**Meet the editor**

Dr. S. Ramakrishnan is a professor and the head of the Department of Information Technology, Dr. Mahalingam College of Engineering and Technology, Pollachi, India. Dr. Ramakrishnan is a reviewer of 25 international journals such as IEEE transactions, IET journals, *ACM Computing Reviews*, Elsevier science journals, Springer journals, Wiley journals, etc. He is in the editorial board

of seven international journals. He is a guest editor of special issues in three international journals including the *Telecommunication Systems Journal* of Springer. He has published 139 papers and 6 books. Dr. S. Ramakrishnan has published a book on wireless sensor networks for CRC Press, USA; four books on speech processing and pattern recognition for InTech Publisher, Croatia; and a book on computational techniques for Lambert Academic Publishing, Germany.

## Contents

## **Preface XIII**



**Section 5 Applications of Fuzzy in Industrial Engineering 345**

Chapter 8 **Robust Adaptive Fuzzy Control for a Class of Switching Power**

Chapter 9 **Fuzzy Optimization Control: From Crisp Optimization 163**

**Section 3 Applications of Fuzzy in Navigation Systems 183**

Chapter 10 **An Approach of Fuzzy Logic H∞ Filter in Mobile Robot**

**Navigation Considering Non-Gaussian Noise 185** Hamzah Ahmad, Nur Aqilah Othman and Saifudin Razali

Chapter 11 **Indoor Mobile Positioning Using Neural Networks and Fuzzy**

Anatoly D. Khomonenko, Sergey E. Adadurov, Alexandr V.

**Based on Fuzzy Logic Denoising of Sensors Signals 257** Teodor Lucian Grigorie and Ruxandra Mihaela Botez

Chapter 12 **A Fuzzy Logic Approach for Separation Assurance and Collision Avoidance for Unmanned Aerial Systems 225** Brandon Cook, Tim Arnett and Kelly Cohen

Chapter 13 **Precision Improvement in Inertial Miniaturized Navigators**

**Section 4 Applications of Fuzzy in Imaging and Healthcare 279**

Chapter 14 **A Fuzzy Belief-Desire-Intention Model for Agent-Based Image**

Chapter 15 **ANFIS Definition of Focal Length for Zoom Lens via Fuzzy Logic**

Chapter 16 **EMG-Controlled Prosthetic Hand with Fuzzy Logic Classification**

Bahadır Ergün, Cumhur Sahin and Ugur Kaplan

**Converters 139** Cheng-Lun Chen

**VI** Contents

Makoto Katoh

**Logic Control 201**

**Analysis 281** Peter Hofmann

**Functions 297**

**Algorithm 321**

Beyda Taşar and Arif Gülten

Krasnovidow and Pavel A. Novikov


## Preface

Control systems play an important role in engineering. Fuzzy logic is the natural choice for designing control applications and is the most popular and appropriate for the control of home and industrial appliances. Academic and industrial experts are constantly researching and proposing innovative and effective fuzzy control systems. This book is an edited vol‐ ume and has twenty-one innovative chapters arranged into five sections covering applica‐ tions of fuzzy control systems in energy and power systems, navigation systems, imaging, and industrial engineering.

Section 1 is on modern fuzzy control systems and has five valuable chapters written by leading researchers. These five chapters provide strong theoretical-cum-application-orientated pre‐ sentations. Improvisations of conventional models such as Takagi-Sugeno and Mamdani mod‐ els along with numerical illustrations and applications are provided in these five chapters.

Section 2 is on energy and power system applications and has four well-written chapters by scholarly authors. All these four chapters provide requisite mathematical models, illustra‐ tions, and experimental analysis.

Section 3 is on navigation systems and has four rich chapters written by eminent authors. Out of these four chapters, two chapters focus on mobile navigations, and the other two chapters focus on inertial and aerial navigations. These four chapters provide not only theo‐ retical stuffs but also experimental results.

Section 4 is on imaging and healthcare applications and comprises three chapters written by renowned authors. This section covers object-based image analysis, camera calibration, and prosthetic hand.

The last section is on industrial engineering applications containing five potential chapters written by distinguished authors. The applications are presented with mathematical models and simulation results. The five applications include control systems for fruit juice-produc‐ ing company, MIMO systems in mechatronics, vibration control, wastewater treatment plant and soil vulnerability testing.

Overall, this book provides a rich set of modern fuzzy control systems and their applications and will be a useful resource for the graduate students, researchers, and practicing engi‐ neers in the field of electrical engineering.

I would like to express my sincere thanks to all authors for their contribution and effort to bring in this wonderful book. My earnest gratitude and appreciation are extended to the InTe‐ chOpen Publisher, in particular Ms. Romina Rovan. I would like to express my heartfelt thanks to the management, secretary, and principal of my institute. Finally, I would like to extend my dearest thanks to my family members and in particular to my cute daughter Abira‐ mi.

**Dr. S. Ramakrishnan**

Professor and Head, Department of Information Technology Dr. Mahalingam College of Engineering and Technology XIV Preface

Pollachi, India

**Modern Fuzzy Control Systems**

Pollachi, India

X Preface

## **Stabilizing Fuzzy Control via Output Feedback**

Dušan Krokavec and Anna Filasová

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68129

#### Abstract

The chapter presents new conditions suitable in design of stabilizing static as well as dynamic output controllers for a class of continuous-time nonlinear systems represented by Takagi-Sugeno models. Taking into account the affine properties of the TS model structure, and applying the fuzzy control scheme relating to the parallel-distributed output compensators, the sufficient design conditions are outlined in the terms of linear matrix inequalities. Depending on the proposed procedures, the Lyapunov matrix can be decoupled from the system parameter matrices using linear matrix inequality techniques or a fuzzy-relaxed approach can be applied to make closed-loop dynamics faster. Numerical examples illustrate the design procedures and demonstrate the performances of the proposed design methods.

Keywords: continuous-time nonlinear systems, Takagi-Sugeno fuzzy systems, linear matrix inequality approach, parallel-distributed compensation, output feedback

### 1. Introduction

Contrarily to the linear framework, nonlinear systems are too complex to be represented by unified mathematical resources and so, a generic method has not been developed yet to design a controller valid for all types of nonlinear systems. An alternative to nonlinear system models is Takagi-Sugeno (TS) fuzzy approach [1], which benefits from the advantages of suitable linear approximation of sector nonlinearities. Using the TS fuzzy model, each rule utilizes the local system dynamics by a linear model and the nonlinear system is represented by a collection of fuzzy rules. Recently, TS model-based fuzzy control approaches are being fast and successfully used in nonlinear control frameworks. As a result, a range of stability analysis conditions [2–5] as well as control design methods have been developed for TS fuzzy systems [6–9], relying mostly on the feasibility of an associated set of linear matrix inequalities

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(LMI) [10]. An important fact is that the design problem is a standard feasibility problem with several LMIs, potentially combined with one matrix equality to overcome the problem of bilinearity. In consequence, the state and output feedback control based on fuzzy TS systems models is mostly realized in such structures, which can be designed using numerical techniques based on LMIs.

The TS fuzzy model-based state control is based on an implicit assumption that all states are available for measurement. If it is impossible, TS fuzzy observers are used to estimate the unmeasurable state variables, and the state controller exploits the system state variable estimate values [11–14]. The nonlinear output feedback design is so formulated as two LMI set problem, and treated by the two-stage procedure using the separation principle, that is, dealing with a set of LMIs for the observer parameters at first and then solving another set of LMIs for the controller parameters [15]. Since, the fuzzy output control does not require the measurement of system state variables and can be formulated as a one LMI set problem, such structure of feedback control is preferred, of course, if the system is stabilizable.

From a relatively wide range of problems associated with the fuzzy output feedback control design for the continuous-time nonlinear MIMO systems approximated by a TS model, the chapter deals with the techniques incorporating the slack matrix application and fuzzy membership-relaxed approaches. The central idea of the TS fuzzy model-based control design, that is, to derive control rules so as to compensate each rule of a fuzzy system and construct the control strategy based on the parallel-distributed compensators (PDC), is reflected in the approach of output control. Motivated by the above mentioned observations, the proposed design method respects the results presented in Refs. [16, 17], and is constructed on an enhanced form of quadratic Lyapunov function. Comparing with the approaches based only on quadratic Lyapunov matrix [18], which are particular in the case of large number of rules, that are very conservative as a common symmetric positive definite matrix, is used to verify all Lyapunov inequalities, presented principle naturally extends the affine TS model properties using slack matrix variables to decouple Lyapunov matrix and the system matrices in LMIs, and gives substantial reducing of conservativeness. Moreover, extra quadratic constraints are included to incorporate fuzzy membership functions relaxes [19, 20] and applied for static as well as dynamic TS fuzzy output controllers design. Note, other constraints with respect to, for example, to decay rate and closed-loop pole clustering can be utilized to extend the proposed design procedures.

The remainder of this chapter is organized as follows. In Section 2, the structure of TS model for considered class of nonlinear systems is briefly described, and some of its properties are outlined. The output feedback control design problem for systems with measurable promise variables is given in Section 3, where the design conditions that guarantees global quadratic stability are formulated and proven. To complete the solutions, Section 4 formulate the static decoupling principle in static TS fuzzy output control, and the method is reformulated in Section 5 in defined criteria for TS fuzzy dynamic output feedback control design. Section 6 gives the numerical examples to illustrate the effectiveness of the proposed approach, and to confirm the validity of the control scheme. The last section, Section 7, draws conclusions and some future directions.

Throughout the chapter, the following notations are used: x<sup>T</sup> , X<sup>T</sup> denotes the transpose of the vector x and matrix X, respectively, for a square matrix X = X<sup>T</sup> > 0 (respectively, X = X<sup>T</sup> < 0) means that X is a symmetric positive definite matrix (respectively, negative definite matrix), the symbol I<sup>n</sup> represents the n-th order unit matrix, IR denotes the set of real numbers, and IR<sup>n</sup> � <sup>r</sup> denotes the set of all <sup>n</sup> � <sup>r</sup> real matrices.

### 2. Takagi-Sugeno fuzzy models

(LMI) [10]. An important fact is that the design problem is a standard feasibility problem with several LMIs, potentially combined with one matrix equality to overcome the problem of bilinearity. In consequence, the state and output feedback control based on fuzzy TS systems models is mostly realized in such structures, which can be designed using numerical tech-

The TS fuzzy model-based state control is based on an implicit assumption that all states are available for measurement. If it is impossible, TS fuzzy observers are used to estimate the unmeasurable state variables, and the state controller exploits the system state variable estimate values [11–14]. The nonlinear output feedback design is so formulated as two LMI set problem, and treated by the two-stage procedure using the separation principle, that is, dealing with a set of LMIs for the observer parameters at first and then solving another set of LMIs for the controller parameters [15]. Since, the fuzzy output control does not require the measurement of system state variables and can be formulated as a one LMI set problem, such

From a relatively wide range of problems associated with the fuzzy output feedback control design for the continuous-time nonlinear MIMO systems approximated by a TS model, the chapter deals with the techniques incorporating the slack matrix application and fuzzy membership-relaxed approaches. The central idea of the TS fuzzy model-based control design, that is, to derive control rules so as to compensate each rule of a fuzzy system and construct the control strategy based on the parallel-distributed compensators (PDC), is reflected in the approach of output control. Motivated by the above mentioned observations, the proposed design method respects the results presented in Refs. [16, 17], and is constructed on an enhanced form of quadratic Lyapunov function. Comparing with the approaches based only on quadratic Lyapunov matrix [18], which are particular in the case of large number of rules, that are very conservative as a common symmetric positive definite matrix, is used to verify all Lyapunov inequalities, presented principle naturally extends the affine TS model properties using slack matrix variables to decouple Lyapunov matrix and the system matrices in LMIs, and gives substantial reducing of conservativeness. Moreover, extra quadratic constraints are included to incorporate fuzzy membership functions relaxes [19, 20] and applied for static as well as dynamic TS fuzzy output controllers design. Note, other constraints with respect to, for example, to decay rate and closed-loop pole clustering can be utilized to extend the proposed design procedures. The remainder of this chapter is organized as follows. In Section 2, the structure of TS model for considered class of nonlinear systems is briefly described, and some of its properties are outlined. The output feedback control design problem for systems with measurable promise variables is given in Section 3, where the design conditions that guarantees global quadratic stability are formulated and proven. To complete the solutions, Section 4 formulate the static decoupling principle in static TS fuzzy output control, and the method is reformulated in Section 5 in defined criteria for TS fuzzy dynamic output feedback control design. Section 6 gives the numerical examples to illustrate the effectiveness of the proposed approach, and to confirm the validity of the control scheme. The last section, Section 7, draws conclusions and some future directions.

structure of feedback control is preferred, of course, if the system is stabilizable.

Throughout the chapter, the following notations are used: x<sup>T</sup>

vector x and matrix X, respectively, for a square matrix X = X<sup>T</sup> > 0 (respectively, X = X<sup>T</sup> < 0)

, X<sup>T</sup> denotes the transpose of the

niques based on LMIs.

4 Modern Fuzzy Control Systems and Its Applications

The systems under consideration are from one class of multi-input and multi-output (MIMO) dynamic systems, which are nonlinear in sectors and represented by TS fuzzy model. Constructing the set of membership functions hi (θ(t)), i = 1, 2, …, s, where

$$\boldsymbol{\theta}(t) = \begin{bmatrix} \theta\_1(t) & \theta\_2(t) & \cdots & \theta\_q(t) \end{bmatrix}, \tag{1}$$

is the vector of premise variables, the final states of the systems are inferred in the TS fuzzy system model as follows

$$\dot{q}(t) = \sum\_{i=1}^{s} h\_i(\boldsymbol{\theta}(t)) \left( \mathbf{A}\_i \boldsymbol{q}(t) + \mathbf{B}\_i \boldsymbol{u}(t) \right), \tag{2}$$

with the output given by the relation

$$\mathbf{y}(t) = \mathbf{C}q(t),\tag{3}$$

where q(t) ∈ IRn , u(t) ∈ IR<sup>r</sup> , y(t) ∈ IRm are vectors of the state, input, and output variables, A<sup>i</sup> ∈ IRn � <sup>n</sup> , B<sup>i</sup> ∈ IRn � <sup>r</sup> , C ∈ IRm � <sup>n</sup> are real finite values matrix, and where hi(θ(t)) is the averaging weight for the i-th rule, representing the normalized grade of membership (membership function).

By definition, the membership functions satisfy the following convex sum properties.

$$0 \le h\_i(\boldsymbol{\theta}(t)) \le 1, \quad \sum\_{i=1}^s h\_i(\boldsymbol{\theta}(t)) = 1 \quad \forall i \in \langle 1, \ldots, s \rangle. \tag{4}$$

It is assumed that the premise variable is a system state variable or a measurable external variable, and none of the premise variables depends on the inputs u(t).

It is evident that a general fuzzy model is achieved by fuzzy amalgamation of the linear system models. Using a TS model, the conclusion part of a single rule consists no longer of a fuzzy set [21], but determines a function with state variables as arguments, and the corresponding function is a local function for the fuzzy region that is described by the premise part of the rule. Thus, using linear functions, a system state is described locally (in fuzzy regions) by linear models, and at the boundaries between regions an interpolation is used between the corresponding local models.

Note, the models, Eqs. (2) and (3), are mostly considered for analysis, control, and state estimation of nonlinear systems.

Assumption 1 Each triplet (Ai, Bi, C) is locally controllable and observable, the matrix C is the same for all local models.

It is supposed in the next that the aforementioned model does not include parameter uncertainties or external disturbances, and the premise variables are measured.

#### 3. Static fuzzy output controller

In the next, the fuzzy output controller is designed using the concept of parallel-distributed compensation, in which the fuzzy controller shares the same sets of normalized membership functions like the TS fuzzy system model.

Definition 1 Considering Eqs. (2) and (3), and using the same set of normalized membership function Eq. (4), the fuzzy static output controller is defined as

$$\mathfrak{u}(t) = \sum\_{j=1}^{s} h\_{\dot{\gamma}}(\mathfrak{G}(t)) \mathbf{K}\_{\dot{\gamma}} \mathfrak{y}(t) = \sum\_{j=1}^{s} h\_{\dot{\gamma}}(\mathfrak{G}(t)) \mathbf{K}\_{\dot{\gamma}} \mathbf{C} \mathbf{q}(t) \,. \tag{5}$$

Note that the fuzzy controller Eq. (5) is in general nonlinear.

Considering the system, Eqs. (2) and (3), and the control law, Eq. (5), yields

$$\dot{\boldsymbol{q}}(t) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \left( \mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_j \mathbf{C} \right) \boldsymbol{q}(t) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \mathbf{A}\_{i\bar{j}} \boldsymbol{q}(t), \qquad (6)$$

$$\mathbf{A}\_{\rm cij} = \mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_j \mathbf{C}, \quad \mathbf{A}\_{\rm cji} = \mathbf{A}\_j + \mathbf{B}\_j \mathbf{K}\_i \mathbf{C} \,. \tag{7}$$

Proposition 1 (standard design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically stable if there exist a positive definite symmetric matrix W ∈ IR<sup>n</sup> � <sup>n</sup> and matrices Y<sup>j</sup> ∈ IRr � <sup>m</sup>, H ∈ IR<sup>m</sup> � <sup>m</sup> such that

$$\mathbf{W} = \mathbf{W}^T > \mathbf{0},\tag{8}$$

$$\mathbf{A}\_{i}\mathbf{W} + \mathbf{W}\mathbf{A}\_{i}^{T} + \mathbf{B}\_{i}\mathbf{Y}\_{i}\mathbf{C} + \mathbf{C}^{T}\mathbf{Y}\_{i}^{T}\mathbf{B}\_{i}^{T} < \mathbf{0},\tag{9}$$

$$\frac{\mathbf{A}\_i \mathbf{W} + \mathbf{W} \mathbf{A}\_i^T}{2} + \frac{\mathbf{A}\_j \mathbf{W} + \mathbf{W} \mathbf{A}\_j^T}{2} + \frac{\mathbf{B}\_i \mathbf{Y}\_j \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_j^T \mathbf{B}\_i^T}{2} + \frac{\mathbf{B}\_j \mathbf{Y}\_i \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_i^T \mathbf{B}\_j^T}{2} < 0,\tag{10}$$

$$\mathbf{CW} = \mathbf{HC} \tag{11}$$

for i = 1, 2, …, s as well as i = 1, 2, …, s � 1, j = i + 1, i + 2, …, s, and hi(θ(t))hj(θ(t)) 6¼ 0. When the above conditions hold, the control law gain matrices are given as

$$\mathbf{K}\_{i} = \mathbf{Y}\_{i}\mathbf{H}^{-1}.\tag{12}$$

Proof. (compare, for example, Ref. [16]) Prescribing the Lyapunov function candidate of the form

$$\nu(\boldsymbol{q}(t)) = \boldsymbol{q}^T(t)\boldsymbol{P}\boldsymbol{q}(t) > 0,\tag{13}$$

where P ∈ IR<sup>n</sup> � <sup>n</sup> is a symmetric positive definite matrix, the time derivative of Eq. (13) along the system trajectory is

$$
\dot{\boldsymbol{\nu}}(\boldsymbol{q}(t)) = \dot{\boldsymbol{q}}^T(t)\mathbf{P}\boldsymbol{q}(t) + \boldsymbol{q}^T(t)\mathbf{P}\dot{\boldsymbol{q}}(t) < 0. \tag{14}
$$

Inserting Eq. (6) into Eq. (14), it has to be satisfied

Assumption 1 Each triplet (Ai, Bi, C) is locally controllable and observable, the matrix C is the same

It is supposed in the next that the aforementioned model does not include parameter uncer-

In the next, the fuzzy output controller is designed using the concept of parallel-distributed compensation, in which the fuzzy controller shares the same sets of normalized membership

Definition 1 Considering Eqs. (2) and (3), and using the same set of normalized membership function

j¼1

i¼1

<sup>i</sup> B<sup>T</sup>

<sup>j</sup> B<sup>T</sup> i 2 þ

Xs j¼1

Acij ¼ A<sup>i</sup> þ BiKjC, Acji ¼ A<sup>j</sup> þ BjKiC: ð7Þ

<sup>W</sup> <sup>¼</sup> <sup>W</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>ð</sup>8<sup>Þ</sup>

<sup>B</sup>jYi<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

CW ¼ HC ð11Þ

hjð Þ θð Þt KjCqð Þt : ð5Þ

hið Þ θð Þt hjð Þ θð Þt Acijqð Þt ; ð6Þ

<sup>i</sup> < 0; ð9Þ

<sup>i</sup> B<sup>T</sup> j

<sup>2</sup> <sup>&</sup>lt; <sup>0</sup>; <sup>ð</sup>10<sup>Þ</sup>

hjð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>K</sup>jyðÞ¼ <sup>t</sup> <sup>X</sup><sup>s</sup>

tainties or external disturbances, and the premise variables are measured.

for all local models.

3. Static fuzzy output controller

6 Modern Fuzzy Control Systems and Its Applications

functions like the TS fuzzy system model.

<sup>q</sup>\_ðÞ¼ <sup>t</sup> <sup>X</sup><sup>s</sup>

i¼1

<sup>A</sup>i<sup>W</sup> <sup>þ</sup> WA<sup>T</sup>

i 2 þ

Xs j¼1

Eq. (4), the fuzzy static output controller is defined as

<sup>u</sup>ðÞ¼ <sup>t</sup> <sup>X</sup><sup>s</sup>

Note that the fuzzy controller Eq. (5) is in general nonlinear.

j¼1

Considering the system, Eqs. (2) and (3), and the control law, Eq. (5), yields

symmetric matrix W ∈ IR<sup>n</sup> � <sup>n</sup> and matrices Y<sup>j</sup> ∈ IRr � <sup>m</sup>, H ∈ IR<sup>m</sup> � <sup>m</sup> such that

<sup>A</sup>i<sup>W</sup> <sup>þ</sup> WA<sup>T</sup>

j 2 þ

When the above conditions hold, the control law gain matrices are given as

for i = 1, 2, …, s as well as i = 1, 2, …, s � 1, j = i + 1, i + 2, …, s, and hi(θ(t))hj(θ(t)) 6¼ 0.

<sup>A</sup>j<sup>W</sup> <sup>þ</sup> WA<sup>T</sup>

hið Þ <sup>θ</sup>ð Þ<sup>t</sup> hjð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �qðÞ¼ <sup>t</sup> <sup>X</sup><sup>s</sup>

Proposition 1 (standard design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically stable if there exist a positive definite

<sup>i</sup> <sup>þ</sup> <sup>B</sup>iYi<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

<sup>B</sup>iYj<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

$$\dot{\nu}(q(t)) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\theta}(t)) h\_j(\boldsymbol{\theta}(t)) q^T(t) \mathbf{P}\_{i\bar{j}} q(t) < 0,\tag{15}$$

$$\mathbf{P}\_{c\ddot{\eta}} = \mathbf{P} \mathbf{A}\_{c\dddot{\eta}} + \mathbf{A}\_{c\dddot{\eta}}^T \mathbf{P}.\tag{16}$$

Since P is positive definite, the state coordinate transform can be defined as

$$\boldsymbol{q}(t) = \mathbf{W}\boldsymbol{p}(t), \quad \mathbf{W} = \mathbf{P}^{-1}, \tag{17}$$

and subsequently, Eqs. (15) and (16) can be rewritten as

$$\dot{\boldsymbol{p}}(\boldsymbol{p}(t)) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\theta}(t)) h\_j(\boldsymbol{\theta}(t)) \boldsymbol{p}^T(t) \mathbf{W}\_{cij} \boldsymbol{p}(t) < 0,\tag{18}$$

$$\mathbf{W}\_{\rm cij} = \mathbf{A}\_{\rm cij}\mathbf{W} + \mathbf{W}\mathbf{A}\_{\rm cij}^T. \tag{19}$$

Permuting the subscripts i and j in Eq. (18), also it can write

$$\dot{\boldsymbol{\nu}}(\boldsymbol{\eta}(t)) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \boldsymbol{p}^T(t) \mathbf{W}\_{\boldsymbol{c}ji} \boldsymbol{p}(t) < 0,\tag{20}$$

$$\mathbf{W}\_{cji} = \mathbf{A}\_{cji}\mathbf{W} + \mathbf{W}\mathbf{A}\_{cji}^T. \tag{21}$$

Thus, adding Eqs. (17) and (19), it yields

$$2\dot{\boldsymbol{\nu}}(\boldsymbol{\mathfrak{p}}(t)) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\varTheta}(t)) h\_j(\boldsymbol{\varTheta}(t)) \boldsymbol{\upnu}^T(t) \left(\boldsymbol{\varTheta}\_{c\bar{\boldsymbol{\upnu}}} + \boldsymbol{\varTheta}\_{c\bar{\boldsymbol{\upnu}}}\right) \boldsymbol{\upnu}(t) < 0 \tag{22}$$

and subsequently,

$$\dot{\boldsymbol{\nu}}(\boldsymbol{p}(t)) = \sum\_{i=1}^{s} h\_i^2(\boldsymbol{\Theta}(t)) \boldsymbol{p}^T(t) \mathbf{W}\_{\text{cil}} \boldsymbol{p}(t) + 2 \sum\_{i=1}^{s-1} \sum\_{j=i+1}^{s} h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \boldsymbol{p}^T(t) \frac{\mathbf{W}\_{\text{cil}} + \mathbf{W}\_{\text{cjl}}}{2} \boldsymbol{p}(t) < 0, \quad (23)$$

which leads to the set of inequalities.

$$(\mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_i \mathbf{C})\mathbf{W} + \mathbf{W}(\mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_i \mathbf{C})^T < 0,\tag{24}$$

$$\frac{\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{j}\mathbf{C}\right)\mathbf{W}}{2} + \frac{\left(\mathbf{A}\_{j} + \mathbf{B}\_{j}\mathbf{K}\_{i}\mathbf{C}\right)\mathbf{W}}{2} + \frac{\mathbf{W}\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{j}\mathbf{C}\right)^{T}}{2} + \frac{\mathbf{W}\left(\mathbf{A}\_{j} + \mathbf{B}\_{j}\mathbf{K}\_{i}\mathbf{C}\right)^{T}}{2} < 0\qquad(25)$$

for i = 1, 2, …, s as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s and hi(θ(t))hj(θ(t)) 6¼ 0.

Thus, setting here

$$\mathbf{K}\_{\circ}\mathbf{C}\mathbf{W} = \mathbf{K}\_{\circ}\mathbf{H}\mathbf{H}^{-1}\mathbf{C}\mathbf{W},\tag{26}$$

where H is a regular square matrix of appropriate dimension and defining

$$\mathbf{H}^{-1}\mathbf{C} = \mathbf{C}\mathbf{W}^{-1}, \quad \mathbf{Y}\_{\rangle} = \mathbf{K}\_{\rangle}\mathbf{H}, \tag{27}$$

the LMI forms of Eqs. (9) and (10) are obtained from Eqs. (24) and (25), respectively, and Eq. (27) implies Eq. (11). This concludes the proof.

Trying to minimize the number of LMIs owing to the limitation of solvers, Proposition 1 is presented in the structure, in which the number of stabilization conditions, used in fuzzy controller design, is equal to N = (s <sup>2</sup> + s)/2 + 1. Evidently, the number of stabilization conditions is substantially reduced if s is large.

Proposition 2 (enhanced design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically stable if for given a positive δ ∈ IR, there exist positive definite symmetric matrices V, S ∈ IR<sup>n</sup> � <sup>n</sup> , and matrices Y<sup>j</sup> ∈ IR<sup>r</sup> � <sup>m</sup>, H ∈ IRm � <sup>m</sup> such that

$$\mathbf{S} = \mathbf{S}^T > 0, \quad \mathbf{V} = \mathbf{V}^T > 0,\tag{28}$$

$$
\begin{bmatrix}
\mathbf{A}\_i \mathbf{S} + \mathbf{S} \mathbf{A}\_i^T + \mathbf{B}\_i \mathbf{Y}\_i \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_i^T \mathbf{B}\_i^T & \* \\
\mathbf{V} - \mathbf{S} + \delta \mathbf{A}\_i \mathbf{S} + \delta \mathbf{B}\_i \mathbf{Y}\_i \mathbf{C} & -2\delta \mathbf{S}
\end{bmatrix} < 0,\tag{29}
$$

$$
\begin{bmatrix}
\mathbf{O}\_{\dot{\boldsymbol{\eta}}} & \* \\
\mathbf{V} - \mathbf{S} + \delta \frac{\mathbf{A}\_{i}\mathbf{S} + \mathbf{A}\_{j}\mathbf{S}}{2} + \delta \frac{\mathbf{B}\_{i}\mathbf{Y}\_{j} + \mathbf{B}\_{j}\mathbf{Y}\_{i}}{2}\mathbf{C} & -2\delta \mathbf{S}
\end{bmatrix} < 0,\tag{30}
$$

$$\mathbf{CS} = \mathbf{HC} \,, \tag{31}$$

for i = 1, 2, …, s, as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s, hi(θ(t))hj(θ(t)) 6¼ 0, and

Stabilizing Fuzzy Control via Output Feedback http://dx.doi.org/10.5772/68129 9

$$\mathbf{O}\_{\dot{\boldsymbol{\eta}}} = \frac{\mathbf{A}\_{i}\mathbf{S} + \mathbf{S}\mathbf{A}\_{i}^{T}}{2} + \frac{\mathbf{A}\_{j}\mathbf{S} + \mathbf{S}\mathbf{A}\_{j}^{T}}{2} + \frac{\mathbf{B}\_{i}\mathbf{Y}\_{j}\mathbf{C} + \mathbf{C}^{T}\mathbf{Y}\_{j}^{T}\mathbf{B}\_{i}^{T}}{2} + \frac{\mathbf{B}\_{j}\mathbf{Y}\_{i}\mathbf{C} + \mathbf{C}^{T}\mathbf{Y}\_{i}^{T}\mathbf{B}\_{j}^{T}}{2}.\tag{32}$$

When the above conditions hold, the control law gain matrices are given as

$$\mathbf{K}\_{i} = \mathbf{Y}\_{i}\mathbf{H}^{-1}.\tag{33}$$

Here and hereafter, ∗ denotes the symmetric item in a symmetric matrix.

