϕ</sup>*<sup>0</sup> (*t*)) (17)

*<sup>q</sup>* (*τ*))*dτ* (18)

(19)

dynamics in case of perfect tracking of the modified reference *x*∗

*E*2

*<sup>ϕ</sup>*<sup>0</sup> <sup>≈</sup> *<sup>R</sup> Emd fm*

*<sup>ϕ</sup>*<sup>0</sup> <sup>≈</sup> *Emd R*

*md* <sup>≥</sup> <sup>4</sup>*R*<sup>2</sup> *fm*

and the right hand side *�*Ψ(*x*∗

following condition is verified

*A*) Given the reference signal *x*∗

condition (18) are

is

is required that

component of *�*Ψ(*x*∗

*dv*2(*t*)

The aim of this section is to present a precise algorithm to properly select the SAF hardware components, two different design objectives are considered, the first is to select the minimal component values dependent on the level of current distortion imposed by the load, while the second is to find the minimum capacitor value necessary to compensate all the possible loads compatible with the maximum current rating of the inverter switching devices. Both the methods are control-oriented, that is they ensure the feasibility of control objectives stated in 2.2 and that control input saturation is avoided under nominal load and line voltage conditions.

The proposed design method is based on the model derived in Section 2, a further approximation is considered with respect to equation (6); the inductors are modeled as pure inductance, that is the parasitic resistance *R* is neglected, while ideal mains voltage tern and converter switches are considered as in the previous section.

### **3.1 Inductance value selection**

The inductance value can be selected regardless the loads, hence this part of the design procedure is the same for both the design objectives previously defined.

The design criterion is based on the maximum current ripple Δ*IMpp* allowed for the filter currents; current ripple is a consequence of the PWM technique applied to obtain the reference command value *u*∗ *abc*, it has to be bounded in order to limit high frequency distortion. The actual command vector *uabc*(*t*) and filter current *i*(*t*) are affected by a ripple component

$$\begin{aligned} \dot{i}(t) &= \dot{i}^\*(t) + \Delta \dot{i}(t) \\\\ \mu\_{abc}(t) &= \mu\_{abc}^\*(t) + \Delta \mu\_{abc}(t) \end{aligned} \tag{22}$$

substituting these expressions in the state equation (6) it turns out

$$L\frac{d\dot{\boldsymbol{x}}(t)}{dt} = -\Delta\boldsymbol{u}\_{abc}(t)\boldsymbol{v}(t)\tag{23}$$

by simple computation it can be showed that the worst ripple case occurs when the desired command value *u*∗ *abc* is in the middle of a feasibility hexagon side (see Fig. 3). In this condition, assuming that the DC-link voltage has constant value *V* in a PWM period, the peak to peak current ripple is

$$
\Delta I\_{pp} = \int\_{t}^{t + T\_s/2} \frac{d\Delta i(t)}{dt} dt = \frac{V}{6f\_{PWM}L} \tag{24}
$$

where the sampling period *Ts* and the PWM frequency *fPWM* are assumed already set before starting the sizing procedure. If the peak to peak ripple must be bounded by the desired

Fig. 3. Current ripple worst case.

maximum value Δ*IMpp*, the following inequality needs to be fulfilled

$$L \ge \frac{v\_M}{6f\_{PWM} \Delta I\_{Mpp}} \Rightarrow L\_{min} = \frac{v\_M}{6f\_{PWM} \Delta I\_{Mpp}} \tag{25}$$

−200 −100 0 100 200

<sup>353</sup> Analysis, Dimensioning and Robust Control

the above equation is obtained by inversion of equation (10) with *R* = 0 and expressing the model in the *synchronous* reference frame in current rather than in power variables, in order to

The constraints on the command inputs need to be considered too, by (27) the inductance

current ripple limitation we select *L* = *Lmin*. As mentioned, the choice of the of the capacitor voltage lower bound value plays a key role to avoid saturation issues on command inputs, this can be easily verified approximating the hexagon in Fig. 2 with the inscribed circle. In order to avoid control action saturation (assuming perfect power tracking) it must be imposed

> *abc*(*t*)|| *vm*

 *n fm* , *n* + 1 *fm*

*abc* = *v*(*t*)*u*<sup>∗</sup>

<sup>≤</sup> *rin* <sup>=</sup> <sup>1</sup>

√3

*dq* feasible, taking into account also the

, ∀*t* (28)

(30)

*abc*. From (28) design equation for

, *n* = 0, 1, . . . (29)

−200

directly consider the load currents in the design approach.

*abc*|| <sup>=</sup> ||*v*<sup>∗</sup>

*vM* <sup>≥</sup> *vm* <sup>≥</sup> ||*v*<sup>∗</sup>

*abc*(*t*)|| *<sup>v</sup>*(*t*) <sup>≤</sup> ||*v*<sup>∗</sup>

*abc*(*t*)|| *rin*

*p filt*(*t*)=[*vdq*(*t*)]*Ti*

, ∀*t* ∈

usually *vm* is oversized with respect the value given by the inequality above, in order to avoid saturation even if non-zero tracking errors are present. If condition (29) cannot be satisfied, some alternatives need to be considered; the capacitor can be changed in order to adopt an higher upper bound *vM*, when the costs of the project have to be limited and the kind of capacitor cannot be substituted, the number of harmonics considered must be reduced until (29) is satisfied. To preserve the number of harmonics to compensate, the inductance value can be reduced, penalizing the current ripple and then tolerating a greater high frequency

The capacitor value can then be selected assuming an ideal converter and writing the balance equation between the instantaneous reference power at the input of the six switches bridge

> ∗ *dq*(*t*) = *<sup>d</sup>*

*dt* 1 2 *Cv*2(*t*) 

value must be as low as possible in order to make *u*∗


with *rin* the radius of the inscribed circle and *v*<sup>∗</sup>

and the power of the DC-link capacitor, hence

Fig. 4. Hexagon of feasible filter current.

that

*vm* can be obtained

distortion.

−100

0

100

200

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

the upper bound of the voltage range *vM* depends only on the kind of capacitor and it can be supposed already chosen before starting the design procedure, hence the minimum inductance value *Lmin* compatible with the desired maximum current ripple can be selected applying equation (25).

#### **3.2 Load-based approach**

Let us now consider the first design algorithm based on the knowledge of the load to be compensated for. The load distortion will be modeled as in equation (12), taking into account the constraint on the maximum current *Imax* of the device implementing the bridge switches. The switching devices sizing depends on the total amount of power (distorted and reactive) *P* = 3*VmRMS ISAFRMS* that the filter has to compensate for (if the load is known then *P* is known), hence by the route mean square value *ISAFRMS*, the maximum current that the switches need to drain can be readily obtained as *Imax* <sup>=</sup> <sup>√</sup> 2*ISAFRMS*.

The desired filter currents (denoted with ∗) necessary to fulfill the tracking objective *A* defined in 2.2 can be effectively imposed by the converter if each component is less than the maximum allowed value, i.e

$$\dot{\mathbf{u}}^{\*}(t) = \begin{bmatrix} \dot{\mathbf{i}}\_{a}^{\*}(t) \ \dot{\mathbf{i}}\_{b}^{\*}(t) \ \dot{\mathbf{i}}\_{c}^{\*}(t) \end{bmatrix}^{T} \le I\_{\max} [1 \ 1 \ 1]^{T}, \ \forall t \tag{26}$$

this feasibility condition can be graphically represented considering that each projection of the filter currents vector must be less then *Imax*, hence the feasibility space is an hexagon similar to that reported in Fig. 4 (obtained taking *P* = 45*kVAR* as filter size, *VmRMS* = 220*V* and then *Imax* = 70*A*). Therefore condition (26) can be readily checked considering the inscribed circle in the feasibility hexagon. If the load currents do not satisfy constraint (26) the number of current harmonics to be compensated for has to be reduced, differently, when the filter performance cannot be decreased, the opportunity to connect two shunt active filters to the same load can be considered.

Assuming that an inductance value such that *L* ≥ *Lmin* has been selected, the voltages at the input of the six switches bridge can be calculated as

$$v\_{dq}^\*(t) = v(t)u\_{dq}^\*(t) = \begin{bmatrix} V\_m \\ 0 \end{bmatrix} - L\frac{di\_{dq}^\*}{dt} + \begin{bmatrix} 0 & \omega\_m \\ -\omega\_m & 0 \end{bmatrix} i\_{dq}^\* \tag{27}$$

Fig. 4. Hexagon of feasible filter current.

10 Will-be-set-by-IN-TECH

*u*2

0 0.2 0.4 0.6

the upper bound of the voltage range *vM* depends only on the kind of capacitor and it can be supposed already chosen before starting the design procedure, hence the minimum inductance value *Lmin* compatible with the desired maximum current ripple can be selected

Let us now consider the first design algorithm based on the knowledge of the load to be compensated for. The load distortion will be modeled as in equation (12), taking into account the constraint on the maximum current *Imax* of the device implementing the bridge switches. The switching devices sizing depends on the total amount of power (distorted and reactive) *P* = 3*VmRMS ISAFRMS* that the filter has to compensate for (if the load is known then *P* is known), hence by the route mean square value *ISAFRMS*, the maximum current that the

The desired filter currents (denoted with ∗) necessary to fulfill the tracking objective *A* defined in 2.2 can be effectively imposed by the converter if each component is less than the maximum

this feasibility condition can be graphically represented considering that each projection of the filter currents vector must be less then *Imax*, hence the feasibility space is an hexagon similar to that reported in Fig. 4 (obtained taking *P* = 45*kVAR* as filter size, *VmRMS* = 220*V* and then *Imax* = 70*A*). Therefore condition (26) can be readily checked considering the inscribed circle in the feasibility hexagon. If the load currents do not satisfy constraint (26) the number of current harmonics to be compensated for has to be reduced, differently, when the filter performance cannot be decreased, the opportunity to connect two shunt active filters to the

Assuming that an inductance value such that *L* ≥ *Lmin* has been selected, the voltages at the

*<sup>c</sup>* (*t*)]*<sup>T</sup>* <sup>≤</sup> *Imax*[111]

*u*∗

*u*1

<sup>⇒</sup> *Lmin* <sup>=</sup> *vM*

6 *fPWM*Δ*IMpp*

2*ISAFRMS*.

 0 *ω<sup>m</sup>* −*ω<sup>m</sup>* 0

 *i* ∗

*<sup>T</sup>*, <sup>∀</sup>*<sup>t</sup>* (26)

*dq* (27)

(25)

0

maximum value Δ*IMpp*, the following inequality needs to be fulfilled

*vM* 6 *fPWM*Δ*IMpp*

*L* ≥

switches need to drain can be readily obtained as *Imax* <sup>=</sup> <sup>√</sup>

*i* ∗(*t*)=[*i* ∗ *<sup>a</sup>* (*t*) *i* ∗ *<sup>b</sup>* (*t*) *i* ∗

input of the six switches bridge can be calculated as

*dq*(*t*) = *v*(*t*)*u*<sup>∗</sup>

*dq*(*t*) =

 *Vm* 0 − *L di*∗ *dq dt* <sup>+</sup>

*v*∗

Fig. 3. Current ripple worst case.

applying equation (25).

allowed value, i.e

same load can be considered.

**3.2 Load-based approach**

0.2

0.4

0.6

the above equation is obtained by inversion of equation (10) with *R* = 0 and expressing the model in the *synchronous* reference frame in current rather than in power variables, in order to directly consider the load currents in the design approach.

The constraints on the command inputs need to be considered too, by (27) the inductance value must be as low as possible in order to make *u*∗ *dq* feasible, taking into account also the current ripple limitation we select *L* = *Lmin*. As mentioned, the choice of the of the capacitor voltage lower bound value plays a key role to avoid saturation issues on command inputs, this can be easily verified approximating the hexagon in Fig. 2 with the inscribed circle. In order to avoid control action saturation (assuming perfect power tracking) it must be imposed that

$$||u^\*\_{abc}|| = \frac{||v^\*\_{abc}(t)||}{v(t)} \le \frac{||v^\*\_{abc}(t)||}{v\_m} \le r\_{in} = \frac{1}{\sqrt{3}}, \quad \forall t \tag{28}$$

with *rin* the radius of the inscribed circle and *v*<sup>∗</sup> *abc* = *v*(*t*)*u*<sup>∗</sup> *abc*. From (28) design equation for *vm* can be obtained

$$v\_M \ge v\_m \ge \frac{||v\_{abc}^\*(t)||}{r\_{in}} \forall t \in \left[\frac{n}{f\_m}, \frac{n+1}{f\_m}\right], n = 0, 1, \dots \tag{29}$$

usually *vm* is oversized with respect the value given by the inequality above, in order to avoid saturation even if non-zero tracking errors are present. If condition (29) cannot be satisfied, some alternatives need to be considered; the capacitor can be changed in order to adopt an higher upper bound *vM*, when the costs of the project have to be limited and the kind of capacitor cannot be substituted, the number of harmonics considered must be reduced until (29) is satisfied. To preserve the number of harmonics to compensate, the inductance value can be reduced, penalizing the current ripple and then tolerating a greater high frequency distortion.

The capacitor value can then be selected assuming an ideal converter and writing the balance equation between the instantaneous reference power at the input of the six switches bridge and the power of the DC-link capacitor, hence

$$p\_{filt}(t) = \left[v\_{dq}(t)\right]^T \dot{i}\_{dq}^\*(t) = \frac{d}{dt}\left(\frac{1}{2}\mathbb{C}v^2(t)\right) \tag{30}$$

the corresponding energy can be calculated as

$$E\_{fill}(t) = \int\_{t\_0}^{t} p\_{fill}(\tau)d\tau \tag{31}$$

Once *Eworst*

reference frame is

the switches-based design approach can be selected.

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

higher than the one calculated under ideal conditions.

expression, that by direct computation can be written as

the ideal situation, in order to accumulate more energy.

**4. Robust controller design**

*max* has been determined, substituting its value in (33), the capacitor value relative to

In the discussion above, ideal mains voltages have been assumed, if also the grid line voltages are distorted, the capacitor has to provide more energy to the load, hence its value must be

<sup>355</sup> Analysis, Dimensioning and Robust Control

In case of ideal mains voltages the load instant power is the one calculated in (12) and the only power term that the filter must deliver is *xldn* <sup>=</sup> <sup>∑</sup>*N*+<sup>1</sup> *<sup>n</sup>*=<sup>1</sup> *Xldncos*(*nωmt* <sup>+</sup> *<sup>ψ</sup>dn*) = *Vmildn* having zero mean value. If the mains voltages are distorted, their representation in the synchronous

*vmdq*(*t*)=[*Vm* + *vmdn*, *vmq*]

line voltages harmonic perturbation produces additive terms in the load instantaneous power

the above equation shows that the filter has to provide more power to the load, furthermore the power mean value in a PWM period can be different from zero. Hence also assuming that the mean value becomes zero in a finite time, the capacitor must be oversized with respect to

In this section the control problem defined in 2.2 is addressed, relying upon a suitable capacitor value given by the procedure described in the previous section, the two interlaced objectives *A* and *B* defined in 2.2 can be approached individually by exploiting the principle of singular perturbation. Two independent controllers (reported in the block diagram of Fig. 5) will be designed. An internal model-based controller (IMC) is proposed in order to deal with the problem of robust reference tracking (defined in objective *A*) for the fast subsystems composed by the power variables dynamics, while an independent voltage controller for the slow DC-link voltage subsystem is designed to produce a reference modification *η* which compensate the unknown power losses term *ϕ*0, allowing to achieve objective *B*. The averaged voltage value is chosen as the controlled variable, and a phasor variables representation is exploited to design the regulator, this choice allows for the necessary voltage oscillation during nominal operation, and improves the voltage dynamics behavior with respect to other proposed solutions (Marconi et al., 2007). Stability analysis is carried out in two steps; the *reduced averaged dynamics*, obtained replacing the steady state of the fast subsystem into the slow voltage dynamics and carrying out the average value to obtain a phasor variables representation, and the *boundary layer system*, obtained considering the SAF currents dynamics and an ideal energy storage element, are proved to be asymptotically stabilized by the proposed controllers. Then practical stability for the overall closed-loop error system is stated exploiting well-established singular perturbation and two time-scale systems theory results. Before detailing the proposed control structure, consider the first preliminary control law

*pladd* = *vmdn*(*t*)*ildn*(*t*) + *vmqilq*(*t*) (37)

*u*¯(*t*) = *v*(*t*)*udq*(*t*) (38)

*<sup>T</sup>* (36)

by the hypothesis of sinusoidal load currents and ideal mains voltages *Efilt*(*t*) is periodic of frequency *fm* and its mean value is zero. Defining

$$\begin{aligned} E\_{\max} &= \max |E\_{fill}(t)| \\ v\_{ref} &= \frac{v\_M + v\_m}{2} \end{aligned} \tag{32}$$

and imposing that the voltage variation corresponding to *Emax* is *vref* − *vm*, the capacitor value design equation can be written as

$$\mathcal{C} = \frac{2E\_{\text{max}}}{v\_{ref}^2 - v\_m^2} \tag{33}$$

#### **3.3 Switches-based approach**

As stated at the beginning of this section, a different design method aims to find the capacitor value that makes the filter able to compensate for the worst load compatible with the switches maximum current. If the resulting capacitor value is not too expensive, this method allows to design the filter only knowing the amount of current that has to be compensated.

During the optimization procedure the load currents need to be the only varying parameters while all the other values must be fixed. The inductance value is chosen equal to the minimum compatible with the allowed ripple, while the minimum capacitor voltage *vm* is supposed sufficiently low to make simple the voltage control, and, at the same time, the resulting capacitor value feasible. Writing the filter currents spectrum in the *d* − *q* synchronous reference frame, an expression similar to (12) can be obtained

$$\dot{r}\_{j}(t) = I\_{j0} + \sum\_{n=1}^{N+1} I\_{jn} \cos(2\pi nf\_{m}t + \psi\_{jn}), \quad j = d, q \tag{34}$$

the parameters to be varied in order to calculate the worst *Emax* are the (2*N* + 1) + 1 magnitudes and the 2*N* + 1 phases, so the following optimization problem

$$E\_{max}^{worst} = \max\_{\boldsymbol{z}} \max\_{\boldsymbol{t}} | \int\_{t\_0}^{t} [v\_{d\boldsymbol{q}}(\boldsymbol{\tau})]^T \boldsymbol{i}\_{d\boldsymbol{q}}(\boldsymbol{\tau}) d\boldsymbol{\tau}| \tag{35}$$

has to be solved with respect to the array *z* of 4(*N* + 1) + 1 variables, taking into account the following constraints


Once *Eworst max* has been determined, substituting its value in (33), the capacitor value relative to the switches-based design approach can be selected.

In the discussion above, ideal mains voltages have been assumed, if also the grid line voltages are distorted, the capacitor has to provide more energy to the load, hence its value must be higher than the one calculated under ideal conditions.

In case of ideal mains voltages the load instant power is the one calculated in (12) and the only power term that the filter must deliver is *xldn* <sup>=</sup> <sup>∑</sup>*N*+<sup>1</sup> *<sup>n</sup>*=<sup>1</sup> *Xldncos*(*nωmt* <sup>+</sup> *<sup>ψ</sup>dn*) = *Vmildn* having zero mean value. If the mains voltages are distorted, their representation in the synchronous reference frame is

$$\left[\boldsymbol{v}\_{mdq}(t) = \left[V\_m + \boldsymbol{v}\_{mdn\nu}\,\,\boldsymbol{v}\_{mq}\right]^T\right.\tag{36}$$

line voltages harmonic perturbation produces additive terms in the load instantaneous power expression, that by direct computation can be written as

$$p\_{ladd} = v\_{mdn}(t)i\_{ldn}(t) + v\_{mq}i\_{lq}(t)\tag{37}$$

the above equation shows that the filter has to provide more power to the load, furthermore the power mean value in a PWM period can be different from zero. Hence also assuming that the mean value becomes zero in a finite time, the capacitor must be oversized with respect to the ideal situation, in order to accumulate more energy.

#### **4. Robust controller design**

12 Will-be-set-by-IN-TECH

 *t t*0

by the hypothesis of sinusoidal load currents and ideal mains voltages *Efilt*(*t*) is periodic of

*Emax* = *max*|*Efilt*(*t*)| *vref* <sup>=</sup> *vM* <sup>+</sup> *vm* 2

and imposing that the voltage variation corresponding to *Emax* is *vref* − *vm*, the capacitor value

*<sup>C</sup>* <sup>=</sup> <sup>2</sup>*Emax v*2 *ref* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *m*

As stated at the beginning of this section, a different design method aims to find the capacitor value that makes the filter able to compensate for the worst load compatible with the switches maximum current. If the resulting capacitor value is not too expensive, this method allows to

During the optimization procedure the load currents need to be the only varying parameters while all the other values must be fixed. The inductance value is chosen equal to the minimum compatible with the allowed ripple, while the minimum capacitor voltage *vm* is supposed sufficiently low to make simple the voltage control, and, at the same time, the resulting capacitor value feasible. Writing the filter currents spectrum in the *d* − *q* synchronous

the parameters to be varied in order to calculate the worst *Emax* are the (2*N* + 1) + 1

has to be solved with respect to the array *z* of 4(*N* + 1) + 1 variables, taking into account the

• switches currents must be less than the maximum allowed, that is the current vector must be inside an hexagon similar to that reported in Fig. 4. This can be easily checked

• the control output must be feasible, that is the vector *uabc* must be inside the hexagon reported in Fig. 2. This can be easily checked approximating the hexagon with its inscribed

• harmonics components phases have to be greater than −*π* and less than *π*.

*<sup>t</sup>* | *t t*0

design the filter only knowing the amount of current that has to be compensated.

*N*+1 ∑ *n*=1

magnitudes and the 2*N* + 1 phases, so the following optimization problem

*max* <sup>=</sup> max*<sup>z</sup>* max

reference frame, an expression similar to (12) can be obtained

*ij*(*t*) = *Ij*<sup>0</sup> +

*Eworst*

approximating the hexagon with its inscribed circle;

*p filt*(*τ*)*dτ* (31)

*Ijncos*(2*πn fmt* + *ψjn*), *j* = *d*, *q* (34)

[*vdq*(*τ*)]*Tidq*(*τ*)*dτ*<sup>|</sup> (35)

(32)

(33)

*Efilt*(*t*) =

the corresponding energy can be calculated as

frequency *fm* and its mean value is zero. Defining

design equation can be written as

**3.3 Switches-based approach**

following constraints

circle;

In this section the control problem defined in 2.2 is addressed, relying upon a suitable capacitor value given by the procedure described in the previous section, the two interlaced objectives *A* and *B* defined in 2.2 can be approached individually by exploiting the principle of singular perturbation. Two independent controllers (reported in the block diagram of Fig. 5) will be designed. An internal model-based controller (IMC) is proposed in order to deal with the problem of robust reference tracking (defined in objective *A*) for the fast subsystems composed by the power variables dynamics, while an independent voltage controller for the slow DC-link voltage subsystem is designed to produce a reference modification *η* which compensate the unknown power losses term *ϕ*0, allowing to achieve objective *B*. The averaged voltage value is chosen as the controlled variable, and a phasor variables representation is exploited to design the regulator, this choice allows for the necessary voltage oscillation during nominal operation, and improves the voltage dynamics behavior with respect to other proposed solutions (Marconi et al., 2007). Stability analysis is carried out in two steps; the *reduced averaged dynamics*, obtained replacing the steady state of the fast subsystem into the slow voltage dynamics and carrying out the average value to obtain a phasor variables representation, and the *boundary layer system*, obtained considering the SAF currents dynamics and an ideal energy storage element, are proved to be asymptotically stabilized by the proposed controllers. Then practical stability for the overall closed-loop error system is stated exploiting well-established singular perturbation and two time-scale systems theory results. Before detailing the proposed control structure, consider the first preliminary control law

$$
\ddot{u}(t) = v(t)u\_{dq}(t)\tag{38}
$$

Fig. 5. Controller structure.

which is always well defined provided that *v*(*t*) ≥ *vm* > 0 for all *t* ≥ 0 according to objective *B*. Replacing (38) into (10) yields

$$\begin{aligned} \dot{\mathbf{x}} &= M(\mathbf{R}, L)\mathbf{x} + \frac{1}{L}\ddot{\mathbf{u}} + d\_0 \\ \frac{d\mathbf{v}^2}{dt} &= \varepsilon \ddot{\mathbf{u}}^T \mathbf{x} \end{aligned} \tag{39}$$

and a constant bias *η* which is needed in order to compensate the power losses and to make

<sup>357</sup> Analysis, Dimensioning and Robust Control

In order to fulfill objective *B* the voltage dynamics need to be stabilized, in this respect the subsystem composed by the capacitor voltage dynamics will be considered, a suitable *reduced averaged system* will be sought, and then a controller for the capacitor voltage DC component

The first step is to average the voltage differential equation to obtain the dynamics in the so-called *phasor-variables*, then, a control law, itself expressed on phasor representation, can be designed following an approach similar to that proposed in (Valderrama et. al, 2001), however in this work the only voltage subsystem is controlled using phasor variables, while the power

The controlled variable is chosen to be the time-window averaged value *z*˜*a* of the square voltage error *z*˜, and the averaging is performed over the time interval [*t* − *T*, *t*]. In terms

*t*−*T*

 *t t*−*T*

*u*¯*T*[*x*˜ + *x*<sup>∗</sup>

*<sup>z</sup>*˜(*τ*)*d<sup>τ</sup>* <sup>=</sup> *<sup>z</sup>*˜(*t*) <sup>−</sup> *<sup>z</sup>*˜(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*)

the fact that *z*˜*a* is a zero-order phasor allows to obtain its derivative by simply applying the

note that the average voltage derivative can also be expressed as the difference over one period

this insight connotes the availability of *z*˜*a* for measurement in real time, and, as it will later

All further elaborations will focus on the integral-differential equation (44) representing the averaged error voltage dynamics. This equation depends on *u*¯ which is actually provided by the power tracking controller, to eliminate *u*¯ consider that the differential equation for *x*˜ in

> *<sup>η</sup>* − *x*˙ ∗

where *D*˜ (*x*˜) collects all the terms depending on the power tracking error *x*˜. The next step is to exploit the fact that the reference term *x*∗ is *T*-periodic (*T* = 1/ *fm*), hence it results in a constant value when averaged over this period, this is a key advantage of the averaging

*<sup>η</sup>* + *<sup>d</sup>*0)*Tx*<sup>∗</sup>

*z*˜(*τ*)*dτ* = *�*

*z*˜(*τ*)*dτ* (43)

*<sup>η</sup>* ]*dτ* (44)

*<sup>T</sup>* (45)

*<sup>η</sup> <sup>d</sup><sup>τ</sup>* + *�LD*˜ (*x*˜) (47)

*<sup>η</sup>* <sup>+</sup> *<sup>M</sup>*(*R*, *<sup>L</sup>*)*x*˜ <sup>+</sup> *<sup>d</sup>*<sup>0</sup> <sup>−</sup> ˙*x*˜) (46)

of (Sanders et al., 1991) this average value is a zero-order phasor defined as

same averaging procedure to its differential equation in (42)

˙ *<sup>z</sup>*˜*<sup>a</sup>* <sup>=</sup> <sup>1</sup> *T t t*−*T*

*d*

*dt*(*z*˜*a*) = *<sup>d</sup>*

*u*¯ = *L*(*M*(*R*, *L*)*x*∗

replacing (46) into (44) the following equation is obtained

˙ *<sup>z</sup>*˜*<sup>a</sup>* <sup>=</sup> *�<sup>L</sup> T t t*−*T* *dt t t*−*T*

clarified, it is of crucial importance for an actual implementation of the controller.