Proof. Writing Eq. (6) in the form

<sup>ν</sup>\_ð Þ¼ <sup>p</sup>ð Þ<sup>t</sup> <sup>X</sup><sup>s</sup>

Thus, setting here

such that

i¼1 h2

which leads to the set of inequalities.

8 Modern Fuzzy Control Systems and Its Applications

<sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>W</sup>

2 þ

Eq. (27) implies Eq. (11). This concludes the proof.

there exist positive definite symmetric matrices V, S ∈ IR<sup>n</sup> � <sup>n</sup>

<sup>A</sup>i<sup>S</sup> <sup>þ</sup> SA<sup>T</sup>

AiS þ AjS <sup>2</sup> <sup>þ</sup> <sup>δ</sup>

V � S þ δ

2 6 4

controller design, is equal to N = (s

is substantially reduced if s is large.

<sup>i</sup> ð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>p</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>W</sup>ciipð Þþ <sup>t</sup> <sup>2</sup>

<sup>A</sup><sup>j</sup> <sup>þ</sup> <sup>B</sup>jKi<sup>C</sup> � �<sup>W</sup>

2 þ

where H is a regular square matrix of appropriate dimension and defining

H�<sup>1</sup>

for i = 1, 2, …, s as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s and hi(θ(t))hj(θ(t)) 6¼ 0.

<sup>K</sup>jCW <sup>¼</sup> <sup>K</sup>jHH�<sup>1</sup>

<sup>C</sup> <sup>¼</sup> CW�<sup>1</sup>

the LMI forms of Eqs. (9) and (10) are obtained from Eqs. (24) and (25), respectively, and

Trying to minimize the number of LMIs owing to the limitation of solvers, Proposition 1 is presented in the structure, in which the number of stabilization conditions, used in fuzzy

Proposition 2 (enhanced design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically stable if for given a positive δ ∈ IR,

<sup>i</sup> <sup>þ</sup> <sup>B</sup>iYi<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

for i = 1, 2, …, s, as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s, hi(θ(t))hj(θ(t)) 6¼ 0, and

" #

V � S þ δAiS þ δBiYiC �2δS

<sup>i</sup> B<sup>T</sup> <sup>i</sup> ∗

Φij ∗

BiY<sup>j</sup> þ BjY<sup>i</sup>

<sup>2</sup> <sup>C</sup> �2δ<sup>S</sup>

Xs�1 i¼1

Xs j¼iþ1

hið Þ <sup>θ</sup>ð Þ<sup>t</sup> hjð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>p</sup><sup>T</sup>ð Þ<sup>t</sup>

W A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>T</sup>

ð Þ <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKi<sup>C</sup> <sup>W</sup> <sup>þ</sup> W Að Þ <sup>i</sup> <sup>þ</sup> <sup>B</sup>iKi<sup>C</sup> <sup>T</sup> <sup>&</sup>lt; <sup>0</sup>; <sup>ð</sup>24<sup>Þ</sup>

2 þ

Wcij þ Wcji

W A<sup>j</sup> <sup>þ</sup> <sup>B</sup>jKi<sup>C</sup> � �<sup>T</sup>

CW ; ð26Þ

, and matrices Y<sup>j</sup> ∈ IR<sup>r</sup> � <sup>m</sup>, H ∈ IRm � <sup>m</sup>

< 0; ð29Þ

<sup>5</sup> <sup>&</sup>lt; <sup>0</sup>; <sup>ð</sup>30<sup>Þ</sup>

, Y<sup>j</sup> ¼ KjH; ð27Þ

<sup>2</sup> + s)/2 + 1. Evidently, the number of stabilization conditions

<sup>S</sup> <sup>¼</sup> <sup>S</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>, <sup>V</sup> <sup>¼</sup> <sup>V</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>ð</sup>28<sup>Þ</sup>

3 7

CS ¼ HC; ð31Þ

<sup>2</sup> <sup>p</sup>ð Þ<sup>t</sup> <sup>&</sup>lt; <sup>0</sup>; <sup>ð</sup>23<sup>Þ</sup>

<sup>2</sup> <sup>&</sup>lt; <sup>0</sup> <sup>ð</sup>25<sup>Þ</sup>

$$\sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \left(\mathbf{A}\_{cij} \boldsymbol{q}(t) - \dot{\boldsymbol{q}}(t)\right) = \mathbf{0},\tag{34}$$

then with an arbitrary symmetric positive definite matrix S ∈ IRn � <sup>n</sup> and a positive scalar δ ∈ IR, it yields

$$\sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\theta}(t)) \mathbf{h}\_{j}(\boldsymbol{\theta}(t)) \left(\boldsymbol{q}^{T}(t)\mathbf{S} + \delta \dot{\boldsymbol{q}}^{T}(t)\mathbf{S}\right) \left(\mathbf{A}\_{i\bar{\boldsymbol{\eta}}}\boldsymbol{q}(t) - \dot{\boldsymbol{q}}(t)\right) = \mathbf{0}.\tag{35}$$

Since S is positive definite, the new state variable coordinate system can be introduced so that

$$\mathbf{p}(t) = \mathbf{S}\mathbf{q}(t), \quad \dot{\mathbf{p}}(t) = \mathbf{S}\dot{\mathbf{q}}(t), \quad \mathbf{V} = \mathbf{S}^{-1}\mathbf{P}\mathbf{S}^{-1}. \tag{36}$$

Therefore, Eq. (14) can be rewritten as

$$
\dot{\mathbf{v}}\left(\mathbf{p}(t)\right) = \dot{\mathbf{p}}^T(t)\mathbf{V}\mathbf{p}(t) + \mathbf{p}^T(t)\mathbf{V}\dot{\mathbf{p}}(t) < 0\tag{37}
$$

and Eq. (35) takes the form

$$\sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\theta}(t)) \mathbf{h}\_{j}(\boldsymbol{\theta}(t)) \left(\boldsymbol{p}^{T}(t) + \delta \boldsymbol{p}^{T}(t)\right) \left(\mathbf{A}\_{c\vec{\boldsymbol{\eta}}} \mathbf{S} \boldsymbol{p}(t) - \mathbf{S} \boldsymbol{\dot{p}}(t)\right) = \mathbf{0}.\tag{38}$$

Thus, adding Eq. (38) as well as the transposition of Eq. (38) to Eq. (37), it yields

$$\begin{split} \dot{\boldsymbol{\eta}}(\boldsymbol{\eta}(t)) &= \dot{\boldsymbol{\eta}}^T(t) \mathbf{V} \boldsymbol{\eta}(t) + \boldsymbol{\eta}^T(t) \mathbf{V} \dot{\boldsymbol{\eta}}(t) \\ &+ \sum\_{i=1}^s \sum\_{j=1}^s h\_i(\boldsymbol{\Theta}(t)) \mathbf{h}\_{\dot{\boldsymbol{\eta}}}(\boldsymbol{\Theta}(t)) \left( \boldsymbol{\eta}^T(t) + \boldsymbol{\delta} \dot{\boldsymbol{\eta}}^T(t) \right) \left( \mathbf{A}\_{c\dot{\boldsymbol{\eta}}} \mathbf{S} \boldsymbol{\eta}(t) - \mathbf{S} \dot{\boldsymbol{\eta}}(t) \right) \\ &+ \sum\_{i=1}^s \sum\_{j=1}^s h\_i(\boldsymbol{\Theta}(t)) \mathbf{h}\_{\dot{\boldsymbol{\eta}}}(\boldsymbol{\Theta}(t)) \left( \mathbf{A}\_{c\dot{\boldsymbol{\eta}}} \mathbf{S} \boldsymbol{\eta}(t) - \mathbf{S} \dot{\boldsymbol{\eta}}(t) \right)^T \left( \mathbf{p}(t) + \boldsymbol{\delta} \dot{\boldsymbol{\eta}}(t) \right) < 0. \end{split} \tag{39}$$

Using the notation

$$\boldsymbol{p}\_c^T(t) = \begin{bmatrix} \boldsymbol{p}^T(t) & \dot{\boldsymbol{p}}^T(t) \end{bmatrix},\tag{40}$$

the inequality Eq. (39) can be written as

$$\dot{\boldsymbol{\nu}}(\boldsymbol{\mathcal{p}}\_c(t)) = \sum\_{i=1}^s \sum\_{j=1}^s \boldsymbol{h}\_i(\boldsymbol{\Theta}(t)) \mathbf{h}\_j(\boldsymbol{\Theta}(t)) \boldsymbol{p}\_c^T(t) \mathbf{S}\_{c\bar{\eta}} \boldsymbol{p}\_c(t) < 0,\tag{41}$$

$$\mathbf{S}\_{c\bar{\eta}} = \begin{bmatrix} \left(\mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_{\bar{\jmath}} \mathbf{C}\right) \mathbf{S} + \mathbf{S} \left(\mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_{\bar{\jmath}} \mathbf{C}\right)^T & \* \\\mathbf{V} - \mathbf{S} + \delta \left(\mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_{\bar{\jmath}} \mathbf{C}\right) \mathbf{S} & -2\delta \mathbf{S} \end{bmatrix} < 0. \tag{42}$$

Permuting the subscripts i and j in Eq. (41), and following the way used above, analogously it can obtain

$$\operatorname{div}(\boldsymbol{\mathfrak{p}}\_{\boldsymbol{c}}(t)) = \sum\_{i=1}^{s} \mathbf{h}\_{i}^{2}(\boldsymbol{\mathfrak{G}}(t)) \boldsymbol{p}\_{\boldsymbol{c}}^{T}(t) \mathbf{S}\_{\boldsymbol{c}\bar{\boldsymbol{u}}} \boldsymbol{p}\_{\boldsymbol{c}}(t) + 2 \sum\_{i=1}^{s-1} \sum\_{j=i+1}^{s} \mathbf{h}\_{i}(\boldsymbol{\mathfrak{G}}(t)) \mathbf{h}\_{j}(\boldsymbol{\mathfrak{G}}(t)) \boldsymbol{p}\_{\boldsymbol{c}}^{T}(t) \frac{\mathbf{S}\_{\boldsymbol{c}\bar{\boldsymbol{u}}} + \mathbf{S}\_{\boldsymbol{c}\bar{\boldsymbol{u}}}}{2} \boldsymbol{p}\_{\boldsymbol{c}}(t) < 0. \quad (43)$$

Since r = m, it is now possible to set

$$\mathbf{K}\_{\rangle}\mathbf{C}\mathbf{S}=\mathbf{K}\_{\rangle}\mathbf{H}\mathbf{H}^{-1}\mathbf{C}\mathbf{S},\tag{44}$$

where H is a regular square matrix of appropriate dimension and introducing

$$H^{-1}\mathbf{C} = \mathbf{C}\mathbf{S}^{-1}, \quad \mathbf{Y}\_{\rangle} = \mathbf{K}\_{\rangle}\mathbf{H} \tag{45}$$

then Eqs. (42) and (45) imply Eqs. (29)–(31). This concludes the proof.

Note, Eq. (42) leads to the set of LMIs only if δ is a prescribed constant. (δ can be considered as a tuning parameter). Considering δ as a LMI variable, Eq. (42) represents the set of bilinear matrix inequalities (BMI).

Theorem 1 (enhanced relaxed design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically stable if for given a positive δ ∈ IR there exist positive definite symmetric matrices V, S ∈ IR<sup>n</sup> � <sup>n</sup> , the matrices <sup>X</sup>ij <sup>¼</sup> <sup>X</sup><sup>T</sup> ji ∈IRr�<sup>n</sup> , and Y<sup>j</sup> ∈ IR<sup>r</sup> � <sup>m</sup>, H ∈ IRm � <sup>m</sup> such that

$$\mathbf{S} = \mathbf{S}^{T} > \mathbf{0}, \quad \mathbf{V} = \mathbf{V}^{T} > \mathbf{0}, \quad \begin{bmatrix} \mathbf{X}\_{11} & \mathbf{X}\_{12} & \cdots & \mathbf{X}\_{1s} \\ \mathbf{X}\_{21} & \mathbf{X}\_{22} & \cdots & \mathbf{X}\_{2s} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{X}\_{s1} & \mathbf{X}\_{s2} & \cdots & \mathbf{X}\_{ss} \end{bmatrix} > \mathbf{0}, \tag{46}$$

$$
\begin{bmatrix}
\mathbf{A}\_i \mathbf{S} + \mathbf{S} \mathbf{A}\_i^T + \mathbf{B}\_i \mathbf{Y}\_i \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_i^T \mathbf{B}\_i^T + \mathbf{X}\_{ii} & \* \\
\mathbf{V} - \mathbf{S} + \delta \mathbf{A}\_i \mathbf{S} + \delta \mathbf{B}\_i \mathbf{Y}\_i \mathbf{C} & -2\delta \mathbf{S}
\end{bmatrix} < 0,\tag{47}
$$

$$
\begin{bmatrix}
\mathbf{O}\_{\circlearrowright} & & \* \\
\mathbf{V} - \mathbf{S} + \delta \frac{\mathbf{A}\_{i}\mathbf{S} + \mathbf{A}\_{j}\mathbf{S}}{2} + \delta \frac{\mathbf{B}\_{i}\mathbf{Y}\_{j} + \mathbf{B}\_{j}\mathbf{Y}\_{i}}{2}\mathbf{C} & -2\delta \mathbf{S}
\end{bmatrix} < 0,\tag{48}
$$

$$\mathbf{CS} = \mathbf{HC},\tag{49}$$

for i = 1, 2, …, s, as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s, hi(θ(t))hj(θ(t)) 6¼ 0 and

$$\mathbf{O}\_{\vec{\eta}} = \frac{\mathbf{A}\_i \mathbf{S} + \mathbf{S} \mathbf{A}\_i^T}{2} + \frac{\mathbf{A}\_{\vec{\eta}} \mathbf{S} + \mathbf{S} \mathbf{A}\_{\vec{\eta}}^T}{2} + \frac{\mathbf{B}\_{\vec{\eta}} \mathbf{Y}\_{\vec{\eta}} \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_{\vec{\eta}}^T \mathbf{B}\_i^T}{2} + \frac{\mathbf{B}\_{\vec{\eta}} \mathbf{Y}\_i \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_{\vec{\eta}}^T \mathbf{B}\_{\vec{\eta}}^T}{2} + \frac{\mathbf{X}\_{\vec{\eta}} + \mathbf{X}\_{\vec{\eta}}}{2}. \tag{50}$$

When the above conditions hold, the control law gain matrices are given as

$$\mathbf{K}\_{i} = \mathbf{Y}\_{i}\mathbf{H}^{-1}.\tag{51}$$

Proof. Introducing the positive real term

$$\nu\_{\boldsymbol{v}}(\boldsymbol{\theta}(t)) = \boldsymbol{q}^{T}(t)\mathbb{Z}(\boldsymbol{\theta}(t))\boldsymbol{q}(t) > 0,\tag{52}$$

$$\mathbf{Z}(\boldsymbol{\theta}(t)) = \mathbf{Z}^{T}(\boldsymbol{\theta}(t)) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\theta}(t)) \mathbf{h}\_{j}(\boldsymbol{\theta}(t)) \mathbf{Z}\_{ij} > 0,\tag{53}$$

where <sup>Z</sup>ij <sup>¼</sup> <sup>Z</sup><sup>T</sup> ji ∈IRn�<sup>n</sup>, i, j = 1, 2, …, s is the set of associated matrices and using the state coordinate transform Eq. (36), then Eq. (53) can be rewritten as

$$\nu\_{\boldsymbol{\nu}}(\boldsymbol{\mathfrak{p}}(t)) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\Theta}(t)) \mathbf{h}\_{j}(\boldsymbol{\Theta}(t)) \boldsymbol{\mathfrak{p}}^{T}(t) \mathbf{X}\_{i\boldsymbol{\overline{\boldsymbol{\eta}}}} \boldsymbol{\mathfrak{p}}(t) > 0, \quad \mathbf{X}\_{i\boldsymbol{\overline{\boldsymbol{\eta}}}} = \mathbf{S}^{-1} \mathbf{Z}\_{i\boldsymbol{\overline{\boldsymbol{\eta}}}} \mathbf{S}^{-1} = \mathbf{X}\_{\boldsymbol{\overline{\boldsymbol{\eta}}}}^{\boldsymbol{T}}, \tag{54}$$

where

Using the notation

can obtain

<sup>ν</sup>\_ <sup>p</sup>cð Þ<sup>t</sup> � � <sup>¼</sup> <sup>X</sup><sup>s</sup>

i¼1 h2 <sup>i</sup> ð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>p</sup><sup>T</sup>

matrix inequalities (BMI).

Y<sup>j</sup> ∈ IR<sup>r</sup> � <sup>m</sup>, H ∈ IRm � <sup>m</sup> such that

Since r = m, it is now possible to set

the inequality Eq. (39) can be written as

10 Modern Fuzzy Control Systems and Its Applications

pT

Xs j¼1

<sup>S</sup>cij <sup>¼</sup> <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>S</sup> <sup>þ</sup> S A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>T</sup> <sup>∗</sup>

Xs�1 i¼1

hið Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>h</sup>jð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>p</sup><sup>T</sup>

" #

Permuting the subscripts i and j in Eq. (41), and following the way used above, analogously it

Xs j¼iþ1

<sup>K</sup>jCS <sup>¼</sup> <sup>K</sup>jHH�<sup>1</sup>

Note, Eq. (42) leads to the set of LMIs only if δ is a prescribed constant. (δ can be considered as a tuning parameter). Considering δ as a LMI variable, Eq. (42) represents the set of bilinear

Theorem 1 (enhanced relaxed design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically stable if for given a positive δ ∈ IR

> > <sup>i</sup> B<sup>T</sup>

Φij ∗

BiY<sup>j</sup> þ BjY<sup>i</sup>

V � S þ δAiS þ δBiYiC �2δS

<sup>i</sup> <sup>þ</sup> <sup>B</sup>iYi<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

AiS þ AjS <sup>2</sup> <sup>þ</sup> <sup>δ</sup>

� �

<sup>C</sup> <sup>¼</sup> CS�<sup>1</sup>

where H is a regular square matrix of appropriate dimension and introducing

H�<sup>1</sup>

then Eqs. (42) and (45) imply Eqs. (29)–(31). This concludes the proof.

there exist positive definite symmetric matrices V, S ∈ IR<sup>n</sup> � <sup>n</sup>

<sup>S</sup> <sup>¼</sup> <sup>S</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>, <sup>V</sup> <sup>¼</sup> <sup>V</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>,

<sup>A</sup>i<sup>S</sup> <sup>þ</sup> SA<sup>T</sup>

V � S þ δ

2 4 <sup>V</sup> � <sup>S</sup> <sup>þ</sup> <sup>δ</sup> <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>S</sup> �2δ<sup>S</sup>

<sup>h</sup>ið Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>h</sup>jð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>p</sup><sup>T</sup>

<sup>c</sup> ð Þt

Scij þ Scji

CS; ð44Þ

, Y<sup>j</sup> ¼ KjH ð45Þ

, the matrices <sup>X</sup>ij <sup>¼</sup> <sup>X</sup><sup>T</sup>

3

X<sup>11</sup> X<sup>12</sup> ⋯ X1<sup>s</sup> X<sup>21</sup> X<sup>22</sup> ⋯ X2<sup>s</sup> ⋮ ⋮⋱⋮ Xs<sup>1</sup> Xs<sup>2</sup> ⋯ Xss

<sup>i</sup> þ Xii ∗

<sup>2</sup> <sup>C</sup> �2δ<sup>S</sup>

i¼1

<sup>c</sup> ð Þt Sciipcð Þþ t 2

<sup>ν</sup>\_ <sup>p</sup>cð Þ<sup>t</sup> � � <sup>¼</sup> <sup>X</sup><sup>s</sup>

<sup>c</sup> ðÞ¼ <sup>t</sup> <sup>p</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>p</sup>\_ <sup>T</sup>ð Þ<sup>t</sup> � �; <sup>ð</sup>40<sup>Þ</sup>

<sup>c</sup> ð Þt Scijpcð Þt < 0; ð41Þ

< 0: ð42Þ

<sup>2</sup> <sup>p</sup>cð Þ<sup>t</sup> <sup>&</sup>lt; <sup>0</sup>: <sup>ð</sup>43<sup>Þ</sup>

ji ∈IRr�<sup>n</sup>

> 0; ð46Þ

< 0; ð47Þ

5 < 0; ð48Þ

, and

$$\mathbf{Z}(\boldsymbol{\theta}(t)) = \begin{bmatrix} \mathbf{h}\_{1}(\boldsymbol{\theta}(t)\boldsymbol{p}(t) & \mathbf{h}\_{2}(\boldsymbol{\theta}(t)\boldsymbol{p}(t) & \cdots & \mathbf{h}\_{s}(\boldsymbol{\theta}(t)\boldsymbol{p}(t)) \end{bmatrix} \begin{bmatrix} \mathbf{X}\_{11} & \mathbf{X}\_{12} & \cdots & \mathbf{X}\_{1s} \\ \mathbf{X}\_{21} & \mathbf{X}\_{22} & \cdots & \mathbf{X}\_{2s} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{X}\_{s1} & \mathbf{X}\_{s2} & \cdots & \mathbf{X}\_{ss} \end{bmatrix} \begin{bmatrix} \mathbf{h}\_{1}(\boldsymbol{\theta}(t)\boldsymbol{p}(t) \\ \mathbf{h}\_{2}(\boldsymbol{\theta}(t)\boldsymbol{p}(t) \\ \vdots \\ \mathbf{h}\_{s}(\boldsymbol{\theta}(t)\boldsymbol{p}(t) \end{bmatrix} \tag{55}$$

is symmetric, an positive definite if Eq. (46) is satisfied. Then, in the sense of the Krasovskii theorem (see, for example, Ref. [22]), it can be set up in Eq. (39)

$$\begin{split} \dot{\boldsymbol{\nu}}(\dot{\boldsymbol{p}}(t)) &= \dot{\boldsymbol{p}}^{T}(t)\boldsymbol{V}\boldsymbol{p}(t) + \boldsymbol{p}^{T}(t)\boldsymbol{V}\dot{\boldsymbol{p}}(t) \\ &+ \sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\theta}(t)) \mathbf{h}\_{j}(\boldsymbol{\theta}(t)) \left( \boldsymbol{p}^{T}(t) + \delta \dot{\boldsymbol{p}}^{T}(t) \right) \left( \mathbf{A}\_{c\bar{\boldsymbol{\eta}}} \mathbf{S} \boldsymbol{p}(t) - \mathbf{S} \dot{\boldsymbol{p}}(t) \right) \\ &+ \sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\theta}(t)) \mathbf{h}\_{j}(\boldsymbol{\theta}(t)) (\mathbf{A}\_{c\bar{\boldsymbol{\eta}}} \mathbf{S} \boldsymbol{p}(t) - \mathbf{S} \dot{\boldsymbol{p}}(t))^{T} \left( \boldsymbol{p}(t) + \delta \dot{\boldsymbol{p}}(t) \right) \\ &< -\sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\theta}(t)) \mathbf{h}\_{j}(\boldsymbol{\theta}(t)) \mathbf{p}^{T}(t) \mathbf{X}\_{i} \boldsymbol{\eta}(t) \\ &< 0, \end{split} \tag{56}$$

which in the consequence, modifies Eq. (42) as follows

$$\mathbf{S}\_{\mathrm{cj}} = \begin{bmatrix} \left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\mathrm{f}}\mathbf{C}\right)\mathbf{S} + \mathbf{S}\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\mathrm{f}}\mathbf{C}\right)^{T} + \mathbf{X}\_{i\natural} & \* \\ \mathbf{V} - \mathbf{S} + \delta\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\mathrm{f}}\mathbf{C}\right)\mathbf{S} & -2\delta\mathbf{S} \end{bmatrix} < 0. \tag{57}$$

Following the same way as in the proof of Proposition 2, then Eqs. (47) and (48) can be derived from Eq. (57), while Eq. (55) implies Eq. (46). This concludes the proof.

This principle naturally exploits the affine TS model properties. Introducing the slack matrix variable S into the LMIs, the system matrices are decoupled from the equivalent Lyapunov matrix <sup>V</sup>. Note, to respect the conditions <sup>X</sup>1<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> ji , the set of inequalities Eqs. (47) and (48) have to be constructed. In the opposite case, constructing a set on s <sup>2</sup> LMIs, the constraint conditions have to be set as <sup>X</sup>1<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> ij > 0, that is, the weighting matrices have to be symmetric positive definite.

Corollary 1 Prescribing S = V and using the Schur complement property, then Eq. (57) implies

$$\mathbf{A}\_{c\dot{\boldsymbol{\eta}}}\mathbf{S} + \mathbf{S}\mathbf{A}\_{c\dot{\boldsymbol{\eta}}}^T + \mathbf{X}\_{\dot{\boldsymbol{\eta}}} + \mathbf{0}.5\delta\mathbf{S}\mathbf{A}\_{c\dot{\boldsymbol{\eta}}}^T\delta^{-1}\mathbf{S}^{-1}\delta\mathbf{A}\_{c\dot{\boldsymbol{\eta}}}\mathbf{S} < \mathbf{0} \tag{58}$$

and for δ = 0 evidently, it has to be

$$\mathbf{A}\_{c\dot{\boldsymbol{\eta}}} \mathbf{S} + \mathbf{S} \mathbf{A}\_{c\dot{\boldsymbol{\eta}}}^{\mathrm{T}} + \mathbf{X}\_{\dot{\boldsymbol{\eta}}} < \mathbf{0}.\tag{59}$$

Evidently, then Eqs. (47) and (48) imply

$$\mathbf{S}(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{i}\mathbf{C})^{T} + (\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{i}\mathbf{C})\mathbf{S} + \mathbf{X}\_{ii} < \mathbf{0},\tag{60}$$

$$\frac{\mathbf{S}\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{i}\mathbf{C}\right)\mathbf{S}}{2} + \frac{\left(\mathbf{A}\_{j} + \mathbf{B}\_{i}\mathbf{K}\_{i}\mathbf{C}\right)\mathbf{S}}{2} + \frac{\mathbf{S}\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{j}\mathbf{C}\right)^{T}}{2} + \frac{\mathbf{S}\left(\mathbf{A}\_{j} + \mathbf{B}\_{i}\mathbf{K}\_{i}\mathbf{C}\right)^{T}}{2} + \frac{\mathbf{X}\_{\bar{\eta}} + \mathbf{X}\_{\bar{\mu}}}{2} < 0. \qquad (61)$$

Considering S = W and comparing with Eqs. (23) and (24), then Eqs. (60) and (61) are the extended set of inequalities Eqs. (23) and (24). The result is that the equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically stable if there exist a positive definite symmetric matrices S ∈ IR<sup>n</sup> � <sup>n</sup> , the matrices <sup>X</sup>1<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> ji ∈ IR<sup>r</sup>�<sup>n</sup>, and Y<sup>j</sup> ∈ IRr � <sup>n</sup> , H ∈ IRm � <sup>m</sup> such that

$$\mathbf{S} = \mathbf{S}^{T} > 0,\quad \begin{bmatrix} \mathbf{X}\_{11} & \mathbf{X}\_{12} & \cdots & \mathbf{X}\_{1s} \\ \mathbf{X}\_{21} & \mathbf{X}\_{22} & \cdots & \mathbf{X}\_{2s} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{X}\_{s1} & \mathbf{X}\_{s2} & \cdots & \mathbf{X}\_{ss} \end{bmatrix} > 0,\tag{62}$$

$$\mathbf{A}\_i \mathbf{S} + \mathbf{S} \mathbf{A}\_i^T + \mathbf{B}\_i \mathbf{Y}\_i \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_i^T \mathbf{B}\_i^T + \mathbf{X}\_{i\bar{i}} < \mathbf{0},\tag{63}$$

$$\frac{\mathbf{A}\_i \mathbf{S} + \mathbf{S} \mathbf{A}\_i^T}{2} + \frac{\mathbf{A}\_j \mathbf{S} + \mathbf{S} \mathbf{A}\_j^T}{2} + \frac{\mathbf{B}\_i \mathbf{Y}\_j \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_j^T \mathbf{B}\_i^T}{2} + \frac{\mathbf{B}\_j \mathbf{Y}\_i \mathbf{C} + \mathbf{C}^T \mathbf{Y}\_i^T \mathbf{B}\_j^T \mathbf{X}\_{ji} + \mathbf{X}\_{ji}}{2} < 0,\tag{64}$$

$$\mathbf{CS} = \mathbf{HC},\tag{65}$$

for i = 1, 2, …, s, as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s, and hi(θ(t))hj(θ(t)) 6¼ 0. Subsequently, if this set of LMIs is satisfied, the set of control law gain matrices is given as

$$\mathbf{K}\_{i} = \mathbf{Y}\_{i}\mathbf{H}^{-1}.\tag{66}$$

These LMIs form relaxed design conditions.

<sup>S</sup>cij <sup>¼</sup> <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>S</sup> <sup>þ</sup> S A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>T</sup> <sup>þ</sup> <sup>X</sup>ij <sup>∗</sup>

from Eq. (57), while Eq. (55) implies Eq. (46). This concludes the proof.

matrix <sup>V</sup>. Note, to respect the conditions <sup>X</sup>1<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup>

have to be set as <sup>X</sup>1<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup>

12 Modern Fuzzy Control Systems and Its Applications

and for δ = 0 evidently, it has to be

Evidently, then Eqs. (47) and (48) imply

<sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>S</sup>

∈ IR<sup>r</sup>�<sup>n</sup>, and Y<sup>j</sup> ∈ IRr � <sup>n</sup>

<sup>A</sup>i<sup>S</sup> <sup>þ</sup> SA<sup>T</sup> i 2 þ

2 þ

definite.

to be constructed. In the opposite case, constructing a set on s

<sup>A</sup>cij<sup>S</sup> <sup>þ</sup> SA<sup>T</sup>

<sup>A</sup><sup>j</sup> <sup>þ</sup> <sup>B</sup>jKi<sup>C</sup> � �<sup>S</sup>

2 þ

stable if there exist a positive definite symmetric matrices S ∈ IR<sup>n</sup> � <sup>n</sup>

<sup>i</sup> <sup>þ</sup> <sup>B</sup>iYi<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

<sup>B</sup>iYj<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

, H ∈ IRm � <sup>m</sup> such that

<sup>S</sup> <sup>¼</sup> <sup>S</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>,

<sup>A</sup>i<sup>S</sup> <sup>þ</sup> SA<sup>T</sup>

<sup>A</sup>j<sup>S</sup> <sup>þ</sup> SA<sup>T</sup> j 2 þ

<sup>V</sup> � <sup>S</sup> <sup>þ</sup> <sup>δ</sup> <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>S</sup> �2δ<sup>S</sup>

< 0: ð57Þ

ji , the set of inequalities Eqs. (47) and (48) have

cij þ Xij < 0: ð59Þ

ij > 0, that is, the weighting matrices have to be symmetric positive

S Að Þ <sup>i</sup> <sup>þ</sup> <sup>B</sup>iKi<sup>C</sup> <sup>T</sup> <sup>þ</sup> ð Þ <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKi<sup>C</sup> <sup>S</sup> <sup>þ</sup> <sup>X</sup>ii <sup>&</sup>lt; <sup>0</sup>; <sup>ð</sup>60<sup>Þ</sup>

S A<sup>j</sup> <sup>þ</sup> <sup>B</sup>jKi<sup>C</sup> � �<sup>T</sup>

<sup>B</sup>jYi<sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>T</sup>Y<sup>T</sup>

2

2 þ

cijδ�<sup>1</sup> S�<sup>1</sup> <sup>2</sup> LMIs, the constraint conditions

δAcijS < 0 ð58Þ

Xij þ Xji

<sup>2</sup> <sup>&</sup>lt; <sup>0</sup>: <sup>ð</sup>61<sup>Þ</sup>

ji

, the matrices <sup>X</sup>1<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup>

<sup>2</sup> <sup>&</sup>lt; <sup>0</sup>; <sup>ð</sup>64<sup>Þ</sup>

> 0; ð62Þ

<sup>i</sup> þ Xii < 0; ð63Þ

Xij þ Xji

<sup>i</sup> B<sup>T</sup> j

CS ¼ HC; ð65Þ

" #

Following the same way as in the proof of Proposition 2, then Eqs. (47) and (48) can be derived

This principle naturally exploits the affine TS model properties. Introducing the slack matrix variable S into the LMIs, the system matrices are decoupled from the equivalent Lyapunov

Corollary 1 Prescribing S = V and using the Schur complement property, then Eq. (57) implies

cij <sup>þ</sup> <sup>X</sup>ij <sup>þ</sup> <sup>0</sup>:5δSA<sup>T</sup>

<sup>A</sup>cij<sup>S</sup> <sup>þ</sup> SA<sup>T</sup>

S A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �<sup>T</sup>

2 þ

X<sup>11</sup> X<sup>12</sup> ⋯ X1<sup>s</sup> X<sup>21</sup> X<sup>22</sup> ⋯ X2<sup>s</sup> ⋮ ⋮⋱⋮ Xs<sup>1</sup> Xs<sup>2</sup> ⋯ Xss

<sup>j</sup> B<sup>T</sup> i 2 þ

<sup>i</sup> B<sup>T</sup>

Considering S = W and comparing with Eqs. (23) and (24), then Eqs. (60) and (61) are the extended set of inequalities Eqs. (23) and (24). The result is that the equilibrium of the fuzzy system Eqs. (2) and (3), controlled by the fuzzy controller Eq. (5), is global asymptotically Note the derived results are linked to some existing finding when the design problem involves additive performance requirements and the relaxed quadratic stability conditions of fuzzy control systems (see, e.g., Refs. [11, 19]) are equivalently steered.

#### 4. Forced mode in static output control

In practice, the plant with r = m (square plants) is often encountered, since in this case, it is possible to associate with each output signal as a reference signal, which is expected to influence this wanted output. Such mode, reflecting nonzero set working points, is called the forced regime.

Definition 2 A forced regime for the TS fuzzy system Eqs. (2) and (3) with the TS fuzzy static output controller Eq. (5) is foisted by the control policy

$$\mathfrak{u}(t) = \sum\_{j=1}^{s} h\_{\rangle}(\mathfrak{G}(t)) \mathbf{K}\_{\slash} \mathfrak{y}(t) + \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_{i}(\mathfrak{G}(t)) h\_{\rangle}(\mathfrak{G}(t)) \mathbf{W}\_{\not\equiv} \mathfrak{w}(t) \,, \tag{67}$$

where r = m, w(i) ∈ IR<sup>m</sup> is desired output signal vector, and Wij ∈ IR<sup>m</sup> � <sup>m</sup>, i, j = 1, 2, … s is the set of the signal gain matrices.

Lemma 1. The static decoupling challenge is solvable if (Ai, Bi) is stabilizable and

$$
rank \begin{bmatrix} A\_i & \mathbf{B}\_i \\ \mathbf{C} & \mathbf{0} \end{bmatrix} = n + m. \tag{68}
$$

Proof. If (Ai, Bi) is stabilizable, it is possible to find K<sup>j</sup> such that matrices Acij = A<sup>i</sup> + BiKjC are Hurwitz. Assuming that for such Kj, it yields

$$\text{rank}\begin{bmatrix} \mathbf{A}\_{i} & \mathbf{B}\_{i} \\ \mathbf{C} & \mathbf{0} \end{bmatrix} = \text{rank}\begin{bmatrix} \mathbf{A}\_{i} & \mathbf{B}\_{i} \\ \mathbf{C} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{I}\_{\text{n}} & \mathbf{0} \\ \mathbf{K}\_{\text{f}}\mathbf{C} & \mathbf{I}\_{\text{m}} \end{bmatrix} = \text{rank}\begin{bmatrix} \mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\text{f}}\mathbf{C} & \mathbf{B}\_{i} \\ \mathbf{C} & \mathbf{0} \end{bmatrix},\tag{69}$$

$$\text{rank}\begin{bmatrix} \mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\uparrow}\mathbf{C} & \mathbf{B}\_{i} \\ \mathbf{C} & \mathbf{0} \end{bmatrix} = \text{rank}\begin{bmatrix} I\_{n} & \mathbf{0} \\ -\mathbf{C}(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\uparrow}\mathbf{C})^{-1} & I\_{m} \end{bmatrix} \begin{bmatrix} A\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\uparrow}\mathbf{C} & \mathbf{B}\_{i} \\ \mathbf{C} & \mathbf{0} \end{bmatrix},\tag{70}$$

respectively, then

$$\text{rank}\begin{bmatrix} \mathbf{A}\_{i} & \mathbf{B}\_{i} \\ \mathbf{C} & \mathbf{0} \end{bmatrix} = \text{rank}\begin{bmatrix} \mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{j}\mathbf{C} & \mathbf{B}\_{i} \\ \mathbf{0} & -\mathbf{C}\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{j}\mathbf{C}\right)^{-1}\mathbf{B}\_{i} \end{bmatrix} = \mathbf{n} + m,\tag{71}$$

since rank(A<sup>i</sup> + BiKjC) = n, and rankB<sup>i</sup> = m.

Thus, evidently, it has to be satisfied

$$\text{rank}\left(\mathbf{C}\left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{j}\mathbf{C}\right)^{-1}\mathbf{B}\_{i}\right) = m.\tag{72}$$

This concludes the proof.

Theorem 2. To reach a forced regime for the TS fuzzy system Eqs. (2) and (3) with the TS fuzzy control policy Eq. (67), the signal gain matrices have to take the forms

$$\mathbf{W}\_{\vec{\eta}} = \left( \mathbf{C} \left( \mathbf{A}\_{i} + \mathbf{B}\_{i} \mathbf{K}\_{j} \mathbf{C} \right)^{-1} \mathbf{B}\_{i} \right)^{-1},\tag{73}$$

where Wij ∈ IRm � <sup>m</sup>, i, j = 1, 2, … s.

Proof. In a steady state, which corresponds to q\_ðÞ¼ t 0, the equality y<sup>o</sup> = w<sup>o</sup> must hold, where q<sup>o</sup> ∈ IRn , θ<sup>o</sup> ∈ IR<sup>q</sup> , yo, w<sup>o</sup> ∈ IRm are the vectors of steady state values of q(t), θ(t), y(t), w(t), respectively.

Substituting Eq. (67) in Eq. (2) yields the expression

$$\sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\Theta}\_{o}) \mathbf{h}\_{j}(\boldsymbol{\Theta}\_{o}) \left( \left( \mathbf{A}\_{i} + \mathbf{B}\_{i} \mathbf{K}\_{j} \mathbf{C} \right) \boldsymbol{\eta}\_{o} + \mathbf{B}\_{i} \mathbf{W}\_{ij} \boldsymbol{\varpi}\_{o} \right) = \mathbf{0},\tag{74}$$

$$-\sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\Theta}\_{o}) \mathbf{h}\_{j}(\boldsymbol{\Theta}\_{o}) \boldsymbol{q}\_{o} = -\boldsymbol{q}\_{o} = \sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_{i}(\boldsymbol{\Theta}\_{o}) \mathbf{h}\_{j}(\boldsymbol{\Theta}\_{o}) \left(\mathbf{A}\_{i} + \mathbf{B}\_{i} \mathbf{K}\_{j} \mathbf{C}\right)^{-1} \mathbf{B}\_{i} \mathbf{W}\_{ij} \boldsymbol{w}\_{o}, \tag{75}$$

respectively, and it can be set

$$\mathfrak{z}\_o = \mathsf{C}\mathfrak{q}\_o = -\sum\_{i=1}^{s} \sum\_{j=1}^{s} \mathbf{h}\_i(\mathsf{\Theta}\_o)\mathbf{h}\_j(\mathsf{\Theta}\_o)\mathbf{C}\left(\mathbf{A}\_i + \mathbf{B}\_i\mathbf{K}\_j\mathbf{C}\right)^{-1}\mathbf{B}\_i\mathbf{W}\_{ij}\mathfrak{w}\_o = \mathbf{I}\_m\mathfrak{w}\_o. \tag{76}$$

Thus, Eq. (76) gives the solution

$$\mathbf{W}\_{ij}^{-1} = -\mathbf{C} \left(\mathbf{A}\_i + \mathbf{B}\_i \mathbf{K}\_j \mathbf{C}\right)^{-1} \mathbf{B}\_i,\tag{77}$$

which implies Eq. (68). Hence, declaredly,

$$\text{rank}\mathbf{W}\_{\dot{j}} = \text{rank}\left(\mathbf{C} \left(\mathbf{A}\_{i} + \mathbf{B}\_{i}\mathbf{K}\_{\dot{j}}\mathbf{C}\right)^{-1}\mathbf{B}\_{i}\right) = m. \tag{78}$$

This concludes the proof.

The forced regime is basically designed for constant references and is very closely related to shift of origin. If the command value w(t) is changed "slowly enough," the above scheme can do a reasonable job of tracking, that is, making y(t) follow w(t) [23].

#### 5. Bi-proper dynamic output controller

The full order biproper dynamic output controller is defined by the equation

$$\dot{\mathbf{p}}(t) = \sum\_{j=1}^{s} \mathbf{h}\_{j}(\boldsymbol{\theta}(t)) \left( \mathbf{J}\_{j}\mathbf{p}(t) + \mathbf{L}\_{j}\mathbf{y}(t) \right), \tag{79}$$

$$\mathfrak{u}(t) = \sum\_{j=1}^{s} \mathbf{h}\_{\dot{\mathbb{M}}}(\boldsymbol{\theta}(t)) \left( \mathbf{M}\_{\dot{\boldsymbol{\beta}}} \mathbf{p}(t) + \mathbf{N}\_{\dot{\boldsymbol{\beta}}} \mathbf{y}(t) \right), \tag{80}$$

where p(t) ∈ IRh is the vector of the controller state variables and the parameter matrix

$$\mathbf{K}\_{j}^{\prime} = \begin{bmatrix} \mathbf{J}\_{j} & \mathbf{L}\_{j} \\ \mathbf{M}\_{j} & \mathbf{N}\_{j} \end{bmatrix},\tag{81}$$

K∘ <sup>j</sup> ∈ IRð Þ� <sup>n</sup>þ<sup>r</sup> ð Þ <sup>h</sup>þ<sup>m</sup> , is considered in this block matrix structure with respect to the matrices J<sup>j</sup> ∈ IR<sup>h</sup> � <sup>h</sup> , L<sup>j</sup> ∈ IRh � <sup>m</sup>, M<sup>j</sup> ∈ IR<sup>r</sup> � <sup>h</sup> , and N<sup>j</sup> ∈ IRr � <sup>m</sup>. For simplicity, the full order p = n controller is considered in the following.

To analyze the stability of the closed-loop system structure with the dynamic output controller, the closed-loop system description implies the following form

$$\dot{\boldsymbol{q}}^{\*}(t) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \mathbf{A}\_{c\dot{\boldsymbol{\eta}}}^{\*} \boldsymbol{\eta}^{\*}(t), \tag{82}$$

$$\mathbf{y}^\*(t) = \mathbf{I}^\* \mathbf{C}^\* \boldsymbol{q}^\*(t),\tag{83}$$

where

Thus, evidently, it has to be satisfied

14 Modern Fuzzy Control Systems and Its Applications

where Wij ∈ IRm � <sup>m</sup>, i, j = 1, 2, … s.

, θ<sup>o</sup> ∈ IR<sup>q</sup>

� Xs i¼1

Xs j¼1

respectively, and it can be set

Thus, Eq. (76) gives the solution

This concludes the proof.

policy Eq. (67), the signal gain matrices have to take the forms

Substituting Eq. (67) in Eq. (2) yields the expression

Xs j¼1

<sup>h</sup>ið Þ <sup>θ</sup><sup>o</sup> <sup>h</sup>jð Þ <sup>θ</sup><sup>o</sup> <sup>q</sup><sup>o</sup> ¼ �q<sup>o</sup> <sup>¼</sup> <sup>X</sup><sup>s</sup>

Xs j¼1

W�<sup>1</sup>

do a reasonable job of tracking, that is, making y(t) follow w(t) [23].

i¼1

Xs i¼1

<sup>y</sup><sup>o</sup> <sup>¼</sup> Cq<sup>o</sup> ¼ �X<sup>s</sup>

which implies Eq. (68). Hence, declaredly,

This concludes the proof.

q<sup>o</sup> ∈ IRn

respectively.

rank C A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � ��<sup>1</sup>

� �

Theorem 2. To reach a forced regime for the TS fuzzy system Eqs. (2) and (3) with the TS fuzzy control

Proof. In a steady state, which corresponds to q\_ðÞ¼ t 0, the equality y<sup>o</sup> = w<sup>o</sup> must hold, where

<sup>h</sup>ið Þ <sup>θ</sup><sup>o</sup> <sup>h</sup>jð Þ <sup>θ</sup><sup>o</sup> <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � �q<sup>o</sup> <sup>þ</sup> <sup>B</sup>iWijw<sup>o</sup>

Xs j¼1

<sup>h</sup>ið Þ <sup>θ</sup><sup>o</sup> <sup>h</sup>jð Þ <sup>θ</sup><sup>o</sup> C A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � ��<sup>1</sup>

ij ¼ �C A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � ��<sup>1</sup>

The forced regime is basically designed for constant references and is very closely related to shift of origin. If the command value w(t) is changed "slowly enough," the above scheme can

� �

rankW<sup>j</sup> <sup>¼</sup> rank C A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � ��<sup>1</sup>

i¼1

� ��<sup>1</sup>

<sup>W</sup>ij <sup>¼</sup> C A<sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � ��<sup>1</sup>

Bi

Bi

, yo, w<sup>o</sup> ∈ IRm are the vectors of steady state values of q(t), θ(t), y(t), w(t),

� � <sup>¼</sup> <sup>0</sup>; <sup>ð</sup>74<sup>Þ</sup>

<sup>h</sup>ið Þ <sup>θ</sup><sup>o</sup> <sup>h</sup>jð Þ <sup>θ</sup><sup>o</sup> <sup>A</sup><sup>i</sup> <sup>þ</sup> <sup>B</sup>iKj<sup>C</sup> � ��<sup>1</sup>

Bi

¼ m: ð72Þ

; ð73Þ

BiWijw<sup>o</sup> ; ð75Þ

BiWijw<sup>o</sup> ¼ Imw<sup>o</sup> : ð76Þ

B<sup>i</sup> ; ð77Þ

¼ m: ð78Þ

$$\boldsymbol{\mathfrak{q}}^{\ast T}(t) = \begin{bmatrix} \boldsymbol{\mathfrak{q}}^{T}(t) & \boldsymbol{\mathfrak{p}}^{T}(t) \end{bmatrix},\tag{84}$$

$$\mathbf{A}\_{cij}^{\*} = \begin{bmatrix} \mathbf{A}\_{i} + \mathbf{B}\_{i} \mathbf{N}\_{j} \mathbf{C} & \mathbf{B}\_{i} \mathbf{M}\_{j} \\ \mathbf{L}\_{j} \mathbf{C} & \mathbf{N}\_{j} \end{bmatrix}, \quad \mathbf{I}^{\*} = \begin{bmatrix} \mathbf{0} & I\_{m} \end{bmatrix}, \quad \mathbf{C}^{\*} = \begin{bmatrix} \mathbf{0} & I\_{n} \\ \mathbf{C} & \mathbf{0} \end{bmatrix} \tag{85}$$

and A<sup>∘</sup> cij ∈ IR<sup>2</sup>n�2<sup>n</sup>, I <sup>∘</sup> ∈ IRm � (<sup>n</sup> <sup>+</sup> <sup>m</sup>) , C<sup>∘</sup> ∈ IR(<sup>n</sup> <sup>+</sup> <sup>m</sup>) � <sup>2</sup><sup>n</sup> .

Introducing the notations

$$\mathbf{A}\_{i}^{\*} = \begin{bmatrix} \mathbf{A}\_{i} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix}, \quad \mathbf{B}\_{i}^{\*} = \begin{bmatrix} \mathbf{0} & \mathbf{B}\_{i} \\ I\_{n} & \mathbf{0} \end{bmatrix}, \tag{86}$$

where A<sup>∘</sup> <sup>i</sup> ∈IR<sup>2</sup>n�2<sup>n</sup>, B<sup>∘</sup> <sup>i</sup> ∈ IR<sup>2</sup>n�ð Þ <sup>n</sup>þ<sup>r</sup> , the closed-loop system matrices take the equivalent forms

$$\mathbf{A}^\*\_{cij} = \mathbf{A}^\*\_i + \mathbf{B}^\*\_i \mathbf{K}^\*\_j \mathbf{C}^\*. \tag{87}$$

In the sequel, it is supposed that A<sup>∘</sup> <sup>i</sup> ; B<sup>∘</sup> i � � is stabilizable, A<sup>∘</sup> <sup>i</sup> ;C<sup>∘</sup> i � � is detectable [24].

Note this kind of controllers can be preferred in fault tolerant control (FTC) structures with virtual actuators [25].

Theorem 3 (relaxed design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3) controlled by the fuzzy dynamic output controller Eqs. (79) and (80) is global asymptotically stable if there exist a positive definite symmetric matrix S<sup>∘</sup> ∈ IR2<sup>n</sup> � <sup>2</sup><sup>n</sup> , symmetric matrices X<sup>∘</sup> ij <sup>¼</sup> <sup>X</sup><sup>∘</sup> ji ∈ IR<sup>2</sup>n�2<sup>n</sup>, a regular matrix H<sup>∘</sup> ∈ IR(<sup>n</sup> <sup>+</sup> <sup>m</sup>) � (<sup>n</sup> <sup>+</sup> <sup>m</sup>) , and matrices Y<sup>∘</sup> <sup>j</sup> ∈IRð Þ� <sup>n</sup>þ<sup>r</sup> ð Þ <sup>n</sup>þ<sup>m</sup> such that

$$\mathbf{S}^\* = \mathbf{S}^T > 0,\quad \begin{bmatrix} \mathbf{X}\_{11}^\* & \mathbf{X}\_{12}^\* & \cdots & \mathbf{X}\_{1s}^\* \\ \mathbf{X}\_{21}^\* & \mathbf{X}\_{22}^\* & \cdots & \mathbf{X}\_{2s}^\* \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{X}\_{s1}^\* & \mathbf{X}\_{s2}^\* & \cdots & \mathbf{X}\_{ss}^\* \end{bmatrix} > 0,\tag{88}$$

$$\mathbf{A}\_i^\ast \mathbf{S}^\ast + \mathbf{S}^\ast \mathbf{A}\_i^{\ast T} + \mathbf{B}\_i^\ast \mathbf{Y}\_i^\ast \mathbf{C}^\ast + \mathbf{C}^{\ast T} \mathbf{Y}\_i^{\ast T} \mathbf{B}\_i^{\ast T} + \mathbf{X}\_{ii}^\ast < 0,\tag{89}$$

$$\frac{\mathbf{A}\_i^\ast + \mathbf{A}\_j^\ast}{2} \mathbf{S}^\ast + \mathbf{S}^\ast \frac{\mathbf{A}\_i^{\ast T} + \mathbf{A}\_j^{\ast T}}{2} + \frac{\mathbf{B}\_i^\ast \mathbf{Y}\_j^\ast + \mathbf{B}\_j^\ast \mathbf{Y}\_i^\ast}{2} \mathbf{C}^\ast + \mathbf{C}^\ast \frac{\mathbf{Y}\_j^{\ast T} \mathbf{B}\_i^{\ast T} + \mathbf{Y}\_i^{\ast T} \mathbf{B}\_j^{\ast T}}{2} + \frac{\mathbf{X}\_{ij}^\ast + \mathbf{X}\_{ji}^\ast}{2} < 0,\tag{90}$$

$$\mathbf{C}^\*\mathbf{S}^\* = H^\*\mathbf{C}^\*,\tag{91}$$

for all i ∈ 〈1, 2, … s〉, i < j ≤ s, i, j ∈ 〈1, 2, … s〉, respectively, and hi(θ(t))hj(θ(t)) 6¼ 0.

When the above conditions hold, the set of control law gain matrices are given as

$$\mathbf{K}\_{j}^{\*} = \mathbf{Y}\_{j}^{\*} (\mathbf{H}^{\*})^{-1}, \quad j = 1, 2, \ldots, s \tag{92}$$

Proof. Defining the Lyapunov function as follows

$$\nu(\boldsymbol{q}^\*(t)) = \boldsymbol{q}^{\*T}(t)\boldsymbol{P}^\*\boldsymbol{q}^\*(t) > 0,\tag{93}$$

where P<sup>∘</sup> ∈ IR2<sup>n</sup> � <sup>2</sup><sup>n</sup> is a positive definite matrix, then the time derivative of ν(q(t)) along a closed-loop system trajectory is

$$
\dot{\boldsymbol{\nu}}\,\dot{\boldsymbol{\nu}}(\boldsymbol{q}^\*(t)) = \dot{\boldsymbol{q}}^{\*T}(t)\boldsymbol{P}^\*\boldsymbol{q}^\*(t) + \boldsymbol{q}^{\*T}(t)\boldsymbol{P}^\*\dot{\boldsymbol{q}}^\*(t) < 0. \tag{94}
$$

Substituting Eq. (87), then Eq. (94) implies

$$\dot{\boldsymbol{\nu}}(\boldsymbol{\eta}^\*(t)) = \sum\_{i=1}^s \sum\_{j=1}^s h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \boldsymbol{\eta}^{\*T}(t) \mathbf{P}\_{cij}^{\boldsymbol{\nu}} \boldsymbol{\eta}^\*(t) < 0,\tag{95}$$

$$\mathbf{P}\_{c\mathbf{i}\mathbf{j}}^{\*} = \mathbf{P}^{\*} \mathbf{A}\_{c\mathbf{i}\mathbf{j}}^{\*} + \mathbf{A}\_{c\mathbf{i}\mathbf{j}}^{\*T} \mathbf{P}^{\*}.\tag{96}$$

Since P<sup>∘</sup> is positive definite, the state coordinate transform can now be defined as

Stabilizing Fuzzy Control via Output Feedback http://dx.doi.org/10.5772/68129 17

$$\mathfrak{q}^\*(t) = \mathbf{S}^\* \mathfrak{p}^\*(t), \quad \mathbf{S}^\* = (\mathbf{P}^\*)^{-1}, \tag{97}$$

and subsequently Eqs. (95) and (96) can be rewritten as

$$\dot{\boldsymbol{\nu}}(\boldsymbol{\mathfrak{p}}^{\*}(t)) = \sum\_{i=1}^{s} \sum\_{j=1}^{s} h\_{i}(\boldsymbol{\varTheta}(t)) h\_{\boldsymbol{\varbeta}}(\boldsymbol{\varTheta}(t)) \boldsymbol{\mathfrak{p}}^{\*T}(t) \mathbf{S}\_{cij}^{\boldsymbol{\mathfrak{e}}} \boldsymbol{\mathfrak{p}}^{\*}(t) < 0,\tag{98}$$

$$\mathbf{S}\_{c\circ j}^{\diamond} = \mathbf{A}\_{c\circ j}^{\ast}\mathbf{S}^{\ast} + \mathbf{S}^{\ast}\mathbf{A}\_{c\circ j}^{\ast T}.\tag{99}$$

Introducing, analogously to Eqs. (54) and (55), the positive term

$$\nu\_\nu(\mathfrak{p}^\*(t)) = \mathfrak{p}^{\*T}(t)\mathbb{Z}^\*(\mathfrak{G}(t))\mathfrak{p}^\*(t) > 0,\tag{100}$$

defined by the set of matrices X<sup>∘</sup> ij <sup>¼</sup> <sup>X</sup><sup>∘</sup><sup>T</sup> ji <sup>∈</sup> IR<sup>n</sup>�n, i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, s n o in the structure Eq. (88) such that

$$\mathbf{Z}^\*(\boldsymbol{\theta}(t)) = \mathbf{Z}^{\*\mathrm{T}}(\boldsymbol{\theta}(t)) = \sum\_{i=1}^s \sum\_{j=1}^s h\_i(\boldsymbol{\theta}(t)) h\_j(\boldsymbol{\theta}(t)) \mathbf{X}\_{ij}^\* > 0,\tag{101}$$

then, in the sense of Krasovskii theorem, it can be set up

$$\dot{\boldsymbol{\nu}}(\boldsymbol{\mathfrak{p}}^\*(t)) = \sum\_{i=1}^s \sum\_{j=1}^s h\_i(\boldsymbol{\varTheta}(t)) h\_j(\boldsymbol{\varTheta}(t)) \boldsymbol{\mathfrak{p}}^{\*T}(t) \mathbf{S}\_{cij}^\* \boldsymbol{\mathfrak{p}}^\*(t) < 0,\tag{102}$$

where

A∘ cij <sup>¼</sup> <sup>A</sup><sup>∘</sup>

<sup>i</sup> ; B<sup>∘</sup> i

, and matrices Y<sup>∘</sup>

A<sup>∘</sup><sup>T</sup> <sup>i</sup> <sup>þ</sup> <sup>B</sup><sup>∘</sup> iY∘

B∘ iY∘ <sup>j</sup> <sup>þ</sup> <sup>B</sup><sup>∘</sup> jY∘ i <sup>2</sup> <sup>C</sup><sup>∘</sup> <sup>þ</sup> <sup>C</sup><sup>∘</sup><sup>T</sup> <sup>Y</sup><sup>∘</sup><sup>T</sup>

<sup>S</sup><sup>∘</sup> <sup>¼</sup> <sup>S</sup><sup>∘</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>,

In the sequel, it is supposed that A<sup>∘</sup>

16 Modern Fuzzy Control Systems and Its Applications

positive definite symmetric matrix S<sup>∘</sup> ∈ IR2<sup>n</sup> � <sup>2</sup><sup>n</sup>

A∘ <sup>i</sup>S<sup>∘</sup> <sup>þ</sup> <sup>S</sup><sup>∘</sup>

<sup>i</sup> <sup>þ</sup> <sup>A</sup><sup>∘</sup><sup>T</sup> j 2 þ

Proof. Defining the Lyapunov function as follows

closed-loop system trajectory is

Substituting Eq. (87), then Eq. (94) implies

virtual actuators [25].

matrix H<sup>∘</sup> ∈ IR(<sup>n</sup> <sup>+</sup> <sup>m</sup>) � (<sup>n</sup> <sup>+</sup> <sup>m</sup>)

A∘ <sup>i</sup> <sup>þ</sup> <sup>A</sup><sup>∘</sup> j <sup>2</sup> <sup>S</sup><sup>∘</sup> <sup>þ</sup> <sup>S</sup><sup>∘</sup> <sup>A</sup><sup>∘</sup><sup>T</sup> <sup>i</sup> <sup>þ</sup> <sup>B</sup><sup>∘</sup> iK∘ jC∘

> <sup>i</sup> ;C<sup>∘</sup> i

, symmetric matrices X<sup>∘</sup>

<sup>j</sup> ∈IRð Þ� <sup>n</sup>þ<sup>r</sup> ð Þ <sup>n</sup>þ<sup>m</sup> such that

⋮ ⋮⋱⋮

<sup>i</sup>C<sup>∘</sup> <sup>þ</sup> <sup>C</sup><sup>∘</sup><sup>T</sup>Y<sup>∘</sup><sup>T</sup>

C∘

q∘

ð Þþ <sup>t</sup> <sup>q</sup><sup>∘</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup><sup>∘</sup>

hið Þ <sup>θ</sup>ð Þ<sup>t</sup> hjð Þ <sup>θ</sup>ð Þ<sup>t</sup> <sup>q</sup><sup>∘</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup><sup>∘</sup>

q\_∘

cijq<sup>∘</sup>

<sup>12</sup> ⋯ X<sup>∘</sup>

<sup>22</sup> ⋯ X<sup>∘</sup>

<sup>s</sup><sup>2</sup> ⋯ X<sup>∘</sup>

<sup>i</sup> B<sup>∘</sup><sup>T</sup> <sup>i</sup> <sup>þ</sup> <sup>X</sup><sup>∘</sup>

<sup>j</sup> B<sup>∘</sup><sup>T</sup> <sup>i</sup> <sup>þ</sup> <sup>Y</sup><sup>∘</sup><sup>T</sup> <sup>i</sup> B<sup>∘</sup><sup>T</sup> j 2 þ

1s

2s

ss

� � is stabilizable, A<sup>∘</sup>

Note this kind of controllers can be preferred in fault tolerant control (FTC) structures with

Theorem 3 (relaxed design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3) controlled by the fuzzy dynamic output controller Eqs. (79) and (80) is global asymptotically stable if there exist a

> X∘ <sup>11</sup> X<sup>∘</sup>

C∘

<sup>j</sup> <sup>H</sup><sup>∘</sup> ð Þ�<sup>1</sup>

<sup>ν</sup> <sup>q</sup><sup>∘</sup> ð Þ¼ ð Þ<sup>t</sup> <sup>q</sup><sup>∘</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup><sup>∘</sup>

where P<sup>∘</sup> ∈ IR2<sup>n</sup> � <sup>2</sup><sup>n</sup> is a positive definite matrix, then the time derivative of ν(q(t)) along a

q∘

for all i ∈ 〈1, 2, … s〉, i < j ≤ s, i, j ∈ 〈1, 2, … s〉, respectively, and hi(θ(t))hj(θ(t)) 6¼ 0.