*<sup>η</sup>* − *x*˙ ∗

(*M*(*R*, *L*)*x*∗

of the actual voltage, hence

(42) can be rewritten as

*<sup>z</sup>*˜*a*(*t*) = *<sup>t</sup>*

the range [*vm vM*] an invariant subspace for the voltage dynamics.

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

**4.1 Averaging voltage controller**

subsystem is controlled in the real time domain.

will be designed.

now consider the modified power reference

$$\mathbf{x}\_{\eta}^{\*} = \mathbf{x}^{\*} + (\eta \ \mathbf{0})^{T} \tag{40}$$

and define the change of variables

$$
\mathfrak{X} = \mathfrak{x} - \mathfrak{x}\_{\eta'}^\* \quad \mathfrak{z} = v^2 - V^{\*2} \tag{41}
$$

where *V*∗<sup>2</sup> = (*v*<sup>2</sup> *<sup>m</sup>* + *v*<sup>2</sup> *<sup>M</sup>*)/(2) is the reference value for the square DC-link voltage. Note that the requirement B of having *v*(*t*) ∈ [*vm vM*] for all *t* ≥ *t*<sup>0</sup> can be equivalently formulated in the error variable *z*˜ requiring *z*˜(*t*) ∈ [−*l* ∗ *l* <sup>∗</sup>] for all *t* ≥ *t*0, with *l* <sup>∗</sup> = (*v*<sup>2</sup> *<sup>M</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *<sup>m</sup>*)/2. The complete system (39) can be then expressed in the error variables defined in (41), the transformed model results

$$\begin{aligned} \dot{\mathfrak{x}} &= M(\mathbf{R}, \mathbf{L}) \mathfrak{x} - \frac{1}{L} \mathfrak{u} + d\_0 - \dot{\mathfrak{x}}\_{\eta}^\* + M \mathfrak{x}\_{\eta}^\* \\ \dot{\mathfrak{z}} &= \mathfrak{e} \mathfrak{u}^T [\mathfrak{x} + \mathfrak{x}\_{\eta}^\*]. \end{aligned} \tag{42}$$

The controller design will be carried out considering the error dynamics in (42), in summary the idea is to steer the closed loop dynamics toward a steady state in which *z*˜ is free to oscillate within the admissible region, but its mean value is steered to zero (i.e the DC-link voltage mean value is steered to *V*∗), and *x*˜ is steered to zero, i.e the power *x* follows a reference which is the sum of the term *x*∗, which takes into account the undesired harmonic load components, and a constant bias *η* which is needed in order to compensate the power losses and to make the range [*vm vM*] an invariant subspace for the voltage dynamics.

#### **4.1 Averaging voltage controller**

14 Will-be-set-by-IN-TECH

which is always well defined provided that *v*(*t*) ≥ *vm* > 0 for all *t* ≥ 0 according to objective

the requirement B of having *v*(*t*) ∈ [*vm vM*] for all *t* ≥ *t*<sup>0</sup> can be equivalently formulated in the

system (39) can be then expressed in the error variables defined in (41), the transformed model

The controller design will be carried out considering the error dynamics in (42), in summary the idea is to steer the closed loop dynamics toward a steady state in which *z*˜ is free to oscillate within the admissible region, but its mean value is steered to zero (i.e the DC-link voltage mean value is steered to *V*∗), and *x*˜ is steered to zero, i.e the power *x* follows a reference which is the sum of the term *x*∗, which takes into account the undesired harmonic load components,

*η* ].

<sup>∗</sup>] for all *t* ≥ *t*0, with *l*

*<sup>L</sup> <sup>u</sup>*¯ <sup>+</sup> *<sup>d</sup>*<sup>0</sup> <sup>−</sup> *<sup>x</sup>*˙

1 *<sup>L</sup> <sup>u</sup>*¯ <sup>+</sup> *<sup>d</sup>*<sup>0</sup>

*<sup>M</sup>*)/(2) is the reference value for the square DC-link voltage. Note that

∗ *<sup>η</sup>* + *Mx*<sup>∗</sup> *η*

*<sup>η</sup>* = *<sup>x</sup>*<sup>∗</sup> + (*<sup>η</sup>* <sup>0</sup>)*<sup>T</sup>* (40)

*<sup>η</sup>*, *<sup>z</sup>*˜ <sup>=</sup> *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>V</sup>*∗<sup>2</sup> (41)

*<sup>M</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>m</sup>*)/2. The complete

<sup>∗</sup> = (*v*<sup>2</sup>

(39)

(42)

*x*˙ = *M*(*R*, *L*)*x* +

*dv*<sup>2</sup>

*dt* <sup>=</sup> *�u*¯*Tx*

*x*∗

*x*˜ = *x* − *x*<sup>∗</sup>

∗ *l*

˙*x*˜ <sup>=</sup> *<sup>M</sup>*(*R*, *<sup>L</sup>*)*x*˜ <sup>−</sup> <sup>1</sup>

*z*˜ = *�u*¯*T*[*x*˜ + *x*<sup>∗</sup>

˙

Fig. 5. Controller structure.

*B*. Replacing (38) into (10) yields

and define the change of variables

*<sup>m</sup>* + *v*<sup>2</sup>

error variable *z*˜ requiring *z*˜(*t*) ∈ [−*l*

where *V*∗<sup>2</sup> = (*v*<sup>2</sup>

results

now consider the modified power reference

In order to fulfill objective *B* the voltage dynamics need to be stabilized, in this respect the subsystem composed by the capacitor voltage dynamics will be considered, a suitable *reduced averaged system* will be sought, and then a controller for the capacitor voltage DC component will be designed.

The first step is to average the voltage differential equation to obtain the dynamics in the so-called *phasor-variables*, then, a control law, itself expressed on phasor representation, can be designed following an approach similar to that proposed in (Valderrama et. al, 2001), however in this work the only voltage subsystem is controlled using phasor variables, while the power subsystem is controlled in the real time domain.

The controlled variable is chosen to be the time-window averaged value *z*˜*a* of the square voltage error *z*˜, and the averaging is performed over the time interval [*t* − *T*, *t*]. In terms of (Sanders et al., 1991) this average value is a zero-order phasor defined as

$$
\tilde{z}\_a(t) = \int\_{t-T}^t \tilde{z}(\tau)d\tau \tag{43}
$$

the fact that *z*˜*a* is a zero-order phasor allows to obtain its derivative by simply applying the same averaging procedure to its differential equation in (42)

$$\dot{\tilde{z}}\_{a} = \frac{1}{T} \int\_{t-T}^{t} \tilde{z}(\tau) d\tau = \epsilon \int\_{t-T}^{t} \bar{u}^{T} [\tilde{x} + x\_{\eta}^{\*}] d\tau \tag{44}$$

note that the average voltage derivative can also be expressed as the difference over one period of the actual voltage, hence

$$\frac{d}{dt}(\tilde{z}\_d) = \frac{d}{dt} \int\_{t-T}^t \tilde{z}(\tau)d\tau = \frac{\tilde{z}(t) - \tilde{z}(t-T)}{T} \tag{45}$$

this insight connotes the availability of *z*˜*a* for measurement in real time, and, as it will later clarified, it is of crucial importance for an actual implementation of the controller.

All further elaborations will focus on the integral-differential equation (44) representing the averaged error voltage dynamics. This equation depends on *u*¯ which is actually provided by the power tracking controller, to eliminate *u*¯ consider that the differential equation for *x*˜ in (42) can be rewritten as

$$\vec{u} = L(M(R,L)x\_{\eta}^{\*} - \dot{x}\_{\eta}^{\*} + M(R,L)\tilde{x} + d\_{0} - \dot{\tilde{x}}) \tag{46}$$

replacing (46) into (44) the following equation is obtained

$$\dot{\tilde{\mathbf{z}}}\_{a} = \frac{\epsilon \mathcal{L}}{T} \int\_{t-T}^{t} \left( M(\mathcal{R}, L) \mathbf{x}\_{\eta}^{\*} - \dot{\mathbf{x}}\_{\eta}^{\*} + d\_{0} \right)^{T} \mathbf{x}\_{\eta}^{\*} d\tau + \epsilon L \tilde{D}(\tilde{\mathbf{x}}) \tag{47}$$

where *D*˜ (*x*˜) collects all the terms depending on the power tracking error *x*˜. The next step is to exploit the fact that the reference term *x*∗ is *T*-periodic (*T* = 1/ *fm*), hence it results in a constant value when averaged over this period, this is a key advantage of the averaging approach for the voltage system. The *T*-periodic terms in (47) can be summarized to

$$D^\* = \frac{1}{T} \int\_{t-T}^t [(M(\mathbb{R}, L)\mathbf{x}^\* - \dot{\mathbf{x}}^\* + d\_0)^T \mathbf{x}^\*] d\tau \tag{48}$$

can be written as

˙

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

˙

Now it is possible to design the control input *ηa* in order to stabilize the origin of system (53),

<sup>359</sup> Analysis, Dimensioning and Robust Control

*η<sup>a</sup>* = −*KPz*˜*<sup>a</sup>* + *θ*

*θ* = −*�KIz*˜*<sup>a</sup>*

it is further to notice that the *�* factor in the integral action of the controller is introduced to keep the voltage controller speed in scale with the voltage subsystem to control, thus

In order to prove the asymptotic stability of the closed-loop system resulting by the

since *�*, *Emd* are positive, the matrix in (55) is Hurwitz for all *KP* > 0, *KI* > 0, and system (55)

The problem with implementing the regulator (55) is that the resulting control signal is the average value of the actual control input *η*, thus some procedure is required to synthesize a real-world control signal whose mean value satisfies the above conditions. In the SAF specific

it can be rewritten on the left side as the difference over one period, while the right side is

using (45), the derivative of the averaged square voltage error is actually measurable, thus the above formula is implementable. However, while the interconnection between voltage subsystem and controller is stable in sense of the averaged value, a further step is required. In the incremental implementation (58) there is no more an integral action, the control input history is kept in memory for one period, still the controller provides stability for the averaged voltage error *z*˜*a*. Consider now that for the phasor variables system, a stable steady-state guarantees that all the variables have a constant average value, while being allowed to oscillate freely. This property is desired for what concern the capacitor voltage and it is the main motivation for applying the averaging procedure, however implementation according to (58) can introduce undesired periodic oscillation in the control input *η*, moreover oscillation

*<sup>z</sup>*˜*<sup>a</sup>* <sup>+</sup> ˜˙

*<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*KP* ˙

*z*˜*a*(*t*) − *�TKIz*˜*a*(*t*) + *η*(*t* − *T*) (58)

−*EmdKP Emd* −*KI* 0

 *z*˜*<sup>a</sup>* ˜ *θ* 

where, as mentioned, *ϕ*<sup>0</sup> is the smallest solution of equation (16).

a standard PI regulator in the averaged variables is proposed

maintaining the two-time scale behavior of the overall system.

*d dt z*˜*a* ˜ *θ* = *�* 

result asymptotically stable despite the unknown disturbance *ϕ*0.

case this problem can be solved, consider the derivative of signal *ηa*

replaced with the derivative of *ηa* expressed in (54);

1

solving for *η*(*t*) yields

*d dt <sup>η</sup><sup>a</sup>* <sup>=</sup> *<sup>d</sup> dt* 1 *T t t*−*T*

*<sup>T</sup>* [*η*(*t*) <sup>−</sup> *<sup>η</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*)] = <sup>−</sup>*KP* ˙

*<sup>η</sup>*(*t*) = <sup>−</sup>*TKP* ˙

in the closed-loop error dynamics

interconnection of (54) and (53) consider the change of coordinates ˜

*z*˜*<sup>a</sup>* = *�Emd*[*η<sup>a</sup>* − *ϕ*0] (53)

(54)

(55)

*θ* = *θ* − *ϕ*0, which results

*η*(*τ*)*dτ* (56)

*z*˜*<sup>a</sup>* − *�KIz*˜*<sup>a</sup>* (57)

since *x*∗ is periodic in *T*, *D*∗ is a constant disturbance, and, due to power losses induced by the parasitic resistance *R*, it also follows that *D*∗ < 0. For further simplification the integral operator can be applied to the occurring derivative terms. Using definitions (11), (40), after some computations the averaged error voltage dynamics can be expressed completely in phasor variables

$$\dot{\tilde{z}}\_a = \varepsilon [\mathbf{E}\_{md} \eta\_a - 2\mathbf{R} \nu\_a - \mathbf{L} \dot{\nu}\_a + \mathbf{L} \mathbf{D}^\* + \mathbf{L} \tilde{D}] \tag{49}$$

where the following nonlinear term has been defined

$$\boldsymbol{\nu}(t) = \boldsymbol{\eta}(t) \left( \frac{1}{2} \boldsymbol{\eta}(t) + \boldsymbol{x}\_d^\* \right) \tag{50}$$

which enters (49) with its average and its averaged derivative

$$\begin{aligned} \nu\_a(t) &= \frac{1}{T} \int\_{t-T}^t \nu(\tau) d\tau\\ \dot{\nu}\_a(t) &= \frac{\nu\_a(t) - \nu\_a(t-T)}{T} \end{aligned} \tag{51}$$

the averaged error voltage system is thus controlled by means of the averaged control input

$$
\eta\_a(t) = \frac{1}{T} \int\_{t-T}^t \eta(\tau)d\tau. \tag{52}
$$

According to singular perturbation theory, the voltage controller design can be carried out considering only the *reduced dynamics*, obtained confusing the value of *x*˜ with its steady state value *x*˜ = 0.

As previously remarked, this approximation can be introduced thanks to the small value of *�* which, multiplying the voltage dynamics in the second of (42), makes the voltage subsystem much slower with respect to the power dynamics in the first of (42) (this phenomenon is usually referred as two time-scale system behavior) that will approach the steady state much faster then *z*˜. Thus reduced voltage dynamics can be obtained by (49) simply dropping the coupling term *D*˜ , because by definition *D*˜ (0) = 0.

The nonlinear terms *νa*, and *ν*˙*a* cannot be managed easily, beside non-linearity they contain an integral, a time delay and a time-varying term *x*∗ *<sup>d</sup>*. In order to simplify the mathematical treatment, a sort of linearized version of system (49) will be considered. This linear approximation is motivated by several facts; since the parasitic resistance *R* and the filter inductance value *L* are usually very small with respect to the term *Emd* in every realistic setup, nonlinear term are much smaller than the linear ones. Furthermore the component *x*∗ *<sup>d</sup>* has no influence at all in averaging terms if *η* is constant, thanks to the fact that it is *T*-periodic with zero mean value. Hence it will influence the averaged system only while *η* is varying, and also in this case its oscillatory part will be filtered by the averaging procedure. As a result of the previous steps and considerations, the linearized averaged model for the reduced dynamics can be written as

16 Will-be-set-by-IN-TECH

[(*M*(*R*, *L*)*x*<sup>∗</sup> − *x*˙

since *x*∗ is periodic in *T*, *D*∗ is a constant disturbance, and, due to power losses induced by the parasitic resistance *R*, it also follows that *D*∗ < 0. For further simplification the integral operator can be applied to the occurring derivative terms. Using definitions (11), (40), after some computations the averaged error voltage dynamics can be expressed completely in

> 1 2

*<sup>ν</sup>*˙*a*(*t*) = *<sup>ν</sup>a*(*t*) <sup>−</sup> *<sup>ν</sup>a*(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*) *T*

the averaged error voltage system is thus controlled by means of the averaged control input

According to singular perturbation theory, the voltage controller design can be carried out considering only the *reduced dynamics*, obtained confusing the value of *x*˜ with its steady state

As previously remarked, this approximation can be introduced thanks to the small value of *�* which, multiplying the voltage dynamics in the second of (42), makes the voltage subsystem much slower with respect to the power dynamics in the first of (42) (this phenomenon is usually referred as two time-scale system behavior) that will approach the steady state much faster then *z*˜. Thus reduced voltage dynamics can be obtained by (49) simply dropping the

The nonlinear terms *νa*, and *ν*˙*a* cannot be managed easily, beside non-linearity they contain an

treatment, a sort of linearized version of system (49) will be considered. This linear approximation is motivated by several facts; since the parasitic resistance *R* and the filter inductance value *L* are usually very small with respect to the term *Emd* in every realistic setup, nonlinear term are much smaller than the linear ones. Furthermore the component *x*∗

influence at all in averaging terms if *η* is constant, thanks to the fact that it is *T*-periodic with zero mean value. Hence it will influence the averaged system only while *η* is varying, and also in this case its oscillatory part will be filtered by the averaging procedure. As a result of the previous steps and considerations, the linearized averaged model for the reduced dynamics

*T t t*−*T*

*η*(*t*) + *x*∗ *d* 

*ν*(*τ*)*dτ*

*<sup>z</sup>*˜*<sup>a</sup>* <sup>=</sup> *�*[*Emdη<sup>a</sup>* <sup>−</sup> <sup>2</sup>*Rν<sup>a</sup>* <sup>−</sup> *<sup>L</sup>ν*˙*<sup>a</sup>* <sup>+</sup> *LD*<sup>∗</sup> <sup>+</sup> *LD*˜ ] (49)

<sup>∗</sup> + *d*0)*Tx*∗]*dτ* (48)

*η*(*τ*)*dτ*. (52)

*<sup>d</sup>*. In order to simplify the mathematical

(50)

(51)

*<sup>d</sup>* has no

approach for the voltage system. The *T*-periodic terms in (47) can be summarized to

*ν*(*t*) = *η*(*t*)

*<sup>ν</sup>a*(*t*) = <sup>1</sup> *T t t*−*T*

*<sup>η</sup>a*(*t*) = <sup>1</sup>

*<sup>D</sup>*<sup>∗</sup> <sup>=</sup> <sup>1</sup> *T t t*−*T*

˙

where the following nonlinear term has been defined

coupling term *D*˜ , because by definition *D*˜ (0) = 0.

integral, a time delay and a time-varying term *x*∗

which enters (49) with its average and its averaged derivative

phasor variables

value *x*˜ = 0.

$$
\dot{\bar{z}}\_a = \epsilon \to\_{md} [\eta\_a - \varphi\_0] \tag{53}
$$

where, as mentioned, *ϕ*<sup>0</sup> is the smallest solution of equation (16). Now it is possible to design the control input *ηa* in order to stabilize the origin of system (53), a standard PI regulator in the averaged variables is proposed

$$\begin{aligned} \eta\_a &= -K\_P \tilde{z}\_a + \theta \\ \dot{\theta} &= -\varepsilon K\_I \tilde{z}\_a \end{aligned} \tag{54}$$

it is further to notice that the *�* factor in the integral action of the controller is introduced to keep the voltage controller speed in scale with the voltage subsystem to control, thus maintaining the two-time scale behavior of the overall system.

In order to prove the asymptotic stability of the closed-loop system resulting by the interconnection of (54) and (53) consider the change of coordinates ˜ *θ* = *θ* − *ϕ*0, which results in the closed-loop error dynamics

$$
\frac{d}{dt} \begin{bmatrix} \tilde{\boldsymbol{z}}\_a \\ \tilde{\theta} \end{bmatrix} = \epsilon \begin{bmatrix} -\boldsymbol{E}\_{md} \boldsymbol{K}\_P \ \boldsymbol{E}\_{md} \\ -\boldsymbol{K}\_I \ \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \tilde{\boldsymbol{z}}\_a \\ \tilde{\theta} \end{bmatrix} \tag{55}
$$

since *�*, *Emd* are positive, the matrix in (55) is Hurwitz for all *KP* > 0, *KI* > 0, and system (55) result asymptotically stable despite the unknown disturbance *ϕ*0.

The problem with implementing the regulator (55) is that the resulting control signal is the average value of the actual control input *η*, thus some procedure is required to synthesize a real-world control signal whose mean value satisfies the above conditions. In the SAF specific case this problem can be solved, consider the derivative of signal *ηa*

$$\frac{d}{dt}\eta\_{\!\!t} = \frac{d}{dt}\frac{1}{T} \int\_{t-T}^{t} \eta(\tau)d\tau \tag{56}$$

it can be rewritten on the left side as the difference over one period, while the right side is replaced with the derivative of *ηa* expressed in (54);

$$\frac{1}{T}[\eta(t) - \eta(t - T)] = -K\_P \dot{\tilde{z}}\_a + \dot{\tilde{\theta}} = -K\_P \dot{\tilde{z}}\_a - \epsilon K\_I \tilde{z}\_a \tag{57}$$

solving for *η*(*t*) yields

$$\eta(t) = -TK\_P \dot{\tilde{z}}\_a(t) - \epsilon TK\_I \tilde{z}\_a(t) + \eta(t - T) \tag{58}$$

using (45), the derivative of the averaged square voltage error is actually measurable, thus the above formula is implementable. However, while the interconnection between voltage subsystem and controller is stable in sense of the averaged value, a further step is required. In the incremental implementation (58) there is no more an integral action, the control input history is kept in memory for one period, still the controller provides stability for the averaged voltage error *z*˜*a*. Consider now that for the phasor variables system, a stable steady-state guarantees that all the variables have a constant average value, while being allowed to oscillate freely. This property is desired for what concern the capacitor voltage and it is the main motivation for applying the averaging procedure, however implementation according to (58) can introduce undesired periodic oscillation in the control input *η*, moreover oscillation will persist being remembered through the time delay term. In summary, while *ηa* will approach the constant power loss value *ϕ*0, the actual input *η* might be any periodic signal with average value equal to *ϕ*0. Recalling that *η* modifies the real power reference value *x*∗ *d*, any oscillation will result in a non-zero error for the power tracking controller. In order to avoid this situation the following term can be added to (58)

$$d\_{\eta}(t) = \eta(t - T) - \eta\_{a}(t - T/2) \tag{59}$$

*d*(*t*) can be seen as the output of the following linear system

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

Ω*<sup>j</sup>* =

˙*x*˜ = (*M*(*R*, *<sup>L</sup>*) <sup>−</sup> <sup>1</sup>

*ξ* + *Qx*˜.

˙*x*˜ = (*M*(*R*, *<sup>L</sup>*) <sup>−</sup> <sup>1</sup>

*ξ* + *Qx*˜.

following internal model-based controller can be designed

as Ω = *blkdiag*(Ω*j*) with Ω<sup>0</sup> = 0 and

the internal model error variables as ˜

dynamics, hence by (47), (54) and ˙

the controller matrices as

closed-loop error dynamics can be rewritten as

˙ ˜ *ξ* = Φ ˜

˜˙ *ξ* = Φ ˜

> *K* = *k kd* 0 0 *kq* , *Q* =

non-singular solutions of the following Sylvester equations:

*<sup>w</sup>*˙ *<sup>i</sup>*(*t*) = <sup>Ω</sup>*wi*(*t*), *wi* <sup>∈</sup> **<sup>R</sup>**2*N*+<sup>1</sup> *dim*(*t*) = Γ*iwi*(*t*), *i* = *d*, *q*

<sup>361</sup> Analysis, Dimensioning and Robust Control

where <sup>Γ</sup>*<sup>i</sup>* <sup>∈</sup> **<sup>R</sup>**(1×2*N*+1) are suitably defined vectors and matrix <sup>Ω</sup> <sup>∈</sup> **<sup>R</sup>**(2*N*+1)×(2*N*+1) is defined

with the pairs (Γ*i*,Ω) observable. Defining Φ = *blkdiag*(Ω, Ω) and Γ = *blkdiag*(Γ*d*, Γ*q*), the

*ξ* = Φ*ξ* + *Qx*˜

where matrices *Q* and *K* need to be properly assigned. Once chosen *u*¯ as in (67) and defined

*<sup>K</sup>*)*x*˜ <sup>−</sup> <sup>1</sup> *L* Γ ˜

According to the general two time-scale averaging theory, the power tracking problem can be studied focusing on the *boundary layer system*, obtained by putting *�* = 0 into the overall error

*<sup>L</sup> <sup>K</sup>*)*x*˜ <sup>−</sup> <sup>1</sup>

Now matrices *K*, *Q* need to be selected such that asymptotic stability is provided for the boundary layer system. Define two arbitrary Hurwitz matrices *Fd*, *Fq* ∈ **R**(2*N* + 1) × (2*N* + 1), and two arbitrary vectors *Gd*, *Gq* such that the pairs (*Fd*, *Gd*), (*Fq*, *Gq*) are controllable, taking

> *E*−<sup>1</sup> *<sup>d</sup>* 0 0 *E*−<sup>1</sup> *q*

with *kd*, *kq* two arbitrary positive scalars, *k* a positive design parameter, and *Ed*, *Eq* defined as

*FdEd* − *Ed*Ω*<sup>d</sup>* = −*Gd*Γ*<sup>d</sup> FqEq* − *Eq*Ω*<sup>q</sup>* = −*Gq*Γ*<sup>q</sup>*

*L* Γ ˜

*<sup>ξ</sup>* <sup>=</sup> *<sup>ξ</sup>* <sup>−</sup> *Lw*, where *<sup>w</sup>* := [*w<sup>T</sup>*

*ξ* + *f*(*z*˜*a*, ˜

*θ* = 0, thus system (68) becomes

*ξ* + *f*(*z*˜*a*, ˜

 *Gd* 0 0 *Gq*

, *j* = 1, . . . , *N* (66)

*<sup>u</sup>*¯ <sup>=</sup> <sup>Γ</sup>*<sup>ξ</sup>* <sup>+</sup> *Kx*˜ (67)

*<sup>d</sup>* , *<sup>w</sup><sup>T</sup> q* ]

*θ*, 0, 0, 0)

*θ*, ˙ *<sup>z</sup>*˜*a*, ˜˙ *θ*, *�*)

*<sup>T</sup>*, the power subsystem

*K* (70)

 0 *jω<sup>m</sup>* −*jω<sup>m</sup>* 0

˙

*L*

*<sup>z</sup>*˜*<sup>a</sup>* <sup>=</sup> 0, ˙ ˜ (65)

(68)

(69)

(71)

the reason of this modification is to cancel the oscillations stored in memory, by correcting the stored signal towards its own mean value *ηa*(*t* − *T*/2). It is important to remark that the averaged value is not the actual mean value of its corresponding signal, the mean value *sm* of a signal *s*(*t*) is defined as

$$s\_m = \frac{1}{T} \int\_{t - T/2}^{t + T/2} s(\tau) d\tau \tag{60}$$

the above equation is identical to the zero-order phasor definition, except for a time shift of *T*/2. For this reason the mean value of the stored signal *η*(*t* − *T*) has been expressed as its time shifted average value, note that the mean value of this stored signal can be computed because also its "future" values are available. The additive term *d<sup>η</sup>* is a zero mean value signal, because it is obtained removing its DC-value to a periodic signal. Since the control input *η* enters the averaged system (55) after being averaged itself, any modification having zero mean value will not affect the behavior of the averaged system dynamics. Hence the final implementation of control input together with (59) is

$$\eta(t) = -T\mathcal{K}\_P \dot{\tilde{z}}\_a - \epsilon T\mathcal{K}\_I \tilde{z}\_a + \eta\_a(t - T/2) \tag{61}$$

this controller will not introduce undesired oscillation because it depends solely on averaged signals, whose simplified dynamics (55) cannot give oscillations.