When the above conditions hold, the set of control law gain matrices are given as K∘ <sup>j</sup> <sup>¼</sup> <sup>Y</sup><sup>∘</sup>

<sup>ν</sup>\_ <sup>q</sup><sup>∘</sup> ð Þ¼ ð Þ<sup>t</sup> <sup>q</sup>\_<sup>∘</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup><sup>∘</sup>

i¼1

Xs j¼1

> P∘ cij <sup>¼</sup> <sup>P</sup><sup>∘</sup> A∘ cij <sup>þ</sup> <sup>A</sup><sup>∘</sup><sup>T</sup> cijP<sup>∘</sup>

Since P<sup>∘</sup> is positive definite, the state coordinate transform can now be defined as

<sup>ν</sup>\_ <sup>q</sup><sup>∘</sup> ð Þ¼ ð Þ<sup>t</sup> <sup>X</sup><sup>s</sup>

<sup>S</sup><sup>∘</sup> <sup>¼</sup> <sup>H</sup><sup>∘</sup>

X∘ <sup>21</sup> X<sup>∘</sup>

X∘ <sup>s</sup><sup>1</sup> X<sup>∘</sup> : ð87Þ

� � is detectable [24].

ij <sup>¼</sup> <sup>X</sup><sup>∘</sup>

X∘ ij <sup>þ</sup> <sup>X</sup><sup>∘</sup> ji

; ð91Þ

ð Þt > 0; ð93Þ

ð Þt < 0: ð94Þ

ð Þt < 0; ð95Þ

: ð96Þ

, j ¼ 1, 2, …, s ð92Þ

ji ∈ IR<sup>2</sup>n�2<sup>n</sup>, a regular

<sup>2</sup> <sup>&</sup>lt; <sup>0</sup>; <sup>ð</sup>90<sup>Þ</sup>

> 0; ð88Þ

ii < 0; ð89Þ

$$\mathbf{S}\_{c\circ j}^{\*} = \mathbf{A}\_{c\circ j}^{\*}\mathbf{S}^{\*} + \mathbf{S}^{\*}\mathbf{A}\_{c\circ j}^{\*T} + \mathbf{X}\_{ij}^{\*}.\tag{103}$$

Therefore, Eq. (102) can be factorized as follows

$$\dot{\mathbf{p}}\cdot\mathbf{p}^\*(t) = \sum\_{i=1}^s h\_i^2(\boldsymbol{\Theta}(t)) \mathbf{p}^{\*T}(t) \mathbf{S}\_{c\bar{\mathbf{u}}}^\* \mathbf{p}^\*(t) + 2 \sum\_{i=1}^{s-1} \sum\_{j=i+1}^s h\_i(\boldsymbol{\Theta}(t)) h\_j(\boldsymbol{\Theta}(t)) \mathbf{p}^{\*T}(t) \frac{\mathbf{S}\_{c\bar{\mathbf{u}}}^\* + \mathbf{S}\_{c\bar{\mathbf{u}}}^\*}{2} \mathbf{p}^\*(t) < 0, \quad (104)$$

which, using Eq. (87), leads to the following sets of inequalities

$$\mathbf{A}\_i^\* \mathbf{S}^\* + \mathbf{S}^\* \mathbf{A}\_i^{\*T} + \mathbf{B}\_i^\* \mathbf{K}\_j^\* \mathbf{C}^\* \mathbf{S}^\* + \mathbf{S}^\* \mathbf{C}^T \mathbf{K}\_j^T \mathbf{B}\_i^{\*T} + \mathbf{X}\_{ij}^\* < 0,\tag{105}$$

$$\begin{split} \frac{\left(\mathbf{A}\_{i}^{\ast} + \mathbf{B}\_{i}^{\ast}\mathbf{K}\_{j}^{\ast}\mathbf{C}^{\ast}\right)\mathbf{S}^{\ast}}{2} &+ \frac{\left(\mathbf{A}\_{j}^{\ast} + \mathbf{B}\_{j}^{\ast}\mathbf{K}\_{i}^{\ast}\mathbf{C}^{\ast}\right)\mathbf{S}^{\ast}}{2} + \frac{\mathbf{S}^{\ast}\left(\mathbf{A}\_{i}^{\ast} + \mathbf{B}\_{i}^{\ast}\mathbf{K}\_{j}^{\ast}\mathbf{C}^{\ast}\right)^{\top}}{2} \\ &+ \frac{\mathbf{S}^{\ast}\left(\mathbf{A}\_{j}^{\ast} + \mathbf{B}\_{j}^{\ast}\mathbf{K}\_{i}^{\ast}\mathbf{C}^{\ast}\right)^{\top}}{2} + \frac{\mathbf{X}\_{ij}^{\ast} + \mathbf{X}\_{ji}^{\ast}}{2} < 0, \end{split} \tag{106}$$

for i = 1, 2, …, s, as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s, and hi(θ(t))hj(θ(t)) 6¼ 0.

Analyzing the product B<sup>∘</sup> iK∘ jC∘ S∘ , it can set

$$\mathbf{B}\_i^\* \mathbf{K}\_j^\* \mathbf{C}^\* \mathbf{S}^\* = \mathbf{B}\_i^\* \mathbf{K}\_j^\* \mathbf{H}^\*(\mathbf{H}^\*)^{-1} \mathbf{C}^\* \mathbf{S}^\* = \mathbf{B}\_i^\* \mathbf{Y}\_j^\* \mathbf{C}^\*,\tag{107}$$

where

$$\mathbf{K}\_{\dot{\gamma}}^{\*}\mathbf{H}^{\*} = \mathbf{Y}\_{\dot{\gamma}}^{\*} \quad (\mathbf{H}^{\*})^{-1}\mathbf{C} = \mathbf{C}^{\*}(\mathbf{S}^{\*})^{-1} \tag{108}$$

and H<sup>∘</sup> ∈ IR(<sup>m</sup> <sup>+</sup> <sup>n</sup>) � (<sup>m</sup> <sup>+</sup> <sup>n</sup>) is a regular square matrix. Thus, with Eq. (108), then Eqs. (105) and (106) implies Eqs. (89) and (90) and Eq. (108) gives Eq. (91). This concludes the proof.

This theorem provides the sufficient condition under LMIs and LME formulations for the synthesis of the dynamic output controller reflecting the membership function properties.

For the same reasons as in Theorem 1, the following theorem is proven.

Theorem 4 (enhanced relaxed design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3) controlled by the fuzzy dynamic output controller Eqs. (79) and (80) is global asymptotically stable if for given a positive δ ∈ IR there exist positive definite symmetric matrices V<sup>∘</sup> , S<sup>∘</sup> ∈ IR<sup>n</sup> � <sup>n</sup> , and matrices Y∘ <sup>j</sup> <sup>∈</sup> IRr�n, <sup>H</sup><sup>∘</sup> <sup>∈</sup> IR<sup>m</sup> � <sup>m</sup> such that

$$\mathbf{S}^\* = \mathbf{S}^{\*T} > 0, \quad \mathbf{V}^\* = \mathbf{V}^{\*T} > 0,\quad \begin{bmatrix} \mathbf{X}\_{11}^\* & \mathbf{X}\_{12}^\* & \cdots & \mathbf{X}\_{1s}^\* \\ \mathbf{X}\_{21}^\* & \mathbf{X}\_{22}^\* & \cdots & \mathbf{X}\_{2s}^\* \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{X}\_{s1}^\* & \mathbf{X}\_{s2}^\* & \cdots & \mathbf{X}\_{ss}^\* \end{bmatrix} > 0,\tag{109}$$

$$
\begin{bmatrix}
\mathbf{A}\_i^\* \mathbf{S}^\* + \mathbf{S}^\* \mathbf{A}\_i^{\*T} + \mathbf{B}\_i^\* \mathbf{Y}\_i \mathbf{C}^\* + \mathbf{C}^{\*T} \mathbf{Y}\_i^{\*T} \mathbf{B}\_i^{\*T} & \* \\
\mathbf{V}^\* - \mathbf{S}^\* + \delta \mathbf{A}\_i^\* \mathbf{S}^\* + \delta \mathbf{B}\_i^\* \mathbf{Y}\_i \mathbf{C}^\* & -2\delta \mathbf{S}^\*
\end{bmatrix} < 0,\tag{110}
$$

$$
\begin{bmatrix}
\mathbf{O}^\*\_{ij} \\
\mathbf{V}^\* - \mathbf{S}^\* + \delta \frac{A^\*\_i \mathbf{S}^\* + A^\*\_j \mathbf{S}^\*}{2} + \delta \frac{\mathbf{B}^\*\_i \mathbf{Y}^\*\_j + \mathbf{B}^\*\_j \mathbf{Y}^\*\_i}{2} \mathbf{C}^\* & -2\delta \mathbf{S}^\*
\end{bmatrix} < 0,\tag{111}
$$

$$\mathbf{C}^\*\mathbf{S}^\* = H^\*\mathbf{C}^\*,\tag{112}$$

for i = 1, 2, …, s, as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s, hi(θ(t))hj(θ(t)) 6¼ 0, and

$$\mathbf{O}\_{\vec{\eta}}^{\*} = \frac{\mathbf{A}\_{\vec{i}}^{\*}\mathbf{S}^{\*} + \mathbf{S}^{\*}\mathbf{A}\_{\vec{i}}^{\*T}}{2} + \frac{\mathbf{A}\_{\vec{j}}^{\*}\mathbf{S}^{\*} + \mathbf{S}^{\*}\mathbf{A}\_{\vec{j}}^{T}}{2} + \frac{\mathbf{B}\_{\vec{i}}^{\*}\mathbf{Y}\_{\vec{j}}^{\*}\mathbf{C}^{\*} + \mathbf{C}^{T}\mathbf{Y}\_{\vec{j}}^{T}\mathbf{B}\_{\vec{i}}^{T}}{2} + \frac{\mathbf{B}\_{\vec{j}}^{\*}\mathbf{Y}\_{\vec{i}}^{\*}\mathbf{C}^{\*} + \mathbf{C}^{T}\mathbf{Y}\_{\vec{j}}^{T}\mathbf{B}\_{\vec{j}}^{T}}{2}.\tag{113}$$

When the above conditions hold, the control law gain matrices are given as

$$\mathbf{K}\_i^\circ = \mathbf{Y}\_i^\circ (\mathbf{H}^\circ)^{-1}. \tag{114}$$

Proof. Since Eq. (82), Eq. (87) takes formally the same structure as Eqs. (6) and (7), following the same way as in the proof of Theorem 1, the conditions given in Theorem 4 can be obtained. From this reason, the proof is omitted. Compare, for example, Ref. [17].

Following the presented results, it is evident that the standard as well as the enhanced conditions for biproper dynamic output controller design can be derived from Theorem 3 and Theorem 4 in a simple way.

### 6. Illustrative example

Analyzing the product B<sup>∘</sup>

18 Modern Fuzzy Control Systems and Its Applications

<sup>j</sup> <sup>∈</sup> IRr�n, <sup>H</sup><sup>∘</sup> <sup>∈</sup> IR<sup>m</sup> � <sup>m</sup> such that

where

Y∘

Φ<sup>∘</sup> ij <sup>¼</sup> <sup>A</sup><sup>∘</sup>

a simple way.

iK∘ jC∘ S∘

> B∘ iK∘ jC∘

> > K∘ <sup>j</sup>H<sup>∘</sup> <sup>¼</sup> <sup>Y</sup><sup>∘</sup>

, it can set

<sup>S</sup><sup>∘</sup> <sup>¼</sup> <sup>B</sup><sup>∘</sup> iK∘

For the same reasons as in Theorem 1, the following theorem is proven.

given a positive δ ∈ IR there exist positive definite symmetric matrices V<sup>∘</sup>

<sup>S</sup><sup>∘</sup> <sup>¼</sup> <sup>S</sup><sup>∘</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>, <sup>V</sup><sup>∘</sup> <sup>¼</sup> <sup>V</sup><sup>∘</sup><sup>T</sup> <sup>&</sup>gt; <sup>0</sup>,

A<sup>∘</sup><sup>T</sup> <sup>i</sup> <sup>þ</sup> <sup>B</sup><sup>∘</sup> iY∘

<sup>V</sup><sup>∘</sup> � <sup>S</sup><sup>∘</sup> <sup>þ</sup> <sup>δ</sup>A<sup>∘</sup>

A∘ <sup>i</sup>S<sup>∘</sup> <sup>þ</sup> <sup>A</sup><sup>∘</sup> jS∘ <sup>2</sup> <sup>þ</sup> <sup>δ</sup>

A∘ <sup>i</sup>S<sup>∘</sup> <sup>þ</sup> <sup>S</sup><sup>∘</sup>

2 4

<sup>i</sup>S<sup>∘</sup> <sup>þ</sup> <sup>S</sup><sup>∘</sup>

A<sup>∘</sup><sup>T</sup> i 2 þ

<sup>V</sup><sup>∘</sup> � <sup>S</sup><sup>∘</sup> <sup>þ</sup> <sup>δ</sup>

A∘ <sup>j</sup>S<sup>∘</sup> <sup>þ</sup> <sup>S</sup><sup>∘</sup> jH<sup>∘</sup> <sup>H</sup><sup>∘</sup> ð Þ�<sup>1</sup>

<sup>j</sup> , <sup>H</sup><sup>∘</sup> ð Þ�<sup>1</sup>

(106) implies Eqs. (89) and (90) and Eq. (108) gives Eq. (91). This concludes the proof.

and H<sup>∘</sup> ∈ IR(<sup>m</sup> <sup>+</sup> <sup>n</sup>) � (<sup>m</sup> <sup>+</sup> <sup>n</sup>) is a regular square matrix. Thus, with Eq. (108), then Eqs. (105) and

This theorem provides the sufficient condition under LMIs and LME formulations for the synthesis of the dynamic output controller reflecting the membership function properties.

Theorem 4 (enhanced relaxed design conditions). The equilibrium of the fuzzy system Eqs. (2) and (3) controlled by the fuzzy dynamic output controller Eqs. (79) and (80) is global asymptotically stable if for

> X∘ <sup>11</sup> X<sup>∘</sup>

<sup>i</sup>C<sup>∘</sup> <sup>þ</sup> <sup>C</sup><sup>∘</sup><sup>T</sup>Y<sup>∘</sup><sup>T</sup>

B∘ iY∘ <sup>j</sup> <sup>þ</sup> <sup>B</sup><sup>∘</sup> jY∘ i <sup>2</sup> <sup>C</sup><sup>∘</sup> �2δS<sup>∘</sup>

<sup>S</sup><sup>∘</sup> <sup>¼</sup> <sup>H</sup><sup>∘</sup>

iY∘

<sup>i</sup>S<sup>∘</sup> <sup>þ</sup> <sup>δ</sup>B<sup>∘</sup>

Φ∘

C∘

for i = 1, 2, …, s, as well as i = 1, 2, …, s � 1, j = 1 + 1, i + 2, …, s, hi(θ(t))hj(θ(t)) 6¼ 0, and

B∘ iY∘

K∘ <sup>i</sup> <sup>¼</sup> <sup>Y</sup><sup>∘</sup>

A<sup>∘</sup><sup>T</sup> j 2 þ

When the above conditions hold, the control law gain matrices are given as

From this reason, the proof is omitted. Compare, for example, Ref. [17].

� �

X∘ <sup>21</sup> X<sup>∘</sup>

X∘ <sup>s</sup><sup>1</sup> X<sup>∘</sup>

> <sup>i</sup> B<sup>∘</sup><sup>T</sup> <sup>i</sup> ∗

ij ∗

C∘

<sup>j</sup>C<sup>∘</sup> <sup>þ</sup> <sup>C</sup><sup>∘</sup><sup>T</sup>Y<sup>∘</sup><sup>T</sup>

<sup>i</sup> <sup>H</sup><sup>∘</sup> ð Þ�<sup>1</sup>

Proof. Since Eq. (82), Eq. (87) takes formally the same structure as Eqs. (6) and (7), following the same way as in the proof of Theorem 1, the conditions given in Theorem 4 can be obtained.

Following the presented results, it is evident that the standard as well as the enhanced conditions for biproper dynamic output controller design can be derived from Theorem 3 and Theorem 4 in

iC<sup>∘</sup> �2δS<sup>∘</sup>

<sup>j</sup> B<sup>∘</sup><sup>T</sup> i 2 þ B∘ jY∘

<sup>12</sup> ⋯ X<sup>∘</sup>

<sup>22</sup> ⋯ X<sup>∘</sup>

<sup>s</sup><sup>2</sup> ⋯ X<sup>∘</sup>

⋮ ⋮⋱⋮

1s

2s

ss

3

; ð112Þ

<sup>i</sup>C<sup>∘</sup> <sup>þ</sup> <sup>C</sup><sup>∘</sup><sup>T</sup>Y<sup>∘</sup><sup>T</sup>

: ð114Þ

C∘ <sup>S</sup><sup>∘</sup> <sup>¼</sup> <sup>B</sup><sup>∘</sup> iY∘ jC∘

; ð107Þ

<sup>C</sup><sup>∘</sup> <sup>¼</sup> <sup>C</sup><sup>∘</sup> <sup>S</sup><sup>∘</sup> ð Þ�<sup>1</sup> <sup>ð</sup>108<sup>Þ</sup>

, S<sup>∘</sup> ∈ IR<sup>n</sup> � <sup>n</sup>

, and matrices

> 0; ð109Þ

< 0; ð110Þ

5 < 0; ð111Þ

<sup>i</sup> B<sup>∘</sup><sup>T</sup> j <sup>2</sup> : <sup>ð</sup>113<sup>Þ</sup> The nonlinear dynamics of the system is represented by TS model with s = 3 and the system model parameters [20]

$$\begin{aligned} A\_{1} &= \begin{bmatrix} -1.0522 & -1.8666 & 0.5102 \\ -0.4380 & -5.4335 & 0.9205 \\ -0.5522 & 0.1334 & -0.4898 \end{bmatrix}, A\_{2} = \begin{bmatrix} -1.0565 & -1.8661 & 0.5116 \\ -0.4380 & -5.4359 & 0.9214 \\ -0.5565 & 0.1339 & -0.4884 \end{bmatrix}, \\\ A\_{3} &= \begin{bmatrix} -1.0602 & -1.8657 & 0.5133 \\ -0.4381 & -5.4353 & 0.9216 \\ -0.5602 & 0.1343 & -0.4867 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} -0.1765 & 0.0000 \\ 0.0000 & 0.0000 \\ 0.1176 & 0.4721 \end{bmatrix}, \mathbf{C} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}. \end{aligned}$$

To the state vector q(t) are associated the premise variables and themembership functions as follows

$$\boldsymbol{\theta}(t) = \begin{bmatrix} \theta\_1(t) \\ \theta\_2(t) \\ \theta\_3(t) \end{bmatrix}, \boldsymbol{\theta}\_i(t) = \begin{cases} \theta\_1(t) \text{ if } q\_1(t) \text{ is about} & 2.5, \\ \theta\_2(t) \text{ if } q\_1(t) \text{ is about} & 0, \quad h\_2(\theta\_1(t)) = 1 - \frac{1}{2.5} |\theta\_1(t)| \\ \theta\_3(t) \text{ if } q\_1(t) \text{ is about } -2.5, & h\_3(\theta\_3(t)) = 1 - \frac{1}{2.5} |\theta\_3(t)| \\ & h\_3(\theta\_3(t)) = 1 - \frac{1}{2.5} |\theta\_3(t) + 2.5| \end{cases}$$

while the generalized premise variable is θ(t) = q1(t).

Thus, solving Eqs. (46)–(49) for prescribed δ = 1.2 with respect to the LMI matrix variables S, V H, Yi, j = 1, 2, 3, and Xij, i, j = 1, 2, 3 using Self–Dual–Minimization (SeDuMi) package for Matlab [26], then the feedback gain matrix design problem was feasible with the results

S ¼ 0:3899 �0:0102 �0:0000 �0:0102 0:1596 �0:0000 �0:0000 �0:0000 0:4099 2 6 4 3 7 <sup>5</sup>,<sup>V</sup> <sup>¼</sup> 0:9280 0:1235 �0:1525 0:1235 1:1533 �0:3979 �0:1525 �0:3979 0:7574 2 6 4 3 7 5, <sup>H</sup> <sup>¼</sup> <sup>0</sup>:<sup>3899</sup> �0:<sup>0102</sup> �0:0102 0:<sup>1596</sup> � �, X ¼ 0:4567 0:0983 �0:0517 0:0694 0:0463 �0:0174 0:0694 0:0463 �0:0174 0:0983 0:7153 �0:1118 0:0463 0:1906 �0:0441 0:0463 0:1905 �0:0440 �0:0517 �0:1118 0:1883 �0:0175 �0:0442 0:0143 �0:0175 �0:0442 0:0142 0:0694 0:0463 �0:0175 0:4573 0:0981 �0:0515 0:0695 0:0463 �0:0174 0:0463 0:1906 �0:0442 0:0981 0:7154 �0:1115 0:0463 0:1905 �0:0440 �0:0174 �0:0441 0:0143 �0:0515 �0:1115 0:1876 �0:0175 �0:0441 0:0142 0:0694 0:0463 �0:0175 0:0695 0:0463 �0:0175 0:4578 0:0978 �0:0514 0:0463 0:1905 �0:0442 0:0463 0:1905 �0:0441 0:0978 0:7152 �0:1111 �0:0174 �0:0440 0:0142 �0:0174 �0:0440 0:0142 �0:0514 �0:1111 0:1868 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 , <sup>Y</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:<sup>5607</sup> �0:<sup>4590</sup> <sup>0</sup>:<sup>1544</sup> �0:<sup>1191</sup> � �, <sup>Y</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>5558</sup> �0:<sup>4577</sup> <sup>0</sup>:<sup>1579</sup> �0:<sup>1207</sup> � �, <sup>Y</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup>:<sup>5518</sup> �0:<sup>4566</sup> <sup>0</sup>:<sup>1606</sup> �0:<sup>1222</sup> � �;

Substituting the above parameters into Eq. (51) to solve the controller parameters, the following gain matrices are obtained

Figure 1. TS fuzzy static output control structure in a forced mode.

$$\begin{aligned} \mathbf{K}\_{1} &= \begin{bmatrix} 1.3653 & -2.7895 \\ 0.3772 & -0.7224 \end{bmatrix}, \mathbf{K}\_{2} = \begin{bmatrix} 1.3530 & -2.7823 \\ 0.3860 & -0.7318 \end{bmatrix}, \mathbf{K}\_{3} = \begin{bmatrix} 1.3428 & -2.7761 \\ 0.3925 & -0.7406 \end{bmatrix}, \\\ \mathbf{A}\_{d2} &= \begin{bmatrix} -1.2953 & -1.3750 & 0.5116 \\ -0.4380 & -5.4359 & 0.9214 \\ -0.2152 & -0.5388 & -0.4884 \end{bmatrix}, \mathbf{A}\_{c31} = \begin{bmatrix} -1.3012 & -1.3734 & 0.5133 \\ -0.4381 & -5.4353 & 0.9216 \\ -0.2216 & -0.5348 & -0.4867 \end{bmatrix}, \end{aligned}$$

For simplicity, other closed-loop matrices of subsystem dynamics are not listed here.

Since the diagonal elements of Acij, i, j = 1, 2, 3, are dominant, in terms of Gerschgorin theorem [27, 28], all eigenvalues of Acij are real, resulting in the aperiodic dynamics, that is,

$$\begin{split} \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl1}1}) &= \{-0.6751, \ -1.0816, \ -5.4598\}, \quad \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl2}1}) = \{-0.6756, \ -1.0842, \ -5.4620\}, \\ \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl3}1}) &= \{-0.6757, \ -1.0861, \ -5.4613\}, \quad \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl2}}) = \{-0.6745, \ -1.0805, \ -5.4593\}, \\ \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl2}}) &= \{-0.6750, \ -1.0831, \ -5.4615\}, \quad \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl2}}) = \{-0.6751, \ -1.0851, \ -5.4609\}, \\ \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl3}}) &= \{-0.6742, \ -1.0795, \ -5.4588\}, \quad \tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl2}}) = \{-0.6748, \ -1.0820, \ -5.4610\}, \end{split}$$

$$\tilde{\mathbf{n}}(\mathbf{A}\_{\mathrm{cl3}}) = \{-0.6748, \ -1.0840, \ -5.4604\}.$$

Figure 1 gives the associated TS fuzzy static output control structure in a forced mode.

For Eqs. (88)–(91), it can find the following feasible solutions by using the given design procedure

<sup>S</sup><sup>∘</sup> <sup>¼</sup> :6194 �0:0614 0:0000 0:0000 0:0000 0:0000 �0:0614 0:1305 0:0000 0:0000 0:0000 0:0000 :0000 0:0000 0:8724 0:0000 0:0000 0:0000 :0000 0:0000 0:0000 0:7066 0:0000 0:0000 :0000 0:0000 0:0000 0:0000 0:7066 0:0000 :0000 0:0000 0:0000 0:0000 0:0000 0:7066 , <sup>H</sup><sup>∘</sup> <sup>¼</sup> :7066 0:0000 0:0000 0:0000 0:0000 :0000 0:7066 0:0000 0:0000 0:0000 :0000 0:0000 0:7066 0:0000 0:0000 :0000 0:0000 0:0000 0:6808 �0:0614 :0000 0:0000 0:0000 0:4889 0:0691 , Y<sup>1</sup> ¼ �0:5668 0:0000 0:0000 0:0000 0:0000 :0000 �0:5668 0:0000 �0:0001 0:0000 :0000 0:0000 �0:5667 0:0000 0:0000 :0000 0:0000 0:0000 0:3612 �0:2783 :0000 0:0000 0:0000 0:7396 �1:1397 , Y<sup>2</sup> ¼ �0:5668 0:0000 0:0000 0:0000 0:0000 :0000 �0:5668 0:0000 �0:0001 0:0000 :0000 0:0000 �0:5667 0:0000 0:0000 :0000 0:0000 0:0000 0:3615 �0:2784 :0000 0:0000 0:0000 0:7397 �1:1397 , Y<sup>3</sup> ¼ �0:5667 �0:0001 0:0000 0:0000 0:0000 �0:0001 �0:5667 0:0000 �0:0001 0:0000 :0000 0:0000 �0:5668 0:0000 0:0000 :0000 0:0000 0:0000 0:3519 �0:2859 �0:0001 0:0000 �0:0001 0:7486 �1:1421 

and, computing the biproper dynamic output controller parameters, then

<sup>K</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>:<sup>3653</sup> �2:<sup>7895</sup>

Modern Fuzzy Control Systems and Its Applications

Ac<sup>22</sup> ¼

 

:3772 �0:7224

�1:2953 �1:3750 0:5116 �0:4380 �5:4359 0:9214 �0:2152 �0:5388 �0:4884

" #

Figure 1. TS fuzzy static output control structure in a forced mode.