#### **4.2 Power tracking controller**

The voltage controller output reported in (61) can be replaced into the filter error power dynamics in (42), recalling also equation (54), it turns out

$$\dot{\tilde{\mathbf{x}}} = M(\mathcal{R}, L)\tilde{\mathbf{x}} - \frac{1}{L}\bar{\boldsymbol{u}} + d(t) + f(\varepsilon, \tilde{z}\_{a\prime}\tilde{\theta}, \dot{\tilde{z}}\_{a\prime}\dot{\tilde{\theta}}) \tag{62}$$

where

$$d(t) = d\_0 + M(\mathcal{R}, L)\mathbf{x}^\* - \dot{\mathbf{x}}^\* + M(\mathcal{R}, L)\boldsymbol{\varrho}\_0 \tag{63}$$

is a T-periodic term composed by the sum of a constant term and sinusoids having known frequency, while

$$\begin{split} f(\tilde{z}\_a, \tilde{\theta}, \dot{\tilde{z}}\_a, \dot{\tilde{\theta}}, \varepsilon) &= \text{TK}\_p \ddot{\tilde{z}}\_a + \varepsilon \mathcal{K}\_l \dot{\tilde{z}}\_a + \mathcal{K}\_p \dot{\tilde{z}}\_a (t - \mathcal{T}/2) - \dot{\tilde{\theta}} (t - \mathcal{T}/2) \\ &+ M(\mathcal{R}, L) [-\mathcal{T} \mathcal{K}\_p \dot{\tilde{z}}\_a - \varepsilon \mathcal{K}\_l \tilde{z}\_a - \mathcal{K}\_p \tilde{z}\_a (t - \mathcal{T}/2) + \tilde{\theta} (t - \mathcal{T}/2)]. \end{split} \tag{64}$$

The problem of forcing *x*˜ in (62) clearly requires the ability of the control law to compensate for the signal *d*(*t*), perfect tracking cannot be achieved by a feedforward action since SAF parameters and *d*(*t*) are not fully known. To comply with uncertainties and provide robustness we propose an internal model-based controller. Each component of the vector *d*(*t*) can be seen as the output of the following linear system

18 Will-be-set-by-IN-TECH

will persist being remembered through the time delay term. In summary, while *ηa* will approach the constant power loss value *ϕ*0, the actual input *η* might be any periodic signal with average value equal to *ϕ*0. Recalling that *η* modifies the real power reference value *x*∗

the reason of this modification is to cancel the oscillations stored in memory, by correcting the stored signal towards its own mean value *ηa*(*t* − *T*/2). It is important to remark that the averaged value is not the actual mean value of its corresponding signal, the mean value *sm* of

> *t*+*T*/2 *t*−*T*/2

the above equation is identical to the zero-order phasor definition, except for a time shift of *T*/2. For this reason the mean value of the stored signal *η*(*t* − *T*) has been expressed as its time shifted average value, note that the mean value of this stored signal can be computed because also its "future" values are available. The additive term *d<sup>η</sup>* is a zero mean value signal, because it is obtained removing its DC-value to a periodic signal. Since the control input *η* enters the averaged system (55) after being averaged itself, any modification having zero mean value will not affect the behavior of the averaged system dynamics. Hence the final implementation

this controller will not introduce undesired oscillation because it depends solely on averaged

The voltage controller output reported in (61) can be replaced into the filter error power

is a T-periodic term composed by the sum of a constant term and sinusoids having known

The problem of forcing *x*˜ in (62) clearly requires the ability of the control law to compensate for the signal *d*(*t*), perfect tracking cannot be achieved by a feedforward action since SAF parameters and *d*(*t*) are not fully known. To comply with uncertainties and provide robustness we propose an internal model-based controller. Each component of the vector

*z*˜*<sup>a</sup>* + *Kp* ˙

*<sup>z</sup>*˜*<sup>a</sup>* <sup>−</sup> *�KIz*˜*<sup>a</sup>* <sup>−</sup> *Kpz*˜*a*(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*/2) + ˜

*<sup>L</sup> <sup>u</sup>*¯ <sup>+</sup> *<sup>d</sup>*(*t*) + *<sup>f</sup>*(*�*, *<sup>z</sup>*˜*a*, ˜

*d<sup>η</sup>* (*t*) = *η*(*t* − *T*) − *ηa*(*t* − *T*/2) (59)

*s*(*τ*)*dτ* (60)

*z*˜*<sup>a</sup>* − *�TKIz*˜*<sup>a</sup>* + *ηa*(*t* − *T*/2) (61)

*θ*, ˙ *<sup>z</sup>*˜*a*, ˙ ˜

*<sup>z</sup>*˜*a*(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*/2) <sup>−</sup> ˙

*θ*) (62)

*<sup>θ</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*/2)]. (64)

<sup>∗</sup> + *M*(*R*, *L*)*ϕ*<sup>0</sup> (63)

˜ *θ*(*t* − *T*/2)

any oscillation will result in a non-zero error for the power tracking controller.

*sm* <sup>=</sup> <sup>1</sup> *T*

*<sup>η</sup>*(*t*) = <sup>−</sup>*TKP* ˙

signals, whose simplified dynamics (55) cannot give oscillations.

˙*x*˜ <sup>=</sup> *<sup>M</sup>*(*R*, *<sup>L</sup>*)*x*˜ <sup>−</sup> <sup>1</sup>

*θ*, *�*) = *TKp* ¨

<sup>+</sup>*M*(*R*, *<sup>L</sup>*)[−*TKp* ˙

*d*(*t*) = *d*<sup>0</sup> + *M*(*R*, *L*)*x*<sup>∗</sup> − *x*˙

*z*˜*<sup>a</sup>* + *�KI* ˙

dynamics in (42), recalling also equation (54), it turns out

In order to avoid this situation the following term can be added to (58)

a signal *s*(*t*) is defined as

of control input together with (59) is

**4.2 Power tracking controller**

*f*(*z*˜*a*, ˜ *θ*, ˙ *<sup>z</sup>*˜*a*, ˙ ˜

where

frequency, while

*d*,

$$\begin{aligned} \dot{w}\_i(t) &= \Omega w\_i(t), \; w\_i \in \mathbb{R}^{2N+1} \\ d\_{im}(t) &= \Gamma\_i w\_i(t), \; i = d, \; q \end{aligned} \tag{65}$$

where <sup>Γ</sup>*<sup>i</sup>* <sup>∈</sup> **<sup>R</sup>**(1×2*N*+1) are suitably defined vectors and matrix <sup>Ω</sup> <sup>∈</sup> **<sup>R</sup>**(2*N*+1)×(2*N*+1) is defined as Ω = *blkdiag*(Ω*j*) with Ω<sup>0</sup> = 0 and

$$
\Omega\_j = \begin{bmatrix} 0 & j\omega\_m \\ -j\omega\_m & 0 \end{bmatrix}, \ j = 1, \ldots, N \tag{66}
$$

with the pairs (Γ*i*,Ω) observable. Defining Φ = *blkdiag*(Ω, Ω) and Γ = *blkdiag*(Γ*d*, Γ*q*), the following internal model-based controller can be designed

$$\begin{aligned} \dot{\xi} &= \Phi \xi + Q \mathfrak{X} \\ \mathfrak{U} &= \Gamma \underline{\mathfrak{x}} + K \mathfrak{X} \end{aligned} \tag{67}$$

where matrices *Q* and *K* need to be properly assigned. Once chosen *u*¯ as in (67) and defined the internal model error variables as ˜ *<sup>ξ</sup>* <sup>=</sup> *<sup>ξ</sup>* <sup>−</sup> *Lw*, where *<sup>w</sup>* := [*w<sup>T</sup> <sup>d</sup>* , *<sup>w</sup><sup>T</sup> q* ] *<sup>T</sup>*, the power subsystem closed-loop error dynamics can be rewritten as

$$\begin{split} \dot{\tilde{\mathfrak{x}}} &= (M(\mathbb{R}, L) - \frac{1}{L} \mathbb{K}) \tilde{\mathfrak{x}} - \frac{1}{L} \Gamma \tilde{\xi} + f(\tilde{z}\_{a\nu} \tilde{\theta}, \dot{\tilde{z}}\_{a\nu} \dot{\tilde{\theta}}, \epsilon) \\ \dot{\tilde{\xi}} &= \Phi \tilde{\xi} + Q \tilde{\mathfrak{x}}. \end{split} \tag{68}$$

According to the general two time-scale averaging theory, the power tracking problem can be studied focusing on the *boundary layer system*, obtained by putting *�* = 0 into the overall error dynamics, hence by (47), (54) and ˙ *<sup>z</sup>*˜*<sup>a</sup>* <sup>=</sup> 0, ˙ ˜ *θ* = 0, thus system (68) becomes

$$\begin{aligned} \dot{\tilde{\mathbf{x}}} &= (M(\mathbf{R}, L) - \frac{1}{L} K) \tilde{\mathbf{x}} - \frac{1}{L} \Gamma \tilde{\xi} + f(\tilde{z}\_d, \tilde{\theta}, 0, 0, 0) \\ \dot{\tilde{\xi}} &= \Phi \tilde{\xi} + Q \tilde{\mathbf{x}}. \end{aligned} \tag{69}$$

Now matrices *K*, *Q* need to be selected such that asymptotic stability is provided for the boundary layer system. Define two arbitrary Hurwitz matrices *Fd*, *Fq* ∈ **R**(2*N* + 1) × (2*N* + 1), and two arbitrary vectors *Gd*, *Gq* such that the pairs (*Fd*, *Gd*), (*Fq*, *Gq*) are controllable, taking the controller matrices as

$$K = k \begin{bmatrix} k\_d & 0 \\ 0 & k\_q \end{bmatrix}, \ Q = \begin{bmatrix} E\_d^{-1} & 0 \\ 0 & E\_q^{-1} \end{bmatrix} \begin{bmatrix} G\_d & 0 \\ 0 & G\_q \end{bmatrix} K \tag{70}$$

with *kd*, *kq* two arbitrary positive scalars, *k* a positive design parameter, and *Ed*, *Eq* defined as non-singular solutions of the following Sylvester equations:

$$\begin{aligned} F\_d E\_d - E\_d \Omega\_d &= -G\_d \Gamma\_d \\ F\_q E\_q - E\_q \Omega\_q &= -G\_q \Gamma\_q \end{aligned} \tag{71}$$

asymptotic stability of the boundary layer system can be stated. In order to prove this claim let us define the vector

$$R\_{\xi} = \left[ -\frac{R}{\Gamma\_{d1}} \begin{array}{c} 0\_{2N} \ \ -\frac{\omega\_{m}L}{\Gamma\_{q1}} \ \ 0\_{2N} \end{array} \right]^{T} \tag{72}$$

it is only local with respect to the set <sup>A</sup>*<sup>z</sup>* for the voltage initial condition (*z*˜(0), ˜

control when the capacitor voltage is inside the admissible range, as regards ˜

values of *ϕ*<sup>0</sup> are usually very small, the restriction on the initial state ˜

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

fulfilled taking ˜

safe range.

**5. Simulation results**

nonlinear load scenario.

*KI* = 3.7.

frequency *fm* = 50*Hz* have been modeled.

compensation performance are summarized.

*θ*(0) = 0.

as remarked in 2.2, this is not a constraint for *z*˜(0), since it is always possible to switch on the

<sup>363</sup> Analysis, Dimensioning and Robust Control

In summary, even if asymptotic stability of the complete system has not been stated, and formally the tracking error is not asymptotically null, in practice the two control objectives defined in 2.2 can be considered achieved, in fact, by properly tuning the control parameters and dimensioning the hardware components, we can ideally (assuming no cost or technology limitations) improve the filter tracking performance keeping the capacitor voltage value in a

Simulation tests have been performed in order to validate the proposed control solution. Two different scenarios have been adopted; first model (6) has been implemented in MATLAB/Simulink and a load scenario with two harmonics at 7*ωm* and 13*ωm* has been chosen. Then, in order to validate the controller performance in a situation closer to a real setup, the proposed continuous-time regulator has been discretized adopting a sampling frequency *fs* = 7*KHz*, then the SAF converter components have been modeled by using Simulink/SimPowerSystems toolbox, and a suitable PWM technique with a carrier frequency equal to *fs* has been implemented. Finally a three-phase diode bridge has been selected as

The following system parameters have been set, according to the procedure illustrated in Section 3; *C* = 4400*μF*, *L* = 3.3*mH*, *R* = 0.12Ω, while the DC-link voltage limits have been set to *vm* = 700*V*, *vM* = 900*V*. Ideal three-phase mains voltages with amplitude *Vm* = 310*V* and

For what concern the simulations in time continuous domain, the internal-model based controller has been tuned to the load disturbances, according to the procedure described in 4.2. As regards the diode rectifier load scenario, the most relevant power disturbances, that is the 6*th* and the 12*th* load current harmonics expressed in the *synchronous d* <sup>−</sup> *<sup>q</sup>* reference frame (corresponding respectively to the 5*th* and the 7*th*, and to the 11*th* and the 13*th* in the fixed reference frame), have been considered. Then the IMC controller has been discretized according to the procedure reported in (Ronchi et al., 2003), thus the following matrices have been selected; Ω = *blkdiag*(Ω0, Ω6, Ω12), Γ*<sup>d</sup>* = Γ*<sup>q</sup>* = (1, 1, 0, 1, 0)*T*, *K* = *diag*(200, 200) and *Q* = 103*diag*(*Qd*, *Qq*), where *Qd* = *Qq* = (40.6, 80.7, 7.15, 78.7, 17.6)*T*. For what concerns the voltage stabilizer described in 4.1, the following parameters have been selected *KP* = 0.3,

Consider now the performance obtained on the first simulation scenario, with ideal SAF model and the 7*th*, 13*th* disturbance harmonics; in Fig. 6 the tracking error on both real and imaginary power variables is reported, as expected, asymptotic tracking is achieved and the vector *x*˜ is steered arbitrary close to the origin. This ideal behavior is confirmed by Fig. 7, 8; the two harmonics currents are totally canceled out by the filter currents, while a small current component oscillating at the first-order harmonic frequency and aligned to the corresponding voltage, arises on the line side due to the voltage controller action. In table 2 the harmonics

For what concerns the voltage controller, in order to validate the stability properties, a value

*θ*(0)). However,

*θ*, since typical

*θ*(0) is always in practice

where Γ*d*1, Γ*q*<sup>1</sup> denote the first element of vectors Γ*d*, Γ*<sup>q</sup>* respectively and 02*<sup>N</sup>* is a zero raw vector having dimension 2*N*. Consider now the change of variables

$$\tilde{\chi} = E\_{\tilde{\xi}}^{\tilde{\Xi}} - E R\_{\tilde{\xi}}(\tilde{\theta}(t - T/2) - K\_p \tilde{z}\_a(t - T/2)) + LG\tilde{x} \tag{73}$$

where *E* = *blkdiag*(*Ed*, *Eq*), *G* = *blkdiag*(*Gd*, *Gq*), in this coordinates system (69) results

$$\begin{aligned} \dot{\mathfrak{X}} &= (M(\mathbb{R}, L) - \frac{1}{L} \mathbb{K} + \Gamma L^{-1} G) \mathfrak{x} - \frac{1}{L} \Gamma E^{-1} \tilde{\chi} \\ \dot{\tilde{\chi}} &= F \tilde{\chi} - L(FG - GM(\mathbb{R}, L)) \tilde{\mathfrak{x}} \end{aligned} \tag{74}$$

where *<sup>F</sup>* <sup>=</sup> *blkdiag*(*Fd*, *Fq*). Using standard linear system tools it can be verified that a value ¯ *<sup>k</sup>* exists, such that <sup>∀</sup> *<sup>k</sup>* <sup>≥</sup> ¯ *k* the state matrix of the system in the new coordinates is Hurwitz, hence asymptotic stability of the boundary layer system can be stated.

#### **4.3 Overall system stability**

Asymptotic stability has been stated for the boundary layer system and a linearized version of the averaged reduced voltage dynamics. Exploiting the main results of the two time-scale averaged systems theory, it can be proved that the two separately designed controllers are able to provide practical stability for the complete system (42), i.e it is possible to claim that the set

$$\{ (\mathfrak{x}, \tilde{\xi}) : \mathfrak{x} = 0, \,\, \tilde{\xi} = 0 \} \times \mathcal{A}\_{\Sigma} \quad \text{with} \quad \mathcal{A}\_{\tilde{\omega}} = \{ (\tilde{\omega}, \tilde{\theta}) : |\tilde{\omega}| \le l^\*, \tilde{\theta} = 0 \}.$$

is *practically stable* (Khalil, 1996) for the closed-loop trajectories of the complete error system. More precisely we can define the positive values *�*∗, ¯ *k*, *l* ∗ *<sup>s</sup>* , such that for all positive *ls* ≤ *l* ∗ *<sup>s</sup>* , *<sup>k</sup>* <sup>≥</sup> ¯ *<sup>k</sup>*, *�* <sup>≤</sup> *�*∗, the trajectories of the overall closed loop system with initial condition (*x*˜(0), ˜ *ξ*(0)) belonging to an arbitrary compact set, and (*z*˜(0), ˜ *θ*(0)), such that *dist*((*z*˜(0), ˜ *θ*(0)), A*z*) ≤ *ls* (*dist* denotes the distance of the initial state vector from the set A*z*), are bounded. Moreover there exist positive *M*, *λ*, and a class KL function (Khalil, 1996) *β* such that

$$\begin{aligned} |\tilde{\mathbf{x}}(t)| \le Me^{-\lambda t} |\tilde{\mathbf{x}}(0)| + \nu \\ dist((\tilde{\boldsymbol{z}}, \tilde{\boldsymbol{\theta}}), \mathcal{A}\_{\boldsymbol{z}}) \le \beta (dist((\tilde{\boldsymbol{z}}(0), \tilde{\boldsymbol{\theta}}(0)), \mathcal{A}\_{\boldsymbol{z}}), \boldsymbol{\epsilon}t) + \nu \end{aligned} \tag{75}$$

for all *ν* > 0. Proof of this result is omitted owing to space limitation, it relies upon the results for two-time scale averaged systems given in (Teel et al., 2003) (see in particular Theorem 1). Analyzing the previous result, it can be clarified how the proposed controller satisfies the objectives in a practical way; by the second of (75) we deduce that ˜ *θ* tends arbitrary close to the power loss term *ϕ*0, while by the first of (75), it can be seen that the power vector *x* is steered arbitrary close to the reference value *x*∗ *<sup>ϕ</sup>*<sup>0</sup> . In particular the asymptotic tracking error can be arbitrary reduced by taking a smaller value for *�*, namely by increasing the capacitor value *C*.

It's further to notice that the practical stability result is semi-global for what concerns the power variables, i.e the initial state (*x*˜(0), ˜ *ξ*(0)) can belong to an arbitrary compact set, while it is only local with respect to the set <sup>A</sup>*<sup>z</sup>* for the voltage initial condition (*z*˜(0), ˜ *θ*(0)). However, as remarked in 2.2, this is not a constraint for *z*˜(0), since it is always possible to switch on the control when the capacitor voltage is inside the admissible range, as regards ˜ *θ*, since typical values of *ϕ*<sup>0</sup> are usually very small, the restriction on the initial state ˜ *θ*(0) is always in practice fulfilled taking ˜ *θ*(0) = 0.

In summary, even if asymptotic stability of the complete system has not been stated, and formally the tracking error is not asymptotically null, in practice the two control objectives defined in 2.2 can be considered achieved, in fact, by properly tuning the control parameters and dimensioning the hardware components, we can ideally (assuming no cost or technology limitations) improve the filter tracking performance keeping the capacitor voltage value in a safe range.

### **5. Simulation results**

20 Will-be-set-by-IN-TECH

asymptotic stability of the boundary layer system can be stated. In order to prove this claim

where Γ*d*1, Γ*q*<sup>1</sup> denote the first element of vectors Γ*d*, Γ*<sup>q</sup>* respectively and 02*<sup>N</sup>* is a zero raw

where *E* = *blkdiag*(*Ed*, *Eq*), *G* = *blkdiag*(*Gd*, *Gq*), in this coordinates system (69) results

*L*

where *<sup>F</sup>* <sup>=</sup> *blkdiag*(*Fd*, *Fq*). Using standard linear system tools it can be verified that a value ¯

Asymptotic stability has been stated for the boundary layer system and a linearized version of the averaged reduced voltage dynamics. Exploiting the main results of the two time-scale averaged systems theory, it can be proved that the two separately designed controllers are able to provide practical stability for the complete system (42), i.e it is possible to claim that

*<sup>ξ</sup>* <sup>=</sup> <sup>0</sup>}×A*z*, *with* <sup>A</sup>*<sup>z</sup>* <sup>=</sup> {(*z*˜, ˜

is *practically stable* (Khalil, 1996) for the closed-loop trajectories of the complete error system.

*<sup>s</sup>* , *<sup>k</sup>* <sup>≥</sup> ¯ *<sup>k</sup>*, *�* <sup>≤</sup> *�*∗, the trajectories of the overall closed loop system with initial condition (*x*˜(0), ˜

(*dist* denotes the distance of the initial state vector from the set A*z*), are bounded. Moreover

for all *ν* > 0. Proof of this result is omitted owing to space limitation, it relies upon the results for two-time scale averaged systems given in (Teel et al., 2003) (see in particular Theorem 1). Analyzing the previous result, it can be clarified how the proposed controller satisfies the

the power loss term *ϕ*0, while by the first of (75), it can be seen that the power vector *x* is

can be arbitrary reduced by taking a smaller value for *�*, namely by increasing the capacitor

It's further to notice that the practical stability result is semi-global for what concerns the

there exist positive *M*, *λ*, and a class KL function (Khalil, 1996) *β* such that

objectives in a practical way; by the second of (75) we deduce that ˜

<sup>|</sup>*x*˜(*t*)| ≤ *Me*−*λ<sup>t</sup>*

*<sup>θ</sup>*), <sup>A</sup>*z*) <sup>≤</sup> *<sup>β</sup>*(*dist*((*z*˜(0), ˜

*k*, *l* ∗


˙*χ*˜ <sup>=</sup> *<sup>F</sup>χ*˜ <sup>−</sup> *<sup>L</sup>*(*FG* <sup>−</sup> *GM*(*R*, *<sup>L</sup>*))*x*˜

<sup>02</sup>*<sup>N</sup>* <sup>−</sup> *<sup>ω</sup>mL* Γ*q*<sup>1</sup>

*<sup>K</sup>* <sup>+</sup> <sup>Γ</sup>*L*−1*G*)*x*˜ <sup>−</sup> <sup>1</sup>

02*<sup>N</sup> T*

*θ*(*t* − *T*/2) − *Kpz*˜*a*(*t* − *T*/2)) + *LGx*˜ (73)

*θ*) : |*z*˜| ≤ *l*

*θ*(0)), such that *dist*((*z*˜(0), ˜

*θ*(0)), A*z*), *�t*) + *ν*

<sup>∗</sup>, ˜ *θ* = 0}

*<sup>s</sup>* , such that for all positive *ls* ≤ *l*

*<sup>ϕ</sup>*<sup>0</sup> . In particular the asymptotic tracking error

*ξ*(0)) can belong to an arbitrary compact set, while

*L* Γ*E*−1*χ*˜

*k* the state matrix of the system in the new coordinates is Hurwitz,

(72)

(74)

∗

*θ*(0)), A*z*) ≤ *ls*

*θ* tends arbitrary close to

*ξ*(0))

(75)

*R<sup>ξ</sup>* =

*χ*˜ = *E* ˜

vector having dimension 2*N*. Consider now the change of variables

˙*x*˜ = (*M*(*R*, *<sup>L</sup>*) <sup>−</sup> <sup>1</sup>

hence asymptotic stability of the boundary layer system can be stated.

*<sup>ξ</sup>* <sup>−</sup> *ER<sup>ξ</sup>* (˜

 − *R* Γ*d*<sup>1</sup>

let us define the vector

*<sup>k</sup>* exists, such that <sup>∀</sup> *<sup>k</sup>* <sup>≥</sup> ¯

**4.3 Overall system stability**

{(*x*˜, ˜

*ξ*) : *x*˜ = 0, ˜

More precisely we can define the positive values *�*∗, ¯

belonging to an arbitrary compact set, and (*z*˜(0), ˜

*dist*((*z*˜, ˜

steered arbitrary close to the reference value *x*∗

power variables, i.e the initial state (*x*˜(0), ˜

the set

value *C*.

Simulation tests have been performed in order to validate the proposed control solution. Two different scenarios have been adopted; first model (6) has been implemented in MATLAB/Simulink and a load scenario with two harmonics at 7*ωm* and 13*ωm* has been chosen. Then, in order to validate the controller performance in a situation closer to a real setup, the proposed continuous-time regulator has been discretized adopting a sampling frequency *fs* = 7*KHz*, then the SAF converter components have been modeled by using Simulink/SimPowerSystems toolbox, and a suitable PWM technique with a carrier frequency equal to *fs* has been implemented. Finally a three-phase diode bridge has been selected as nonlinear load scenario.

The following system parameters have been set, according to the procedure illustrated in Section 3; *C* = 4400*μF*, *L* = 3.3*mH*, *R* = 0.12Ω, while the DC-link voltage limits have been set to *vm* = 700*V*, *vM* = 900*V*. Ideal three-phase mains voltages with amplitude *Vm* = 310*V* and frequency *fm* = 50*Hz* have been modeled.

For what concern the simulations in time continuous domain, the internal-model based controller has been tuned to the load disturbances, according to the procedure described in 4.2. As regards the diode rectifier load scenario, the most relevant power disturbances, that is the 6*th* and the 12*th* load current harmonics expressed in the *synchronous d* <sup>−</sup> *<sup>q</sup>* reference frame (corresponding respectively to the 5*th* and the 7*th*, and to the 11*th* and the 13*th* in the fixed reference frame), have been considered. Then the IMC controller has been discretized according to the procedure reported in (Ronchi et al., 2003), thus the following matrices have been selected; Ω = *blkdiag*(Ω0, Ω6, Ω12), Γ*<sup>d</sup>* = Γ*<sup>q</sup>* = (1, 1, 0, 1, 0)*T*, *K* = *diag*(200, 200) and *Q* = 103*diag*(*Qd*, *Qq*), where *Qd* = *Qq* = (40.6, 80.7, 7.15, 78.7, 17.6)*T*. For what concerns the voltage stabilizer described in 4.1, the following parameters have been selected *KP* = 0.3, *KI* = 3.7.

Consider now the performance obtained on the first simulation scenario, with ideal SAF model and the 7*th*, 13*th* disturbance harmonics; in Fig. 6 the tracking error on both real and imaginary power variables is reported, as expected, asymptotic tracking is achieved and the vector *x*˜ is steered arbitrary close to the origin. This ideal behavior is confirmed by Fig. 7, 8; the two harmonics currents are totally canceled out by the filter currents, while a small current component oscillating at the first-order harmonic frequency and aligned to the corresponding voltage, arises on the line side due to the voltage controller action. In table 2 the harmonics compensation performance are summarized.

For what concerns the voltage controller, in order to validate the stability properties, a value


Table 2. Compensation performance for the two harmonics disturbance scenario.

quite far from the mean voltage reference value (*v*<sup>2</sup> *<sup>m</sup>* + *v*<sup>2</sup> *<sup>M</sup>*)/2 = 800*V* has been chosen as initial condition for the capacitor voltage. As showed in Fig. 9, even though the average value is initialized at zero and needs one period before representing the actual voltage, the controller reacts immediately, thanks to its dependence on the averaged derivative ˙ *z*˜*a*. Hence the voltage averaged error is successfully steered to zero, and the capacitor voltage is brought back to the middle of the safe interval, without exceeding the upper and lower bounds. The initial nonlinear behavior of the voltage error trajectories is originated by the neglected nonlinearities and also by the coupling term *D*˜ (*x*˜), although it has been discarded due to two time-scale behavior hypothesis, it is excited by the internal model controller transient when harmonics compensation starts.

1 1.025 1.05

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

1 1.025 1.05

1 1.025 1.05

time [s]

time [s]

1 1.025 1.05

time [s]

1 1.025 1.05

1 1.025 1.05

time [s]

time [s]

(b) Virtual power component tracking

−5

−2.5

˜*xq* [W]

<sup>365</sup> Analysis, Dimensioning and Robust Control

error.

−25

25

−25

−10

*ia* [A]

Fig. 7. Current and line voltage waveforms on phase *a*: two harmonics load scenario.

0 10

−10

*ila* [A]

0 10

25

0

2.5

5 x 10−5

time [s]

(a) Real power component tracking

Fig. 6. Error variables *x*˜*d*, *x*˜*q*: two harmonics load scenario.

−5

−350

−0.5

−0.25

*ima* [A]

0

0.25

0.5

−175

*vma* [V]

0

175

350

−2.5

˜*xd* [W]

error.

0

2.5

5 x 10−5

As regards the second simulation scenario, carried out in discrete time domain and with a more detailed filter physical model, the power tracking performance are reported in Fig. 10, 11, 12. In this case the power error variables *x*˜*d*, *x*˜*<sup>q</sup>* are higher with respect to the previous situation, due to the fact that the AC/DC rectifier high order harmonics are not compensated by the internal model, furthermore the discretization effects have to be taken into account. However the load currents harmonics for which the controller has been tuned are strongly reduced at the line side, as the currents magnitude spectrum reported in Fig. 12 shows. Analyzing the currents waveform in the time domain (Fig. 11), it can be verified that the mains currents are almost sinusoidal and perfectly aligned with the corresponding phase voltages, hence also the load imaginary power is almost totally compensated. The ripple introduced by the pulse with modulation can be noted on the filter current, it affects also the mains currents, however thanks to a correct inductance sizing, the high frequency distortion is properly bounded. Quantitative performance of the power-tracking controller obtained with this scenario are summarized in Tab. 3.

The current component corresponding to the line frequency oscillation is slightly larger at the line side than at the load side, due to the additional active power drained to compensate for the filter losses.

As regards the averaging voltage controller, a discrete time version has been implemented, while the same initial conditions of the first scenario have been reproduced. In Fig. 13 the squared voltage error, its averaged value and the actual capacitor voltage are reported, also in this case the objective relative to the voltage dynamics behavior is accomplished, similar considerations to those made for the previous scenario can be made.


Table 3. Compensation performance for the diode bridge load scenario.

22 Will-be-set-by-IN-TECH

Harmonic frequency (Hz) *ima* (A) *ila* (A) Compensation percentage 350 0.0039 10 99.96% 650 0.0038 10 99.96%

initial condition for the capacitor voltage. As showed in Fig. 9, even though the average value is initialized at zero and needs one period before representing the actual voltage, the controller

averaged error is successfully steered to zero, and the capacitor voltage is brought back to the middle of the safe interval, without exceeding the upper and lower bounds. The initial nonlinear behavior of the voltage error trajectories is originated by the neglected nonlinearities and also by the coupling term *D*˜ (*x*˜), although it has been discarded due to two time-scale behavior hypothesis, it is excited by the internal model controller transient when harmonics

As regards the second simulation scenario, carried out in discrete time domain and with a more detailed filter physical model, the power tracking performance are reported in Fig. 10, 11, 12. In this case the power error variables *x*˜*d*, *x*˜*<sup>q</sup>* are higher with respect to the previous situation, due to the fact that the AC/DC rectifier high order harmonics are not compensated by the internal model, furthermore the discretization effects have to be taken into account. However the load currents harmonics for which the controller has been tuned are strongly reduced at the line side, as the currents magnitude spectrum reported in Fig. 12 shows. Analyzing the currents waveform in the time domain (Fig. 11), it can be verified that the mains currents are almost sinusoidal and perfectly aligned with the corresponding phase voltages, hence also the load imaginary power is almost totally compensated. The ripple introduced by the pulse with modulation can be noted on the filter current, it affects also the mains currents, however thanks to a correct inductance sizing, the high frequency distortion is properly bounded. Quantitative performance of the power-tracking controller obtained with

The current component corresponding to the line frequency oscillation is slightly larger at the line side than at the load side, due to the additional active power drained to compensate

As regards the averaging voltage controller, a discrete time version has been implemented, while the same initial conditions of the first scenario have been reproduced. In Fig. 13 the squared voltage error, its averaged value and the actual capacitor voltage are reported, also in this case the objective relative to the voltage dynamics behavior is accomplished, similar

> Harmonic frequency (Hz) *ima*(A) *ila*(A) Compensation percentage 0.03 3.88 99.2% 0.04 1.91 97.9% 0.03 1.57 98.1% 0.02 1.08 98.1%

considerations to those made for the previous scenario can be made.

Table 3. Compensation performance for the diode bridge load scenario.

*<sup>m</sup>* + *v*<sup>2</sup>

*<sup>M</sup>*)/2 = 800*V* has been chosen as

*z*˜*a*. Hence the voltage

Table 2. Compensation performance for the two harmonics disturbance scenario.

reacts immediately, thanks to its dependence on the averaged derivative ˙

quite far from the mean voltage reference value (*v*<sup>2</sup>

compensation starts.

for the filter losses.

this scenario are summarized in Tab. 3.

Fig. 6. Error variables *x*˜*d*, *x*˜*q*: two harmonics load scenario.

Fig. 7. Current and line voltage waveforms on phase *a*: two harmonics load scenario.

(b) Load current magnitude spectrum.

1 1.025 1.05

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

1 1.025 1.05

1 1.025 1.05

time [s]

0 200 400 600 800 1000

frequency [Hz]

(b) Load current magnitude spectrum.

time [s]

−25

−20

0

5

10

*ila* [A]

Fig. 12. FFT of the a-phase main current and of the corresponding load current: diode bridge

15

20

−10

*ia* [A]

Fig. 11. Current and line voltage waveforms on phase *a*: diode bridge load scenario.

0

10

20

−10

*ila* [A]

<sup>367</sup> Analysis, Dimensioning and Robust Control

0 10

25

time [s]

1 1.025 1.05

time [s]

0 200 400 600 800 1000

frequency [Hz]

(a) Main current magnitude spectrum.

−350

25

−25

0

5

10

*ima* [A]

load scenario.

15

20

−10

*ima* [A]

0 10

−175

*vma* [V]

0

175

350

Fig. 8. FFT of the a-phase main current and of the corresponding load current: two harmonics load scenario.

(a) Main current magnitude spectrum.

Fig. 9. Voltage controller performance: two harmonics load scenario.

computed average value (bold).

Fig. 10. Error variables *x*˜*d*, *x*˜*q*: diode bridge load scenario.

24 Will-be-set-by-IN-TECH

0

770

−20

error.

−10

˜*xq* [W]

0

10

20

785

800

*v* [*V*] 815

830

5

*ila* [A]

Fig. 8. FFT of the a-phase main current and of the corresponding load current: two

10

15

200 400 600 800 1000

frequency [Hz]

0 0.2 0.4 0.6

time [s]

1 1.025 1.05

time [s]

(b) Virtual power component tracking

(b) Actual capacitor voltage value.

(b) Load current magnitude spectrum.

200 400 600 800 1000

frequency [Hz]

0 0.2 0.4 0.6

time [s]

1 1.025 1.05

time [s]

(a) Real power component tracking

Fig. 10. Error variables *x*˜*d*, *x*˜*q*: diode bridge load scenario.

Fig. 9. Voltage controller performance: two harmonics load scenario.

(a) Square capacitor voltage error and computed average value (bold).

(a) Main current magnitude spectrum.

0

−1

−20

error.

−10

˜*xd* [W]

0

10

20

−0.5

˜*z*, ˜*za* [*V*2]

0

0.5

1 x 10<sup>5</sup>

0.005

*ima* [A]

harmonics load scenario.

0.01

0.015

0.02

Fig. 11. Current and line voltage waveforms on phase *a*: diode bridge load scenario.

Fig. 12. FFT of the a-phase main current and of the corresponding load current: diode bridge load scenario.

Akagi, H. (1996). "New trends in active filters for power conditioning",*IEEE Transactions on*

<sup>369</sup> Analysis, Dimensioning and Robust Control

Bhattacharya, S., Veltman, A., Divan, D. M. & Lorentz, R. D. (1995). "Flux based active filter controller, *IEEE-IAS Annual Meeting*, pp. 2483-2491, Orlando, Florida, USA, 1995. Cavini, A., Rossi, C., & Tilli, A. (2004). "Digital implementation of decoupled nonlinear

Cavini, A., Rossi, C. & Tilli, A. (2004). "Shunt Active Filters controller with new saturation

Chandra, A., Singh, B. & Al.Haddad, K. (2000). "An improved control algorithm of shunt

Gyugy, L. & Strycula, E. (1976). "Active ac power filters",*IEEE-IAS Annual Meeting*, pp.

Hanschke, J., Marconi, L. & Tilli, A. (2006). "Averaging control of the DC-link voltage in shunt

Jeong,S. G. & Woo, M. H (1997). "Dsp-based active power filter with predictive current control", *IEEE Transactions on Industrial Electronics*,vol. 44, pp 329-336, 1997. Kazmierkowski, M. & Malesani, L.(1998). "Current control techniques for three-phase

Krause, P., Wasynczuk, O. & Sudhoff, S.D. (1995). *Analysis of Electric Machinery*, IEEE Press,

Marconi, L., Ronchi, F. & Tilli, A. (2004). "Robust control of shunt active filter based on the

Marconi, L., Ronchi, F. & Tilli, A. (2004). "Robust perfect compensation of load harmonics in

Marconi, L., Ronchi, F. & Tilli, A. (2007). "Robust nonlinear control of shunt active filters for harmonic current compensation", *Automatica 2007*, vol. 43, pp. 252-263, 2007. Mohan, N., Undeland, T. M. & Robbins, W. P. (1989). *Power Electronics.Converters, applications*

Rastogi, M., Mohan, N. & Edris, A. A. (1995). "Hybrid-active filtering of harmonic currents in power systems", *IEEE Transactions on Power Delivery*, vol. 10, pp 1994-2000, 1995. Ronchi, F. & Tilli, A. (2002). "Design methodology for shunt active filters",*EPE-PEMC, 10th*

Ronchi, F., Tilli, A. & Marconi, L. (2003). "Control of an active filter based on the internal model

Sanders, S, Novorolsky, M., Liu, X. & Verghese, G. (1991). "Generalized averaging method

principle: tuning procedure and experimental results",*European control conference*,

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*and design*, Wiley, 2nd Edition, New York, NY (USA), 1989.

*International power electronics and motion control conference*,2002.

Khalil, H. (1996). *Nonlinear Systems*, McMillan, 2nd Edition, New York (USA), 1996.

control strategies for shunt active filters", *IEEE International Conference on Industrial*

strategy", *Annual Conference of the IEEE Industrial Electronics Society*, pp. 546-551,

active filter for voltage regulation, harmonic elimination, power-factor correction, and balancing of nonlinear loads", *IEEE Transactions on Power Electronics*,vol. 15, pp

active filters", *IEEE Conference on Decision and Control*, pp 6211-6216, San Diego, CA,

voltage-source pwm converters: a survey", *IEEE Transactions on Industrial*

internal model principle", *American Control Conference*,vol. 5, pp. 3943-3948, Denver,

shunt active filters", *IEEE Conference on Decision and Control*, pp. 2978-2983, Paradise

*Industrial Applications*,vol. 32, pp. 1312-1332, 1996.

of Shunt Active Filter for Harmonic Currents Compensation in Electrical Mains

*Technology*, pp. 364-369, Hammamet, Tunisia, 2004.

529-535, Cincinnati, Ohio, USA, 1976.

*Electronics*,vol. 45, pp 691-703, 1998.

Piscataway, NY (USA), 1995.

Colorado, USA, 2003.

Island, Bahamas, 2004.

2003.

251-259, 1991.

Busan, Korea, 2004.

495-507, 2000.

USA, 2006.

Fig. 13. Voltage controller performance: diode bridge load scenario.

### **6. Conclusions**

In this chapter a nonlinear robust control solution for a shunt active filter has been proposed, the focus has been firstly put on the hardware components design issue, providing a suitable algorithm, based on the structural system properties, which gives guarantees on the feasibility of the control problem and allows to obtain a crucial time-scale separation between the power and voltage dynamics. Then exploiting nonlinear systems analysis well established tools, such as averaging and singular perturbation theory, an averaging capacitor voltage controller and a power tracking controller based on the internal model principle, have been presented. The former exploits the insight that, regulating the averaged voltage value, makes it possible to ignore the necessary oscillations for a proper filter operation, and improves the voltage dynamics behavior. The second is chosen in order to ensure asymptotic tracking of undesired load current components, providing also robustness with respect to disturbances and model uncertainties.

Saturation issues have not been explicitly addressed in this work, owing to space limitation, however it is of utmost importance to deal with these phenomena for an actual industrial implementation with stability and performance guarantees. Some solutions, for the SAF specific case, have been proposed (see Cavini et al. (2004), Cavini et al. (2004)), however this is still an open research topic. Future effort will thus be devoted to improve the filter performance under control input saturation, analyzing the problem in the context of modern anti-windup approaches, hence providing a rigorous characterization of the system under saturation constraints. Moreover discretization issues relative to the nonlinear controller here discussed will be further analyzed, in order to improve the discrete-time controller performance with respect to that obtained applying standard discretization techniques.

### **7. References**

Akagi, H., Kanagawa, Y. & Nabae, A. (1984). "Instantaneous reactive power compensator comprising switching devices without energy storage components", *IEEE Transactions on Industry Application*,20, 1984.

26 Will-be-set-by-IN-TECH

770

785

800

*v* [*V*]

In this chapter a nonlinear robust control solution for a shunt active filter has been proposed, the focus has been firstly put on the hardware components design issue, providing a suitable algorithm, based on the structural system properties, which gives guarantees on the feasibility of the control problem and allows to obtain a crucial time-scale separation between the power and voltage dynamics. Then exploiting nonlinear systems analysis well established tools, such as averaging and singular perturbation theory, an averaging capacitor voltage controller and a power tracking controller based on the internal model principle, have been presented. The former exploits the insight that, regulating the averaged voltage value, makes it possible to ignore the necessary oscillations for a proper filter operation, and improves the voltage dynamics behavior. The second is chosen in order to ensure asymptotic tracking of undesired load current components, providing also robustness with respect to disturbances and model

Saturation issues have not been explicitly addressed in this work, owing to space limitation, however it is of utmost importance to deal with these phenomena for an actual industrial implementation with stability and performance guarantees. Some solutions, for the SAF specific case, have been proposed (see Cavini et al. (2004), Cavini et al. (2004)), however this is still an open research topic. Future effort will thus be devoted to improve the filter performance under control input saturation, analyzing the problem in the context of modern anti-windup approaches, hence providing a rigorous characterization of the system under saturation constraints. Moreover discretization issues relative to the nonlinear controller here discussed will be further analyzed, in order to improve the discrete-time controller performance with respect to that obtained applying standard discretization techniques.

Akagi, H., Kanagawa, Y. & Nabae, A. (1984). "Instantaneous reactive power compensator

*Transactions on Industry Application*,20, 1984.

comprising switching devices without energy storage components", *IEEE*

815

830

0 0.2 0.4 0.6

time [s]

(b) Actual capacitor voltage value.

0 0.2 0.4 0.6

time [s]

Fig. 13. Voltage controller performance: diode bridge load scenario.

(a) Square capacitor voltage error and computed average value (bold).

−1

−0.5

˜*z*, ˜*za* [*V*2]

**6. Conclusions**

uncertainties.

**7. References**

0

0.5

1 x 10<sup>5</sup>


Singh, B. & Al-Haddad, K. (1999). "A review of active filters for power quality improvement" *IEEE Trans. Ind. Electron.*, vol. 46, pp. 960-971, 1999.

**19** 

*Algeria* 

Achour Abdelyazid

*A. Mira University, Bejaia,* 

*Department of Electrical Engineering,* 

**Passivity Based Control for** 

**Permanent-Magnet Synchronous Motors** 

The Passivity based control (PBC) is a well established technique which has proved very powerful to design robust control for physical system, especially electrical machinery. The PBC have clear physical interpretation in terms of interconnection system with its environment, and are robust overlooked non dissipative effects modelled. These features are extremely valuable in practical implementations of controllers. In this chapter, we show how the PBC can be used to control the speed of permanents magnets synchronous motor (PMSM). In first part, we consider the Euler-Lagrange model in the -referential to design the Passivity Based Voltage Controller. The dq-model of the PMSM is considered to design

The idea of Passivity Based Control (PBC) design is to reshape the natural energy of the system and inject the required damping in such a way that the control objective is achieved. Expected advantages of this approach are the enhanced robustness properties, which stem

The technique has its roots in classical mechanics (Arnold, 1989) and was introduced in the control theory in the seminal paper (Takegaki & Arimoto, 1981). This method has been instrumented as the solution of several robot manipulator (Ailon & Ortega, 1993; Ortega & Spong; Takegaki & Arimoto, 1981) induction motor (Gökder & Simaan, 1997; Kim et al., 1997; Ortega et al., 1996, 1997; Ortega & Loria), and power electronics (Sira-Ramirez et al.,

PBC was also combined with other techniques (Achour & Mendil, 2007; Ortega & García-Canseco 2004a, 2004b; Qiu & Zhao, 2006; Petrović et al., 2001; Travieso-Torres et al., 2006, 2008). The design of two single-input single-output controllers for induction motors based on adaptive passivity is presented in (Travieso-Torres et al., 2008). Given their nature, the two controllers work together with field orientation block. In ((Travieso-Torres et al., 2006), a cascade passivity-based control scheme for speed tracking purposes is proposed. The scheme is valid for a certain class of nonlinear system even with unstable zero dynamic, and it is also useful for regulation and stabilization purposes. A methodology based on energy shaping and passivation principles has been applied to a PMSM in (Petrović et al., 2001). The interconnection and damping structures of the system were assigned using the Port-Controlled Hamiltonian (PCH) structure. The resulting scheme consists of a steady state feedback to which a nonlinear observer is added to estimate the unknown load torque. The

the Passivity Based current Controller in the second part.

from the fact that conciliation of system nonlinearities is avoided.

1995), which were intractable with other stabilization techniques.

**1. Introduction** 


## **Passivity Based Control for Permanent-Magnet Synchronous Motors**

### Achour Abdelyazid

*Department of Electrical Engineering, A. Mira University, Bejaia, Algeria* 

### **1. Introduction**

28 Will-be-set-by-IN-TECH

370 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Singh, B. & Al-Haddad, K. (1999). "A review of active filters for power quality improvement"

Teel, A. R., Moreau, L. & Nesic, D. (2003). "A unified framework for input-to-state stability

Tilli, A., Ronchi, F. & Tonielli, A. (2002). "Shunt active filters: selective compensation of current

Valderrama, G., Mattavelli, P. & Stankovic, A. (2001). "Reactive power and unbalance

Van Harmelen, G. L. & Enslin, J. H. R. (1993). "Real-time dynamic control of dynamic power

in systems with two time scales", *IEEE Transactions on Automatic Control*, vol. 48, pp.

harmonics via state observer", *IEEE-IECON, 28th Annual Conference of the Industrial*

compensation using STATCOM with dissipativity-based control", *IEEE Transactions*

filters in supplies with high contamination", *IEEE Transactions on Power Electronics*,

*IEEE Trans. Ind. Electron.*, vol. 46, pp. 960-971, 1999.

*on Control Systems Technology*, vol.9, no.5, pp. 718-727, 2001.

*Electronics Society*, vol. 2, pp. 874-879, 2002.

15261544, 2003.

vol. 8, pp. 301-308, 1993.

The Passivity based control (PBC) is a well established technique which has proved very powerful to design robust control for physical system, especially electrical machinery. The PBC have clear physical interpretation in terms of interconnection system with its environment, and are robust overlooked non dissipative effects modelled. These features are extremely valuable in practical implementations of controllers. In this chapter, we show how the PBC can be used to control the speed of permanents magnets synchronous motor (PMSM). In first part, we consider the Euler-Lagrange model in the -referential to design the Passivity Based Voltage Controller. The dq-model of the PMSM is considered to design the Passivity Based current Controller in the second part.

The idea of Passivity Based Control (PBC) design is to reshape the natural energy of the system and inject the required damping in such a way that the control objective is achieved. Expected advantages of this approach are the enhanced robustness properties, which stem from the fact that conciliation of system nonlinearities is avoided.

The technique has its roots in classical mechanics (Arnold, 1989) and was introduced in the control theory in the seminal paper (Takegaki & Arimoto, 1981). This method has been instrumented as the solution of several robot manipulator (Ailon & Ortega, 1993; Ortega & Spong; Takegaki & Arimoto, 1981) induction motor (Gökder & Simaan, 1997; Kim et al., 1997; Ortega et al., 1996, 1997; Ortega & Loria), and power electronics (Sira-Ramirez et al., 1995), which were intractable with other stabilization techniques.

PBC was also combined with other techniques (Achour & Mendil, 2007; Ortega & García-Canseco 2004a, 2004b; Qiu & Zhao, 2006; Petrović et al., 2001; Travieso-Torres et al., 2006, 2008). The design of two single-input single-output controllers for induction motors based on adaptive passivity is presented in (Travieso-Torres et al., 2008). Given their nature, the two controllers work together with field orientation block. In ((Travieso-Torres et al., 2006), a cascade passivity-based control scheme for speed tracking purposes is proposed. The scheme is valid for a certain class of nonlinear system even with unstable zero dynamic, and it is also useful for regulation and stabilization purposes. A methodology based on energy shaping and passivation principles has been applied to a PMSM in (Petrović et al., 2001). The interconnection and damping structures of the system were assigned using the Port-Controlled Hamiltonian (PCH) structure. The resulting scheme consists of a steady state feedback to which a nonlinear observer is added to estimate the unknown load torque. The

Passivity Based Control for Permanent-Magnet Synchronous Motors 373

where the outputs of the forward subsystem are the regulated outputs. Second, we design an inner feedback loop that, via the injection of a nonlinear damping term, ensures the controlled subsystem defines a strictly passive map from control signals to regulated outputs. Third, the passivity-based technique is applied to this subsystem leaving the feedback subsystem as a "passive perturbation". This last step involves the definition of the desired closed loop energy function whose associated "target" dynamics evolves on a

The main contribution is in the design of an observer that utilizes the high quality position information and voltage for reconstructing the velocity and current signals. The proposed observer is inspired from the passivity based controller design concept. The problem is tackled by constructing an observer that forces the estimated error to match a desired energy function, thereby preserving the passivity property. In addition, for asymptotic stabilization, damping has to be included in the loop. The main feature of this approach is in the fact that it establishes a concept duality between the controller and observer design strategy. Using passivity concept solves stability of the combined controller-observer design. We will introduce a desired energy function that consists of two parts, one for the closed

The organization of this Section is as follows: In Subsection. 1.2 we present the two phases model of PMSM described by Euler-Lagrange (EL) equations, and his properties. The design procedure and the stability problem of the combined controller-observer are given in Subsection. 1.3. Simulation results are presented in Subsection. 1.4. Finally, concluding

The PMSM uses surface mounted rare earth magnets. We consider the following assumptions: -No significant saliency effects; -negligible damping effects in the rotor; negligible saturation effects; -ideal symmetrical phases and sinusoidal distributed phase windings; -negligible capacity effects in stator windings, considering rigid shaft and not magnetic material in stator. Under the assumptions above, the standard two phases model of PMSM obtained in (Ortega et al., 1997) via direct application of EL equation is

> *Dq R q q q mm m m e m L* (, )

> > <sup>2</sup> (, ) ( ) *<sup>T</sup> em m e*

*e d q e aa*

 is stator current vector; <sup>2</sup> (,) *q q m m* are the rotor angular position and velocity respectively; (q m ) is the flux linkages due to permanent magnets; Ld ,Lq are

*D diag L L R diag R R*

,; ,

*d q W q dq*

( ) () .

*<sup>m</sup> <sup>m</sup> m*

2

*Dq W q q q U ee m m e* 2 e () R (1)

*q q W q q* (3)

(2)

loop controller dynamic and the other for the closed loop observer dynamic.

subspace of the state space ensuring zero error tracking.

remarks are given in Subsection. 1.5.

**2.1.1 Model** 

given by:

where

*e ee q qq* 

<sup>2</sup> , *<sup>T</sup>*

**2.1 Permanent-magnet synchronous motor model** 

authors in (Qiu & Zhao, 2006) developed a PMSM speed control law based on PCH that achieves stabilization via system passivity. In particular, the PCH interconnection and damping matrices were shaped so that the physical (Hamiltonian) system structure is preserved at the closed-loop level. The difference between the physical energy of the system and the energy supplied by the controller forms the closed-loop energy function. A review of the fundamental theory of the Interconnection and Damping Assignment Passivity Based Control technique (IDA-PBC) can be found in (Ortega & García-Canseco 2004a, 2004b). In the concerned papers it was showed the role played by the three matrices (i.e. interconnection, damping, Kernel of system input) of the PCH model in the IDA-PBC design.