, <sup>K</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>:<sup>3530</sup> �2:<sup>7823</sup>

 , <sup>A</sup>c<sup>31</sup> <sup>¼</sup>

Since the diagonal elements of Acij, i, j = 1, 2, 3, are dominant, in terms of Gerschgorin theorem [27, 28], all eigenvalues of Acij are real, resulting in the aperiodic dynamics, that is,

For simplicity, other closed-loop matrices of subsystem dynamics are not listed here.

:3860 �0:7318

 

" #

, <sup>K</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>:<sup>3428</sup> �2:<sup>7761</sup>

�1:3012 �1:3734 0:5133 �0:4381 �5:4353 0:9216 �0:2216 �0:5348 �0:4867

:3925 �0:7406

,

 ;

" #

J<sup>1</sup> ¼ �0:8022 0:0000 0:0000 0:0000 �0:8021 0:0000 0:0000 0:0000 �0:8021 2 6 4 3 7 <sup>5</sup>, <sup>L</sup><sup>1</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup> 0:0394 0:0048 �0:3143 0:3318 0:0221 �0:0508 2 6 4 3 7 5, <sup>M</sup><sup>1</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> �0:2041 0:<sup>1504</sup> �0:<sup>0600</sup> �0:5318 0:<sup>0915</sup> �0:<sup>3275</sup> " #, <sup>N</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>:<sup>0889</sup> �2:<sup>1701</sup> <sup>7</sup>:<sup>8914</sup> �9:<sup>4765</sup> " #, J<sup>2</sup> ¼ �0:8022 0:0001 0:0000 0:0001 �0:8022 0:0000 0:0000 0:0000 �0:8021 2 6 4 3 7 <sup>5</sup>, <sup>L</sup><sup>2</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup> �0:2009 0:3531 0:0453 �0:2017 �0:1903 0:1765 2 6 4 3 7 5, <sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> �0:2022 0:1779 0:<sup>1796</sup> �0:4575 0:0413 0:<sup>2985</sup> " #, <sup>N</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>:<sup>0897</sup> �2:<sup>1707</sup> <sup>7</sup>:<sup>8915</sup> �9:<sup>4766</sup> " #, J<sup>3</sup> ¼ �0:8020 �0:0001 0:0000 �0:0001 �0:8021 0:0000 0:0000 0:0000 �0:8022 2 6 4 3 7 <sup>5</sup>, <sup>L</sup><sup>3</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup> �0:0641 0:0139 �0:1516 0:2382 �0:2116 0:2630 2 6 4 3 7 5, <sup>M</sup><sup>3</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> �0:0135 0:<sup>0020</sup> �0:<sup>0238</sup> �0:0917 0:<sup>0218</sup> �0:<sup>1102</sup> " #, <sup>N</sup><sup>3</sup> <sup>¼</sup> <sup>2</sup>:<sup>1286</sup> �2:<sup>2445</sup> <sup>7</sup>:<sup>9148</sup> �9:<sup>4907</sup> " #:

It is evident that all matrices Ji, i = 1.2.3 are Hurwitz, which rise up a TS fuzzy stable dynamic output controller, and based on the solutions obtained, the TS fuzzy dynamic controller can be designed via the concept of PDC.

Verifying the closed-loop stability, it can compute the eigenvalue spectra as follows

ñð Þ¼ � Ac<sup>11</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2804i , ñð Þ¼ � Ac<sup>21</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2804i , ñð Þ¼ � Ac<sup>21</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2804i , ñð Þ¼ � Ac<sup>12</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2805i , ñð Þ¼ � Ac<sup>22</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2805i , ñð Þ¼ � Ac<sup>32</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3788, � 1:2946 � 0:2963i , ñð Þ¼ � Ac<sup>13</sup> f g 0:8020, � 0:8023, � 0:8023, � 4:3713, � 1:2919 � 0:2797i , ñð Þ¼ � Ac<sup>23</sup> f g 0:8020, � 0:8023, � 0:8023, � 4:3713, � 1:2919 � 0:2797i , ñð Þ¼ � Ac<sup>33</sup> f g 0:8020, � 0:8023, � 0:8023, � 4:3728, � 1:2945 � 0:2958i :

#### 7. Concluding remarks

New approach for static and dynamic output feedback control design, taking into account the affine properties of the TS fuzzy model structure, is presented in the chapter. Applying the fuzzy output control schemes relating to the parallel-distributed output compensators, the method presented methods that significantly reduces the conservativeness in the control design conditions. Sufficient existence conditions of the both output controller realization, manipulating the global stability of the system, implies the parallel decentralized control framework which stabilizes the nonlinear system in the sense of Lyapunov, and the design of controller parameters, resulting directly from these conditions, is a feasible numerical problem. An additional benefit of the method is that controllers use minimum feedback information with respect to desired system output and the approach is flexible enough to allow the inclusion of additional design conditions. The validity and applicability of the approach is demonstrated through numerical design examples.

## Acknowledgement

J<sup>1</sup> ¼

J<sup>2</sup> ¼

J<sup>3</sup> ¼

2 6 4

designed via the concept of PDC.

7. Concluding remarks

2 6 4

22 Modern Fuzzy Control Systems and Its Applications

2 6 4 �0:8022 0:0000 0:0000 0:0000 �0:8021 0:0000 0:0000 0:0000 �0:8021

<sup>M</sup><sup>1</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> �0:2041 0:<sup>1504</sup> �0:<sup>0600</sup>

�0:8022 0:0001 0:0000 0:0001 �0:8022 0:0000 0:0000 0:0000 �0:8021

<sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> �0:2022 0:1779 0:<sup>1796</sup>

�0:8020 �0:0001 0:0000 �0:0001 �0:8021 0:0000 0:0000 0:0000 �0:8022

<sup>M</sup><sup>3</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> �0:0135 0:<sup>0020</sup> �0:<sup>0238</sup>

�0:5318 0:0915 �0:3275

�0:4575 0:0413 0:2985

�0:0917 0:0218 �0:1102

Verifying the closed-loop stability, it can compute the eigenvalue spectra as follows

" #

" #

" #

3 7

3 7

> 3 7

It is evident that all matrices Ji, i = 1.2.3 are Hurwitz, which rise up a TS fuzzy stable dynamic output controller, and based on the solutions obtained, the TS fuzzy dynamic controller can be

ñð Þ¼ � Ac<sup>11</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2804i , ñð Þ¼ � Ac<sup>21</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2804i , ñð Þ¼ � Ac<sup>21</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2804i , ñð Þ¼ � Ac<sup>12</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2805i , ñð Þ¼ � Ac<sup>22</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3774, � 1:2919 � 0:2805i , ñð Þ¼ � Ac<sup>32</sup> f g 0:8022, � 0:8021, � 0:8021, � 4:3788, � 1:2946 � 0:2963i , ñð Þ¼ � Ac<sup>13</sup> f g 0:8020, � 0:8023, � 0:8023, � 4:3713, � 1:2919 � 0:2797i , ñð Þ¼ � Ac<sup>23</sup> f g 0:8020, � 0:8023, � 0:8023, � 4:3713, � 1:2919 � 0:2797i , ñð Þ¼ � Ac<sup>33</sup> f g 0:8020, � 0:8023, � 0:8023, � 4:3728, � 1:2945 � 0:2958i :

New approach for static and dynamic output feedback control design, taking into account the affine properties of the TS fuzzy model structure, is presented in the chapter. Applying the fuzzy output control schemes relating to the parallel-distributed output compensators, the method presented methods that significantly reduces the conservativeness in the control

<sup>5</sup>, <sup>L</sup><sup>1</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup>

<sup>5</sup>, <sup>L</sup><sup>2</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup>

<sup>5</sup>, <sup>L</sup><sup>3</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup>

2 6 4

2 6 4

, <sup>N</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>:<sup>0889</sup> �2:<sup>1701</sup>

, <sup>N</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>:<sup>0897</sup> �2:<sup>1707</sup>

, <sup>N</sup><sup>3</sup> <sup>¼</sup> <sup>2</sup>:<sup>1286</sup> �2:<sup>2445</sup>

2 6 4

0:0394 0:0048 �0:3143 0:3318 0:0221 �0:0508

7:8914 �9:4765

�0:2009 0:3531 0:0453 �0:2017 �0:1903 0:1765

7:8915 �9:4766

�0:0641 0:0139 �0:1516 0:2382 �0:2116 0:2630

7:9148 �9:4907

" #

" #

" #

3 7 5,

,

3 7 5,

,

3 7 5,

:

The work presented in this chapter was supported by VEGA, the Grant Agency of Ministry of Education and Academy of Science of Slovak Republic, under Grant No. 1/0608/17. This support is very gratefully acknowledged.

## Author details

Dušan Krokavec\* and Anna Filasová

\*Address all correspondence to: dusan.krokavec@tuke.sk

Department of Cybernetics, Artificial Intelligence, Faculty of Electrical Engineering, Informatics, Technical University of Košice, Košice, Slovakia

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24 Modern Fuzzy Control Systems and Its Applications

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## **Non-Fragile Guaranteed Cost Control of Nonlinear Systems with Different State and Input Delays Based on T-S Fuzzy Local Bilinear Models**

Junmin Li, Jinsha Li and Ruirui Duan

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69777

#### Abstract

This paper focuses on the non-fragile guaranteed cost control problem for a class of Takagi-Sugeno (T-S) fuzzy time-varying delay systems with local bilinear models and different state and input delays. A non-fragile guaranteed cost state-feedback controller is designed such that the closed-loop T-S fuzzy local bilinear control system is delay-dependent asymptotically stable, and the closed-loop fuzzy system performance is constrained to a certain upper bound when the additive controller gain perturbations exist. By employing the linear matrix inequality (LMI) technique, sufficient conditions are established for the existence of desired non-fragile guaranteed cost controllers. The simulation examples show that the proposed approach is effective and feasible.

Keywords: fuzzy control, non-fragile guaranteed cost control, delay-dependent, linear matrix inequality (LMI), T-S fuzzy bilinear model

### 1. Introduction

In recent years, T-S (Takagi-Sugeno) model-based fuzzy control has attracted wide attention, essentially because the fuzzy model is an effective and flexible tool for the control of nonlinear systems [1–8]. Through the application of sector nonlinearity approach, local approximation in fuzzy partition spaces or other different approximation methods, T-S fuzzy models will be used to approximate or exactly represent a nonlinear system in a compact set of state variables. The merit of the model is that the consequent part of a fuzzy rule is a linear dynamic subsystem, which makes it possible to apply the classical and mature linear systems theory to nonlinear systems. Further, by using the fuzzy inference method, the overall fuzzy model will

be obtained. A fuzzy controller is designed via the method titled 'parallel distributed compensation (PDC)' [3–6], the main idea of which is that for each linear subsystem, the corresponding linear controller is carried out. Finally, the overall nonlinear controller is obtained via fuzzy blending of each individual linear controller. Based on the above content, T-S fuzzy model has been widely studied, and many results have been obtained [1–8]. In practical applications, time delay often occurs in many dynamic systems such as biological systems, network systems, etc. It is shown that the existence of delays usually becomes the source of instability and deteriorating performance of systems [3–8]. In general, when delay-dependent results were calculated, the emergence of the inner product between two vectors often makes the process of calculation more complicated. In order to avoid it, some model transformations were utilized in many papers, unfortunately, which will arouse the generation of an inequality, resulting in possible conservatism. On the other hand, due to the influence of many factors such as finite word length, truncation errors in numerical computation and electronic component parameter change, the parameters of the controller in a certain degree will change, which lead to imprecision in controller implementation. In this case, some small perturbations of the controllers' coefficients will make the designed controllers sensitive, even worse, destabilize the closedloop control system [9]. So the problem of non-fragile control has been important issues. Recently, the research of non-fragile control has been paid much attention, and a series of productions have been obtained [10–13].

As we know, bilinear models have been widely used in many physical systems, biotechnology, socioeconomics and dynamical processes in other engineering fields [14, 15]. Bilinear model is a special nonlinear model, the nonlinear part of which consists of the bilinear function of the state and input. Compared with a linear model, the bilinear models have two main advantages. One is that the bilinear model can better approximate a nonlinear system. Another is that because of nonlinearity of it, many real physical processes may be appropriately modeled as bilinear systems. A famous example of a bilinear system is the population of biological species, which can be showed by <sup>d</sup><sup>θ</sup> dt <sup>¼</sup> <sup>θ</sup>v. In this equation, <sup>v</sup> is the birth rate minus death rate, and θ denotes the population. Obviously, the equation cannot be approximated by a linear model [14].

Most of the existing results focus on the stability analysis and synthesis based on T-S fuzzy model with linear local model. However, when a nonlinear system has of complex nonlinearities, the constructed T-S model will consist of a number of fuzzy local models. This will lead to very heavy computational burden. According to the advantages of bilinear systems and T-S fuzzy control, so many researchers paid their attentions to the T-S fuzzy models with bilinear rule consequence [16–18]. From these papers, it is evident that the T-S fuzzy bilinear model may be suitable for some classes of nonlinear plants. In Ref. [16], a nonlinear system was transformed into a bilinear model via Taylor's series expansion, and the stability of T-S fuzzy bilinear model was studied. Moreover, the result was stretched into the complex fuzzy system with state time delay [17]. Ref. [18] presented robust stabilization for a class of discrete-time fuzzy bilinear system. Very recently, a class of nonlinear systems is described by T-S fuzzy models with nonlinear local models in Ref. [19], and in this paper, the scholars put forward a new fuzzy control scheme with local nonlinear feedbacks, the advantage of which over the existing methods is that a fewer fuzzy rules and less computational burden. The non-fragile guaranteed cost controller was designed for a class of T-S discrete-time fuzzy bilinear systems in Ref. [20]. However, in Refs. [19, 20], the time-delay effects on the system is not considered. Ref. [17] is only considered the fuzzy system with the delay in the state and the derivatives of time-delay, \_ dðtÞ < 1 is required. Refs. [21–23] dealt with the uncertain fuzzy systems with timedelay in different ways. It should be pointed out that all the aforementioned works did not take into account the effect of the control input delays on the systems. The results therein are not applicable to systems with input delay. Recently, some controller design approaches have been presented for systems with input delay, see [2, 3, 4, 18, 24–32] for fuzzy T-S systems and [8, 15, 33, 34] for non-fuzzy systems and the references therein. All of these results are required to know the exact delay values in the implementation. T-S fuzzy stochastic systems with state time-vary or distributed delays were studied in Refs. [35–39]. The researches of fractional order T-S fuzzy systems on robust stability, stability analysis about "0 < α < 1", and decentralized stabilization in multiple time delays were presented in Refs. [40–42], respectively. For different delay types, the corresponding adaptive fuzzy controls for nonlinear systems were proposed in Refs. [33, 43, 44]. In Refs. [45, 46], to achieve small control amplitude, a new T-S fuzzy hyperbolic model was developed, moreover, Ref. [46] considered the input delay of the novel model. In Ref. [25, 47], the problems of observer-based fuzzy control design for T-S fuzzy systems were concerned.

be obtained. A fuzzy controller is designed via the method titled 'parallel distributed compensation (PDC)' [3–6], the main idea of which is that for each linear subsystem, the corresponding linear controller is carried out. Finally, the overall nonlinear controller is obtained via fuzzy blending of each individual linear controller. Based on the above content, T-S fuzzy model has been widely studied, and many results have been obtained [1–8]. In practical applications, time delay often occurs in many dynamic systems such as biological systems, network systems, etc. It is shown that the existence of delays usually becomes the source of instability and deteriorating performance of systems [3–8]. In general, when delay-dependent results were calculated, the emergence of the inner product between two vectors often makes the process of calculation more complicated. In order to avoid it, some model transformations were utilized in many papers, unfortunately, which will arouse the generation of an inequality, resulting in possible conservatism. On the other hand, due to the influence of many factors such as finite word length, truncation errors in numerical computation and electronic component parameter change, the parameters of the controller in a certain degree will change, which lead to imprecision in controller implementation. In this case, some small perturbations of the controllers' coefficients will make the designed controllers sensitive, even worse, destabilize the closedloop control system [9]. So the problem of non-fragile control has been important issues. Recently, the research of non-fragile control has been paid much attention, and a series of

As we know, bilinear models have been widely used in many physical systems, biotechnology, socioeconomics and dynamical processes in other engineering fields [14, 15]. Bilinear model is a special nonlinear model, the nonlinear part of which consists of the bilinear function of the state and input. Compared with a linear model, the bilinear models have two main advantages. One is that the bilinear model can better approximate a nonlinear system. Another is that because of nonlinearity of it, many real physical processes may be appropriately modeled as bilinear systems. A famous example of a bilinear system is the population of biological

and θ denotes the population. Obviously, the equation cannot be approximated by a linear

Most of the existing results focus on the stability analysis and synthesis based on T-S fuzzy model with linear local model. However, when a nonlinear system has of complex nonlinearities, the constructed T-S model will consist of a number of fuzzy local models. This will lead to very heavy computational burden. According to the advantages of bilinear systems and T-S fuzzy control, so many researchers paid their attentions to the T-S fuzzy models with bilinear rule consequence [16–18]. From these papers, it is evident that the T-S fuzzy bilinear model may be suitable for some classes of nonlinear plants. In Ref. [16], a nonlinear system was transformed into a bilinear model via Taylor's series expansion, and the stability of T-S fuzzy bilinear model was studied. Moreover, the result was stretched into the complex fuzzy system with state time delay [17]. Ref. [18] presented robust stabilization for a class of discrete-time fuzzy bilinear system. Very recently, a class of nonlinear systems is described by T-S fuzzy models with nonlinear local models in Ref. [19], and in this paper, the scholars put forward a new fuzzy control scheme with local nonlinear feedbacks, the advantage of which over the

dt <sup>¼</sup> <sup>θ</sup>v. In this equation, <sup>v</sup> is the birth rate minus death rate,

productions have been obtained [10–13].

28 Modern Fuzzy Control Systems and Its Applications

species, which can be showed by <sup>d</sup><sup>θ</sup>

model [14].

So far, the problem of non-fragile guaranteed cost control for fuzzy system with local bilinear model with different time-varying state and input delays has not been discussed.

In this paper, the problem of delay-dependent non-fragile guaranteed cost control is studied for the fuzzy time-varying delay systems with local bilinear model and different state and input delays. Based on the PDC scheme, new delay-dependent stabilization conditions for the closed-loop fuzzy systems are derived. No model transformation is involved in the derivation. The merit of the proposed conditions lies in its reduced conservatism, which is achieved by circumventing the utilization of some bounding inequalities for the cross-product between two vectors as in Ref. [17]. The three main contributions of this paper are the following: (1) a nonfragile guaranteed cost controller is presented for the fuzzy system with time-varying delay in both state and input; (2) some free-weighting matrices are introduced in the derivation process, where the constraint of the derivatives of time-delay, \_ <sup>d</sup>ðt<sup>Þ</sup> <sup>&</sup>lt; 1 and \_ hðtÞ < 1, is eliminated; and (3) the delay-dependent stability conditions for the fuzzy system are described by LMIs. Finally, simulation examples are given to illustrate the effectiveness of the obtained results.

The paper is organized as follows. Section 2 introduces the fuzzy delay system with local bilinear model, and non-fragile controller law for such system is designed based on the parallel distributed compensation approach in Section 3. Results of non-fragile guaranteed cost control are given in Section 4. Two simulation examples are used to illustrate the effectiveness of the proposed method in Section 5, which is followed by conclusions in Section 6.

Notation: Throughout this paper, the notation P > 0(P ≥ 0) stands for P being real symmetric and positive definite (or positive semi-definite). In symmetric block matrices, the asterisk (\*) refers to a term that is induced by symmetry, and diag{….} denotes a block-diagonal matrix. The superscript T means matrix transposition. The notion P<sup>s</sup> i,j¼<sup>1</sup> is an abbreviation of P<sup>s</sup> i¼1 P<sup>s</sup> <sup>j</sup>¼<sup>1</sup>. Matrices, if the dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

#### 2. System description and assumptions

In this section, we introduce the T-S fuzzy time-delay system with local bilinear model. The ith rule of the fuzzy system is represented by the following form:

$$\begin{aligned} \text{Plant Rule } i: \\ \text{IF } \ $1(t) \text{ is } F\_{\text{il}} \text{ and } \dots \text{ and } \$ \text{s}(t) \text{ is } F\_{\text{ir}}, \text{ THEN} \\ \dot{\mathbf{x}}(t) = A\_i \mathbf{x}(t) + A\_{\text{il}} \mathbf{x}(t - d(t)) + B\_i \mathbf{u}(t) + B\_{\text{il}} \mathbf{u}(t - h(t)) + \mathbf{N}\_i \mathbf{x}(t) \mathbf{u}(t) + \mathbf{N}\_{\vec{d}} \mathbf{x}(t - d(t)) \mathbf{u}(t - h(t)) \\ \mathbf{x}(t) = \phi(t), \quad t \in [-\tau\_1, 0], \ i = 1, 2, \dots, s \end{aligned} \tag{1}$$

where Fij is the fuzzy set, s is the number of fuzzy rules, x(t) ∈ Rn is the state vector, and u(t) ∈ R is the control input, ϑ1(t), ϑ2(t),…,ϑv(t) are the premise variables. It is assumed that the premise variables do not depend on the input u(t). Ai, Adi, Ni, Ndi ∈R<sup>n</sup>�n, Bi, Bhi ∈ Rn�<sup>1</sup> denote the system matrices with appropriate dimensions. d(t) is a time-varying differentiable function that satisfies 0 ≤ d(t) ≤ τ1, 0 ≤ h(t) ≤ τ2, where τ1, τ<sup>2</sup> are real positive constants as the upper bound of the timevarying delay. It is also assumed that \_ <sup>d</sup>ðt<sup>Þ</sup> <sup>≤</sup> <sup>σ</sup>1, \_ hðtÞ ≤ σ2, and σ1, σ<sup>2</sup> are known constants. The initial conditions φ(t), ϕ(t) are continuous functions of t, t∈½�τ, 0�, τ ¼ minðτ1, τ2Þ.

Remark 1: The fuzzy system with time-varying state and input delays will be investigated in this paper, which is different from the system in Ref. [17]. In Ref. [17], only state time-varying delay is considered. And also, here, we assume that the derivative of time-varying delay is less than or equal to a known constant that may be greater than 1; the assumption on time-varying delay in Ref. [17] is relaxed.

By using singleton fuzzifier, product inferred and weighted defuzzifier, the fuzzy system can be expressed by the following globe model:

$$\begin{split} \dot{\mathbf{x}}(t) &= \sum\_{i=1}^{s} h\_i(\mathbf{\dot{s}}(t)) [A\_i \mathbf{x}(t) + A\_{di} \mathbf{x}(t - d(t)) + B\_i \mathbf{u}(t) + B\_{hi} \mathbf{u}(t - h(t)) + N\_i \mathbf{x}(t) \mathbf{u}(t) \\ &+ N\_{di} \mathbf{x}(t - d(t)) \mathbf{u}(t - h(t))] \end{split} \tag{2}$$

where

hiðϑðtÞÞ ¼ ωiðϑðtÞÞ= X<sup>s</sup> <sup>i</sup>¼<sup>1</sup> <sup>ω</sup>iðϑðtÞÞ, <sup>ω</sup>iðϑðtÞÞ ¼ <sup>Y</sup><sup>v</sup> <sup>j</sup>¼<sup>1</sup> <sup>μ</sup>ijðϑðtÞÞ, <sup>μ</sup>ij (ϑ(t)) is the grade of membership of ϑi(t) in Fij. In this paper, it is assumed that ωiðϑðtÞÞ ≥ 0, X<sup>s</sup> <sup>i</sup>¼<sup>1</sup> <sup>ω</sup>iðϑðtÞÞ <sup>&</sup>gt; 0 for all <sup>t</sup>. Then, we have the following conditions hi(ϑ(t)) <sup>≥</sup> 0, <sup>X</sup><sup>s</sup> <sup>i</sup>¼<sup>1</sup> hiðϑðtÞÞ ¼ 1 for all <sup>t</sup>. In the consequent, we use abbreviation hi, hhi, xd(t), ud(t), xh(t), uh(t), to replace hi(ϑ(t)), hi(ϑ(t � h(t))), x(t � d(t)), u(t � d (t)), x(t � h(t)), u(t � h(t)), respectively, for convenience.

The objective of this paper is to design a state-feedback non-fragile guaranteed cost control law for the fuzzy system (2).

#### 3. Non-fragile guaranteed cost controller design

Extending the design concept in Ref. [17], we give the following non-fragile fuzzy control law:

$$\begin{aligned} \text{IF } \mathfrak{S}\_1(t) \text{ is } \boldsymbol{F}\_1^i \text{ and } \dots \text{ and } \mathfrak{S}\_v(t) \text{ is } \boldsymbol{F}\_v^i\\ \text{THEN } \boldsymbol{u}(t) = \frac{\rho(\mathcal{K}\_i + \Delta \mathcal{K}\_i) \mathbf{x}(t)}{\sqrt{1 + \mathbf{x}^T(\mathcal{K}\_i + \Delta \mathcal{K}\_i)^T(\mathcal{K}\_i + \Delta \mathcal{K}\_i)\mathbf{x}}} = \rho \sin \theta\_i = \rho \cos \theta\_i (\mathcal{K}\_i + \Delta \mathcal{K}\_i)\mathbf{x}(t) \end{aligned} \tag{3}$$

where ρ > 0 is a scalar to be assigned, and Ki ∈ Rl�<sup>n</sup> is a local controller gain to be determined. ΔKi represents the additive controller gain perturbations of the form ΔKi = HiFi(t)Eki with Hi and Eki being known constant matrices, and Fi(t) the uncertain parameter matrix satisfying Fi <sup>T</sup>ðtÞFiðt<sup>Þ</sup> <sup>≤</sup> <sup>I</sup>. sin <sup>θ</sup><sup>i</sup> <sup>¼</sup> Kixðt<sup>Þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>þxTK T <sup>i</sup> Kix <sup>q</sup> , cos θ<sup>i</sup> ¼ <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>þx<sup>T</sup>K T <sup>i</sup> Kix <sup>q</sup> , θ<sup>i</sup> ∈ ½�<sup>π</sup> <sup>2</sup> , <sup>π</sup> <sup>2</sup>�, Ki ¼ Ki þ ΔKiðtÞ ¼ Ki þ HiFiðtÞEki.

The overall fuzzy control law can be represented by

$$u(t) = \sum\_{i=1}^{s} h\_i \frac{\rho \overline{\mathbf{K}}\_i \mathbf{x}(t)}{\sqrt{1 + \mathbf{x}^T \overline{\mathbf{K}}\_i^T \overline{\mathbf{K}}\_i \mathbf{x}}} = \sum\_{i=1}^{s} h\_i \rho \sin \theta\_i = \sum\_{i=1}^{s} h\_i \rho \cos \theta\_i \overline{\mathbf{K}}\_i \mathbf{x}(t) \tag{4}$$

When there exists an input delay h(t), we have that

$$\mu\_{\hbar}(t) = \sum\_{l=1}^{s} h\_{\hbar l} \rho \sin \varphi\_{l} = \sum\_{l=1}^{s} h\_{\hbar l} \rho \cos \varphi\_{l} \tilde{\mathbf{K}}\_{l} \mathbf{x}\_{\hbar}(t) \tag{5}$$

where sinϕ<sup>l</sup> <sup>¼</sup> <sup>K</sup><sup>~</sup> lxhðt<sup>Þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>þx<sup>T</sup> hK~ T <sup>l</sup> K~ lxh <sup>q</sup> , cosϕ<sup>l</sup> ¼ <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>þx<sup>T</sup> hK~ T <sup>l</sup> K~ lxh <sup>q</sup> , ϕ<sup>l</sup> ∈ ½�<sup>π</sup> <sup>2</sup> , <sup>π</sup> <sup>2</sup>�, <sup>K</sup><sup>~</sup> <sup>l</sup> <sup>¼</sup> Kl <sup>þ</sup> <sup>Δ</sup>Klð<sup>t</sup> � <sup>h</sup>ðtÞÞ ¼ Kl<sup>þ</sup> HlFlðt � hðtÞÞEkl.

So, it is natural and necessary to make an assumption that the functions hi are well defined all t ∈ [�τ2, 0], and satisfy the following properties:

$$h\_i(\mathfrak{G}(t - h(t))) \ge 0, \text{ for } i = 1, 2, \dots, s, \text{ and } \sum\_{i=1}^s h\_i(\mathfrak{G}(t - h(t))) = 1.$$

By substituting Eq. (5) into Eq. (2), the closed-loop system can be given by

$$\dot{\mathbf{x}}(t) = \sum\_{i,j,l=1}^{s} h\_i h\_j h\_{li} (\Lambda\_{ij} \mathbf{x}(t) + \Lambda\_{dij} \mathbf{x}\_d(t) + \Lambda\_{lil} \mathbf{x}\_h(t)) \tag{6}$$

where

The superscript T means matrix transposition. The notion P<sup>s</sup>

rule of the fuzzy system is represented by the following form:

ible dimensions for algebraic operations.