The permanent-magnet synchronous motor (PMSM) has numerous advantages over other machines that are conventionally used for ac servo drives. It has a higher torque to inertia ratio and power density when compared to the Induction Motion or the wound-rotor Synchronous Motor, which makes it preferable for certain high-performance applications like robotics and aerospace actuators. However, it presents a difficult control problem. This is due to the following reasons: first, the dynamical model of PMSM is nonlinear. Second, the motor parameters (e.g., stator resistance) can vary considerably from the nominal values. Also, the state variable (velocity and current) measurements are often contaminated with a considerable amount of noise. Generally, velocity and current sensors are omitted due to the considerable saving in cost, and volume.

In Section 2, we propose a design strategy that utilizes the passivity concept in order to develop a combined controller-observer system for Permanent-Magnet Synchronous Motors (PMSM) speed control using only rotor position measurement and voltages applied to the stator windings. To this end, first a desired energy function for the closed loop system is introduced, and then a combined controller-observer system is constructed such that the closed loop system matches this energy function. A damping term is included to ensure asymptotic stability of the closed loop system. The interesting feature of this approach is the fact that it establishes a duality concept between the controller and observer design strategy. Such a duality feature is unique for nonlinear systems. Simulation tests on the combined controller-observer design are provided to show the feasibility and the performances of this method.

The work of Section 3 is related with previous work concerning the voltage control of PMSM (Achour & Mendil, 2007). The PBC has been combined with a variable structure compensator (VSC) in order to deal with important parameter uncertainties plant, without raising the damping values of the controller. The dynamics of the PMSM were represented as feedback interconnection of a passive electrical and mechanical subsystem. The PBC is applied only to the electrical subsystem while the mechanical subsystem has been treated as a passive perturbation. A new passivity based current controller (PBCC) designed using the dq-model of PMSM is proposed in this Section 3.

### **2. Passivity based controller-observer design for permanent magnet synchronous motors**

In this part, we develop a control algorithm based on the passivity concept that forces the PMSM to track desired velocity and torque vectors without the need for velocity and stator current measurements, but using only rotor position and stator voltage measurements.

The passivity-based controller design proceeds as follows. First, we carry out a decomposition of the system dynamics as a feedback interconnection of passive subsystems, where the outputs of the forward subsystem are the regulated outputs. Second, we design an inner feedback loop that, via the injection of a nonlinear damping term, ensures the controlled subsystem defines a strictly passive map from control signals to regulated outputs. Third, the passivity-based technique is applied to this subsystem leaving the feedback subsystem as a "passive perturbation". This last step involves the definition of the desired closed loop energy function whose associated "target" dynamics evolves on a subspace of the state space ensuring zero error tracking.

The main contribution is in the design of an observer that utilizes the high quality position information and voltage for reconstructing the velocity and current signals. The proposed observer is inspired from the passivity based controller design concept. The problem is tackled by constructing an observer that forces the estimated error to match a desired energy function, thereby preserving the passivity property. In addition, for asymptotic stabilization, damping has to be included in the loop. The main feature of this approach is in the fact that it establishes a concept duality between the controller and observer design strategy. Using passivity concept solves stability of the combined controller-observer design. We will introduce a desired energy function that consists of two parts, one for the closed loop controller dynamic and the other for the closed loop observer dynamic.

The organization of this Section is as follows: In Subsection. 1.2 we present the two phases model of PMSM described by Euler-Lagrange (EL) equations, and his properties. The design procedure and the stability problem of the combined controller-observer are given in Subsection. 1.3. Simulation results are presented in Subsection. 1.4. Finally, concluding remarks are given in Subsection. 1.5.

### **2.1 Permanent-magnet synchronous motor model 2.1.1 Model**

The PMSM uses surface mounted rare earth magnets. We consider the following assumptions: -No significant saliency effects; -negligible damping effects in the rotor; negligible saturation effects; -ideal symmetrical phases and sinusoidal distributed phase windings; -negligible capacity effects in stator windings, considering rigid shaft and not magnetic material in stator. Under the assumptions above, the standard two phases model of PMSM obtained in (Ortega et al., 1997) via direct application of EL equation is given by:

$$\text{Rel}\ddot{q}\_{\ell} + \text{V} \,\text{V} \,\text{2} (q\_{m}) \dot{q}\_{m} + \text{Re}\_{\ell} \dot{q}\_{\ell} = \text{LI} \tag{1}$$

$$D\_m \ddot{q}\_m + R\_m \dot{q}\_m = \pi \left(\dot{q}\_{e'} q\_m\right) - \pi\_L \tag{2}$$

$$
\pi \left( \dot{q}\_{e'} q\_m \right) = \mathsf{VV}\_2^T \left( q\_m \right) \dot{q}\_e \tag{3}
$$

where

372 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

authors in (Qiu & Zhao, 2006) developed a PMSM speed control law based on PCH that achieves stabilization via system passivity. In particular, the PCH interconnection and damping matrices were shaped so that the physical (Hamiltonian) system structure is preserved at the closed-loop level. The difference between the physical energy of the system and the energy supplied by the controller forms the closed-loop energy function. A review of the fundamental theory of the Interconnection and Damping Assignment Passivity Based Control technique (IDA-PBC) can be found in (Ortega & García-Canseco 2004a, 2004b). In the concerned papers it was showed the role played by the three matrices (i.e. interconnection, damping, Kernel of system input) of the PCH model in the IDA-PBC

The permanent-magnet synchronous motor (PMSM) has numerous advantages over other machines that are conventionally used for ac servo drives. It has a higher torque to inertia ratio and power density when compared to the Induction Motion or the wound-rotor Synchronous Motor, which makes it preferable for certain high-performance applications like robotics and aerospace actuators. However, it presents a difficult control problem. This is due to the following reasons: first, the dynamical model of PMSM is nonlinear. Second, the motor parameters (e.g., stator resistance) can vary considerably from the nominal values. Also, the state variable (velocity and current) measurements are often contaminated with a considerable amount of noise. Generally, velocity and current sensors are omitted due to the

In Section 2, we propose a design strategy that utilizes the passivity concept in order to develop a combined controller-observer system for Permanent-Magnet Synchronous Motors (PMSM) speed control using only rotor position measurement and voltages applied to the stator windings. To this end, first a desired energy function for the closed loop system is introduced, and then a combined controller-observer system is constructed such that the closed loop system matches this energy function. A damping term is included to ensure asymptotic stability of the closed loop system. The interesting feature of this approach is the fact that it establishes a duality concept between the controller and observer design strategy. Such a duality feature is unique for nonlinear systems. Simulation tests on the combined controller-observer design are provided to show the feasibility and the performances of this

The work of Section 3 is related with previous work concerning the voltage control of PMSM (Achour & Mendil, 2007). The PBC has been combined with a variable structure compensator (VSC) in order to deal with important parameter uncertainties plant, without raising the damping values of the controller. The dynamics of the PMSM were represented as feedback interconnection of a passive electrical and mechanical subsystem. The PBC is applied only to the electrical subsystem while the mechanical subsystem has been treated as a passive perturbation. A new passivity based current controller (PBCC) designed using the

In this part, we develop a control algorithm based on the passivity concept that forces the PMSM to track desired velocity and torque vectors without the need for velocity and stator current measurements, but using only rotor position and stator voltage measurements. The passivity-based controller design proceeds as follows. First, we carry out a decomposition of the system dynamics as a feedback interconnection of passive subsystems,

**2. Passivity based controller-observer design for permanent magnet** 

design.

method.

considerable saving in cost, and volume.

dq-model of PMSM is proposed in this Section 3.

**synchronous motors** 

$$\begin{aligned} D\_e &= \operatorname{diag} \left\{ L\_d \, \biguplus L\_q \right\} ; R\_e = \operatorname{diag} \left\{ R\_a \, \biguplus R\_a \right\} \\ \mathcal{W}\_2 \left( q\_m \right) &= \frac{d \, \mu(q\_m)}{d \, q\_m} . \end{aligned}$$

<sup>2</sup> , *<sup>T</sup> e ee q qq* is stator current vector; <sup>2</sup> (,) *q q m m* are the rotor angular position and velocity respectively; (q m ) is the flux linkages due to permanent magnets; Ld ,Lq are

Passivity Based Control for Permanent-Magnet Synchronous Motors 375

2 e 2 e

2e , L2e are the spaces of 3 and 1 dimension respectively of square integral,

*<sup>T</sup> T T W q W q q qW q m mm e m* (8)

*m*

 

( )- ( ) *<sup>T</sup> Cq C q m m* (10)

(11)

(6)

(7)

(9)

*Hm m mm q D q*

 *qm*

 *m*

essentially bounded functions and their extensions.

to prove the passivity property.

the following compact form:

"workless forces" are given by:

(i.e., C (qm ) is a skew symmetric matrix.)

as C (qm ) verifies:

Remark

Where

Where, *D dia gD D R dia e m* , ; , *gR R e m*

2.1.2.3 Workless forces

Considering the total energy He of the electric subsystem e, that is:

where L3

Proof

L : L L

<sup>1</sup> (, ) ( ) <sup>2</sup> *T T <sup>H</sup> qq q <sup>D</sup> q qq e m e ee m e <sup>e</sup>*

A similar procedure used above to prove the passivity of PMSM can be used to establish the passivity of e, and for mechanical m we consider the energy function <sup>1</sup> <sup>2</sup> () 2

In order to introduce the third property, we note that the model (1)-(3) can be written under

*D q W q q Rq MU* ()

2 12 21 L ,0 ; , ; 0 , - *T T <sup>T</sup> <sup>T</sup> M I q qq e m*

2 2 ( ) ( ) , ( )

Based on the passivity property of the PMSM and the relations (1)-(3), we deduce that the

0 () () () 0

*W q C q W q* 

*m T*

In the present of the saliency effects, the "workless forces" are given by:

( ,) *<sup>m</sup>*

*Cq q C C*

11 1 m <sup>1</sup> (q ) <sup>2</sup> *CWq <sup>m</sup>*

22 2 2 1 1

> 11 12 21 22

*C C*

 

*m*

the direct and quadrate stator inductance respectively; Dm is the rotor inertia; Rm 0 is the mechanical friction; , *<sup>T</sup> U uu* is stator voltage vector; and ,L are the generated and load torque respectively. The subscripts (.)e, (.)m, (.)T denotes the electrical, mechanical and vector transposition respectively.

### **2.1.2 Properties**

In this subsection, we present three properties of the PMSM model, which are useful for the methodology of control design.

2.1.2.1 Passivity property of permanent-magnet synchronous motor

Lemma 1

The PMSM represents a passive system, if , *<sup>T</sup> <sup>T</sup> <sup>U</sup> <sup>L</sup>* and , *<sup>T</sup> <sup>T</sup> q qq e m* are considered as inputs and outputs respectively.

Proof

The total energy H of the PMSM is:

$$H(\dot{q}\_{e'}, \dot{q}\_{m'}, q\_{m}) = \frac{1}{2} \dot{q}\_{e}^{T} D\_{e} \dot{q}\_{e} + \mu^{T} \left(q\_{m}\right) \dot{q}\_{e} + \frac{1}{2} D\_{m} \dot{q}\_{m}^{2} \tag{4}$$

Taking the time derivative of H along the trajectory (1)-(3), we get:

$$\dot{H}(\dot{q}\_{\mathcal{E}}, \dot{q}\_{\mathcal{III}}, q\_{\mathcal{III}}) = -\dot{\eta}^T R \dot{\eta} + \dot{\eta}^T \nu + \frac{d}{dt} \Big(\mu^T (q\_{\mathcal{III}}) \dot{q}\_{\mathcal{E}}\Big) \tag{5}$$

Integrating *H* from zero to > 0, and setting 0 (0) ( ) *<sup>T</sup> H qq m e* , proves the passivity of the PMSM.

2.1.2.2 Passive Feedback Decomposition

Lemma 2

The PMSM can be represented as the negative feedback interconnection of the electrical and mechanical passive subsystems.

Fig. 1. Passive subsystem decomposition.

$$\begin{aligned} \Sigma\_{\varepsilon}: \qquad & \mathcal{L}\_{2\text{-}\varepsilon}^{\mathcal{3}} \to \quad \mathcal{L}\_{2\text{-}\varepsilon}^{\mathcal{3}}\\ & \begin{bmatrix} \mathcal{U} \\ -\dot{q}\_{m} \end{bmatrix} \mapsto \begin{bmatrix} \dot{q}\_{\varepsilon} \\ \tau \end{bmatrix} \end{aligned}$$

$$\begin{aligned} \Sigma\_{\mathfrak{m}} &: \quad \mathrm{L}\_{2\,\mathrm{e}} \to \quad \mathrm{L}\_{2\,\mathrm{e}}\\ (\tau \cdot \tau\_{\mathrm{L}}) &\mapsto \dot{q}\_{\mathrm{m}} \end{aligned}$$

where L3 2e , L2e are the spaces of 3 and 1 dimension respectively of square integral, essentially bounded functions and their extensions.

Proof

374 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

the direct and quadrate stator inductance respectively; Dm is the rotor inertia; Rm 0 is the

load torque respectively. The subscripts (.)e, (.)m, (.)T denotes the electrical, mechanical and

In this subsection, we present three properties of the PMSM model, which are useful for the

1 1 <sup>2</sup> (, , ) ( ) 2 2 *T T Hq q q q Dq q q D q e m m ee m e mm <sup>e</sup>* 

(, , ) ( ) *TT T <sup>d</sup> H q q q q Rq q q q emm m e dt*

The PMSM can be represented as the negative feedback interconnection of the electrical and

3 3 2 e 2 e : L L

*U q*

 

*m*

*q*

<sup>e</sup>

*e*

 

m <sup>L</sup>

*e q*

 

Integrating *H* from zero to > 0, and setting 0 (0) ( ) *<sup>T</sup> H qq m e*

*e*

 *<sup>U</sup> <sup>L</sup>* and ,

*<sup>T</sup> <sup>T</sup> q qq e m* are considered

(4)

(5)

, proves the

  is stator voltage vector; and ,L are the generated and

mechanical friction; , *<sup>T</sup>*

vector transposition respectively.

methodology of control design.

as inputs and outputs respectively.

The total energy H of the PMSM is:

passivity of the PMSM.

Lemma 2

2.1.2.2 Passive Feedback Decomposition

Fig. 1. Passive subsystem decomposition.

U

*qm*

mechanical passive subsystems.

**2.1.2 Properties** 

Lemma 1

Proof

*U uu* 

2.1.2.1 Passivity property of permanent-magnet synchronous motor

Taking the time derivative of H along the trajectory (1)-(3), we get:

The PMSM represents a passive system, if , *<sup>T</sup> <sup>T</sup>*

Considering the total energy He of the electric subsystem e, that is:

$${}^{H}\!\_{e}\langle\phi\_{e},\eta\_{m}\rangle = \frac{1}{2}\dot{\phi}\_{e}^{T}D\_{e}\dot{\eta}\_{e} + \mu^{T}(\eta\_{m})\dot{\eta}\_{e} \tag{6}$$

A similar procedure used above to prove the passivity of PMSM can be used to establish the passivity of e, and for mechanical m we consider the energy function <sup>1</sup> <sup>2</sup> () 2 *Hm m mm q D q* to prove the passivity property.

2.1.2.3 Workless forces

In order to introduce the third property, we note that the model (1)-(3) can be written under the following compact form:

$$
\vec{D}\cdot\vec{q} + \mathcal{W}(q)\dot{q} + \mathcal{R}\dot{q} = M\mathcal{U} + \xi\tag{7}
$$

Where, *D dia gD D R dia e m* , ; , *gR R e m*

$$\begin{aligned} \mathcal{M} &= \begin{bmatrix} I\_2 \ \mathbf{0}\_{1 \times 2} \end{bmatrix}^T \; ; \; \dot{\boldsymbol{q}} = \begin{bmatrix} \dot{\boldsymbol{q}}\_e^T \ \boldsymbol{\mu}\_m \end{bmatrix}^T \; ; \; \boldsymbol{\xi} = \begin{bmatrix} \mathbf{0}\_{2 \times 1} \ \boldsymbol{\mu} \end{bmatrix}^T \\\\ \mathcal{W} & \begin{bmatrix} \boldsymbol{q}\_m \end{bmatrix} = \begin{bmatrix} \mathbf{V} \boldsymbol{\mathcal{V}}\_2^T \ \{\boldsymbol{q}\_m\} \ \dot{\boldsymbol{q}}\_m \ \boldsymbol{\mu} \end{bmatrix} - \dot{\boldsymbol{q}}\_e^T \boldsymbol{\mathcal{W}}\_2 \{\boldsymbol{q}\_m\} \ \boldsymbol{\mu} \end{aligned} \tag{8}$$

Based on the passivity property of the PMSM and the relations (1)-(3), we deduce that the "workless forces" are given by:

$$\mathbf{C} \cdot \begin{pmatrix} q\_m \end{pmatrix} = \begin{pmatrix} 0\_{2 \times 2} & \mathcal{W}\_2 \ \{q\_m\} \\ -\mathcal{W}\_2^T \ \{q\_m\} & 0\_{1 \times 1} \end{pmatrix} \tag{9}$$

as C (qm ) verifies:

$$\mathbf{C} \begin{pmatrix} \mathbf{q}\_m \ \end{pmatrix} = \mathbf{-C}^T \begin{pmatrix} \mathbf{q}\_m \ \end{pmatrix} \tag{10}$$

(i.e., C (qm ) is a skew symmetric matrix.)

Remark

In the present of the saliency effects, the "workless forces" are given by:

$$\mathbf{C} \cdot \begin{pmatrix} q\_{m'} \dot{q} \end{pmatrix} = \begin{pmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} \\ \mathbf{C}\_{21} & \mathbf{C}\_{22} \end{pmatrix} \tag{11}$$

Where

$$\mathbf{C}\_{11} = \frac{1}{2} \mathcal{W}\_1 \text{ (}\mathbf{q}\_{\text{m}}\text{)} \dot{q}\_{\text{m}}$$

$$\mathbf{C}\_{12} = \left(\frac{1}{2}\mathcal{W}\_1 \; \begin{pmatrix} q\_m \end{pmatrix} \dot{q}\_e + \mathcal{W}\_2 \; \begin{pmatrix} q\_m \end{pmatrix} \right)$$

$$\mathbf{C}\_{21} = -\left(\frac{1}{2}\dot{q}\_e^T \mathcal{W}\_1 \; \begin{pmatrix} q\_m \end{pmatrix} + \mathcal{W}\_2^T \; \begin{pmatrix} q\_m \end{pmatrix} \right)$$

$$\mathbf{C}\_{22} = 0$$

as *C* (q ,q ) <sup>m</sup> verifies: ( ) ( , ) ( , ) *<sup>T</sup> <sup>D</sup> <sup>q</sup> <sup>C</sup> q q m m <sup>C</sup> q q*

(i.e., ( ( ) - 2 ( , )) *<sup>D</sup> <sup>q</sup> <sup>C</sup> q q <sup>m</sup>* is a skew symmetric matrix).

The previous identification of the workless forces permitted us to write the relation (7) under the following form:

$$D\ \ddot{q} + \mathcal{C}(q\_m)\dot{q} + R\dot{q} = MLI + \xi \tag{12}$$

Passivity Based Control for Permanent-Magnet Synchronous Motors 377

<sup>1</sup> () e <sup>2</sup>

\* \*

 \* <sup>2</sup> e min () e () , t *<sup>T</sup> He eR ee e e* 

> max min e min max

 

min max . , . are the minimum and maximum eigenvalues respectively.

Considering the quadratic function (17), and using the same procedure, we get:

e1

<sup>e</sup> ( ) m (0) *<sup>e</sup> <sup>t</sup> e e et e e*

> min 1

 0 *<sup>e</sup> e R K D*

max

1. Since, we can not control the magnetic fields from the permanent magnets; it is reasonable to expect that we must eliminate the effect on electric subsystem e of the flux linkages due

<sup>e</sup> ( ) m (0) *<sup>e</sup> <sup>t</sup> e e et e e*

, along the trajectory (15), we get:

e

0 , 0 *e e*

 

 

*e e D R*

*D D*

.

 

*<sup>e</sup> q* is asymptotically attainable. We have the following result:

1

*<sup>T</sup> He eD ee e e* (17)

*U U* (19)

(21)

\* *U U Ke* <sup>1</sup> *<sup>e</sup>* (22)

(23)

(24)

2 1 ( ) *U Dee m m e e e <sup>q</sup> <sup>W</sup> q q <sup>R</sup> <sup>q</sup> K e* (25)

*R et e e* (20)

<sup>e</sup> ( ) ( (U-U )) e *<sup>T</sup> He e R ee e e* (18)

\*

In order to ensure the convergence of the ee to zero, we take:

*e*

*m*

where 1 2e , k 0 *K kI <sup>e</sup>* , I2 identity matrix 2x2.

Then the convergence to the desired state trajectory is faster.

Taking the time derivative of He

Since 0 *<sup>T</sup> R R e e* , we has

we conclude that:

 

Proposition 1

Hence the desired current \*

Where,

Let,

Proof

Where,

Remarks

The control law is:

It is with noting that, these properties have been already derived for Induction machine in (Ortega et al., 1996).

### **2.2 Problem formulation and design procedure 2.2.1 Problem formulation**

The control problem can be formulated as follows: Consider the PMSM model (1)-(3) with state vector , , *<sup>T</sup> <sup>T</sup> q qq q e mm* ; inputs U2; regulated outputs , *q<sup>m</sup>* ; measurable output qm; immeasurable outputs ( , ) *T T q q e m* . The problem consists of constructing an observerbased controller such that for all smooth desired output function ( ) L *t* , with known derivative ( ) L *t* , global torque tracking with internal stability is achieved

#### **2.2.2 Design procedure**

The steps to follow are mentioned in section 1. We consider the ideal case to simplify the procedure, where all outputs are supposed available from measurement, then we design an observer to reconstruct the states that we not available.

2.2.2.1 Passivity approach to controller design

The desired dynamics must be compatible with the bounded constraints of the PMSM. From equations (1)-(3), we deduce the following desired dynamics:

$$D\_e \ddot{q}\_e^\* + \mathcal{V} \mathcal{V}\_2(q\_m) \dot{q}\_m + R\_e \dot{q}\_e^\* = \mathcal{U}^\* \tag{13}$$

$$\left(D\_m \ddot{q}\_m^\* - \mathcal{W}\_2^T(q\_m)\dot{q}\_e^\* + R\_m \dot{q}\_m^\* = -\tau\_L\right) \tag{14}$$

Where \* \* , *q q e m* is the desired current and desired rotor velocity respectively. The error dynamic are described by:

$$\left[D\_e \dot{e}\_e + R\_e \, e\_e = \mathcal{U} - \mathcal{U}^\* \tag{15}$$

$$\left(D\_m \, \dot{e}\_m - \mathcal{W}\_2^T \left(q\_m \right) \; e\_e + R\_m \; e\_m = 0\right) \tag{16}$$

Where \* \* , *ee e em m m q q e q q* are the current error and rotor speed error respectively. The problem is to find a control law U, which ensures () 0 *Lim e t <sup>t</sup>* , where , *<sup>T</sup> <sup>T</sup> e ee e m* . To this end, we shape the energy of the closed loop to match a desired energy function, as:

$$H\_{\epsilon}^{\ast}(\mathbf{e}\_{\epsilon}) = \frac{1}{2} \mathbf{e}\_{\epsilon}^{T} \ D\_{\epsilon} \ \mathbf{e}\_{\epsilon} \tag{17}$$

Taking the time derivative of He , along the trajectory (15), we get:

$$\left(\dot{H}\_e^\*(e\_e) = -e\_e^T \left(\left(R\_e + \left(\mathbf{U} \cdot \mathbf{U}^\*\right)\right)\right) e\_e\right.\tag{18}$$

In order to ensure the convergence of the ee to zero, we take:

$$
\mathcal{U} = \mathcal{U}^\* \tag{19}
$$

Since 0 *<sup>T</sup> R R e e* , we has

$$\left\| \dot{H}\_{\varepsilon}^{\*} (e\_{\varepsilon}) = -e\_{\varepsilon}^{T} \; R\_{\varepsilon} \; \mathbf{e}\_{\varepsilon} \le -\mathcal{J}\_{\text{min}} \left\{ R\_{\varepsilon} \right\} \left\| e\_{\varepsilon} (t) \right\|^{2} \; \forall \; \mathbf{t} \tag{20}$$

we conclude that:

$$\left\|\boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon}}(t)\right\| \leq \mathbf{m}\_{\boldsymbol{\varepsilon}} \left\|\boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon}}(0)\right\| \left\|\boldsymbol{e}^{-\rho\_{\boldsymbol{\varepsilon}}\left|t\right|}\right\| \tag{21}$$

Where,

376 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

( ( ) ( ))

*T T em m*

*me m*

0

The previous identification of the workless forces permitted us to write the relation (7)

*D q C q q Rq MU* () *<sup>m</sup>*

It is with noting that, these properties have been already derived for Induction machine in

The control problem can be formulated as follows: Consider the PMSM model (1)-(3) with

qm; immeasurable outputs ( , ) *T T q q e m* . The problem consists of constructing an observer-

*t* , global torque tracking with internal stability is achieved

The steps to follow are mentioned in section 1. We consider the ideal case to simplify the procedure, where all outputs are supposed available from measurement, then we design an

The desired dynamics must be compatible with the bounded constraints of the PMSM. From

\* \*\* <sup>2</sup> ( ) *<sup>T</sup> Dq W q q R q mm m e m m L*

, *ee e em m m q q e q q* are the current error and rotor speed error respectively.

The problem is to find a control law U, which ensures () 0 *Lim e t <sup>t</sup>* , where ,

To this end, we shape the energy of the closed loop to match a desired energy function, as:

, *q q e m* is the desired current and desired rotor velocity respectively.

; inputs U2; regulated outputs

based controller such that for all smooth desired output function ( ) L

22

\* <sup>2</sup> ( ) *Dee m m e e <sup>q</sup> <sup>W</sup> q q <sup>R</sup> <sup>q</sup> <sup>U</sup>* (13)

\* *De R e U U ee e e* (15)

<sup>2</sup> ( ) 0 *<sup>T</sup> De W q e Re mm m e mm* (16)

(12)

, *q<sup>m</sup>* ; measurable output

(14)

*<sup>T</sup> <sup>T</sup> e ee e m* .