30 Modern Fuzzy Control Systems and Its Applications

2. System description and assumptions

IF ϑ1ðtÞ is Fi<sup>1</sup> and … and ϑvðtÞ is Fiv; THEN

varying delay. It is also assumed that \_

be expressed by the following globe model:

X<sup>s</sup>

we have the following conditions hi(ϑ(t)) <sup>≥</sup> 0, <sup>X</sup><sup>s</sup>

(t)), x(t � h(t)), u(t � h(t)), respectively, for convenience.

þ Ndixðt � dðtÞÞuðt � hðtÞÞ�

<sup>i</sup>¼<sup>1</sup> <sup>ω</sup>iðϑðtÞÞ, <sup>ω</sup>iðϑðtÞÞ ¼ <sup>Y</sup><sup>v</sup>

ship of ϑi(t) in Fij. In this paper, it is assumed that ωiðϑðtÞÞ ≥ 0,

delay in Ref. [17] is relaxed.

i¼1

<sup>x</sup>\_ðtÞ ¼X<sup>s</sup>

hiðϑðtÞÞ ¼ ωiðϑðtÞÞ=

where

xðtÞ ¼ φðtÞ, t ∈½�τ1, 0� ; i ¼ 1, 2, …, s

<sup>j</sup>¼<sup>1</sup>. Matrices, if the dimensions are not explicitly stated, are assumed to have compat-

In this section, we introduce the T-S fuzzy time-delay system with local bilinear model. The ith

x\_ðtÞ ¼ AixðtÞ þ Adixðt � dðtÞÞ þ BiuðtÞ þ Bhiuðt � hðtÞÞ þ NixðtÞuðtÞ þ Ndixðt � dðtÞÞuðt � hðtÞÞ

where Fij is the fuzzy set, s is the number of fuzzy rules, x(t) ∈ Rn is the state vector, and u(t) ∈ R is the control input, ϑ1(t), ϑ2(t),…,ϑv(t) are the premise variables. It is assumed that the premise variables do not depend on the input u(t). Ai, Adi, Ni, Ndi ∈R<sup>n</sup>�n, Bi, Bhi ∈ Rn�<sup>1</sup> denote the system matrices with appropriate dimensions. d(t) is a time-varying differentiable function that satisfies 0 ≤ d(t) ≤ τ1, 0 ≤ h(t) ≤ τ2, where τ1, τ<sup>2</sup> are real positive constants as the upper bound of the time-

<sup>d</sup>ðt<sup>Þ</sup> <sup>≤</sup> <sup>σ</sup>1, \_

Remark 1: The fuzzy system with time-varying state and input delays will be investigated in this paper, which is different from the system in Ref. [17]. In Ref. [17], only state time-varying delay is considered. And also, here, we assume that the derivative of time-varying delay is less than or equal to a known constant that may be greater than 1; the assumption on time-varying

By using singleton fuzzifier, product inferred and weighted defuzzifier, the fuzzy system can

use abbreviation hi, hhi, xd(t), ud(t), xh(t), uh(t), to replace hi(ϑ(t)), hi(ϑ(t � h(t))), x(t � d(t)), u(t � d

hiðϑðtÞÞ½AixðtÞ þ Adixðt � dðtÞÞ þ BiuðtÞ þ Bhiuðt � hðtÞÞ þ NixðtÞuðtÞ

initial conditions φ(t), ϕ(t) are continuous functions of t, t∈½�τ, 0�, τ ¼ minðτ1, τ2Þ.

P<sup>s</sup> i¼1 P<sup>s</sup>

Plant Rule i :

i,j¼<sup>1</sup> is an abbreviation of

hðtÞ ≤ σ2, and σ1, σ<sup>2</sup> are known constants. The

<sup>j</sup>¼<sup>1</sup> <sup>μ</sup>ijðϑðtÞÞ, <sup>μ</sup>ij (ϑ(t)) is the grade of member-

<sup>i</sup>¼<sup>1</sup> hiðϑðtÞÞ ¼ 1 for all <sup>t</sup>. In the consequent, we

<sup>i</sup>¼<sup>1</sup> <sup>ω</sup>iðϑðtÞÞ <sup>&</sup>gt; 0 for all <sup>t</sup>. Then,

X<sup>s</sup>

(1)

(2)

$$A\_{\vec{\eta}} = A\_i + \rho \sin \theta\_{\vec{\jmath}} N\_i + \rho \cos \theta\_{\vec{\jmath}} B\_i \overline{K}\_{\vec{\jmath}\iota} \Lambda\_{\vec{\text{all}}} = A\_{\vec{\text{all}}} + \rho \sin \phi\_{\vec{\eta}} N\_{\vec{\text{all}}} \Lambda\_{\vec{\text{all}}} = \rho \cos \phi\_{\vec{\eta}} B\_{\vec{\text{li}}} \bar{K}\_{\vec{\text{li}}}.$$

Given positive-definite symmetric matrices S ∈ Rn�<sup>n</sup> and W ∈ R, we take the cost function

$$J = \int\_0^\infty [\mathbf{x}^\mathrm{T}(t)\mathbf{S}\mathbf{x}(t) + \mathbf{u}^\mathrm{T}(t)\mathbf{W}\mathbf{u}(t)]dt\tag{7}$$

Definition 1. The fuzzy non-fragile control law u(t) is said to be non-fragile guaranteed cost if for the system (2), there exist control laws (4) and (5) and a scalar J<sup>0</sup> such that the closedloop system (6) is asymptotically stable and the closed-loop value of the cost function (7) satisfies J ≤ J0.

#### 4. Analysis of stability for the closed-loop system

Firstly, the following lemmas are presented which will be used in the paper.

Lemma 1 [20]: Given any matrices M and N with appropriate dimensions such that ε > 0, we have MT N+N<sup>T</sup> M ≤ ε MT M + ε�<sup>1</sup> N<sup>T</sup> N.

Lemma 2 [21]: Given constant matrices G, E and a symmetric constant matrix S of appropriate dimensions. The inequality S + GFE + ET FT G<sup>T</sup> < 0 holds, where F(t) satisfies F<sup>T</sup> (t) F(t) ≤ I if and only if, for some ε > 0, S + εGG<sup>T</sup> + ε�<sup>1</sup> ET E < 0.

The following theorem gives the sufficient conditions for the existence of the non-fragile guaranteed cost controller for system (6) with additive controller gain perturbations.

Theorem 1. Consider system (6) associated with cost function (7). For given scalars ρ > 0, τ<sup>1</sup> > 0, τ<sup>2</sup> > 0, σ1 > 0, σ<sup>2</sup> > 0, if there exist matrices P > 0, Q<sup>1</sup> > 0, Q2 > 0, R<sup>1</sup> > 0, R<sup>2</sup> > 0, Ki, i = 1, 2,…, s, X1, X2, X3, X4, Y1, Y2, Y3, Y4, and scalar ε > 0 satisfying the inequalities (8), the system (6) is asymptotically stable and the control law (5) is a fuzzy non-fragile guaranteed cost control law, moreover,

$$\begin{aligned} \mathbf{J} & \approx \mathbf{x}^{\mathrm{T}}(0)\mathbf{P}\mathbf{x}(0) + \int\_{-d(0)}^{0} \mathbf{x}^{\mathrm{T}}(s)Q\_{1}\mathbf{x}(s)ds + \int\_{-\tau\_{1}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathrm{T}}(s)R\_{1}\dot{\mathbf{x}}(s)dsd\theta\\ & + \int\_{-h(0)}^{0} \mathbf{x}^{\mathrm{T}}(s)Q\_{2}\mathbf{x}(s)ds + \int\_{-\tau\_{2}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathrm{T}}(s)R\_{2}\dot{\mathbf{x}}(s)dsd\theta = \mathbf{J}\_{0} \end{aligned}$$
 
$$\begin{bmatrix} T\_{\ddot{\boldsymbol{\mu}}} & \ast & \ast\\ \tau\_{1}\mathbf{x}^{\mathrm{T}} & -\tau\_{1}R\_{1} & \ast\\ \tau\_{2}Z^{\mathrm{T}} & 0 & -\tau\_{2}R\_{2} \end{bmatrix} < 0, \quad i, j, l = 1, 2, \ldots, s \tag{8}$$

$$where\ T\_{ijl} = \begin{bmatrix} T\_{11,ij} & \* & \* & \* \\ T\_{21,i} & T\_{22,i} & \* & \* \\ T\_{31,i} & T\_{32,ij} & T\_{33,il} & \* \\ T\_{41,i} & T\_{42,i} & T\_{43} & T\_{44} \end{bmatrix}'$$

Non-Fragile Guaranteed Cost Control of Nonlinear Systems with Different State and Input Delays Based on T-S… http://dx.doi.org/10.5772/intechopen.69777 33

<sup>T</sup>11,ij <sup>¼</sup> <sup>Q</sup><sup>1</sup> <sup>þ</sup> <sup>Q</sup><sup>2</sup> <sup>þ</sup> <sup>X</sup><sup>1</sup> <sup>þ</sup> <sup>X</sup><sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>Y</sup>1Ai <sup>þ</sup> <sup>A</sup><sup>T</sup> <sup>i</sup> <sup>Y</sup><sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>S</sup> <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>Y1Y<sup>1</sup> <sup>T</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup>ðN<sup>T</sup> <sup>i</sup> Ni þ ðBiKj Þ T ðBiKj ÞÞ <sup>þ</sup> ZT <sup>1</sup> <sup>þ</sup> Z1 <sup>þ</sup> <sup>ρ</sup><sup>2</sup>K<sup>T</sup> <sup>i</sup> WKi , <sup>T</sup>21,i ¼ �X<sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>X</sup><sup>2</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>Y</sup>2Ai <sup>þ</sup> <sup>A</sup><sup>T</sup> diY<sup>T</sup> <sup>1</sup> , T31,i ¼ Z<sup>3</sup> � Z<sup>1</sup> þ X<sup>3</sup> þ Y3Ai , <sup>T</sup>22,ij ¼ �ð<sup>1</sup> � <sup>σ</sup>1ÞQ<sup>1</sup> � <sup>X</sup><sup>2</sup> � <sup>X</sup><sup>T</sup> <sup>2</sup> <sup>þ</sup> <sup>Y</sup>2Adi <sup>þ</sup> <sup>A</sup><sup>T</sup> diY<sup>T</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>Y2Y<sup>2</sup> <sup>T</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup>N<sup>T</sup> diNdi, <sup>T</sup>32,i ¼ �X<sup>3</sup> <sup>þ</sup> <sup>Y</sup>3Adi � <sup>Z</sup><sup>T</sup> <sup>2</sup> , T33,il ¼ �ð<sup>1</sup> � <sup>σ</sup>2ÞQ<sup>2</sup> � <sup>Z</sup><sup>3</sup> � <sup>Z</sup><sup>T</sup> <sup>3</sup> <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>Y3Y<sup>3</sup> <sup>T</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup>ðBhiK<sup>~</sup> <sup>l</sup> Þ T BhiK<sup>~</sup> <sup>l</sup> <sup>T</sup>41,i <sup>¼</sup> <sup>P</sup> <sup>þ</sup> <sup>X</sup><sup>4</sup> <sup>þ</sup> <sup>Z</sup><sup>4</sup> <sup>þ</sup> <sup>Y</sup>4Ai � <sup>Y</sup><sup>T</sup> <sup>1</sup> , T42,i ¼ �X<sup>4</sup> <sup>þ</sup> <sup>Y</sup>4Ai � <sup>Y</sup><sup>T</sup> 2 , <sup>T</sup><sup>43</sup> ¼ �Z<sup>4</sup> � <sup>Y</sup><sup>T</sup> <sup>3</sup> , T<sup>44</sup> ¼ τ1R<sup>1</sup> þ τ2R<sup>2</sup> � Y<sup>4</sup> � Y<sup>4</sup> <sup>T</sup> <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>Y4Y<sup>T</sup> 4 :

Proof: Take a Lyapunov function candidate as

Given positive-definite symmetric matrices S ∈ Rn�<sup>n</sup> and W ∈ R, we take the cost function

Definition 1. The fuzzy non-fragile control law u(t) is said to be non-fragile guaranteed cost if for the system (2), there exist control laws (4) and (5) and a scalar J<sup>0</sup> such that the closedloop system (6) is asymptotically stable and the closed-loop value of the cost function (7)

Lemma 1 [20]: Given any matrices M and N with appropriate dimensions such that ε > 0, we have

Lemma 2 [21]: Given constant matrices G, E and a symmetric constant matrix S of appropriate

The following theorem gives the sufficient conditions for the existence of the non-fragile

Theorem 1. Consider system (6) associated with cost function (7). For given scalars ρ > 0, τ<sup>1</sup> > 0, τ<sup>2</sup> > 0, σ1 > 0, σ<sup>2</sup> > 0, if there exist matrices P > 0, Q<sup>1</sup> > 0, Q2 > 0, R<sup>1</sup> > 0, R<sup>2</sup> > 0, Ki, i = 1, 2,…, s, X1, X2, X3, X4, Y1, Y2, Y3, Y4, and scalar ε > 0 satisfying the inequalities (8), the system (6) is asymptotically

<sup>x</sup><sup>T</sup>ðsÞQ1xðsÞds <sup>þ</sup>

ð0 θ x\_

ð0 �τ<sup>2</sup>

3 7 ð0 �τ<sup>1</sup> ð0 θ x\_

<sup>T</sup>ðsÞR2x\_ðsÞdsd<sup>θ</sup> <sup>¼</sup>J<sup>0</sup>

<sup>5</sup> <sup>&</sup>lt; <sup>0</sup>; i, j, l <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, s (8)

<sup>T</sup>ðsÞR1x\_ðsÞdsd<sup>θ</sup>

G<sup>T</sup> < 0 holds, where F(t) satisfies F<sup>T</sup>

(t) F(t) ≤ I if and only

<sup>½</sup>x<sup>T</sup>ðtÞSxðtÞ þ <sup>u</sup><sup>T</sup>ðtÞWuðtÞ�dt (7)

J ¼ ð∞ 0

4. Analysis of stability for the closed-loop system

Firstly, the following lemmas are presented which will be used in the paper.

FT

guaranteed cost controller for system (6) with additive controller gain perturbations.

stable and the control law (5) is a fuzzy non-fragile guaranteed cost control law, moreover,

�dð0Þ

<sup>x</sup><sup>T</sup>ðsÞQ2xðsÞds <sup>þ</sup>

ð0

ET E < 0.

<sup>J</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>ð0ÞPxð0Þ þ

�hð0Þ

Tijl � � <sup>τ</sup>1X<sup>T</sup> �τ1R<sup>1</sup> � <sup>τ</sup>2Z<sup>T</sup> <sup>0</sup> �τ2R<sup>2</sup>

þ ð0

T11,ij � �� T21,i T22,i � � T31,i T32,ij T33,il � T41,i T42,i T<sup>43</sup> T<sup>44</sup>

2 6 4

where Tijl ¼

satisfies J ≤ J0.

MT N+N<sup>T</sup> M ≤ ε MT M + ε�<sup>1</sup> N<sup>T</sup> N.

32 Modern Fuzzy Control Systems and Its Applications

dimensions. The inequality S + GFE + ET

if, for some ε > 0, S + εGG<sup>T</sup> + ε�<sup>1</sup>

$$\begin{split} V(\mathbf{x}(t),t) &= \mathbf{x}^{\mathrm{T}}(t)P\mathbf{x}(t) + \int\_{t-d(t)}^{t} \mathbf{x}^{\mathrm{T}}(s)Q\_{1}\mathbf{x}(s)ds + \int\_{-\tau\_{1}}^{0} \int\_{t+\theta}^{t} \dot{\mathbf{x}}^{\mathrm{T}}(s)R\_{1}\dot{\mathbf{x}}(s)dsd\theta \\ &+ \int\_{t-h(t)}^{t} \mathbf{x}^{\mathrm{T}}(s)Q\_{2}\mathbf{x}(s)ds + \int\_{-\tau\_{2}}^{0} \int\_{t+\theta}^{t} \dot{\mathbf{x}}^{\mathrm{T}}(s)R\_{2}\dot{\mathbf{x}}(s)dsd\theta \end{split} \tag{9}$$

The time derivatives of V(x(t),t), along the trajectory of the system (6), are given by

$$\begin{split} \dot{V}(\mathbf{x}(t),t) &= 2\mathbf{x}^{\mathrm{T}}(t)\mathbf{P}\dot{\mathbf{x}}(t) + \mathbf{x}^{\mathrm{T}}(t)(\mathbf{Q}\_{1} + \mathbf{Q}\_{2})\mathbf{x}(t) \\ &- (1 - \dot{d}(t))\mathbf{x}\_{d}^{\mathrm{T}}(t)\mathbf{Q}\_{1}\mathbf{x}\_{d}(t) + \dot{\mathbf{x}}^{\mathrm{T}}(t)(\tau\_{1}\mathbf{R}\_{1} + \tau\_{2}\mathbf{R}\_{2})\dot{\mathbf{x}}(t) \\ &- \int\_{t-\tau\_{1}}^{t} \dot{\mathbf{x}}^{\mathrm{T}}(s)\mathbf{R}\_{1}\dot{\mathbf{x}}(s)ds - (1 - \dot{h}(t))\mathbf{x}\_{h}^{\mathrm{T}}(t)\mathbf{Q}\_{2}\mathbf{x}\_{h}(t) - \int\_{t-\tau\_{2}}^{t} \dot{\mathbf{x}}^{\mathrm{T}}(s)\mathbf{R}\_{2}\dot{\mathbf{x}}(s)ds \end{split} \tag{10}$$

Define the free-weighting matrices as <sup>X</sup> ¼ ½ <sup>X</sup><sup>T</sup> <sup>1</sup> X<sup>T</sup> <sup>2</sup> X<sup>T</sup> <sup>3</sup> X<sup>T</sup> 4 � T , <sup>Y</sup> ¼ ½Y<sup>T</sup> <sup>1</sup> <sup>Y</sup><sup>T</sup> <sup>2</sup> <sup>Y</sup><sup>T</sup> <sup>3</sup> <sup>Y</sup><sup>T</sup> 4 � T, <sup>Z</sup> ¼ ½Z<sup>T</sup> <sup>1</sup> Z<sup>T</sup> <sup>2</sup> Z<sup>T</sup> <sup>3</sup> Z<sup>T</sup> 4 � T , where Xk ∈ Rn�<sup>n</sup> , Yk ∈ Rn�<sup>n</sup> , Zk ∈ Rn�<sup>n</sup> , k = 1, 2, 3, 4 will be determined later.

Using the Leibniz-Newton formula and system equation (6), we have the following identical equations:

$$\left[\mathbf{x}^{T}(t)\mathbf{X}\_{1} + \mathbf{x}\_{d}^{T}(t)\mathbf{X}\_{2} + \mathbf{x}\_{h}^{T}(t)\mathbf{X}\_{3} + \dot{\mathbf{x}}^{T}(t)\mathbf{X}\_{4}\right] \left[\mathbf{x}(t) - \mathbf{x}\_{d}(t) - \int\_{t-d(t)}^{t} \dot{\mathbf{x}}(s)ds\right] \equiv \mathbf{0},$$

$$\left[\mathbf{x}^{T}(t)\mathbf{Z}\_{1} + \mathbf{x}\_{d}^{T}(t)\mathbf{Z}\_{2} + \mathbf{x}\_{h}^{T}(t)\mathbf{Z}\_{3} + \dot{\mathbf{x}}^{T}(t)\mathbf{Z}\_{4}\right] \left[\mathbf{x}(t) - \mathbf{x}\_{h}(t) - \int\_{t-h(t)}^{t} \dot{\mathbf{x}}(s)ds\right] \equiv \mathbf{0},\tag{11}$$

$$\sum\_{i,j=1}^{s} h\_{i}h\_{j}h\_{i}[\mathbf{x}^{T}(t)\mathbf{Y}\_{1} + \mathbf{x}\_{d}^{T}(t)\mathbf{Y}\_{2} + \mathbf{x}\_{h}^{T}(t)\mathbf{Y}\_{4} + \dot{\mathbf{x}}^{T}(t)\mathbf{Y}\_{4}] \left[\Lambda\_{ij}\mathbf{x}(t) + \Lambda\_{\text{dif}}\mathbf{x}\_{d}(t) + \Lambda\_{\text{hif}}\mathbf{x}\_{h}(t) - \dot{\mathbf{x}}(t)\right] \equiv \mathbf{0}$$

Then, substituting Eq. (12) into Eq. (11) yields

<sup>V</sup>\_ <sup>ð</sup>xðtÞ, tÞ ¼ <sup>2</sup>x<sup>T</sup>ðtÞPx\_ðtÞ þ <sup>x</sup><sup>T</sup>ðtÞðQ<sup>1</sup> <sup>þ</sup> <sup>Q</sup>2ÞxðtÞ þ <sup>x</sup>\_ <sup>T</sup>ðtÞðτ1R<sup>1</sup> <sup>þ</sup> <sup>τ</sup>2R2Þx\_ðt<sup>Þ</sup> �ð<sup>1</sup> � \_ <sup>d</sup>ðtÞÞx<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞQ1xdðtÞ�ð<sup>1</sup> � \_ <sup>h</sup>ðtÞÞx<sup>T</sup> <sup>h</sup> ðtÞQ2xhðtÞ � ðt t�τ<sup>1</sup> x\_ <sup>T</sup>ðsÞR1x\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞX½xðtÞ � xdðtÞ � <sup>ð</sup><sup>t</sup> t�dðtÞ x\_ðsÞds� � ðt t�τ<sup>2</sup> x\_ <sup>T</sup>ðsÞR2x\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞZ½xðtÞ � xhðtÞ � <sup>ð</sup><sup>t</sup> t�hðtÞ x\_ðsÞds� <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞ<sup>Y</sup> <sup>X</sup><sup>s</sup> i, j, <sup>l</sup>¼<sup>1</sup> hihjhhl½ΛijxðtÞ þ ΛdilxdðtÞ þ ΛhilxhðtÞ � x\_ðtÞ� <sup>≤</sup> <sup>2</sup>x<sup>T</sup>ðtÞPx\_ðtÞ þ <sup>x</sup><sup>T</sup>ðtÞðQ<sup>1</sup> <sup>þ</sup> <sup>Q</sup>2ÞxðtÞ þ <sup>x</sup>\_ <sup>T</sup>ðtÞðτ1R<sup>1</sup> <sup>þ</sup> <sup>τ</sup>2R2Þx\_ðt<sup>Þ</sup> �ð<sup>1</sup> � <sup>σ</sup>1Þx<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞQ1xdðtÞ�ð<sup>1</sup> � <sup>σ</sup>2Þx<sup>T</sup> <sup>h</sup> ðtÞQ2xhðtÞ � ðt t�dðtÞ x\_ <sup>T</sup>ðsÞR1ðsÞx\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞX½xðtÞ � xdðtÞ � <sup>ð</sup><sup>t</sup> t�dðtÞ x\_ðsÞds� � ðt t�hðtÞ x\_ <sup>T</sup>ðsÞR2ðsÞx\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞZ½xðtÞ � xhðtÞ � <sup>ð</sup><sup>t</sup> t�hðtÞ x\_ðsÞds� <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞ<sup>Y</sup> <sup>X</sup><sup>s</sup> i, j, <sup>l</sup>¼<sup>1</sup> hihjhhl½ΛijxðtÞ þ <sup>Λ</sup>dilxdðtÞ þ <sup>Λ</sup>hilxhðtÞ � <sup>x</sup>\_ðtÞ� þ <sup>x</sup><sup>T</sup>ðtÞSxðt<sup>Þ</sup> þ Xs i, <sup>j</sup>¼<sup>1</sup> hihjρ<sup>2</sup>x<sup>T</sup>ðtÞK<sup>T</sup> <sup>i</sup> cos <sup>θ</sup>iWKj cos <sup>θ</sup>jxðtÞ�½x<sup>T</sup>ðtÞSxðtÞ þ <sup>u</sup><sup>T</sup>ðtÞWuðtÞ� (12)

where <sup>η</sup>ðtÞ¼½x<sup>T</sup>ðtÞ, <sup>x</sup><sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞ, <sup>x</sup><sup>T</sup> <sup>h</sup> ðtÞ, x\_ <sup>T</sup>ðtÞ�<sup>T</sup> .

Applying Lemma 1, we have the following inequalities:

<sup>2</sup>x<sup>T</sup>ðtÞY1Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>ðtÞY1AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup>ðtÞY1Y<sup>1</sup> <sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup> <sup>i</sup> Ni þ ðBi Kj Þ <sup>T</sup>ðBi Kj ÞÞxðtÞ, <sup>2</sup>x<sup>T</sup>ðtÞY1Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>ðtÞY1AdixdðtÞ þ ερ<sup>2</sup> sin <sup>2</sup>φ<sup>l</sup> <sup>x</sup><sup>T</sup>ðtÞY1Y<sup>1</sup> <sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup> diNdixdðtÞ, <sup>2</sup>x<sup>T</sup>ðtÞY1Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup> <sup>x</sup><sup>T</sup>ðtÞY1Y<sup>1</sup> <sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup> Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup> ÞxhðtÞ, 2x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞY2Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞY2AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup> <sup>d</sup> ðtÞY2Y<sup>2</sup> <sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup> <sup>i</sup> Ni þ ðBi Kj Þ T ðBi Kj ÞÞxðtÞ, <sup>2</sup>x<sup>T</sup>ðtÞY1Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup> <sup>x</sup><sup>T</sup>ðtÞY1Y<sup>1</sup> <sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup> Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup> ÞxhðtÞ, 2x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞY2Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞY2AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup> <sup>d</sup> ðtÞY2Y<sup>2</sup> <sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup> <sup>i</sup> Ni þ ðBi Kj Þ T ðBi Kj ÞÞxðtÞ, 2x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞY2Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞY2AdixdðtÞ þ ερ<sup>2</sup> sin <sup>2</sup>φ<sup>l</sup> xT <sup>d</sup> ðtÞY2Y<sup>2</sup> <sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup> diNdixdðtÞ, 2x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞY2Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup> xT <sup>d</sup> ðtÞY2Y<sup>2</sup> <sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup> Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup> ÞxhðtÞ, 2x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞY3Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞY3AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup> <sup>h</sup> ðtÞY3Y<sup>3</sup> <sup>T</sup>xhðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup> <sup>i</sup> Ni þ ðBi Kj Þ T ðBi Kj ÞÞxðtÞ, 2xT <sup>h</sup> <sup>ð</sup>tÞY3Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>xT <sup>d</sup> <sup>ð</sup>tÞY3AdixdðtÞ þ ερ<sup>2</sup> sin <sup>2</sup>φ<sup>l</sup> xT <sup>h</sup> ðtÞY3Y<sup>3</sup> TxhðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup> diNdixdðtÞ, 2x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞY3Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup> xT <sup>h</sup> ðtÞY3Y<sup>3</sup> <sup>T</sup>xhðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup> Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup> ÞxhðtÞ, 2x\_ <sup>T</sup>ðtÞY4Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x\_ <sup>T</sup>ðtÞY4AixðtÞ þ ερ<sup>2</sup>x\_ <sup>T</sup>ðtÞY4Y<sup>4</sup> <sup>T</sup>x\_ðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup> <sup>i</sup> Ni þ ðBi Kj Þ T ðBi Kj ÞÞxðtÞ, 2x\_ <sup>T</sup>ðtÞY4Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x\_ <sup>T</sup>ðtÞY4AdixdðtÞ þ ερ<sup>2</sup>x\_ <sup>T</sup>ðtÞY4Y<sup>4</sup> <sup>T</sup>x\_ðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup> diNdixdðtÞ, 2x\_ <sup>T</sup>ðtÞY4Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup> x\_ <sup>T</sup>ðtÞY4Y<sup>4</sup> <sup>T</sup>x\_ðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup> <sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup> Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup> ÞxhðtÞ (13)