*t* , with known

*C*

12 1 2

*C Wqq W q*

1 ( ( ) ( )) <sup>2</sup>

21 1 2

*C qW q W q*

1

2

as *C* (q ,q ) <sup>m</sup> verifies: ( ) ( , ) ( , ) *<sup>T</sup> <sup>D</sup> <sup>q</sup> <sup>C</sup> q q m m <sup>C</sup> q q* (i.e., ( ( ) - 2 ( , )) *<sup>D</sup> <sup>q</sup> <sup>C</sup> q q <sup>m</sup>* is a skew symmetric matrix).

**2.2 Problem formulation and design procedure** 

observer to reconstruct the states that we not available.

equations (1)-(3), we deduce the following desired dynamics:

2.2.2.1 Passivity approach to controller design

The error dynamic are described by:

Where \* \*

under the following form:

**2.2.1 Problem formulation** 

state vector , , *<sup>T</sup> <sup>T</sup> q qq q e mm*

(Ortega et al., 1996).

derivative ( ) L 

Where \* \*

**2.2.2 Design procedure** 

 max min e min max 0 , 0 *e e e e e D R m D D* .

 min max . , . are the minimum and maximum eigenvalues respectively. Hence the desired current \* *<sup>e</sup> q* is asymptotically attainable. We have the following result: Proposition 1

Let,

$$
\mathcal{U}I = \mathcal{U}^\* - \mathcal{K}\_1 \ e\_e \tag{22}
$$

where 1 2e , k 0 *K kI <sup>e</sup>* , I2 identity matrix 2x2. Then the convergence to the desired state trajectory is faster.

Proof

Considering the quadratic function (17), and using the same procedure, we get:

$$\left\| e\_{\varepsilon}(t) \right\| \le \left\| \mathbf{m}\_{\mathbf{e}} \left\| e\_{\varepsilon}(0) \right\| \right\| e^{-\rho\_{\varepsilon 1} \left\| t} \tag{23}$$

Where,

$$\rho\_{e1} = \frac{\lambda\_{\text{min}} \left\{ R\_e + K\_1 \right\}}{\lambda\_{\text{max}} \left\{ D\_e \right\}} \; \; \; 0 \tag{24}$$

The control law is:

$$\text{LLI} = D\_e \dot{q}\_e^\* + \mathcal{V} \mathbf{V}\_2(q\_m) \dot{q}\_m + \mathcal{R}\_e \dot{q}\_e^\* - \mathcal{K}\_1 e\_e \tag{25}$$

Remarks

1. Since, we can not control the magnetic fields from the permanent magnets; it is reasonable to expect that we must eliminate the effect on electric subsystem e of the flux linkages due to the permanent magnets. Which is seen from (25), the term from the permanent magnets must be concealed out a drawback of the scheme. However, this term is a vector in a measurable quantity (position).

2. In the closed loop system, the positive definite matrix K1 increases the convergence of the tracking error and overcome the imprecise knowledge of system parameters, if we choose high gain ke.

### 2.2.2.2 Desired current and desired torque

The PMSM operating under maximum torque if the direct current id in the general reference frame d-q (direct-quadrate) equals to zero.

Under the above condition, the desired current in reference frame is chosen as:

$$\dot{q}\_e^\* = \frac{2}{3} \frac{\tau^\*}{\text{n}\_p \text{ $\mathcal{X}\_m$ }} \begin{pmatrix} -\text{sin} \ (q\_m) \\ \cos \ (q\_m) \end{pmatrix} \tag{26}$$

Passivity Based Control for Permanent-Magnet Synchronous Motors 379

<sup>2</sup> ˆ ˆˆ ( ) *<sup>T</sup> D q W q q R q Lq mm m e m m L vm*

estimated velocity respectively; ˆ ˆ , *q q qq q q e e em m m* are the estimated current error and

() ˆ ˆˆ *D q C q q Rq MU Lq <sup>m</sup>*

In order to prove the asymptotic stability of the observer estimated error; we choose the

\* <sup>1</sup> (q) q <sup>2</sup>

<sup>o</sup> ( ) m (0) , t. *<sup>o</sup> <sup>t</sup> qt q e*

We conclude that, the observer (34) reconstructs asymptotically the current and velocity

We can notice that the gain matrix L has the same effect than that of matrix K1 in (25), i. e; L is the damping that is injected in the observer system to ensure the asymptotic stability of

 

, along the trajectory (35), we get:

where ˆ ˆˆ , *<sup>T</sup> <sup>T</sup> q qq e m* is the observer state; ˆ ˆ , *q q e m*

Where e v , and L L , L *<sup>T</sup> <sup>T</sup> q qq e m diag*

following desired energy error function:

Since 0 , q 0 *<sup>T</sup> L L* is asymptotically stable.

max min o min max 0 , 0 *<sup>o</sup> D R L*

*D D*

 

The desired dynamics, when only rotor position is measurable are:

 

2.2.2.4 Combined Controller-Observer Design

Taking the time derivative of Ho

where

the observation error.

*m*

signals. Remark

The model (31), (32) can be written under the following form:

From the equation (12) and (34), we deduce the observer error dynamic:

\*

Following the same procedure used in section II.2.1, we conclude that:

estimated velocity error; where:

<sup>2</sup> ˆ ˆˆ ( ) *Dq W q q R q U Lq ee m m e e ee* (31)

represents the estimated current and

0 , 0 *<sup>T</sup> LL L ee v* (33)

(34)

3 1 *Dq C q q R L q* () ( ) 0 *<sup>m</sup>* (35)

*<sup>T</sup> H qD <sup>o</sup>* (36)

(q) ( ) q *<sup>T</sup> H q RL <sup>o</sup>* (37)

(38)

\* <sup>2</sup> ( ) <sup>ˆ</sup> *Dee m m e e <sup>q</sup> <sup>W</sup> q q <sup>R</sup> <sup>q</sup> <sup>U</sup>* (39)

(32)

where is the desired torque; np is the number of pole pairs, and m is the amplitude of the flux linkage established by the permanent magnet.

The desired torque is deduced from the desired mechanical dynamic (14), we have:

$$
\tau^\* = D\_m \ddot{q}\_m^\* + R\_m \dot{q}\_m^\* + \tau\_L \tag{27}
$$

It has been proved in (Kim et al., 1997), that this scheme has two drawbacks, it is an open loop scheme (in the speed tracking error), and its convergence rate is limited by the mechanical constant time (Dm / Rm ). In (14) is defined as:

$$
\tau^\* = D\_m \ddot{q}\_m^\* - z + \tau\_L \tag{28}
$$

$$
\dot{z} = -a \ z + b \ e\_{\mathbf{m}'} \quad \text{and} \ a, \mathbf{b} \; \text{>}\; 0 \; . \tag{29}
$$

With this choice, the convergence rate of the speed error *m m* does not depend only on the natural mechanical damping. This rate can be adjusted by means of the positives gains *b* and *a* have the same role of proportional-derivative (PD) control law.

#### Remark

If, v and *<sup>e</sup> q* are considered as input and output, then it is easy to prove the strict passivity of the closed loop system.

$$
\dot{\boldsymbol{\sigma}} \cdot \boldsymbol{\upsilon} = \boldsymbol{D}\_e \ddot{\boldsymbol{q}}\_e^\* + \boldsymbol{R}\_e \dot{\boldsymbol{q}}\_e^\* \tag{30}
$$

### 2.2.2.3 A passivity Approach to observer design

The problem is to construct an auxiliary dynamic system that asymptotically reconstructs the current and velocity signals from input-output measurements, i.e., stator voltage U and rotor position qm, respectively. To this end we will use a passivity approach. An interesting feature of this approach is that it establishes a conceptual duality, between the strategies of PMSM controller and observer design. Such a duality feature is rather unique for nonlinear systems.

Based on the physical structure of the PMSM model (1)-(3) and the controller structure (25), we introduce the current and velocity observer systems as follows:

$$D\_e \ddot{\hat{q}}\_e + \mathcal{W}\_2(q\_m) \dot{\hat{q}}\_m + R\_e \dot{\hat{q}}\_e = \mathcal{U} - L\_e \dot{\tilde{q}}\_e \tag{31}$$

$$D\_m \ddot{\hat{q}}\_m - \mathcal{W}\_2^T (q\_m) \dot{\hat{q}}\_e + R\_m \dot{\hat{q}}\_m = -\tau\_L - L\_v \dot{\hat{q}}\_m \tag{32}$$

where ˆ ˆˆ , *<sup>T</sup> <sup>T</sup> q qq e m* is the observer state; ˆ ˆ , *q q e m* represents the estimated current and estimated velocity respectively; ˆ ˆ , *q q qq q q e e em m m* are the estimated current error and estimated velocity error; where:

$$L\_e = L\_e^T \ \ \ \ \ 0 \ \ \ \ \ \ L\_v \ \ \ \ \ \ \ \ 0 \tag{33}$$

The model (31), (32) can be written under the following form:

$$D\stackrel{\ddot{\varphi}}{\dot{q}} + \mathcal{C}(q\_m)\dot{\hat{q}} + \mathcal{R}\dot{\hat{q}} = M\mathcal{U} + \mathcal{\xi} - L\dot{\tilde{q}}\tag{34}$$

Where e v , and L L , L *<sup>T</sup> <sup>T</sup> q qq e m diag*

From the equation (12) and (34), we deduce the observer error dynamic:

$$D\stackrel{\ddot{\bullet}}{\ddot{q}} + \mathcal{C}(q\_m)\dot{\tilde{q}} + (R+L)\dot{\tilde{q}} = \mathbf{0}\_{3\times 1} \tag{35}$$

In order to prove the asymptotic stability of the observer estimated error; we choose the following desired energy error function:

$$H\_o^\*(\dot{\tilde{\mathbf{q}}}) = \frac{1}{2} \dot{\tilde{q}}^T D\_i \dot{\tilde{\mathbf{q}}} \tag{36}$$

Taking the time derivative of Ho , along the trajectory (35), we get:

$$
\dot{H}\_o^\*(\dot{\tilde{\mathbf{q}}}) = -\dot{\tilde{q}}^T \text{ ( $R + L$ ) } \dot{\tilde{\mathbf{q}}}\tag{37}
$$

Since 0 , q 0 *<sup>T</sup> L L* is asymptotically stable.

Following the same procedure used in section II.2.1, we conclude that:

$$\left\|\dot{\vec{q}}(t)\right\| \le \mathbf{m}\_{\diamond} \left\|\dot{\vec{q}}(0)\right\| e^{-\rho\_{\diamond}t}, \forall \, t. \tag{38}$$
  $\text{where} \ m\_{\diamond} = \sqrt{\frac{\lambda\_{\max}\{D\}}{\lambda\_{\min}\{D\}}} \;> 0 \;> \ \rho\_{\diamond} = \frac{\lambda\_{\min}\{R+L\}}{\lambda\_{\max}\{D\}} \;> 0$ 

We conclude that, the observer (34) reconstructs asymptotically the current and velocity signals.

Remark

*m*

378 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

to the permanent magnets. Which is seen from (25), the term from the permanent magnets must be concealed out a drawback of the scheme. However, this term is a vector in a

2. In the closed loop system, the positive definite matrix K1 increases the convergence of the tracking error and overcome the imprecise knowledge of system parameters, if we choose

The PMSM operating under maximum torque if the direct current id in the general reference

2 sin ( ) 3 n cos ( )

\* \* *Dq R q mm m m L*

\* *Dq z mm L*

the natural mechanical damping. This rate can be adjusted by means of the positives gains *b*

If, v and *<sup>e</sup> q* are considered as input and output, then it is easy to prove the strict passivity of

The problem is to construct an auxiliary dynamic system that asymptotically reconstructs the current and velocity signals from input-output measurements, i.e., stator voltage U and rotor position qm, respectively. To this end we will use a passivity approach. An interesting feature of this approach is that it establishes a conceptual duality, between the strategies of PMSM controller and observer design. Such a duality feature is rather unique for nonlinear

Based on the physical structure of the PMSM model (1)-(3) and the controller structure (25),

It has been proved in (Kim et al., 1997), that this scheme has two drawbacks, it is an open loop scheme (in the speed tracking error), and its convergence rate is limited by the

where is the desired torque; np is the number of pole pairs, and m is the amplitude of the

*m m*

*m*

(26)

(27)

(28)

*ee e e v Dq R q* (30)

does not depend only on

*z az b* e , and a ,b 0 . <sup>m</sup> (29)

*m m* 

*q*

Under the above condition, the desired current in reference frame is chosen as:

*e*

and *a* have the same role of proportional-derivative (PD) control law.

we introduce the current and velocity observer systems as follows:

mechanical constant time (Dm / Rm ). In (14) is defined as:

With this choice, the convergence rate of the speed error

2.2.2.3 A passivity Approach to observer design

p

The desired torque is deduced from the desired mechanical dynamic (14), we have:

*<sup>q</sup> <sup>q</sup>* 

measurable quantity (position).

2.2.2.2 Desired current and desired torque

frame d-q (direct-quadrate) equals to zero.

flux linkage established by the permanent magnet.

high gain ke.

Remark

systems.

the closed loop system.

We can notice that the gain matrix L has the same effect than that of matrix K1 in (25), i. e; L is the damping that is injected in the observer system to ensure the asymptotic stability of the observation error.

#### 2.2.2.4 Combined Controller-Observer Design

The desired dynamics, when only rotor position is measurable are:

$$\left(D\_c \ddot{q}\_e^\* + \mathcal{W}\_2(q\_m)\dot{\hat{q}}\_m + \mathcal{R}\_e \dot{q}\_e^\* = \mathcal{U}^\*\tag{39}$$

$$D\_m \ddot{q}\_m^\* - \mathcal{V} \boldsymbol{\eta}\_2^T (\boldsymbol{q}\_m) \dot{\boldsymbol{q}}\_e^\* + R\_m \dot{\boldsymbol{q}}\_m^\* = -\boldsymbol{\tau}\_L - k\_m \boldsymbol{e}\_m \tag{40}$$

Passivity Based Control for Permanent-Magnet Synchronous Motors 381

Then, if matrix Q is positive, we can conclude that the closed loop system is asymptotically

<sup>1</sup> ( ) (R L)- ( ) ( ) 0 <sup>4</sup>

*T*

1 ( ) (R L)- ( ) ( ) <sup>4</sup>

2

A block diagram representing the passivity-based method is show in Fig. 2.

a PMSM model, whose parameters are given in the Appendix 1.

tracks its reference, and the estimation error converges.

the rotor speed converges, but the setting time is increased lightly.

*m m m*

*G q Nq N q F F*

11 e 2 e 2 2 2 2 2 e 2 e 2 2 2 2 2 1 1 F ( R )( R ) ( )R ( R ) ( ) 4 4 *T T K L L L WW K L L L WW e ee e ee*

> 2 12 2 4 *vl F W*

21 2 2 e 2 ( R ) <sup>4</sup> *v T T*

22 m 2 m 2 ( R )( R ) <sup>4</sup>

*<sup>l</sup> F kl*

2 22 2 22 2 e2 , , where k 0; l 0 . *L l I K kI e e* We note that if conditions see that the matrix Q is positive definite if conditions (43) are

The performance of the controller-observer system was investigated by simulation. We used

The filter and damping parameters taken in the simulation are; a=100; b=87.5; ke2=100; le2=1000 and lv2=1500. We have limited the desired stator current and chosen the initial

Fig. 3 shows the time response, of the motor, where a load torque L of 1.35 Nm is applied to the PMSM at the starting phase and we take a speed reference of 150 rad/s. The rotor speed converges with of setting time of 0.4s. The estimated observer current and speed errors

Fig. 4 illustrate the time response of the closed loop system without load torque, and speed reference of (150 rad/s if t<=0.65 and –150 rad/s if t>0.65). We can see that the rotor velocity

In Fig. 5, we show the robustness of the combined controller-observer system. We take these uncertainties in the parameters of PMSM (3Ra, 2Rm, 2Ld, 2Lq, 1.5Dm, 0.75m,). We note that,

*<sup>v</sup> m v*

*<sup>l</sup> F WW L*

*<sup>T</sup> G qm Nq N q m m* (47)

11 12 21 22

*F F*

*e*

2 2

Matrix Q is positive if and only if the following inequality is satisfied:

which can be written after calculations;

for simplicity, we have chosen:

**2.3 Simulation results** 

converge to zero.

observer conditions equal to zero.

stable.

Where,

satisfied.

Where, km > 0.

We have the following result: The controller law becomes:

$$\mathcal{L}I = D\_e \dot{q}\_e^\* + \mathcal{V} \mathcal{l}\_2(q\_m) \dot{\hat{q}}\_m + \mathcal{R}\_e \dot{q}\_e^\* - \mathcal{K}\_2 e\_e \tag{41}$$

In order to establish the stability of the closed loop system with presence of the observer, we consider equation of state error (35). We get from (25), (16), (40) and (41):

$$D\ \dot{e} + G(q\_m)e + N(q\_m)\dot{\tilde{q}} = 0\tag{42}$$

Where,

$$\begin{aligned} G(q\_m) &= \begin{pmatrix} \{R\_c + K\_2\} & 0\_{2 \times 1} \\ -\mathcal{W}\_2^T \{q\_m\} & \{R\_m + k\_{m2}\} \end{pmatrix} \\ N(q\_m) &= \begin{pmatrix} L\_{c2} & -\mathcal{W}\_2 \{q\_m\} \\ 0\_{1 \times 2} & \{R\_m + l\_{m2}\} \end{pmatrix} \end{aligned}$$

Proposition 2

Consider the PMSM model (1)-(3) in closed loop with the observer-controller (32)-(33) and (41)-(43). Then, the closed loop system is asymptotically stable provided that:

$$\begin{aligned} k\_{\epsilon 2} &> \frac{1\_{\text{c2}}}{4} - R\_{\text{a}}^2\\ k\_m &> \frac{1\_{\text{v2}}}{4} - R\_{\text{m}}^2\\ 1\_{\text{v2}} &> 41\_{\text{c2}} + 4R\_{\text{c2}} \end{aligned} \tag{43}$$

Proof

To prove the convergence of the vector error T T e , q *T <sup>o</sup> <sup>z</sup>* , let consider the desired energy function error as:

$$H\_{\alpha}^{cl}(\
e, \dot{\tilde{q}} \; \;) = \frac{1}{2} e^{T} D \; e + \frac{1}{2} \dot{\tilde{q}}^{T} D \; \dot{\tilde{q}} \tag{44}$$

The time derivative of cl *H*co along the trajectory (35), (42), gives:

$$\dot{H}\_{\rm co}^{\rm cl} = -e^{T} \mathbf{G} \text{ (}\boldsymbol{q}\_{m}\text{)}\text{ e-}e^{T} \text{N} \text{ (}\boldsymbol{q}\_{m}\text{)}\dot{\tilde{\mathbf{q}}} \text{-}\dot{\tilde{q}}^{T} \text{(}\boldsymbol{R} + \boldsymbol{L}\text{)}\text{ \dot{\tilde{\mathbf{q}}}}\tag{45}$$

Which can be written as,

$$\dot{H}\_{\alpha o}^{cl} = -z\_o^T \mathbf{Q} \ z\_o \tag{46}$$

Where,

$$Q = \begin{pmatrix} G \ \left(q\_m\right) & \frac{1}{2}N \ \left(q\_m\right) \\\\ \frac{1}{2}N^T \ \left(q\_m\right) & \left(R+L\right) \end{pmatrix}$$

Then, if matrix Q is positive, we can conclude that the closed loop system is asymptotically stable.

Matrix Q is positive if and only if the following inequality is satisfied:

$$\text{G } (q\_m) \text{ (R + L)-} \frac{1}{4} N(q\_m) N^T(q\_m) > 0 \tag{47}$$

which can be written after calculations;

$$G \text{ (}\!\!\!-q\_m\text{) (}\text{R}+\text{L)}\text{-}\frac{1}{4}N(q\_m)N^T(q\_m) = \begin{pmatrix} F\_{11} & F\_{12} \\ F\_{21} & F\_{22} \end{pmatrix}$$

Where,

380 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

In order to establish the stability of the closed loop system with presence of the observer, we

() () 0 *De Gq e N q q m m* (42)

( )0 ( ) ( )( )

 

*e m T*

*R K G q Wq R k*

*m*

(41)-(43). Then, the closed loop system is asymptotically stable provided that:

To prove the convergence of the vector error T T e , q

The time derivative of cl *H*co along the trajectory (35), (42), gives:

*Q*

cl

2

v2 e2

1 1 ( , ) 2 2

<sup>1</sup> () () <sup>2</sup>

*m m*

<sup>1</sup> () ( ) <sup>2</sup>

*N q RL*

*Gq Nq*

 

> *T m*

l 4

*k R*

*e a*

*m m*

 

l 4 l 4l 4

*k R*

( ) () 0( )

*L Wq N q R l*

Consider the PMSM model (1)-(3) in closed loop with the observer-controller (32)-(33) and

e2 2

v2 2

*R*

*T*

co ( ) e- ( ) q- ( ) q *TT T H eG q eN q q R L m m* (45)

*cl <sup>T</sup> <sup>T</sup> H eq e De q Dq co* (44)

*cl T H z Qz co o o* (46)

2 21 2 2 2 2 1 2 2

*m mm e m*

*m m*

2 2 ( ) <sup>ˆ</sup> *U Dee m m e e e <sup>q</sup> <sup>W</sup> q q <sup>R</sup> <sup>q</sup> K e* (41)

*k e* (40)

(43)

*<sup>o</sup> <sup>z</sup>* , let consider the desired energy

\* \*\* <sup>2</sup> ( ) *<sup>T</sup> Dmm m e m m L m m q W q q R q*

consider equation of state error (35). We get from (25), (16), (40) and (41):

We have the following result: The controller law becomes:

Where, km > 0.

Where,

Proposition 2

Proof

Where,

function error as:

Which can be written as,

$$\begin{aligned} \mathbf{F}\_{11} &= \left(\mathbf{R}\_{\mathrm{e}} + K\_{2}\right) \left(\mathbf{R}\_{\mathrm{e}} + L\_{\mathrm{e2}}\right) - \frac{1}{4} (\mathbf{L}\_{\mathrm{e2}} \mathbf{L}\_{\mathrm{e2}} + \mathbf{W}\_{2} \mathbf{W}\_{2}^{T}) \mathbf{R}\_{\mathrm{e}} + K\_{2} \left(\mathbf{R}\_{\mathrm{e}} + L\_{\mathrm{e2}}\right) - \frac{1}{4} (\mathbf{L}\_{\mathrm{e2}} L\_{\mathrm{e2}} + W\_{2} W\_{2}^{T}) \\\\ \mathbf{F}\_{12} &= \frac{l\_{v2}}{4} \mathbf{W}\_{2} \\\\ \mathbf{F}\_{21} &= \frac{l\_{v2}}{4} \mathbf{W}\_{2}^{T} - \mathbf{W}\_{2}^{T} \left(\mathbf{R}\_{\mathrm{e}} + L\_{\mathrm{e2}}\right) \\\\ \mathbf{F}\_{22} &= \left(\mathbf{R}\_{\mathrm{m}} + k\_{m2}\right) \left(\mathbf{R}\_{\mathrm{m}} + l\_{v2}\right) - \frac{l\_{v2}^{2}}{4} \end{aligned}$$

for simplicity, we have chosen:

$$\mathbf{L}\_{e2} = \mathbf{l}\_{e2} \ \mathbf{l}\_2 \ \ \ \ \ \mathbf{K}\_2 = \mathbf{k}\_2 \ \ \mathbf{l}\_{2'} \ \text{ where } \mathbf{k}\_2 > 0; \ \mathbf{l}\_{e2} > 0 \ \ .$$

We note that if conditions see that the matrix Q is positive definite if conditions (43) are satisfied.

A block diagram representing the passivity-based method is show in Fig. 2.

### **2.3 Simulation results**

The performance of the controller-observer system was investigated by simulation. We used a PMSM model, whose parameters are given in the Appendix 1.

The filter and damping parameters taken in the simulation are; a=100; b=87.5; ke2=100; le2=1000 and lv2=1500. We have limited the desired stator current and chosen the initial observer conditions equal to zero.

Fig. 3 shows the time response, of the motor, where a load torque L of 1.35 Nm is applied to the PMSM at the starting phase and we take a speed reference of 150 rad/s. The rotor speed converges with of setting time of 0.4s. The estimated observer current and speed errors converge to zero.

Fig. 4 illustrate the time response of the closed loop system without load torque, and speed reference of (150 rad/s if t<=0.65 and –150 rad/s if t>0.65). We can see that the rotor velocity tracks its reference, and the estimation error converges.

In Fig. 5, we show the robustness of the combined controller-observer system. We take these uncertainties in the parameters of PMSM (3Ra, 2Rm, 2Ld, 2Lq, 1.5Dm, 0.75m,). We note that, the rotor speed converges, but the setting time is increased lightly.

Passivity Based Control for Permanent-Magnet Synchronous Motors 383

Fig. 4. Control of speed with reference (150 rd/s if t<=0.65 and –150 rd/s if t>0.65),

Nominal Parameters

**3. Passivity based controller design for a permanent magnet synchronous** 

Within this Section, a new passivity-based controller designed to force the motor to track time-varying speed and torque trajectories is presented. Its design avoids the using of the Euler-Lagrange model and destructuring since it uses a flux-based dq-modelling, independent of the rotor angular position. This dq-model is obtained through the three phase abc-model of the motor, using Park transform. The proposed control law does not compensate the model workless force terms which appear in the machine dq-model, as they have no effect on the system energy balance and they do not influence the system stability properties. Another feature is that the cancellation of the plant primary dynamics and nonlinearities is not done by exact zeroing, but by imposing a desired damped transient. The effectiveness of the proposed control is illustrated by numerical simulation results.

3Ra 2Rm 2Ld 2Lq 1.5Dm 0.75<sup>m</sup>

a) Estimated current error ; b) Estimated velocity error.

Fig. 5. Robustness test.

a

b

**motor in dq-frame** 

Fig. 2. Block diagrams for the passivity-based method.

Fig. 3. Control of speed with reference 150 rd/s; a) Estimated current error; b) Estimated velocity error.

382 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

<sup>v</sup> Deqe Re qe

<sup>W</sup> (q ) <sup>3</sup> <sup>n</sup>

q <sup>2</sup> <sup>m</sup> p m

U v W (q )qˆ k I (qˆ q ) <sup>2</sup> <sup>m</sup> <sup>m</sup> e2 <sup>2</sup> <sup>e</sup> <sup>e</sup> 

PMSM

U

v

q e

d

q e

Controller

2

<sup>e</sup> 

\*

Desired Dynamics

d dt

Fig. 2. Block diagrams for the passivity-based method.

qm

) <sup>e</sup> (qˆ <sup>e</sup> <sup>q</sup> e2 (qm )qˆm Re qˆ <sup>e</sup> <sup>U</sup> <sup>L</sup> <sup>e</sup> W2 qˆ De <sup>L</sup> <sup>T</sup>

qmˆ 


Dmqˆm <sup>W</sup> (qm )qˆ <sup>e</sup> Rm qˆm Lv(qˆm qm)

 

<sup>2</sup>

qe ˆ 

<sup>L</sup> \* Dmqm <sup>z</sup>

L

<sup>z</sup> qm

dt

L

qm

Observer

s a b 

velocity error.

b

a

Fig. 3. Control of speed with reference 150 rd/s; a) Estimated current error; b) Estimated

Fig. 4. Control of speed with reference (150 rd/s if t<=0.65 and –150 rd/s if t>0.65), a) Estimated current error ; b) Estimated velocity error.

Fig. 5. Robustness test.

### **3. Passivity based controller design for a permanent magnet synchronous motor in dq-frame**

Within this Section, a new passivity-based controller designed to force the motor to track time-varying speed and torque trajectories is presented. Its design avoids the using of the Euler-Lagrange model and destructuring since it uses a flux-based dq-modelling, independent of the rotor angular position. This dq-model is obtained through the three phase abc-model of the motor, using Park transform. The proposed control law does not compensate the model workless force terms which appear in the machine dq-model, as they have no effect on the system energy balance and they do not influence the system stability properties. Another feature is that the cancellation of the plant primary dynamics and nonlinearities is not done by exact zeroing, but by imposing a desired damped transient. The effectiveness of the proposed control is illustrated by numerical simulation results.

The Section 2 is organized as follows. The PMSM dq-model and the inner current loop design are presented at Subsection 2.2. In Subsection 3, the passivity property of the PMSM in the dq-reference frame is introduced. Subsection 2.4 deals with the computation of the current, flux and the torque references. The passivity property of the closed loop system and the resulting control structure are given in Subsections 2.5 and 2.6, respectively. Simulation results are presented in Subsection 2.7. Subsection 2.8 concludes this Section. The proof of the passivity property of the PMSM in the dq frame is given. The analysis and proof of the exponential stability of the flux tracking error is introduced. Subsections 2.5 contain the proof of the passivity property of the closed loop system.

### **3.1 Permanent-magnet synchronous motor model in dq frame**

The PMSM uses buried rare earth magnets. Its electrical behaviour is described here by the well known dq model (Krause et al., 2002), given by Equation (48):

$$L\_{dq}\dot{i}\_{dq} + R\_{dq}i\_{dq} + n\_p o o\_m \mathfrak{T}L\_{dq}i\_{dq} + n\_p o o\_m \mathfrak{T}\varphi\_f = \upsilon\_{dq} \tag{48}$$

Passivity Based Control for Permanent-Magnet Synchronous Motors 385

 

3 <sup>2</sup> *<sup>e</sup> n i <sup>p</sup> dq dq*

*dq p n v Ri*

 

Equations (49), (51), (53) and (54). The reference value of the current vector *idq* is denoted by:

*dq*

*i*

 0

 0

model. This result on the simplified dynamic dq-model of the PMSM given below:

 *dq p n Ri m dq dq dq*

 *m m* 

> 3 2

*T <sup>e</sup> n i <sup>p</sup> dq dq*

 

This simplified form of the PMSM model is further used to design the control input *dq i*

The PMSM represents a strictly passive system if the reference vector, of the stator currents,

*t*

*t*

Assuming that by the proper choice of positive gains kdp, kdi, kqp, kqi, these loops work

*m VF m e L J f*

 

*d*

*i*

*i*

*q*

 

*m d <sup>q</sup> dq dq dq* (53)

d , , 0

d , , 0

can be considered as control input for the PMSM

(60)

*dq* are considered as the input and the output vectors,

(58)

(59)

(57)

*q qp q q qi q q qp qi v k i i k i i tk k* (56)

*d dp d d di d d dp di v k i i k i i tk k* (55)

(54)

*T*

*d q mm* and

*T*

*d q i i* to track the

Current controlled dq-model of PMSM

*T*

satisfactory. Then, the reference vector *dq i*

using the passivity approach.

Lemma 3

respectively.

*dq i*

**3.2 Passivity property of dq-model** 

and the flux linkage vector,

reference

Let us define the state model of the PMSM using the state vector

The proportional-integral (PI) current loops, used to force

*d q i i* , are of the form of equations below:

In this equation the following notations have been employed:

$$\begin{aligned} \begin{bmatrix} L\_{d\boldsymbol{q}} = \begin{bmatrix} L\_{d\boldsymbol{l}} & \mathbf{0} \\ \mathbf{0} & L\_{q} \end{bmatrix}; \ i\_{dq} = \begin{bmatrix} \dot{\mathbf{i}}\_{d} \\ \dot{\mathbf{i}}\_{q} \end{bmatrix}; \ R\_{dq} = \begin{bmatrix} R\_{\mathcal{S}} & \mathbf{0} \\ \mathbf{0} & R\_{\mathcal{S}} \end{bmatrix}; \ \boldsymbol{\nu}\_{f} = \begin{bmatrix} \boldsymbol{\phi}\_{f} \\ \mathbf{0} \end{bmatrix}; \ \boldsymbol{\mathfrak{S}} = \begin{bmatrix} \mathbf{0} & -\mathbf{1} \\ \mathbf{1} & \mathbf{0} \end{bmatrix}; \ \boldsymbol{\upsilon}\_{dq} = \begin{bmatrix} \boldsymbol{\upsilon}\_{d} \\ \boldsymbol{\upsilon}\_{q} \end{bmatrix} \end{aligned}$$

In the above-presented relations, Ld and Lq: are the stator inductances in dq frame, RS: is the stator winding resistance, f: is the flux linkages due to permanent magnets, np: is the number of pole-pairs, m: is the mechanical speed, vd and vq: are the stator voltages in dq frame, id and iq: are the stator currents in dq frame.

The mechanical equation of the PMSM is given by:

$$f\dot{o}\_m + f\_{VF}o\_m = \tau\_e - \tau\_L \tag{49}$$

where J is the rotor moment of inertia, fVF is the viscous friction coefficient, and L is the load torque.

The electromagnetic torque e can be expressed in the dq frame as follows:

$$\tau\_e = \frac{3}{2} n\_p \left( \left( L\_d - L\_q \right) i\_d i\_q + \phi\_f i\_q \right) \tag{50}$$

The rotor position m is given by Equation (51):

$$
\dot{\theta}\_{\perp m} = \alpha\_m \tag{51}
$$

where d and q are the flux linkages in dq frame.

The interdependence between the flux linkage motor dq and the current vector *dq i* can be expressed as follow (Krause et al., 2002):

$$
\begin{bmatrix}
\boldsymbol{\nu}\_d \\
\boldsymbol{\nu}\_q
\end{bmatrix} = \boldsymbol{L}\_{d\boldsymbol{q}} \mathbf{i}\_{d\boldsymbol{q}} + \boldsymbol{\nu}\_f
\tag{52}
$$

where d and q are the flux linkages in dq frame.

Substituting *dq i* value obtained by Relation (52) in Equations (48) and (50), yields:

$$
\dot{\varphi}\_{dq} + \mathfrak{n}\_p \alpha \rho\_m \mathfrak{T} \psi\_{dq} = \upsilon\_{dq} - \mathcal{R}\_{dq} \dot{i}\_{dq} \tag{53}
$$

$$\mathbf{r}\_e = -\frac{3}{2} n\_p \boldsymbol{\mu} \boldsymbol{\nu}\_{dq} \mathfrak{H}\_{dq} \tag{54}$$

Current controlled dq-model of PMSM

384 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

The Section 2 is organized as follows. The PMSM dq-model and the inner current loop design are presented at Subsection 2.2. In Subsection 3, the passivity property of the PMSM in the dq-reference frame is introduced. Subsection 2.4 deals with the computation of the current, flux and the torque references. The passivity property of the closed loop system and the resulting control structure are given in Subsections 2.5 and 2.6, respectively. Simulation results are presented in Subsection 2.7. Subsection 2.8 concludes this Section. The proof of the passivity property of the PMSM in the dq frame is given. The analysis and proof of the exponential stability of the flux tracking error is introduced. Subsections 2.5 contain the

The PMSM uses buried rare earth magnets. Its electrical behaviour is described here by the

*Li Ri n Li n v dq dq dq dq p*

0

*R* 

In the above-presented relations, Ld and Lq: are the stator inductances in dq frame, RS: is the stator winding resistance, f: is the flux linkages due to permanent magnets, np: is the number of pole-pairs, m: is the mechanical speed, vd and vq: are the stator voltages in dq

*m VF m e L J f*

where J is the rotor moment of inertia, fVF is the viscous friction coefficient, and L is the load

<sup>2</sup> *<sup>e</sup> n L L ii i <sup>p</sup> <sup>d</sup> <sup>q</sup> <sup>d</sup> q fq*

 *m m* 

The interdependence between the flux linkage motor dq and the current vector *dq i* can be

*L i*

*dq dq f*

*d*

*q*

Substituting *dq i* value obtained by Relation (52) in Equations (48) and (50), yields:

 

*S*

0 *S*

*R*

*dq*

The electromagnetic torque e can be expressed in the dq frame as follows:

<sup>3</sup>

*R*

 *m dq dq p <sup>m</sup> <sup>f</sup> dq* (48)

; 0

 

(50)

*f*

*f*

0 1 1 0 

(49)

(51)

(52)

;

*dq*

*v*

*d*

*q v*

*v* 

 ;

proof of the passivity property of the closed loop system.

**3.1 Permanent-magnet synchronous motor model in dq frame** 

well known dq model (Krause et al., 2002), given by Equation (48):

In this equation the following notations have been employed:

*d*

*q i <sup>i</sup> <sup>i</sup>* ;

0

*q*

;

*dq*

frame, id and iq: are the stator currents in dq frame. The mechanical equation of the PMSM is given by:

The rotor position m is given by Equation (51):

where d and q are the flux linkages in dq frame.

where d and q are the flux linkages in dq frame.

expressed as follow (Krause et al., 2002):

0 *d*

*L <sup>L</sup> <sup>L</sup>* 

*dq*

torque.

Let us define the state model of the PMSM using the state vector *T d q mm* and Equations (49), (51), (53) and (54). The reference value of the current vector *idq* is denoted by:

$$\dot{I}\_{dq}^{\star} = \begin{bmatrix} \dot{I}\_d^{\star} \\\\ \dot{I}\_q^{\star} \end{bmatrix}$$

The proportional-integral (PI) current loops, used to force *T d q i i* to track the reference *T d q i i* , are of the form of equations below:

$$\boldsymbol{\dot{\sigma}} \cdot \boldsymbol{\dot{\sigma}}\_d = \boldsymbol{k}\_{d\boldsymbol{p}} \left( \dot{\mathbf{i}}\_d^\* - \dot{\mathbf{i}}\_d \right) + \boldsymbol{k}\_{d\boldsymbol{d}} \coprod\_{\mathbf{d}} \left( \dot{\mathbf{i}}\_d^\* - \dot{\mathbf{i}}\_d \right) \cdot \mathbf{d} \boldsymbol{t}\_{\boldsymbol{\nu}} \quad \boldsymbol{k}\_{d\boldsymbol{p}\boldsymbol{\nu}} \; \boldsymbol{k}\_{d\boldsymbol{l}} > \mathbf{0} \tag{55}$$

$$\mathbf{w}\_q = k\_{qp} \left( \dot{i}\_q^\* - \dot{i}\_q \right) + k\_{qi} \int\_0^t \left( \dot{i}\_q^\* - \dot{i}\_q \right) \, \mathbf{d}t\_\prime \quad k\_{qp\prime} \ k\_{qi} > 0 \tag{56}$$

Assuming that by the proper choice of positive gains kdp, kdi, kqp, kqi, these loops work satisfactory. Then, the reference vector *dq i* can be considered as control input for the PMSM model. This result on the simplified dynamic dq-model of the PMSM given below:

$$
\dot{\boldsymbol{\nu}}\_{dq} + \boldsymbol{\text{n}}\_{p} \boldsymbol{\alpha}\_{m} \mathfrak{T} \boldsymbol{\mu}\_{dq} = -\boldsymbol{\mathcal{R}}\_{dq} \mathbf{i}\_{dq}^{\star} \tag{57}
$$

$$J\dot{o}\_m + f\_{\rm VF} o\_m = \tau\_e - \tau\_L \tag{58}$$

$$
\dot{\theta}\_m = \alpha\_m \tag{59}
$$

$$\mathbf{r}\_e = -\frac{\mathbf{3}}{\mathbf{2}} n\_p \boldsymbol{\upmu}\_{dq}^T \mathfrak{T} \mathbf{i}\_{dq}^\* \tag{60}$$

This simplified form of the PMSM model is further used to design the control input *dq i* using the passivity approach.

#### **3.2 Passivity property of dq-model**

#### Lemma 3

The PMSM represents a strictly passive system if the reference vector, of the stator currents, *dq i* and the flux linkage vector, *dq* are considered as the input and the output vectors, respectively.

Proof

First, multiply both sides of Equation (57) by *T dq Rs* , yields

$$\nu \boldsymbol{\nu}\_{dq}^T \boldsymbol{i}\_{dq}^\* = -\frac{1}{2\mathcal{R}\_s} \frac{\mathbf{d}\left(\boldsymbol{\nu}\_{dq}^T \boldsymbol{\nu}\_{dq}\right)}{\mathbf{d}t} \tag{61}$$

Passivity Based Control for Permanent-Magnet Synchronous Motors 387

 *dq p m dq* (68)

which ensures the convergence of error vector *ef* to

*<sup>T</sup> Ve ee f ff* (69)

*e n e Ri n f p*

*<sup>m</sup> <sup>f</sup> dq dq*

<sup>1</sup> ( ) <sup>2</sup>

 *<sup>T</sup> Ve e Ri n f f dq dq* 

 1 1 *dq dq dq p m dq dq ff i R n R Ke*

 

dynamics and the damping term injected to make the closed-loop system strictly passive.

Consider the quadratic Function (69) and its time derivative in Equation (70). Substituting

*Kf* is the minimum eigenvalue of the matrix Kf and . is the standard

<sup>2</sup> 2 2 *<sup>T</sup>*

1 <sup>2</sup> ( ) , t 0 <sup>2</sup>

min *Kf* , leads to

min min ( ) , t 0

 

<sup>2</sup>

 <sup>2</sup> min ( ) , t 0 *<sup>T</sup> Ve eKe K e t f f ff* 

 *dq p m dq*

due to the skew-symmetric property of the matrix .

(70)

(71)

consists of two parts: the term which encloses the reference

*f f* (72)

*f fd fq f f e e e ee* (73)

*<sup>T</sup> Ve ee e f ff f* (74)

*K Ve K e f f f f* (75)

zero. The energy function of the closed-loop system is defined as

Taking the time derivative of *V e <sup>f</sup>* along the Trajectory (17), gives

The convergence to zero of the error vector *ef* is ensured by taking

with 0 *fd k* and 0 *fq k* .

The PBCC ensures the exponential stability of the flux tracking error.

**3.3.1 Proof of the exponential stability of the flux tracking error** 

The square of the standard Euclidian norm of the vector ef is given as:

Substituting Equation (16) in Equation (68), yields

The aim is to find the control input *dq i*

Note that the term 0 *<sup>T</sup> n ee p mf f*

0 *fd*

The control input signal, *dq i*

of (71) in (70), yields

where min 0 

euclidian vector norm.

*k*

*f*

*K*

where

*dq i*

0

*fq*

Which combined with Relation (69), gives

Multiplying both sides of (74) by ( )

*k* 

where *<sup>T</sup> dq* is the transposed of vector *dq* .

Note that the term *p m <sup>T</sup> dq dq s n R* does not appear on the right-hand side of (61), since 0 *<sup>T</sup> dq dq* due to skew-symmetric property of the matrix . Integrating both sides of Equation (61), yields

$$\int\_{0}^{t} \left(\boldsymbol{\nu}\_{dq}^{\top} \boldsymbol{i}\_{dq}^{\star}\right) \mathrm{d}t = -\frac{1}{2\,\mathrm{R}\_{s}} \left(\boldsymbol{\nu}\_{dq}^{\top} \boldsymbol{\nu}\_{dq}\right) \mathrm{(t)} + \frac{1}{2\,\mathrm{R}\_{s}} \left(\boldsymbol{\nu}\_{dq}^{\top} \boldsymbol{\nu}\_{dq}\right) \mathrm{(0)}\tag{62}$$

Consider that the *dq i* is the input vector and *dq* is the output vector. Then, with positive definite function

$$V\_f = \frac{1}{2} \boldsymbol{\nu}\_{d\boldsymbol{q}}^T \boldsymbol{\nu}\_{d\boldsymbol{q}} \tag{63}$$

the energy balance Equation (62) of the PMSM becomes

$$\int\_{0}^{t} \left(\boldsymbol{\nu}\_{dq}^{T} \boldsymbol{i}\_{dq}^{\*}\right) \mathbf{d}t = -\frac{1}{R\_s} V\_f(t) + \frac{1}{R\_s} V\_f(0) \tag{64}$$

This means that the PMSM is a strictly passive system (Ortega et al., 1997). Thus, the term <sup>1</sup> *<sup>T</sup> n R <sup>p</sup>m dq dq dq* has no influence on the energy balance and on the asymptotic stability of the PMSM also; it is identified as the workless forces term.

### **3.3 Analysis of tracking errors convergence using passivity-based method**

The desired value of the flux linkage vector *dq* is:

$$\begin{aligned} \boldsymbol{\nu}\boldsymbol{\nu}\_{d\boldsymbol{q}}^{\star} = \begin{bmatrix} \boldsymbol{\nu}\boldsymbol{\nu}\_{d}^{\star} \\ \boldsymbol{\nu}\boldsymbol{\nu}\_{d}^{\star} \end{bmatrix} \end{aligned} \tag{65}$$

and the difference between *dq* and *dq* representing flux tracking error, as:

$$\left[e\_f = \begin{bmatrix} e\_{fd} \\ e\_{fq} \end{bmatrix} \right] = \left\vert \nu\_{dq} - \left\vert \nu\_{dq}^\* \right\vert \tag{66}$$

Rearranging Equation (66)

$$
\psi\_{dq} = e\_f + \psi\_{dq}^\* \tag{67}
$$

Substituting Equation (16) in Equation (68), yields

386 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

*T dq Rs* 

*dq dq T*

*s*

*dq dq* due to skew-symmetric property of the matrix . Integrating both sides of

1 2 *<sup>T</sup> Vf dq* 

*dq dq f f*

This means that the PMSM is a strictly passive system (Ortega et al., 1997). Thus, the term

*dq* is:

*dq*

*fd f dq dq fq*

*dq*

*<sup>e</sup> <sup>e</sup> <sup>e</sup>*

*e*

**3.3 Analysis of tracking errors convergence using passivity-based method** 

*i t Vt V R R*

*T TT dq dq dq dq dq dq s s*

*i t t*

1 1 <sup>d</sup> ( ) (0) 2 2

1 1 d ( ) (0)

has no influence on the energy balance and on the asymptotic stability of

*d*

*d* 

representing flux tracking error, as:

 

*dq f dq*

*s s*

*R R*

*dq dq*

*dq* .

*dq dq*

is the input vector and

*i*

, yields

1 d 2 d

*R t* 

*T*

(61)

does not appear on the right-hand side of (61), since

*dq* is the output vector. Then, with positive

*dq* (63)

(65)

(66)

(67)

 (62)

(64)

Proof

where *<sup>T</sup>* 

0 *<sup>T</sup>*

Equation (61), yields

Consider that the *dq i*

definite function

<sup>1</sup> *<sup>T</sup> n R <sup>p</sup>m dq dq dq*

 

First, multiply both sides of Equation (57) by

*dq* is the transposed of vector

*n R* 

*s*

0

the energy balance Equation (62) of the PMSM becomes

0

the PMSM also; it is identified as the workless forces term.

*dq* and

The desired value of the flux linkage vector

and the difference between

Rearranging Equation (66)

*t T*

*t*

 

Note that the term *p m <sup>T</sup>*

$$\dot{\mathcal{C}}\_f + \mathfrak{n}\_p \rho \partial\_m \mathfrak{S} \mathfrak{e}\_f = -\mathcal{R}\_{dq} \mathfrak{i}\_{dq}^\* - \left(\dot{\psi}\_{dq}^\* + \mathfrak{n}\_p \rho \partial\_m \mathfrak{S} \,\psi\_{dq}^\*\right) \tag{68}$$

The aim is to find the control input *dq i* which ensures the convergence of error vector *ef* to zero. The energy function of the closed-loop system is defined as

$$V(e\_f) = \frac{1}{2} e\_f^T e\_f \tag{69}$$

Taking the time derivative of *V e <sup>f</sup>* along the Trajectory (17), gives

$$\dot{V}\left(e\_f\right) = -e\_f^T \left(R\_{d\eta}\dot{\mathbf{i}}\_{d\eta}^\* + \dot{\boldsymbol{\nu}}\_{d\eta}^\* \boldsymbol{n}\_p \boldsymbol{o}\_m \mathfrak{T} \boldsymbol{\nu}\_{d\eta}^\*\right) \tag{70}$$

Note that the term 0 *<sup>T</sup> n ee p mf f* due to the skew-symmetric property of the matrix . The convergence to zero of the error vector *ef* is ensured by taking

$$\mathbf{i}\_{dq}^{\*} = -\mathbf{R}\_{dq}^{-1} \left( \dot{\mathbf{v}}\_{dq}^{\*} + \mathbf{n}\_{p} \boldsymbol{\alpha}\_{m} \mathbf{G} \boldsymbol{\nu}\_{dq}^{\*} \right) + \mathbf{R}\_{dq}^{-1} \mathbf{K}\_{f} \boldsymbol{e}\_{f} \tag{71}$$
  $\text{where } \mathbf{K}\_{f} = \begin{bmatrix} k\_{fl} & 0 \\ 0 & k\_{fq} \end{bmatrix} \text{ with } k\_{fl} > 0 \text{ and } \boldsymbol{k}\_{fq} > 0 \text{ .}$  
$$\text{The control limit circuit is small. } \boldsymbol{i}^{\*} \text{, consider of two parts: the term which reduces the reference mass.}$$

The control input signal, *dq i* consists of two parts: the term which encloses the reference dynamics and the damping term injected to make the closed-loop system strictly passive. The PBCC ensures the exponential stability of the flux tracking error.

#### **3.3.1 Proof of the exponential stability of the flux tracking error**

Consider the quadratic Function (69) and its time derivative in Equation (70). Substituting *dq i* of (71) in (70), yields

$$\dot{V}\left(e\_f\right) = -e\_f^T K\_f e\_f \le -\lambda\_{\text{min}}\left(K\_f\right) \left\|e\_f(t)\right\|^2, \text{ } \forall \text{ t} \ge 0 \tag{72}$$

where min 0 *Kf* is the minimum eigenvalue of the matrix Kf and . is the standard euclidian vector norm.

The square of the standard Euclidian norm of the vector ef is given as:

$$\left\|\mathbf{e}\_f\right\|^2 = \mathbf{e}\_{f\mathbf{d}}^2 + \mathbf{e}\_{f\boldsymbol{q}}^2 = \mathbf{e}\_f^T \mathbf{e}\_f \tag{73}$$

Which combined with Relation (69), gives

$$V(e\_f) = \frac{1}{2} e\_f^T e\_f \le \left\| e\_f \right\|^2, \forall \ t \ge 0 \tag{74}$$

Multiplying both sides of (74) by ( ) min *Kf* , leads to

$$\left(-\lambda\_{\min}\left\{\mathcal{K}\_f\right\}\right)V(e\_f) \ge \left(-\lambda\_{\min}\left\{\mathcal{K}\_f\right\}\right)\left\|e\_f\right\|^2, \forall \ t \ge 0\tag{75}$$

which combined with Relation (72), gives

$$
\dot{V}\left(e\_f\right) \le -\mathbb{A}\_{\text{min}}\left(K\_f\right)V(e\_f), \forall \text{ t } \ge 0\tag{76}
$$

Integrating both sides of the Inequality (76), yields

$$V(e\_f) \le V(0)e^{-\rho\_f t}, \forall \ t \ge 0 \tag{77}$$

Passivity Based Control for Permanent-Magnet Synchronous Motors 389

In practical applications, the load torque is unknown, therefore it must be estimated. For

<sup>L</sup> ˆ ( ), k 0

The closed loop system represents a strictly passive system if the desired dynamic output

<sup>1</sup>

*R n dq dq p m dq*

 *dq p n R Ke* 

> *T dq Rs*

1 d 2 d

*R t* 

disappears from (88), since 0 *<sup>T</sup>*

 

 0

 

*t T*


*dq dq T dq R t <sup>s</sup>*

*R R*

Let us consider the positive definite function *Vf* from Relation (67). The Energy Balance (90)

*dq f f*

*TTT dq dq dq dq dq s s*

*t t*

 

*s*

*T dq dq T T dq dq ff*

 

*dq* are considered as input and output, respectively.

from (71) in Equation (57), gives

 

**3.4 Passivity property of the closed loop system in the general dq reference frame** 

 

*L Lm m <sup>k</sup>* (85)

(86)

*m d <sup>q</sup> dq ff* (87)

*K e*

*dq dq* due to skew-

*dq ff K e* becomes insignificant. And Equation (88)

(89)

(88)

 

1 d 2 d

 

1 1 <sup>d</sup> ( ) (0) 2 2

1 1 d ( ) (0)

*s s t Vt V R R*

 (90)

(91)

*T*

 

that purpose, an adaptive law (Kim et al., 1997) has been used:

Lemma 4

Proof

where 

The term *p m <sup>T</sup>*

is writes as

*n R* 

*s*

 

vector given by

and the flux linkage vector

Substituting the control input vector *dq i*

is given by Relation (86).

Multiplying both sides of Equation (87) by

*dq dq*

exponentially decreasing. Thus, the term *<sup>T</sup>*

Integrating both sides of Equation (45), yields

0

of the closed loop system becomes

*t*

 

where min 0 *f f <sup>K</sup>* . Considering the Relation (74) at t=0, and multiplying it by *<sup>f</sup> <sup>t</sup> <sup>e</sup>* , gives

$$V(0)e^{-\rho\_f t} \le \left\| e\_f(0) \right\|^2 e^{-\rho\_f t} \tag{78}$$

which combined with Relation (77), leads to the following inequality:

$$V(e\_f) \le \left\| e\_f(0) \right\|^2 e^{-\rho\_f t}, \forall \text{ t } \ge 0 \tag{79}$$

The Inequalities (74) and (79) give that:

$$\left\| e\_f(t) \right\| = \left\| e\_f(0) \right\| e^{-\frac{\rho\_f}{2}t} \tag{80}$$

The Equation (80) shows that, the flux tracking error ef is exponentially decreasing with a rate of convergence of f/2.