Substituting Eq. (13) into Eq. (12) results in

<sup>V</sup>\_ <sup>ð</sup>xðtÞ, t<sup>Þ</sup> <sup>≤</sup> <sup>X</sup><sup>s</sup> i, j, <sup>l</sup>¼<sup>1</sup> hihjhhlη<sup>T</sup>ðtÞTijηðtÞ � <sup>ð</sup><sup>t</sup> t�dðtÞ x\_ <sup>T</sup>ðsÞR1x\_ðsÞds � ðt t�hðtÞ x\_ <sup>T</sup>ðsÞR2x\_ðsÞds �2η<sup>T</sup>ðtÞ<sup>X</sup> ðt t�dðtÞ <sup>x</sup>\_ðsÞds � <sup>2</sup>η<sup>T</sup>ðtÞ<sup>Z</sup> ðt t�hðtÞ <sup>x</sup>\_ðsÞds � ½x<sup>T</sup>ðtÞSxðtÞ þ <sup>u</sup><sup>T</sup>ðtÞWuðtÞ� <sup>≤</sup> <sup>X</sup><sup>s</sup> i, j, <sup>l</sup>¼<sup>1</sup> hihjhhlη<sup>T</sup>ðtÞðTijl <sup>þ</sup> <sup>τ</sup>1XR�<sup>1</sup> <sup>1</sup> <sup>X</sup><sup>T</sup> <sup>þ</sup> <sup>τ</sup>2ZR�<sup>1</sup> <sup>2</sup> <sup>Z</sup><sup>T</sup>Þηðt<sup>Þ</sup> � ðt t�dðtÞ � <sup>η</sup><sup>T</sup>ðtÞ<sup>X</sup> <sup>þ</sup> <sup>x</sup>\_ <sup>T</sup>ðsÞR<sup>1</sup> � R�<sup>1</sup> 1 � <sup>η</sup><sup>T</sup>ðtÞ<sup>X</sup> <sup>þ</sup> <sup>x</sup>\_ <sup>T</sup>ðsÞR<sup>1</sup> �T ds � ðt t�hðtÞ � <sup>η</sup><sup>T</sup>ðtÞ<sup>Z</sup> <sup>þ</sup> <sup>x</sup>\_ <sup>T</sup>ðsÞR<sup>2</sup> � R�<sup>1</sup> 2 � <sup>η</sup><sup>T</sup>ðtÞ<sup>X</sup> <sup>þ</sup> <sup>x</sup>\_ <sup>T</sup>ðsÞR<sup>2</sup> �T ds � ½x<sup>T</sup>ðtÞSxðtÞ þ <sup>u</sup><sup>T</sup>ðtÞWuðtÞ� <sup>≤</sup> <sup>X</sup><sup>s</sup> i, j, <sup>l</sup>¼<sup>1</sup> hihjhhlη<sup>T</sup>ðtÞðT<sup>~</sup> ijl <sup>þ</sup> <sup>τ</sup>1XR�<sup>1</sup> <sup>1</sup> <sup>X</sup><sup>T</sup> <sup>þ</sup> <sup>τ</sup>2ZR�<sup>1</sup> <sup>2</sup> <sup>Z</sup><sup>T</sup>ÞηðtÞ�½x<sup>T</sup>ðtÞSxðtÞ þ <sup>u</sup><sup>T</sup>ðtÞWuðtÞ� (14)

where

<sup>V</sup>\_ <sup>ð</sup>xðtÞ, tÞ ¼ <sup>2</sup>x<sup>T</sup>ðtÞPx\_ðtÞ þ <sup>x</sup><sup>T</sup>ðtÞðQ<sup>1</sup> <sup>þ</sup> <sup>Q</sup>2ÞxðtÞ þ <sup>x</sup>\_

i, j, <sup>l</sup>¼<sup>1</sup>

<sup>≤</sup> <sup>2</sup>x<sup>T</sup>ðtÞPx\_ðtÞ þ <sup>x</sup><sup>T</sup>ðtÞðQ<sup>1</sup> <sup>þ</sup> <sup>Q</sup>2ÞxðtÞ þ <sup>x</sup>\_

<sup>d</sup> <sup>ð</sup>tÞQ1xdðtÞ�ð<sup>1</sup> � <sup>σ</sup>2Þx<sup>T</sup>

<sup>d</sup> <sup>ð</sup>tÞQ1xdðtÞ�ð<sup>1</sup> � \_

<sup>T</sup>ðsÞR1x\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞX½xðtÞ � xdðtÞ �

<sup>T</sup>ðsÞR2x\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞZ½xðtÞ � xhðtÞ �

<sup>T</sup>ðsÞR1ðsÞx\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞX½xðtÞ � xdðtÞ �

<sup>T</sup>ðsÞR2ðsÞx\_ðsÞds <sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞZ½xðtÞ � xhðtÞ �

<sup>h</sup>ðtÞÞx<sup>T</sup>

hihjhhl½ΛijxðtÞ þ ΛdilxdðtÞ þ ΛhilxhðtÞ � x\_ðtÞ�

<sup>h</sup> ðtÞQ2xhðtÞ

hihjhhl½ΛijxðtÞ þ <sup>Λ</sup>dilxdðtÞ þ <sup>Λ</sup>hilxhðtÞ � <sup>x</sup>\_ðtÞ� þ <sup>x</sup><sup>T</sup>ðtÞSxðt<sup>Þ</sup>

<sup>i</sup> cos <sup>θ</sup>iWKj cos <sup>θ</sup>jxðtÞ�½x<sup>T</sup>ðtÞSxðtÞ þ <sup>u</sup><sup>T</sup>ðtÞWuðtÞ�

<sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup>

<sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup>

<sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup>

<sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup>

<sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup>

<sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup>

<sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup>

<sup>T</sup>x\_ðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup>

<sup>T</sup>x\_ðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

<sup>h</sup> <sup>ð</sup>tÞðBhiK<sup>~</sup> <sup>l</sup>

<sup>T</sup>xhðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>ðtÞðN<sup>T</sup>

<sup>x</sup><sup>T</sup>ðtÞY1Y<sup>1</sup>

<sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

<sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

<sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

<sup>T</sup>xhðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

<sup>T</sup>ðtÞY4Y<sup>4</sup>

<sup>T</sup>x\_ðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

xT <sup>d</sup> ðtÞY2Y<sup>2</sup>

xT <sup>h</sup> ðtÞY3Y<sup>3</sup>

<sup>d</sup> ðtÞY2Y<sup>2</sup>

<sup>d</sup> ðtÞY2Y<sup>2</sup>

<sup>h</sup> ðtÞY3Y<sup>3</sup>

<sup>T</sup>ðtÞY4Y<sup>4</sup>

<sup>d</sup>ðtÞÞx<sup>T</sup>

�ð<sup>1</sup> � \_

t�τ<sup>1</sup> x\_

t�τ<sup>2</sup> x\_

�ð<sup>1</sup> � <sup>σ</sup>1Þx<sup>T</sup>

t�dðtÞ x\_

t�hðtÞ x\_

<sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞ<sup>Y</sup> <sup>X</sup><sup>s</sup>

<sup>d</sup> <sup>ð</sup>tÞ, <sup>x</sup><sup>T</sup>

<sup>2</sup>x<sup>T</sup>ðtÞY1Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>ðtÞY1AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup>ðtÞY1Y<sup>1</sup>

<sup>2</sup>x<sup>T</sup>ðtÞY1Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>ðtÞY1AdixdðtÞ þ ερ<sup>2</sup> sin <sup>2</sup>φ<sup>l</sup>

i, j, <sup>l</sup>¼<sup>1</sup>

hihjρ<sup>2</sup>x<sup>T</sup>ðtÞK<sup>T</sup>

<sup>h</sup> ðtÞ, x\_

Applying Lemma 1, we have the following inequalities:

<sup>T</sup>ðtÞ�<sup>T</sup> .

<sup>x</sup><sup>T</sup>ðtÞY1Y<sup>1</sup>

<sup>x</sup><sup>T</sup>ðtÞY1Y<sup>1</sup>

<sup>d</sup> <sup>ð</sup>tÞY2AdixdðtÞ þ ερ<sup>2</sup> sin <sup>2</sup>φ<sup>l</sup>

<sup>d</sup> <sup>ð</sup>tÞY3AdixdðtÞ þ ερ<sup>2</sup> sin <sup>2</sup>φ<sup>l</sup>

<sup>d</sup> <sup>ð</sup>tÞY2AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup>

<sup>d</sup> <sup>ð</sup>tÞY2AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup>

xT <sup>d</sup> ðtÞY2Y<sup>2</sup>

<sup>h</sup> <sup>ð</sup>tÞY3AixðtÞ þ ερ<sup>2</sup>x<sup>T</sup>

xT <sup>h</sup> ðtÞY3Y<sup>3</sup>

<sup>T</sup>ðtÞY4AdixdðtÞ þ ερ<sup>2</sup>x\_

<sup>T</sup>ðtÞY4AixðtÞ þ ερ<sup>2</sup>x\_

x\_ <sup>T</sup>ðtÞY4Y<sup>4</sup>

<sup>þ</sup> <sup>2</sup>η<sup>T</sup>ðtÞ<sup>Y</sup> <sup>X</sup><sup>s</sup>

� ðt

34 Modern Fuzzy Control Systems and Its Applications

� ðt

� ðt

� ðt

> þ Xs i, <sup>j</sup>¼<sup>1</sup>

<sup>2</sup>x<sup>T</sup>ðtÞY1Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup>

<sup>2</sup>x<sup>T</sup>ðtÞY1Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup>

<sup>d</sup> <sup>ð</sup>tÞY2Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup>

<sup>h</sup> <sup>ð</sup>tÞY3Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup>

<sup>T</sup>ðtÞY4Λhilxhðt<sup>Þ</sup> <sup>≤</sup> ερ<sup>2</sup> cos <sup>2</sup>φ<sup>l</sup>

<sup>d</sup> <sup>ð</sup>tÞY2Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>

<sup>d</sup> <sup>ð</sup>tÞY2Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>

<sup>d</sup> <sup>ð</sup>tÞY2Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>

<sup>h</sup> <sup>ð</sup>tÞY3Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x<sup>T</sup>

<sup>T</sup>ðtÞY4Λijxðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x\_

<sup>T</sup>ðtÞY4Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>x\_

<sup>h</sup> <sup>ð</sup>tÞY3Λdilxdðt<sup>Þ</sup> <sup>≤</sup> <sup>2</sup>xT

where <sup>η</sup>ðtÞ¼½x<sup>T</sup>ðtÞ, <sup>x</sup><sup>T</sup>

2x<sup>T</sup>

2x<sup>T</sup>

2x<sup>T</sup>

2x<sup>T</sup>

2x<sup>T</sup>

2xT

2x<sup>T</sup>

2x\_

2x\_

2x\_

<sup>T</sup>ðtÞðτ1R<sup>1</sup> <sup>þ</sup> <sup>τ</sup>2R2Þx\_ðt<sup>Þ</sup>

t�dðtÞ

t�hðtÞ

<sup>T</sup>ðtÞðτ1R<sup>1</sup> <sup>þ</sup> <sup>τ</sup>2R2Þx\_ðt<sup>Þ</sup>

ðt

ðt

t�dðtÞ

t�hðtÞ

<sup>i</sup> Ni þ ðBi

<sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup>

ÞxhðtÞ,

<sup>i</sup> Ni þ ðBi

ÞxhðtÞ,

<sup>i</sup> Ni þ ðBi

<sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup>

<sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup>

diNdixdðtÞ,

ÞxhðtÞ,

ÞxhðtÞ,

<sup>i</sup> Ni þ ðBi

ÞxhðtÞ

<sup>i</sup> Ni þ ðBi

<sup>T</sup>xðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

<sup>T</sup>xdðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

TxhðtÞ þ <sup>ε</sup>�<sup>1</sup>x<sup>T</sup>

<sup>d</sup> <sup>ð</sup>tÞN<sup>T</sup>

Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup>

Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup>

> Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup>

> Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup>

Þ T <sup>ð</sup>BhiK<sup>~</sup> <sup>l</sup> Kj Þ <sup>T</sup>ðBi Kj ÞÞxðtÞ,

Kj Þ T ðBi Kj ÞÞxðtÞ,

Kj Þ T ðBi Kj ÞÞxðtÞ,

Kj Þ T ðBi Kj ÞÞxðtÞ,

Kj Þ <sup>T</sup>ðBi Kj ÞÞxðtÞ,

diNdixdðtÞ,

(13)

diNdixdðtÞ,

diNdixdðtÞ,

x\_ðsÞds�

(12)

x\_ðsÞds�

x\_ðsÞds�

x\_ðsÞds�

<sup>h</sup> ðtÞQ2xhðtÞ

ðt

ðt

$$
\tilde{T}\_{\vec{\eta}\vec{l}} = \begin{bmatrix}
\tilde{T}\_{11,\vec{\eta}} & \* & \* & \* \\
T\_{21,i} & T\_{22,i} & \* & \* \\
T\_{31,i} & T\_{32,\vec{\eta}} & T\_{33,\vec{\eta}} & \* \\
T\_{41,i} & T\_{42,i} & T\_{43} & T\_{44}
\end{bmatrix}, \quad \tilde{T}\_{11,\vec{\eta}} = T\_{11,\vec{\eta}} + \rho^2 \overline{\mathbf{K}}\_i^{\mathrm{T}} \cos\theta\_i \mathcal{W} \overline{\mathbf{K}}\_{\vec{\eta}} \cos\theta\_{\vec{\eta}} - \rho^2 \overline{\mathbf{K}}\_i^{\mathrm{T}} \mathcal{W} \overline{\mathbf{K}}\_{\vec{\eta}}.
$$

In light of the inequality K<sup>T</sup> <sup>i</sup> WKj <sup>þ</sup> <sup>K</sup><sup>T</sup> <sup>j</sup> WKi <sup>≤</sup> <sup>K</sup><sup>T</sup> <sup>i</sup> WKi <sup>þ</sup> <sup>K</sup><sup>T</sup> <sup>j</sup> WKj , we have

$$\dot{V}(\mathbf{x}(t),t) \leq \sum\_{i,j,l=1}^{s} h\_i h\_j h\_{li} \eta^{\mathrm{T}}(t) (T\_{ijl} + \tau\_1 X \mathbf{R}\_1^{-1} X^{\mathrm{T}} + \tau\_2 Z \mathbf{R}\_2^{-1} Z^{\mathrm{T}}) \eta(t) - [\mathbf{x}^{\mathrm{T}}(t) \mathbf{S} \mathbf{x}(t) + u^{\mathrm{T}}(t) \mathbf{W}u(t)] \tag{15}$$

Applying the Schur complement to Eq. (8) yields

$$T\_{\vec{\mu}} + \left. + \tau\_1 X \mathbf{R}\_1^{-1} X^{\mathsf{T}} + \tau\_2 Z \mathbf{R}\_2^{-1} Z^{\mathsf{T}} < 0,\\ T\_{\vec{\mu}} + T\_{\vec{\mu}} + 2 \tau\_1 X \mathbf{R}\_1^{-1} X^{\mathsf{T}} + 2 \tau\_2 Z \mathbf{R}\_2^{-1} Z^{\mathsf{T}} < 0.$$

Therefore, it follows from Eq. (15) that

$$\dot{V}(\mathbf{x}(t),t) \le -\left[\mathbf{x}^T(t)\mathbf{S}\mathbf{x}(t) + \boldsymbol{\mu}^T(t)\mathbf{W}\mathbf{u}(t)\right] < 0\tag{16}$$

which implies that the system (6) is asymptotically stable.

Integrating Eq. (16) from 0 to T produces

$$\int\_0^T [\mathbf{x}^\mathbf{T}(t)\mathbf{S}\mathbf{x}(t) + \boldsymbol{\mu}^\mathbf{T}(t)\mathbf{W}\mathbf{u}(t)]dt \le -V(\mathbf{x}(T), T) + V(\mathbf{x}(0), 0) < V(\mathbf{x}(0), 0)$$

Because of <sup>V</sup> (x(t),t) <sup>≥</sup> 0 and <sup>V</sup>\_ <sup>ð</sup>xðtÞ, t<sup>Þ</sup> <sup>&</sup>lt; 0, thus lim T!∞ VðxðTÞ, TÞ ¼ c, where c is a nonnegative constant. Therefore, the following inequality can be obtained:

$$\begin{aligned} \mathbf{J} & \leq \mathbf{x}^{\mathrm{T}}(\mathbf{0}) \mathbf{P} \mathbf{x}(\mathbf{0}) + \int\_{-d(0)}^{0} \mathbf{x}^{\mathrm{T}}(\mathbf{s}) Q\_{1} \mathbf{x}(\mathbf{s}) ds + \int\_{-\tau\_{1}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathrm{T}}(\mathbf{s}) R\_{1} \dot{\mathbf{x}}(\mathbf{s}) ds d\theta + \int\_{-h(0)}^{0} \mathbf{x}^{\mathrm{T}}(\mathbf{s}) Q\_{2} \mathbf{x}(\mathbf{s}) ds \\ & + \int\_{-\tau\_{2}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathrm{T}}(\mathbf{s}) R\_{2} \dot{\mathbf{x}}(\mathbf{s}) ds d\theta = \mathbf{J}\_{0} \end{aligned} \tag{17}$$

This completes the proof.

Remark 2: In the derivation of Theorem 1, the free-weighting matrices Xk ∈ Rn�<sup>n</sup> , Yk ∈ Rn�<sup>n</sup> , k = 1, 2, 3, 4 are introduced, the purpose of which is to reduce conservatism in the existing delaydependent stabilization conditions, see Ref. [17].

In the following section, we shall turn the conditions given in Theorem 1 into linear matrix inequalities (LMIs). Under the assumptions that Y1, Y2, Y3, Y<sup>4</sup> are non-singular, we can define the matrix Yi �<sup>T</sup> <sup>¼</sup> <sup>λ</sup>Z, <sup>i</sup> = 1, 2, 3, 4, <sup>Z</sup> <sup>=</sup> <sup>P</sup>�<sup>1</sup> ,λ > 0.

Pre- and post-multiply (8) and (9) with <sup>Θ</sup> <sup>¼</sup> diag{Y�<sup>1</sup> <sup>1</sup> , Y�<sup>1</sup> <sup>2</sup> , Y�<sup>1</sup> <sup>3</sup> , Y�<sup>1</sup> <sup>4</sup> , Y�<sup>1</sup> <sup>4</sup> , Y�<sup>1</sup> <sup>4</sup> } and <sup>Θ</sup><sup>T</sup> <sup>¼</sup> diag{Y�<sup>T</sup> <sup>1</sup> , Y�<sup>T</sup> <sup>2</sup> , Y�<sup>T</sup> <sup>3</sup> , Y�<sup>T</sup> <sup>4</sup> , Y�<sup>T</sup> <sup>4</sup> , Y�<sup>T</sup> <sup>4</sup> }, respectively, and letting Q<sup>1</sup> ¼ Y<sup>1</sup> �<sup>1</sup>Q1Y<sup>1</sup> �<sup>T</sup>, Q<sup>2</sup> ¼ Y<sup>1</sup> �<sup>1</sup>Q2Y<sup>1</sup> �<sup>T</sup>, Rk <sup>¼</sup> <sup>Y</sup><sup>4</sup> �<sup>1</sup>RkY<sup>4</sup> �<sup>T</sup>, k <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, Xi <sup>¼</sup> Yi �<sup>1</sup>XiYi �<sup>T</sup>, Zi <sup>¼</sup> Yi �<sup>1</sup>ZiYi �T, i = 1, 2, 3, 4, we obtain the following inequality (18), which is equivalent to (8):

$$
\begin{bmatrix}
\overline{\mathbf{T}}\_{11,ij} & \* & \* & \* & \* & \* \\
\overline{\mathbf{T}}\_{21,i} & \overline{\mathbf{T}}\_{22,i} & \* & \* & \* & \* \\
\overline{\mathbf{T}}\_{31,i} & \overline{\mathbf{T}}\_{32,i} & \overline{\mathbf{T}}\_{33,il} & \* & \* & \* \\
\overline{\mathbf{T}}\_{41,i} & \overline{\mathbf{T}}\_{42,i} & \overline{\mathbf{T}}\_{43} & \overline{\mathbf{T}}\_{44} & \* & \* \\
\tau\_{1}X\_{1} & \tau\_{1}X\_{2} & \tau\_{1}X\_{3} & \tau\_{1}X\_{4} & -\tau\_{1}\overline{R}\_{1} & \* \\
\tau\_{2}\overline{Z}\_{1} & \tau\_{2}\overline{Z}\_{2} & \tau\_{2}\overline{Z}\_{3} & \tau\_{2}\overline{Z}\_{4} & 0 & -\tau\_{2}\overline{R}\_{2}
\end{bmatrix} < 0, \quad i, j, l = 1, 2, ..., s \tag{18}
$$

where

<sup>Τ</sup>11,ij <sup>¼</sup> <sup>Q</sup><sup>1</sup> <sup>þ</sup> <sup>Q</sup><sup>2</sup> <sup>þ</sup> <sup>X</sup><sup>1</sup> <sup>þ</sup> <sup>X</sup><sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>λ</sup>AiZ <sup>þ</sup> <sup>λ</sup>ZA<sup>T</sup> <sup>i</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> ZSZ <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup> <sup>I</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup> λ2 ZN<sup>T</sup> <sup>i</sup> Ni Z <sup>þ</sup> <sup>Z</sup><sup>1</sup> <sup>þ</sup> <sup>Z</sup><sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup> λ2 ðBiKj ZÞ T ðBiKj <sup>Z</sup>Þ þ <sup>ρ</sup><sup>2</sup> λ2 ZK<sup>T</sup> <sup>i</sup> WKi Z, <sup>Τ</sup>21,i ¼ �X<sup>T</sup> <sup>1</sup> <sup>þ</sup> <sup>X</sup><sup>2</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup>AiZ <sup>þ</sup> <sup>λ</sup>ZA<sup>T</sup> di, Τ31,i ¼ Z<sup>3</sup> � Z<sup>1</sup> þ X<sup>3</sup> þ λAiZ, <sup>Τ</sup>41,i <sup>¼</sup> <sup>λ</sup><sup>2</sup> Z þ λAiZ � λZ þ X<sup>4</sup> þ Z4, <sup>Τ</sup>22,i ¼ �ð<sup>1</sup> � <sup>σ</sup>1ÞQ<sup>1</sup> � <sup>X</sup><sup>2</sup> � <sup>X</sup><sup>T</sup> <sup>2</sup> <sup>þ</sup> <sup>λ</sup>AdiZ <sup>þ</sup> <sup>λ</sup>ZA<sup>T</sup> di <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup> <sup>I</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup> λ2 ZN<sup>T</sup> diNdiZ, <sup>Τ</sup>32,i ¼ �X<sup>3</sup> � <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup>AdiZ � <sup>λ</sup>ZA<sup>T</sup> di, Τ42,i¼ � X4þλAiZ � λZ, <sup>Τ</sup>33,il ¼ �ð<sup>1</sup> � <sup>σ</sup>2ÞQ<sup>2</sup> � <sup>Z</sup><sup>3</sup> � <sup>Z</sup><sup>T</sup> <sup>3</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup> λ2 <sup>ð</sup>BhiK<sup>~</sup> lZ<sup>Þ</sup> T BhiK<sup>~</sup> lZ <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup> I, <sup>Τ</sup><sup>43</sup> ¼ �Z<sup>4</sup> � <sup>λ</sup>Z, <sup>Τ</sup><sup>44</sup> <sup>¼</sup> <sup>τ</sup>1R<sup>1</sup> <sup>þ</sup> <sup>τ</sup>2R<sup>2</sup> � <sup>λ</sup><sup>Z</sup> � <sup>λ</sup>Z<sup>T</sup> <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup> I:

Applying the Schur complement to Eq. (18) results in

Non-Fragile Guaranteed Cost Control of Nonlinear Systems with Different State and Input Delays Based on T-S… http://dx.doi.org/10.5772/intechopen.69777 37

$$
\Gamma\_{\vec{\eta}\vec{l}} = \begin{bmatrix}
\Phi\_{11,\vec{i}} & \* & \* \\
\Phi\_{21,\vec{\eta}} & \Phi\_{22} & \* \\
\Phi\_{31,\vec{l}} & 0 & \Phi\_{33}
\end{bmatrix} < 0, \quad \mathbf{i}, \mathbf{j}, \mathbf{l} = 1, 2, ..., s \tag{19}
$$

where

Because of <sup>V</sup> (x(t),t) <sup>≥</sup> 0 and <sup>V</sup>\_ <sup>ð</sup>xðtÞ, t<sup>Þ</sup> <sup>&</sup>lt; 0, thus lim

ð0

36 Modern Fuzzy Control Systems and Its Applications

�dð0Þ

dependent stabilization conditions, see Ref. [17].

�<sup>T</sup> <sup>¼</sup> <sup>λ</sup>Z, <sup>i</sup> = 1, 2, 3, 4, <sup>Z</sup> <sup>=</sup> <sup>P</sup>�<sup>1</sup>

<sup>3</sup> , Y�<sup>T</sup>

Pre- and post-multiply (8) and (9) with <sup>Θ</sup> <sup>¼</sup> diag{Y�<sup>1</sup>

<sup>4</sup> , Y�<sup>T</sup>

<sup>4</sup> , Y�<sup>T</sup>

�<sup>1</sup>RkY<sup>4</sup>

we obtain the following inequality (18), which is equivalent to (8):

T11,ij ��� � � T21,i T22,i �� � � T31,i T32,i T33,il �� � T41,i T42,i T43 T44 � � τ1X<sup>1</sup> τ1X<sup>2</sup> τ1X<sup>3</sup> τ1X<sup>4</sup> �τ1R<sup>1</sup> � τ2Z<sup>1</sup> τ2Z<sup>2</sup> τ2Z<sup>3</sup> τ2Z<sup>4</sup> 0 �τ2R<sup>2</sup>

<sup>T</sup>ðsÞR2x\_ðsÞdsd<sup>θ</sup> <sup>¼</sup>J<sup>0</sup>

<sup>J</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>ð0ÞPxð0Þ þ

This completes the proof.

ð0 θ x\_

þ ð0 �τ<sup>2</sup>

the matrix Yi

<sup>Θ</sup><sup>T</sup> <sup>¼</sup> diag{Y�<sup>T</sup>

Q<sup>2</sup> ¼ Y<sup>1</sup>

where

<sup>1</sup> , Y�<sup>T</sup>

�<sup>1</sup>Q2Y<sup>1</sup>

<sup>Τ</sup>21,i ¼ �X<sup>T</sup>

<sup>Τ</sup>41,i <sup>¼</sup> <sup>λ</sup><sup>2</sup>

<sup>2</sup> , Y�<sup>T</sup>

�<sup>T</sup>, Rk <sup>¼</sup> <sup>Y</sup><sup>4</sup>

<sup>Τ</sup>11,ij <sup>¼</sup> <sup>Q</sup><sup>1</sup> <sup>þ</sup> <sup>Q</sup><sup>2</sup> <sup>þ</sup> <sup>X</sup><sup>1</sup> <sup>þ</sup> <sup>X</sup><sup>T</sup>

<sup>þ</sup> <sup>Z</sup><sup>1</sup> <sup>þ</sup> <sup>Z</sup><sup>T</sup>

<sup>Τ</sup>22,i ¼ �ð<sup>1</sup> � <sup>σ</sup>1ÞQ<sup>1</sup> � <sup>X</sup><sup>2</sup> � <sup>X</sup><sup>T</sup>

<sup>Τ</sup>33,il ¼ �ð<sup>1</sup> � <sup>σ</sup>2ÞQ<sup>2</sup> � <sup>Z</sup><sup>3</sup> � <sup>Z</sup><sup>T</sup>

<sup>Τ</sup>32,i ¼ �X<sup>3</sup> � <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup>AdiZ � <sup>λ</sup>ZA<sup>T</sup>

Applying the Schur complement to Eq. (18) results in

<sup>1</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup>

λ2 ðBiKj ZÞ T ðBiKj

<sup>Τ</sup><sup>43</sup> ¼ �Z<sup>4</sup> � <sup>λ</sup>Z, <sup>Τ</sup><sup>44</sup> <sup>¼</sup> <sup>τ</sup>1R<sup>1</sup> <sup>þ</sup> <sup>τ</sup>2R<sup>2</sup> � <sup>λ</sup><sup>Z</sup> � <sup>λ</sup>Z<sup>T</sup> <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>X</sup><sup>2</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup>AiZ <sup>þ</sup> <sup>λ</sup>ZA<sup>T</sup>

Z þ λAiZ � λZ þ X<sup>4</sup> þ Z4,

constant. Therefore, the following inequality can be obtained:

<sup>x</sup><sup>T</sup>ðsÞQ1xðsÞds <sup>þ</sup>

T!∞

<sup>T</sup>ðsÞR1x\_ðsÞdsd<sup>θ</sup> <sup>þ</sup>

ð0 �τ<sup>1</sup>

Remark 2: In the derivation of Theorem 1, the free-weighting matrices Xk ∈ Rn�<sup>n</sup>

ð0 θ x\_

1, 2, 3, 4 are introduced, the purpose of which is to reduce conservatism in the existing delay-

In the following section, we shall turn the conditions given in Theorem 1 into linear matrix inequalities (LMIs). Under the assumptions that Y1, Y2, Y3, Y<sup>4</sup> are non-singular, we can define

,λ > 0.