#### **3.3.2 Flux reference computation**

The computation of the control signal *dq i* requires the desired flux vector *dq* . If the direct current id in the dq frame is maintained equal to zero, then the PMSM operates under maximum torque. Under this condition and using Equation (52), results in

$$
\varphi\_d^\* = \varphi\_f \tag{81}
$$

$$
\boldsymbol{\wp}\boldsymbol{\upnu}\_q^\* = \boldsymbol{L}\_q \mathbf{i}\_q^\* \tag{82}
$$

The torque set-point value *<sup>e</sup>* corresponding to *dq* is given by Equation (54). Substituting *d* from (81) and *qi* from (82) in (54), it results that:

$$
\pi\_e^\* = \frac{3}{2} \frac{n\_p \wp\_f}{L\_q} \wp\_q^\* \tag{83}
$$

Therefore the value of the flux reference is deduced as

$$
\nu \nu\_q^\* = \frac{2}{3} \frac{L\_q}{n\_p \rho\_f} \tau\_e^\* \tag{84}
$$

#### **3.3.3 Torque reference and load torque computation**

The desired torque *<sup>e</sup>* is computed by the expressions (28)-(29). In practical applications, the load torque is unknown, therefore it must be estimated. For that purpose, an adaptive law (Kim et al., 1997) has been used:

$$
\dot{\hat{\tau}}\_{\rm L} = -k\_{\rm L} (\boldsymbol{\alpha}\_{m} - \boldsymbol{\alpha}\_{m}^{\*})\_{\prime} \quad \mathbf{k}\_{\rm L} > 0 \tag{85}
$$

### **3.4 Passivity property of the closed loop system in the general dq reference frame**

#### Lemma 4

388 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

min ( ), t 0 *V e K Ve f ff*

( ) (0) , t 0 *<sup>f</sup> <sup>t</sup> Ve V e <sup>f</sup>* 

(0) (0) *f f t t Ve e e <sup>f</sup>* 

2 ( ) (0) , t 0 *<sup>f</sup> <sup>t</sup> Ve e e f f* 

<sup>2</sup> ( ) (0)

The Equation (80) shows that, the flux tracking error ef is exponentially decreasing with a

current id in the dq frame is maintained equal to zero, then the PMSM operates under

*dq*

*p f e q q*

*q q e p f L n*

 *d f*

3 2

> 2 3

*n L* 

corresponding to

is computed by the expressions (28)-(29).

from (82) in (54), it results that:

maximum torque. Under this condition and using Equation (52), results in

Therefore the value of the flux reference is deduced as

**3.3.3 Torque reference and load torque computation** 

*f f et e e*

which combined with Relation (77), leads to the following inequality:

*f f <sup>K</sup>* . Considering the Relation (74) at t=0, and multiplying it by *<sup>f</sup> <sup>t</sup> <sup>e</sup>*

2

*f t*

requires the desired flux vector

(76)

(77)

(78)

(79)

(80)

(81)

*q qq L i* (82)

(83)

(84)

is given by Equation (54). Substituting

*dq*

. If the direct

,

which combined with Relation (72), gives

The Inequalities (74) and (79) give that:

rate of convergence of f/2.

The torque set-point value *<sup>e</sup>*

from (81) and *qi*

The desired torque *<sup>e</sup>*

*d*

**3.3.2 Flux reference computation**  The computation of the control signal *dq i*

where min 0 

gives

Integrating both sides of the Inequality (76), yields

The closed loop system represents a strictly passive system if the desired dynamic output vector given by

$$\mathcal{G} = -R\_{dq}^{-1} \left( \dot{\boldsymbol{\nu}}\_{dq}^{\*} + \boldsymbol{\nu}\_{p} \boldsymbol{o} \boldsymbol{o}\_{m} \mathfrak{T} \boldsymbol{\nu}\_{dq}^{\*} \right) \tag{86}$$

and the flux linkage vector *dq* are considered as input and output, respectively.

Proof

Substituting the control input vector *dq i* from (71) in Equation (57), gives

$$
\dot{\boldsymbol{\nu}}\prime\_{dq} + \boldsymbol{\mathfrak{n}}\_p \boldsymbol{o}\prime\_m \mathfrak{Z}\prime \boldsymbol{\nu}\_{dq} = -\boldsymbol{R}\_{dq}\mathcal{G} - \boldsymbol{K}\_f \boldsymbol{e}\_f \tag{87}
$$

where is given by Relation (86).

Multiplying both sides of Equation (87) by *T dq Rs* 

$$\boldsymbol{\nu}\_{d\boldsymbol{q}}^{T}\boldsymbol{\mathcal{G}} = -\frac{1}{2\boldsymbol{R}\_{s}}\frac{\mathrm{d}\left(\boldsymbol{\nu}\_{d\boldsymbol{q}}^{T}\boldsymbol{\nu}\_{d\boldsymbol{q}}\right)}{\mathrm{d}\boldsymbol{t}} - \boldsymbol{\nu}\_{d\boldsymbol{q}}^{T}\boldsymbol{K}\_{f}\boldsymbol{e}\_{f} \tag{88}$$

The term *p m <sup>T</sup> dq dq s n R* disappears from (88), since 0 *<sup>T</sup> dq dq* due to skew-


$$\boldsymbol{\nu}\boldsymbol{\nu}\_{d\boldsymbol{\eta}}^T \boldsymbol{\mathcal{G}} = -\frac{1}{2R\_s} \frac{\mathbf{d}\left(\boldsymbol{\nu}\_{d\boldsymbol{\eta}}^T \boldsymbol{\nu}\_{d\boldsymbol{\eta}}\right)}{\mathbf{d}t} \tag{89}$$

Integrating both sides of Equation (45), yields

$$\int\_{0}^{t} \left(\boldsymbol{\nu}\_{dq}^{T}\boldsymbol{\mathcal{S}}\right) \mathrm{d}t = -\frac{1}{2R\_{s}} \left(\boldsymbol{\nu}\_{dq}^{T}\boldsymbol{\nu}\_{dq}\right) \mathrm{(t)} + \frac{1}{2R\_{s}} \left(\boldsymbol{\nu}\_{dq}^{T}\boldsymbol{\nu}\_{dq}\right) \mathrm{(0)}\tag{90}$$

Let us consider the positive definite function *Vf* from Relation (67). The Energy Balance (90) of the closed loop system becomes

$$\int\_{0}^{t} \left(\boldsymbol{\nu}\_{d\boldsymbol{q}}^{T} \boldsymbol{\mathcal{B}}\right) \mathrm{d}t = -\frac{1}{R\_s} V\_f(t) + \frac{1}{R\_s} V\_f(0) \tag{91}$$

Passivity Based Control for Permanent-Magnet Synchronous Motors 391

reference, without overshoot and all other signals are well shaped. The peaks visible on the electromagnetic torque evolution are due to high gradients imposed to the rotational speed. In practice, these peaks can be easily reduced by limiting the speed reference changing rate

d

q

Imposed dynamics

a better presentation of the control law capabilities and performances.

<sup>e</sup> <sup>p</sup> <sup>f</sup> <sup>n</sup> dL 3 2 d

d <sup>f</sup>

Fig. 6. The block diagram for the passivity-based current controller.

<sup>m</sup> <sup>m</sup>

qi value. However, such situation has been chosen for

 

dq R

<sup>f</sup> ef K1

qi

dt d i d

dq <sup>R</sup> dq <sup>i</sup>

di

i di k d

t 0

t 0

Reverse PARK transform

vd vq

PWM Inverter

voltage PMSM

va vb vc

dt qi qi qi k qi qi vq kqp

i d

i dp k dv

Controller

 <sup>p</sup> <sup>m</sup> dq <sup>n</sup> dt dq <sup>d</sup> <sup>1</sup>

<sup>q</sup>

 

dq <sup>f</sup> <sup>i</sup> dq <sup>L</sup> dq

d

id

PARK transform

ia ib ic

DC supply

iq

and by limiting the imposed current

<sup>L</sup> <sup>z</sup> <sup>ˆ</sup>

Lˆ

Load torque estimator

*e* 

dt <sup>J</sup> <sup>d</sup> <sup>m</sup> <sup>e</sup> 

<sup>m</sup> <sup>z</sup>

s a b 


s L k

 dt d .

\* m

The previous relation shows that, the closed-loop system is a strictly passive (Ortega et al., 1997). Thus, the term *p m <sup>T</sup> dq dq s n R* has no influence on the energy balance and the asymptotic stability of the closed-loop system; it is identified as the workless forces term.

### **3.5 Passivity based current controller structure for PMSM**

The design procedure of the passivity-based current controller for PMSM leads to control structure described by the block diagram in Fig. 6. It consists of three main parts: the load torque estimator given by Equation (85), the desired dynamics expressed by the Relations (28)-(29), (81)-(85), and the controller given by Equations (55), (56) and (71). In this design the imposed flux vector, *dq* , is determined from maximum torque operation conditions allowing the computation of the desired currents *dq i* . Furthermore, the load torque is estimated through speed error, and directly taken into account in the desired dynamics.

The inner loops of the PMSM control are based on well known proportional-integral controllers. Park transform is used for passing electrical variables between the three-phase and dq frame.

The actuator used in the control application is based on a PWM voltage source inverter. Voltage, currents, rotational speed and PMSM angular position are considered measurable variables.

### **3.6 Simulation results**

The parameters of the PMSM used for testing the previously exposed control structure are given in Appendix. 2.

The plant and its corresponding control structure of Fig.6 are implemented using Matlab and Simulink software environment. It employs the PMSM model represented by the Equations (48)-(51) whose parameters are given in appendix 2. The chosen solver is based on Runge-Kutta algorithm (ODE4) and employs an integration time step of 10-4 s. The parameter values of the control system are determined using the procedures detailed in Subsections 2.2 and 2.4 as follows. From the imposed pole locations, the gains of the current PI controller are computed as: kdp=95, kdi=0.85, kqp=95, and kdi=0.8. The gains concerning the desired torque are set at a=75 and b=400 using pole placement method also. The damping parameters values have been obtained by using a trial-and-error procedure starting from guess values based on the stability Condition (71); their final values are kfd = kfq = 650. The gain of the load torque adaptive law is set to kL=6, value which ensures the best asymptotic convergence of the speed error.

In all tests performed in this study, the following signals have been considered as representative for performance analysis: rotational speed (Fig. 7(a)), line current (Fig. 7(b)), electromagnetic torque (Fig. 7(c)), the stator voltages in dq frame (Fig. 7(d)), zoom of voltage at the output of the inverter (Fig. 7(e)), and zoom of line current (Fig. 7(f)). Fig. 7 shows the motor response to square speed reference signal with magnitude ±150 rad/s. This study concerns the robustness test of the designed control system to disturbances. To this end, a load torque step of L=10 Nm has been applied at time 0.5 second and has been removed at time 4.5 seconds (see Fig. 7). The results of Fig. 7 show that the response of the rotor speed to the disturbance is quite and the electromagnetic torque, e, have been increased to a value corresponding to the load applied. The rotational speed and line current tracks quickly the 390 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

The previous relation shows that, the closed-loop system is a strictly passive (Ortega et al.,

asymptotic stability of the closed-loop system; it is identified as the workless forces term.

estimated through speed error, and directly taken into account in the desired dynamics. The inner loops of the PMSM control are based on well known proportional-integral controllers. Park transform is used for passing electrical variables between the three-phase

The actuator used in the control application is based on a PWM voltage source inverter. Voltage, currents, rotational speed and PMSM angular position are considered measurable

The parameters of the PMSM used for testing the previously exposed control structure are

The plant and its corresponding control structure of Fig.6 are implemented using Matlab and Simulink software environment. It employs the PMSM model represented by the Equations (48)-(51) whose parameters are given in appendix 2. The chosen solver is based on Runge-Kutta algorithm (ODE4) and employs an integration time step of 10-4 s. The parameter values of the control system are determined using the procedures detailed in Subsections 2.2 and 2.4 as follows. From the imposed pole locations, the gains of the current PI controller are computed as: kdp=95, kdi=0.85, kqp=95, and kdi=0.8. The gains concerning the desired torque are set at a=75 and b=400 using pole placement method also. The damping parameters values have been obtained by using a trial-and-error procedure starting from guess values based on the stability Condition (71); their final values are kfd = kfq = 650. The gain of the load torque adaptive law is set to kL=6, value which ensures the

In all tests performed in this study, the following signals have been considered as representative for performance analysis: rotational speed (Fig. 7(a)), line current (Fig. 7(b)), electromagnetic torque (Fig. 7(c)), the stator voltages in dq frame (Fig. 7(d)), zoom of voltage at the output of the inverter (Fig. 7(e)), and zoom of line current (Fig. 7(f)). Fig. 7 shows the motor response to square speed reference signal with magnitude ±150 rad/s. This study concerns the robustness test of the designed control system to disturbances. To this end, a load torque step of L=10 Nm has been applied at time 0.5 second and has been removed at time 4.5 seconds (see Fig. 7). The results of Fig. 7 show that the response of the rotor speed to the disturbance is quite and the electromagnetic torque, e, have been increased to a value corresponding to the load applied. The rotational speed and line current tracks quickly the

The design procedure of the passivity-based current controller for PMSM leads to control structure described by the block diagram in Fig. 6. It consists of three main parts: the load torque estimator given by Equation (85), the desired dynamics expressed by the Relations (28)-(29), (81)-(85), and the controller given by Equations (55), (56) and (71). In this design

has no influence on the energy balance and the

. Furthermore, the load torque is

, is determined from maximum torque operation conditions

*dq dq*

*s*

*dq*

best asymptotic convergence of the speed error.

allowing the computation of the desired currents *dq i*

 

**3.5 Passivity based current controller structure for PMSM** 

*n R* 

1997). Thus, the term *p m <sup>T</sup>*

the imposed flux vector,

**3.6 Simulation results** 

given in Appendix. 2.

and dq frame.

variables.

reference, without overshoot and all other signals are well shaped. The peaks visible on the electromagnetic torque evolution are due to high gradients imposed to the rotational speed. In practice, these peaks can be easily reduced by limiting the speed reference changing rate and by limiting the imposed current qi value. However, such situation has been chosen for a better presentation of the control law capabilities and performances.

Fig. 6. The block diagram for the passivity-based current controller.

Passivity Based Control for Permanent-Magnet Synchronous Motors 393

d

 - : (Rs; J) **-**- : (1.5Rs; 2J)



Fig. 8. Motor response to step reference with a change of +50% of the stator winding

A test of robustness at parameter changes has been performed. As presented in Fig. 8, a simultaneous change of +50% of the stator winding resistance Rs and +100% of the moment inertia J. The change of the stator winding resistance, Rs, affects slightly the dynamic motor response. This is due to the fact that the electrical time constant f of closed-loop system appearing in Equation (80) is compensated by the imposed damping gain, Kf, from Equation (71). However, a change of +100% inertia moment J increases the mechanical time constant and hence the rotor speed settling time (see Fig. 2.5). The designed PBCC is based only on the electrical part of the PMSM and has no direct compensation effect on the mechanical

resistance Rs and a change of +100% of the inertia moment J.

 - : (Rs; J) **-**- : (1.5Rs; 2J)

 - : (Rs; J) **-**- : (1.5Rs, 2J)

<sup>b</sup> <sup>a</sup>

 - : (Rs; J) **-**- : (1.5Rs; 2J)

part.

e

c

Fig. 7. Motor response to square speed reference signal with a load torque step of 10 Nm from t=0.5s to t=4.5s.

392 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

 **-**: current reference ia  **--**: line current i\*

 **-**: direct voltage Vd

 **-**: current reference i\*

 **--**: line current ia

**--**: quadrature voltage Vq

a

a

Fig. 7. Motor response to square speed reference signal with a load torque step of 10 Nm

f

d

from t=0.5s to t=4.5s.


a

e

c

m

<sup>b</sup> -: actual speed <sup>m</sup>

Fig. 8. Motor response to step reference with a change of +50% of the stator winding resistance Rs and a change of +100% of the inertia moment J.

A test of robustness at parameter changes has been performed. As presented in Fig. 8, a simultaneous change of +50% of the stator winding resistance Rs and +100% of the moment inertia J. The change of the stator winding resistance, Rs, affects slightly the dynamic motor response. This is due to the fact that the electrical time constant f of closed-loop system appearing in Equation (80) is compensated by the imposed damping gain, Kf, from Equation (71). However, a change of +100% inertia moment J increases the mechanical time constant and hence the rotor speed settling time (see Fig. 2.5). The designed PBCC is based only on the electrical part of the PMSM and has no direct compensation effect on the mechanical part.

Passivity Based Control for Permanent-Magnet Synchronous Motors 395

Ailon, A.; Ortega, R. (1993). An observer-based set-point controller for robot manipulators

Arnold, V I. (1989). *Mathematical Methods of Classical mechanics*, Springer, ISBN 0-387-96890-3,

Berghis, H.; Nijmeijer, H. (1993). A passivity approach to controller-observer design for

Gökder, LU.; Simaan, MA. (1997). A passivity-based control method for Induction motor

Kim, KC.; Ortega, R.; Charara, A.; Vilain, JP. (1997). Theoretical and experimental

Ortega, R.; Spong, M. (1989). Adaptive motion control of rigid robots: A tutorial, *Automatica*,

Ortega, R.; Nicklasson, PJ.; Espinoza–Pérez, G. (1996). On speed control of induction motors,

Ortega, R.; Nicklasson, PJ.; Espinoza–Pérez, G. (1997). Passivity-based controller of a Class

Ortega, R.; Loria, A.; Nicklasson, PJ. (1998). *Passivity-based control of Euler-Lagrange systems*, ,

Ortega, R.; García-Canseco, E. (2004). Interconnection and damping assignment passivity-

Ortega, R.; García-Canseco, E. (2004). Interconnection and damping assignment passivity-

Petrović, V.; Ortega, R.; Stanković, AM. (2001). Interconnection and damping assignment

Qiu, J.; Zhao, G. (2006). PMSM control with port-controlled Hamiltonian theory, *Proceedings* 

Sira-Ramirez, H.; Ortega, R.; Perez-Moreno, R. ; Garcia-Esteban, M. (1995). A sliding mode

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vol.25, No.6, (November 1989), pp 877-888

ISBN 1-85233-016-3, New York, USA, 1998

Island, Bahamas, December 14-17, 2004

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7803-2685-7, New Orleans, LA, 13-15 December, 1995

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with flexible joints. *System Control Literature* 1993, vol.21, No.4, (October 1993), pp

robots. *IEEE Transaction on robotic and automatic,* vol.9, No.6, (December 1993), pp-

control. *IEEE Transactions on Industrial Electrical*. Vol.44, No.5, (October 1997), pp

Comparison of two nonlinear controllers for current-fed induction motors*. IEEE Transactions on Control System Techniques*, vol.5, No.5, (May 1997), pp 338-348, ISSN

of Blondel-Park transformable electric machines, *IEEE Transactions on Automatic* 

Based control: Towards a constructive procedure-Part I, *Proceedings of 43rd IEEE conference on Decision and Control*, pp 3412-3417, ISBN 0-7803-8682-5, Atlantis,

Based control: Towards a constructive procedure-Part II, *Proceedings of 43rd IEEE conference on Decision and Control*, pp 3418-3423, ISBN 0-7803-8682-5, Atlantis,

approach to control of Pm synchronous motors, *IEEE Transactions on Control System Techniques*, vol.9, No.6, (November 2001), pp 811-820, ISSN 1063-

*of 1st International Conference on Innovative Computing, Information and Control (ICICIC'06)*, pp 275-278, ISBN 0-7695-2616-0, Ville, Pay, August 30-31,

controller-observer for DC-to-DC power converters: a passivity approach, *Proceedings* 34th IEEE conference on Decision and Control, pp 3379-3384, ISBN 0-

### **4. Conclusion**

In the section 2, a strategy for designing PMSM control system that requires only rotor position and stator voltage measurements was presented. To this end, the passivity approach to design a controller-observer is adopted. It was shown that this strategy can provide asymptotically stabilizing solutions to the output feedback motor tracking problem. It is shown from simulation results that the robustness of the combined controller-observer with respect to the load and model uncertainties. This is mainly due to the fact that both of the controller and observer exploit the physical structure of the PMSM system and the injection of the high damping.

A new passivity-based speed control law for a PMSM has been developed in the section 3. The proposed control law does not compensate the model workless force terms as they have no effect on the system energy balance. Therefore, the identification of these terms is a key issue in the associated control design. Another feature is that the cancellation of the plant primary dynamics is not done by exact zeroing but by imposing a desired damped transient. The design avoids the using of the Euler-Lagrange model and destructuring (singularities effect) since it uses a flux-based dq-modelling, independent of the rotor angular position. The inner current control loops which have been built using classical PI controllers preserve the passivity property of the current-controlled synchronous machine.

Unlike the majority of the nonlinear control methods used in the PMSM field, this control loop compensates the nonlinearities by means of a damped transient. Its computation aims at imposing the currents set-points based on the flux references in the dq-frame. These latter variables are computed based on the load torque estimation by imposing maximum torque operation conditions.

The speed control law contains a damping term ensuring the system stability and the adjustment of the tracking error convergence speed. The obtained closed-loop system allows exponential zeroing of the speed error, also preserving the passivity property.

Simulation studies show the feasibility and the efficiency of the proposed controller. This controller can be easily included into control structures developed for current-fed induction motor commonly used in industrial applications. Its relatively simple structure should not involve significant hardware and software implementation constraints.

Appendix 1

Ra= 2 ; Rm ; 0.00019 Nm/rd/s; m =0.2 Wb ; np=2 ; Ld=3.1 mH; Lq= 3.1 mH; Dm=0.024 Kgm2; In=15 A; Vn=250 V; Pn=3.75 KW; N=4000 r n/mn.

### Appendix 2

Rated power = 6 Kw; Rated speed = 3000 rpm; Stator winding resistance = 173.77 e-3 ; Stator winding direct inductance = 0.8524 e-3 H; Stator winding quadrate inductance = 0.9515 e-3 H; Rotor flux = 0.1112 Wb; Viscous friction = 0.0085 Nm/rad/s; Inertia = 48 e-4 kg.m2; Pairs pole number = 4; Nominal current line = 31 A; Nominal voltage line = 310 V and the machine type is Siemens 1FT6084-8SK71-1TGO.

### **5. References**

Achour, AY.; Mendil, B. (2007). Commande basée sur la passivité associée aux modes de glissements d'un moteur synchrone à aimants permanents. *JESA*, vol.41, No3-4, April 2007), pp 311-332, ISBN 978-2-7462-1854-3

394 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

In the section 2, a strategy for designing PMSM control system that requires only rotor position and stator voltage measurements was presented. To this end, the passivity approach to design a controller-observer is adopted. It was shown that this strategy can provide asymptotically stabilizing solutions to the output feedback motor tracking problem. It is shown from simulation results that the robustness of the combined controller-observer with respect to the load and model uncertainties. This is mainly due to the fact that both of the controller and observer exploit the physical structure of the PMSM system and the

A new passivity-based speed control law for a PMSM has been developed in the section 3. The proposed control law does not compensate the model workless force terms as they have no effect on the system energy balance. Therefore, the identification of these terms is a key issue in the associated control design. Another feature is that the cancellation of the plant primary dynamics is not done by exact zeroing but by imposing a desired damped transient. The design avoids the using of the Euler-Lagrange model and destructuring (singularities effect) since it uses a flux-based dq-modelling, independent of the rotor angular position. The inner current control loops which have been built using classical PI controllers preserve

Unlike the majority of the nonlinear control methods used in the PMSM field, this control loop compensates the nonlinearities by means of a damped transient. Its computation aims at imposing the currents set-points based on the flux references in the dq-frame. These latter variables are computed based on the load torque estimation by imposing maximum torque

The speed control law contains a damping term ensuring the system stability and the adjustment of the tracking error convergence speed. The obtained closed-loop system allows

Simulation studies show the feasibility and the efficiency of the proposed controller. This controller can be easily included into control structures developed for current-fed induction motor commonly used in industrial applications. Its relatively simple structure should not

Ra= 2 ; Rm ; 0.00019 Nm/rd/s; m =0.2 Wb ; np=2 ; Ld=3.1 mH; Lq= 3.1 mH; Dm=0.024

Achour, AY.; Mendil, B. (2007). Commande basée sur la passivité associée aux modes de

glissements d'un moteur synchrone à aimants permanents. *JESA*, vol.41, No3-4,

Rated power = 6 Kw; Rated speed = 3000 rpm; Stator winding resistance = 173.77 e-3 ; Stator winding direct inductance = 0.8524 e-3 H; Stator winding quadrate inductance = 0.9515 e-3 H; Rotor flux = 0.1112 Wb; Viscous friction = 0.0085 Nm/rad/s; Inertia = 48 e-4 kg.m2; Pairs pole number = 4; Nominal current line = 31 A; Nominal voltage line = 310 V

the passivity property of the current-controlled synchronous machine.

exponential zeroing of the speed error, also preserving the passivity property.

involve significant hardware and software implementation constraints.

Kgm2; In=15 A; Vn=250 V; Pn=3.75 KW; N=4000 r n/mn.

and the machine type is Siemens 1FT6084-8SK71-1TGO.

April 2007), pp 311-332, ISBN 978-2-7462-1854-3

**4. Conclusion** 

injection of the high damping.

operation conditions.

Appendix 1

Appendix 2

**5. References** 


396 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Sudhoff, S.; Wasynczuk, O.; SD Krause, PC. (2002). *Analysis of electric machinery* 

Takegaki, M.; Arimoto, S. (1981). A new feedback for dynamic control of manipulators,

Travieso-Torres, JC.; Duarte Mermoud, MA. ; Estrada, JL. (2006). Tracking control of cascade

Travieso-Torres, JC.; Duarte Mermoud, MA. (2008). Two simple and novel SISO controllers

Van der Schaft, A. (2000). *L2-Gain and Passivity Techniques in Nonlinear Control*, Springer,

2002.

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(January 2008), pp 60-79

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ISBN 1-85233-073-2, London, King Doom, 2000

*and drive Systems*, Wiley-IEEE Press, ISBN 9780470544167, New York, USA,

*Transaction of the ASME, Journal of Dynamic Systems Measurements Control*, vol.

Systems based on passivity: The non-adaptive and adaptive cases, *ISA Transactions*,

for induction motors based on adaptive passivity, *ISA Transactions*, vol.47, No.1,

## *Edited by Andreas Mueller*

Robust control has been a topic of active research in the last three decades culminating in H\_2/H\_\infty and \mu design methods followed by research on parametric robustness, initially motivated by Kharitonov's theorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes of Recent Advances in Robust Control give a selective overview of recent theoretical developments and present selected application examples. The volumes comprise 39 contributions covering various theoretical aspects as well as different application areas. The first volume covers selected problems in the theory of robust control and its application to robotic and electromechanical systems. The second volume is dedicated to special topics in robust control and problem specific solutions. Recent Advances in Robust Control will be a valuable reference for those interested in the recent theoretical advances and for researchers working in the broad field of robotics and mechatronics.

Photo by Ociacia / Shutterstock

Recent Advances in Robust Control -

Theory and Applications in Robotics and Electromechanics

Recent Advances in

Robust Control

Theory and Applications in

Robotics and Electromechanics

*Edited by Andreas Mueller*