�<sup>T</sup>, k <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, Xi <sup>¼</sup> Yi

<sup>1</sup> <sup>þ</sup> <sup>λ</sup>AiZ <sup>þ</sup> <sup>λ</sup>ZA<sup>T</sup>

<sup>3</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>λ</sup>AdiZ <sup>þ</sup> <sup>λ</sup>ZA<sup>T</sup>

λ2

<sup>i</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

<sup>Z</sup>Þ þ <sup>ρ</sup><sup>2</sup>

λ2 ZK<sup>T</sup> <sup>i</sup> WKi Z,

di <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>

di, Τ42,i¼ � X4þλAiZ � λZ,

<sup>ð</sup>BhiK<sup>~</sup> lZ<sup>Þ</sup> T

ZSZ <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>

di, Τ31,i ¼ Z<sup>3</sup> � Z<sup>1</sup> þ X<sup>3</sup> þ λAiZ,

<sup>I</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup>

BhiK<sup>~</sup> lZ <sup>þ</sup> <sup>2</sup>ερ<sup>2</sup>

I:

VðxðTÞ, TÞ ¼ c, where c is a nonnegative

<sup>x</sup><sup>T</sup>ðsÞQ2xðsÞds

, Yk ∈ Rn�<sup>n</sup>

(17)

, k =

<sup>4</sup> } and

�<sup>T</sup>,

�<sup>1</sup>Q1Y<sup>1</sup>

�T, i = 1, 2, 3, 4,

ð0

<sup>1</sup> , Y�<sup>1</sup> <sup>2</sup> , Y�<sup>1</sup> <sup>3</sup> , Y�<sup>1</sup> <sup>4</sup> , Y�<sup>1</sup> <sup>4</sup> , Y�<sup>1</sup>

�<sup>T</sup>, Zi <sup>¼</sup> Yi

�<sup>1</sup>ZiYi

< 0, i, j, l ¼ 1, 2, …, s (18)

λ2 ZN<sup>T</sup> <sup>i</sup> Ni Z

<sup>I</sup> <sup>þ</sup> <sup>4</sup>ε�<sup>1</sup>

λ2 ZN<sup>T</sup> diNdiZ,

I,

<sup>4</sup> }, respectively, and letting Q<sup>1</sup> ¼ Y<sup>1</sup>

�<sup>1</sup>XiYi

�hð0Þ

$$
\boldsymbol{\Phi}\_{11,i} = \begin{bmatrix}
\overline{\mathbf{T}}\_{11,i} & \* & \* & \* & \* & \* \\
\overline{\mathbf{T}}\_{21,i} & \overline{\mathbf{T}}\_{22,i} & \* & \* & \* & \* \\
\overline{\mathbf{T}}\_{31,i} & \overline{\mathbf{T}}\_{32,i} & \overline{\mathbf{T}}\_{33,il} & \* & \* & \* \\
\overline{\mathbf{T}}\_{41,i} & \overline{\mathbf{T}}\_{42,i} & \overline{\mathbf{T}}\_{43} & \overline{\mathbf{T}}\_{44} & \* & \* \\
\tau\_{1}\mathbf{X}\_{1} & \tau\_{1}\mathbf{X}\_{2} & \tau\_{1}\mathbf{X}\_{3} & \tau\_{1}\mathbf{X}\_{4} & -\tau\_{1}\overline{\mathbf{R}}\_{1} & \* \\
\tau\_{2}\overline{\mathbf{Z}}\_{1} & \tau\_{2}\overline{\mathbf{Z}}\_{2} & \tau\_{2}\overline{\mathbf{Z}}\_{3} & \tau\_{2}\overline{\mathbf{Z}}\_{4} & 0 & -\tau\_{2}\overline{\mathbf{R}}\_{2}
\end{bmatrix}.
$$

With

$$\begin{array}{l} \overline{\overline{T}}\_{11,i} = \overline{Q}\_{1} + \overline{Q}\_{2} + \overline{X}\_{1} + \overline{X}\_{1}^{\mathrm{T}} + \lambda A\_{i} \overline{Z} + \lambda Z A\_{i}^{\mathrm{T}} + Z\_{1} + Z\_{1}^{\mathrm{T}} + 2 \varepsilon \rho^{2} I, \\\overline{\overline{T}}\_{22,i} = -(1 - \sigma\_{1}) \overline{Q}\_{1} - \overline{X}\_{2} - \overline{X}\_{2}^{\mathrm{T}} + \lambda A\_{di} \overline{Z} + \lambda Z A\_{di}^{\mathrm{T}} + 2 \varepsilon \rho^{2} I, \\\overline{\overline{T}}\_{33} = -(1 - \sigma\_{2}) \overline{Q}\_{2} - \overline{Z}\_{3} - \overline{Z}\_{3}^{\mathrm{T}} + 2 \varepsilon \rho^{2} I. \end{array}$$

$$\begin{aligned} \Phi\_{21,\vec{\eta}} &= \begin{bmatrix} \lambda Z & 0 & 0 & 0 \\ \lambda N\_i Z & 0 & 0 & 0 \\ \lambda B\_i \overline{K}\_{\vec{\eta}} Z & 0 & 0 & 0 \\ \rho \lambda \overline{K}\_{\vec{\eta}} Z & 0 & 0 & 0 \end{bmatrix} \\ \Phi\_{31,\vec{\iota}} &= \begin{bmatrix} 0 & \lambda B\_{\vec{\eta}i} \bar{K}\_{\vec{\iota}} Z & 0 & 0 \\ 0 & 0 & \lambda N\_{\vec{\alpha}i} Z & 0 \end{bmatrix} \\ \Phi\_{22} &= \begin{bmatrix} -\mathcal{S}^{-1} & 0 & 0 & 0 \\ 0 & -\frac{\mathcal{E}}{4}I & 0 & 0 \\ 0 & 0 & -\frac{\mathcal{E}}{4}I & 0 \\ 0 & 0 & 0 & -W^{-1} \end{bmatrix} \\ \Phi\_{33} &= \begin{bmatrix} -\frac{\mathcal{E}}{4}I & 0 \\ 0 & -\frac{\mathcal{E}}{4}I \end{bmatrix} \end{aligned}$$

Obviously, the closed-loop fuzzy system (6) is asymptotically stable, if for some scalars λ > 0, there exist matrices Z > 0, Q > 0, R > 0 and X1, X2, X3, Ki, i ¼ 1, 2, ::, s satisfying the inequality (19).

Theorem 2. Consider the system (6) associated with cost function (7). For given scalars ρ > 0, τ<sup>1</sup> > 0, τ<sup>2</sup> > 0, σ<sup>1</sup> > 0, σ<sup>2</sup> > 0 and λ > 0, δ > 0, if there exist matrices Z > 0, Q<sup>1</sup> > 0,R<sup>1</sup> > 0, Q<sup>2</sup> > 0,R<sup>2</sup> > 0 and X1, X2, X3, X4, Mi, i = 1,2,…,s and scalar ε > 0 satisfying the following LMI (20), the system (6) is asymptotically stable and the control law (5) is a fuzzy non-fragile guaranteed cost control law

$$
\begin{bmatrix}
\Theta\_{1,ijl} & \* \\
\Theta\_{2,ijl} & \Theta\_3
\end{bmatrix} < 0, \quad i, j, l = 1, 2, \dots, s \tag{20}
$$

Moreover, the feedback gains are given by

$$K\_i = M\_i Z^{-1}, \text{i } = 1, 2, \dots, \text{s}.$$

$$\begin{split} \mathbf{J} & \leq \mathbf{x}^{\mathrm{T}}(\mathbf{0})\mathbf{P}\mathbf{x}(\mathbf{0}) + \int\_{-d(\mathbf{0})}^{0} \mathbf{x}^{\mathrm{T}}(\mathbf{s})\mathbf{Q}\_{1}\mathbf{x}(\mathbf{s})d\mathbf{s} + \int\_{-\mathbf{r}\_{1}}^{0} \int\_{\theta}^{\mathbf{0}} \dot{\mathbf{x}}^{\mathrm{T}}(\mathbf{s})\mathbf{R}\_{1}\dot{\mathbf{x}}(\mathbf{s})d\mathbf{s}d\theta + \int\_{-h(\mathbf{0})}^{0} \mathbf{x}^{\mathrm{T}}(\mathbf{s})\mathbf{Q}\_{2}\mathbf{x}(\mathbf{s})d\mathbf{s} \\ & \quad + \int\_{-\mathbf{r}\_{2}}^{0} \int\_{\theta}^{\mathbf{0}} \dot{\mathbf{x}}^{\mathrm{T}}(\mathbf{s})\mathbf{R}\_{2}\dot{\mathbf{x}}(\mathbf{s})d\mathbf{s}d\theta = \mathbf{J}\_{0} \end{split}$$

$$
\Theta\_{2,\vec{\eta}\vec{l}} = \begin{bmatrix}
\lambda E\_{\vec{k}\vec{l}}Z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\lambda E\_{\vec{k}\vec{l}}Z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \lambda E\_{\vec{k}\vec{l}}Z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & (B\_{\vec{i}}H\_{\vec{j}})^{\mathrm{T}} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho H\_i^{\mathrm{T}} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & (B\_{\mathrm{li}}H\_{\vec{l}})^{\mathrm{T}} \\
\Theta\_3 = \mathrm{diag}\{-\delta I\_{\vec{r}} - \delta I\_{\vec{r}} - \delta I\_{\vec{r}} - \delta^{-1}I\_{\vec{r}} - \delta^{-1}I\_{\vec{r}} - \delta^{-1}I\_{\vec{r}}
\end{bmatrix},
$$

$$+\delta^{-1}\begin{bmatrix}\langle\lambda\boldsymbol{E}\_{\dot{\boldsymbol{\mu}}}\boldsymbol{Z}\rangle^{\mathrm{T}} & \langle\lambda\boldsymbol{E}\_{\dot{\boldsymbol{\mu}}}\boldsymbol{Z}\rangle^{\mathrm{T}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \langle\lambda\boldsymbol{E}\_{\dot{\boldsymbol{\mu}}}\boldsymbol{Z}\rangle^{\mathrm{T}} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \lambda\boldsymbol{E}\_{\dot{\boldsymbol{\mu}}}\boldsymbol{Z} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \lambda\boldsymbol{E}\_{\dot{\boldsymbol{\mu}}}\boldsymbol{Z} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \lambda\boldsymbol{E}\_{\dot{\boldsymbol{\mu}}}\boldsymbol{Z} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{$$

$$
\Phi\_{1,\overline{\eta}} + \begin{bmatrix}
0 & \* & \* & \* & \* & \* & \* & \* & \* & \* \\
0 & 0 & \* & \* & \* & \* & \* & \* & \* & \* \\
0 & 0 & 0 & \* & \* & \* & \* & \* & \* & \* \\
0 & 0 & 0 & 0 & \* & \* & \* & \* & \* & \* \\
0 & 0 & 0 & 0 & 0 & \* & \* & \* & \* & \* \\
0 & 0 & 0 & 0 & 0 & 0 & \* & \* & \* & \* \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \* & \* & \* \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \* & \* & \* \\
\Lambda \Lambda \overline{K}\_{\overline{i}}Z & 0 & 0 & 0 & 0 & 0 & 0 & \* & \* & \* \\
\rho \Lambda \Lambda \overline{K}\_{\overline{i}}Z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \* & \* \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \* \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \Lambda B\_{\overline{i}} \Lambda \overline{K}\_{\overline{i}}Z & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} < 0 \tag{22}
$$

$$\begin{aligned} \mathbf{J} & \leq \mathbf{x}^{\mathrm{T}}(\mathbf{0}) \mathbf{P} \mathbf{x}(\mathbf{0}) + \int\_{-d(0)}^{0} \mathbf{x}^{\mathrm{T}}(\mathbf{s}) \mathbf{Q}\_{1} \mathbf{x}(\mathbf{s}) d\mathbf{s} + \int\_{-\tau\_{1}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathrm{T}}(\mathbf{s}) \mathbf{R}\_{1} \dot{\mathbf{x}}(\mathbf{s}) d\mathbf{s} d\theta + \int\_{-h(0)}^{0} \mathbf{x}^{\mathrm{T}}(\mathbf{s}) \mathbf{Q}\_{2} \mathbf{x}(\mathbf{s}) d\mathbf{s} \\ & \quad + \int\_{-\tau\_{1}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathrm{T}}(\mathbf{s}) \mathbf{R}\_{2} \dot{\mathbf{x}}(\mathbf{s}) d\mathbf{s} d\theta = \mathbf{J}\_{0} \end{aligned}$$

$$
\begin{split} & \left[ \begin{array}{ccc} -\alpha\_{1}I & I \\ I & -Z \end{array} \right] \leq 0, \left[ \begin{array}{ccc} -\alpha\_{2}I & \frac{1}{\lambda}P \\ \frac{1}{\lambda}P & S\_{Q1} \end{array} \right] \leq 0, \left[ \begin{array}{ccc} -\alpha\_{3}I & \frac{1}{\lambda}P \\ \frac{1}{\lambda}P & S\_{Q2} \end{array} \right] \leq 0, \left[ \begin{array}{ccc} -\alpha\_{4}I & \frac{1}{\lambda}P \\ \frac{1}{\lambda}P & S\_{R1} \end{array} \right] \leq 0, \\ & \left[ \begin{array}{ccc} -\alpha\_{5}I & \frac{1}{\lambda}P \\ \frac{1}{\lambda}P & S\_{R2} \end{array} \right] \leq 0, \left[ \begin{array}{ccc} S\_{Q1} & I \\ I & \overline{Q\_{1}} \end{array} \right] \geq 0, \left[ \begin{array}{ccc} S\_{Q2} & I \\ I & \overline{Q\_{2}} \end{array} \right] \geq 0, \\ & \left[ \begin{array}{ccc} \mathbb{S}\_{R1} & I \\ I & \overline{R\_{1}} \end{array} \right] \geq 0, \left[ \begin{array}{ccc} \mathbb{S}\_{R2} & I \\ I & \overline{R\_{2}} \end{array} \right] \geq 0, \end{split} \tag{23}
$$

Using the idea of the cone complement linear algorithm in Ref. [24], we can obtain the solution of the minimization problem of upper bound of the value of the cost function as follows:

$$\begin{aligned} \text{minimize} & \left\{ \text{trace} (P\mathbf{Z} + S\_{Q1}\overline{Q}\_{1} + S\_{Q2}\overline{Q}\_{2} + S\_{R1}\overline{R}\_{1} + S\_{R2}\overline{R}\_{2} + \alpha\_{1}\mathbf{x}^{\mathsf{T}}(0)\mathbf{x}(0) + \alpha\_{2} \right\}\_{-d(0)}^{0} \mathbf{x}^{\mathsf{T}}(s)\mathbf{x}(s)ds \\ & + \alpha\_{4} \int\_{-\tau\_{1}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathsf{T}}(s)\dot{\mathbf{x}}(s)dsd\theta + \alpha\_{3} \int\_{-h(0)}^{0} \mathbf{x}^{\mathsf{T}}(s)\mathbf{x}(s)ds + \alpha\_{5} \int\_{-\tau\_{2}}^{0} \int\_{\theta}^{0} \dot{\mathbf{x}}^{\mathsf{T}}(s)\dot{\mathbf{x}}(s)dsd\theta \right\} \end{aligned} \tag{24}$$
  $\text{subject to } (20), (23), \ \varepsilon > 0, \overline{Q}\_{1} > 0, \overline{Q}\_{2} > 0, \overline{R}\_{1} > 0, \overline{R}\_{2} > 0, \overline{Z} > 0, a\_{i} > 0, i = 1, ..., 5$ 

Using the following cone complement linearization (CCL) algorithm [24] can iteratively solve the minimization problem (24). □

### 5. Simulation examples

In this section, the proposed approach is applied to the Van de Vusse system to verify its effectiveness.

Example: Consider the dynamics of an isothermal continuous stirred tank reactor for the Van de Vusse

$$\dot{\mathbf{x}}\_1 = -50\mathbf{x}\_1 - 10\mathbf{x}\_1^3 + u(10 - \mathbf{x}\_1) + u(t - h) + u(t - h)(0.5\mathbf{x}\_1(t - d) + 0.2\mathbf{x}\_2(t - d)) + 5\mathbf{x}\_2(t - d) \tag{25}$$

$$\dot{\mathbf{x}}\_2 = 50\mathbf{x}\_1 - 100\mathbf{x}\_2 - u(t - h) + u(t - h)(0.3\mathbf{x}\_1(t - d) - 0.2\mathbf{x}\_2(t - d)) + 10\mathbf{x}\_2(t - d) - 5\mathbf{x}\_1(t - d)$$

From the system equation (25), some equilibrium points are tabulated in Table 1. According to these equilibrium points, [xe ue], which are also chosen as the desired operating points, ½x<sup>0</sup> <sup>e</sup> u<sup>0</sup> e�, we can use the similar modeling method that is described in Ref. [16].


Table 1. Data for equilibrium points.

#### Thus, the system (25) can be represented by

R<sup>1</sup> : if x<sup>1</sup> is about 2:0422 then

$$\dot{\mathbf{x}}\_{\delta}(t) = A\_1 \mathbf{x}\_{\delta}(t) + A\_{d1} \mathbf{x}\_{d\delta}(t) + B\_1 \boldsymbol{u}\_{\delta}(t) + B\_{h1} \boldsymbol{u}\_{d\delta}(t) + N\_1 \mathbf{x}\_{\delta}(t) \boldsymbol{u}\_{\delta}(t) + N\_{d1} \mathbf{x}\_{d\delta}(t) \boldsymbol{u}\_{h\delta}(t)$$

$$\begin{array}{l} \text{R}^2: \text{ if } \mathbf{x}\_1 \text{ is about } 3.6626, \text{ then} \\ \dot{\mathbf{x}}\_\delta(t) = A\_2 \mathbf{x}\_\delta(t) + A\_{d2} \mathbf{x}\_{d\delta}(t) + B\_2 \boldsymbol{u}\_\delta(t) + B\_{d2} \boldsymbol{u}\_{h\delta}(t) + \mathbf{N}\_2 \mathbf{x}\_\delta(t) \boldsymbol{u}\_\delta(t) + \mathbf{N}\_{d2} \mathbf{x}\_{d\delta}(t) \boldsymbol{u}\_{h\delta}(t) \\ \text{R}^3: \text{ if } \mathbf{x}\_1 \text{ is about } 5.9543, \text{ then} \\ \dot{\mathbf{x}}\_\delta(t) = A\_3 \mathbf{x}\_\delta(t) + A\_{d3} \mathbf{x}\_{d\delta}(t) + B\_3 \mathbf{u}\_\delta(t) + B\_{d3} \mathbf{u}\_{h\delta}(t) + N\_3 \mathbf{x}\_\delta(t) \boldsymbol{u}\_\delta(t) + N\_{d3} \mathbf{x}\_{d\delta}(t) \boldsymbol{u}\_{h\delta}(t) \end{array} \tag{26}$$

where

�α1I I I �Z

�α5I <sup>1</sup> λP

SR<sup>1</sup> I I R<sup>1</sup>

" #

" #

1 <sup>λ</sup>P SR<sup>2</sup>

þα<sup>4</sup> ð0 �τ<sup>1</sup> ð0 θ x\_

the minimization problem (24). □

5. Simulation examples

effectiveness.

<sup>x</sup>\_ <sup>1</sup> ¼ �50x<sup>1</sup> � <sup>10</sup>x<sup>3</sup>

<sup>e</sup> xT

Table 1. Data for equilibrium points.

de Vusse

xT

" #

40 Modern Fuzzy Control Systems and Its Applications

<sup>≤</sup> <sup>0</sup>, �α2<sup>I</sup> <sup>1</sup>

1 <sup>λ</sup>P SQ<sup>1</sup>

I R<sup>2</sup>

<sup>T</sup>ðsÞx\_ðsÞdsd<sup>θ</sup> <sup>þ</sup> <sup>α</sup><sup>3</sup>

" #

<sup>≤</sup> <sup>0</sup>, Z I I Z " #

<sup>≥</sup> <sup>0</sup>, SR<sup>2</sup> <sup>I</sup>

λP

≥ 0,

minimizeftraceðPZ <sup>þ</sup> SQ1Q<sup>1</sup> <sup>þ</sup> SQ2Q<sup>2</sup> <sup>þ</sup> SR1R<sup>1</sup> <sup>þ</sup> SR2R<sup>2</sup> <sup>þ</sup> <sup>α</sup>1x<sup>T</sup>ð0Þxð0Þ þ <sup>α</sup><sup>2</sup>

ð0 �hð0Þ

subject to ð20Þ;ð23Þ; ε> 0;Q<sup>1</sup> > 0, Q<sup>2</sup> > 0,R<sup>1</sup> > 0, R<sup>2</sup> > 0, Z > 0, α<sup>i</sup> > 0, i ¼ 1,…, 5

<sup>≥</sup> <sup>0</sup>, SQ<sup>1</sup> <sup>I</sup>

<sup>≤</sup> <sup>0</sup>, �α3<sup>I</sup> <sup>1</sup>

1 <sup>λ</sup>P SQ<sup>2</sup>

I Q<sup>1</sup>

Using the idea of the cone complement linear algorithm in Ref. [24], we can obtain the solution of the minimization problem of upper bound of the value of the cost function as follows:

Using the following cone complement linearization (CCL) algorithm [24] can iteratively solve

In this section, the proposed approach is applied to the Van de Vusse system to verify its

Example: Consider the dynamics of an isothermal continuous stirred tank reactor for the Van

<sup>x</sup>\_ <sup>2</sup> <sup>¼</sup> <sup>50</sup>x<sup>1</sup> � <sup>100</sup>x<sup>2</sup> � <sup>u</sup>ð<sup>t</sup> � <sup>h</sup>Þ þ <sup>u</sup>ð<sup>t</sup> � <sup>h</sup>Þð0:3x1ð<sup>t</sup> � <sup>d</sup>Þ � <sup>0</sup>:2x2ð<sup>t</sup> � <sup>d</sup>ÞÞ þ <sup>10</sup>x2ð<sup>t</sup> � <sup>d</sup>Þ � <sup>5</sup>x1ð<sup>t</sup> � <sup>d</sup><sup>Þ</sup> (25)

From the system equation (25), some equilibrium points are tabulated in Table 1. According to these equilibrium points, [xe ue], which are also chosen as the desired operating points, ½x<sup>0</sup>

[2.0422 1.2178] [2.0422 1.2178] 20.3077 20.3077 [3.6626 2.5443] [3.6626 2.5443] 77.7272 77.7272 [5.9543 5.5403] [5.9543 5.5403] 296.2414 296.2414

we can use the similar modeling method that is described in Ref. [16].

<sup>1</sup> þ uð10 � x1Þ þ uðt � hÞ þ uðt � hÞð0:5x1ðt � dÞ þ 0:2x2ðt � dÞÞ þ 5x2ðt � dÞ

de ue ude

<sup>x</sup><sup>T</sup>ðsÞxðsÞds <sup>þ</sup> <sup>α</sup><sup>5</sup>

" #

λP

<sup>≥</sup> <sup>0</sup>, SQ<sup>2</sup> <sup>I</sup>

ð0 �τ<sup>2</sup> ð0 θ x\_

I Q<sup>2</sup>

" #

<sup>≤</sup> <sup>0</sup>, �α4<sup>I</sup> <sup>1</sup>

1 <sup>λ</sup>P SR<sup>1</sup>

λP

≤ 0,

<sup>x</sup><sup>T</sup>ðsÞxðsÞds

(23)

(24)

<sup>e</sup> u<sup>0</sup> e�,

" #

≥ 0,

ð0 �dð0Þ

<sup>T</sup>ðsÞx\_ðsÞdsdθ<sup>g</sup>

" #

" #

$$\begin{aligned} A\_{1} &= \begin{bmatrix} -75.2383 & 7.7946 \\ 50 & -100 \end{bmatrix}, A\_{2} = \begin{bmatrix} -98.3005 & 11.7315 \\ 50 & -100 \end{bmatrix}, A\_{3} = \begin{bmatrix} -122.1228 & 8.8577 \\ 50 & -100 \end{bmatrix}, \\\ N\_{1} = N\_{2} = N\_{3} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}; B\_{1} = B\_{2} = B\_{3} = \begin{bmatrix} 10 \\ 0 \end{bmatrix}; A\_{d1} = A\_{d2} = A\_{d3} = \begin{bmatrix} 0 & 5 \\ 10 & -5 \end{bmatrix}, \\\ N\_{d1} = N\_{d2} = N\_{d3} = \begin{bmatrix} 0.5 & 0.2 \\ 0.3 & -0.2 \end{bmatrix}, B\_{b1} = B\_{b2} = B\_{b3} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{x}\_{\delta} = \mathbf{x}(t) - \mathbf{x}\_{\varepsilon'}^{\prime} \\\ \mathbf{u}\_{\delta} = \mathbf{u}(t) - \mathbf{u}\_{d\varepsilon}^{\prime}, \mathbf{u}\_{\delta\delta} = \mathbf{x}(t - d) - \mathbf{x}\_{\delta\varepsilon}^{\prime}. \end{aligned}$$

The cost function associated with this system is given with <sup>S</sup> <sup>¼</sup> 1 0 0 1 , W <sup>¼</sup> 1. The controller gain perturbation ΔK of the additive form is give with H<sup>1</sup> = H<sup>2</sup> = H<sup>3</sup> = 0.1, Ek<sup>1</sup> = [0.05 �0.01], Ek<sup>2</sup> = [0.02 0.01], Ek<sup>3</sup> = [�0.01 0].

Figure 1. Membership functions.

Figure 2. State responses of x1(t).

Figure 3. State responses of x2(t).

Non-Fragile Guaranteed Cost Control of Nonlinear Systems with Different State and Input Delays Based on T-S… http://dx.doi.org/10.5772/intechopen.69777 43

Figure 4. Control trajectory of system.

0 0.1 0.2 0.3 0.4

0 0.1 0.2 0.3 0.4

**time**

**time**

0

1


Figure 3. State responses of x2(t).




0

0.5

**x 2** 1

1.5

2

2.5

3

Figure 2. State responses of x1(t).

1.5

2

2.5

**x 1** 3

3.5

4

42 Modern Fuzzy Control Systems and Its Applications

The membership functions of state x<sup>1</sup> are shown in Figure 1.

Then, solving LMIs (23) and (24) for ρ = 0.45, λ = 1.02 and δ ¼ 0:11, τ<sup>1</sup> ¼ τ<sup>2</sup> ¼ 2, σ<sup>1</sup> ¼ 0, σ<sup>2</sup> ¼ 0 gives the following feasible solution:

$$\begin{aligned} P &= \begin{bmatrix} 4.2727 & -1.3007 \\ -1.3007 & 6.4906 \end{bmatrix}, Q\_1 = \begin{bmatrix} 14.1872 & -1.9381 \\ -1.9381 & 13.0104 \end{bmatrix}, Q\_2 = \begin{bmatrix} 3.1029 & 1.2838 \\ 1.2838 & 2.0181 \end{bmatrix}, \\\ R\_1 &= \begin{bmatrix} 8.3691 & -1.3053 \\ -1.3053 & 7.0523 \end{bmatrix}, R\_1 = \begin{bmatrix} 5.2020 & 2.2730 \\ 2.2730 & 1.0238 \end{bmatrix}, \varepsilon = 1.8043, \\\ K\_1 &= \begin{bmatrix} -0.4223 & -0.5031 \end{bmatrix}, K\_2 = \begin{bmatrix} -0.5961 & -0.7049 \end{bmatrix}, K\_1 = \begin{bmatrix} -0.4593 & -0.3874 \end{bmatrix}. \end{aligned}$$

Figures 2–4 illustrate the simulation results of applying the non-fragile fuzzy controller to the system (25) with x<sup>0</sup> <sup>e</sup> ¼ ½ 3:6626 2:5443 � <sup>T</sup> and u<sup>0</sup> <sup>e</sup> ¼ 77:7272 under initial condition ϕ(t) = [1.2 �1.8]T , t ∈ [�2 0]. It can be seen that with the fuzzy control law, the closed-loop system is asymptotically stable and an upper bound of the guaranteed cost is J<sup>0</sup> = 292.0399. The simulation results show that the fuzzy non-fragile guaranteed controller proposed in this paper is effective.

#### 6. Conclusions

In this paper, the problem of non-fragile guaranteed cost control for a class of fuzzy timevarying delay systems with local bilinear models has been explored. By utilizing the Lyapunov stability theory and LMI technique, sufficient conditions for the delay-dependent asymptotically stability of the closed-loop T-S fuzzy local bilinear system have been obtained. Moreover, the designed fuzzy controller has guaranteed the cost function-bound constraint. Finally, the effectiveness of the developed approach has been demonstrated by the simulation example. The robust non-fragile guaranteed cost control and robust non-fragile H-infinite control based on fuzzy bilinear model will be further investigated in the future work.

## Acknowledgements

This work is supported by NSFC Nos. 60974139 and 61573013.

## Author details

Junmin Li\*, Jinsha Li and Ruirui Duan

\*Address all correspondence to: jmli@mail.xidian.edu.cn

School of Mathematics and Statistics, Xidian University, Xi'an, PR China

## References


stability theory and LMI technique, sufficient conditions for the delay-dependent asymptotically stability of the closed-loop T-S fuzzy local bilinear system have been obtained. Moreover, the designed fuzzy controller has guaranteed the cost function-bound constraint. Finally, the effectiveness of the developed approach has been demonstrated by the simulation example. The robust non-fragile guaranteed cost control and robust non-fragile H-infinite control based

on fuzzy bilinear model will be further investigated in the future work.

This work is supported by NSFC Nos. 60974139 and 61573013.

\*Address all correspondence to: jmli@mail.xidian.edu.cn

School of Mathematics and Statistics, Xidian University, Xi'an, PR China

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2004;34:1288–1292


## **Fuzzy Interpolation Systems and Applications**

Longzhi Yang, Zheming Zuo, Fei Chao and Yanpeng Qu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68051

#### Abstract

Fuzzy inference systems provide a simple yet effective solution to complex non-linear problems, which have been applied to numerous real-world applications with great success. However, conventional fuzzy inference systems may suffer from either too sparse, too complex or imbalanced rule bases, given that the data may be unevenly distributed in the problem space regardless of its volume. Fuzzy interpolation addresses this. It enables fuzzy inferences with sparse rule bases when the sparse rule base does not cover a given input, and it simplifies very dense rule bases by approximating certain rules with their neighbouring ones. This chapter systematically reviews different types of fuzzy interpolation approaches and their variations, in terms of both the interpolation mechanism (inference engine) and sparse rule base generation. Representative applications of fuzzy interpolation in the field of control are also revisited in this chapter, which not only validate fuzzy interpolation approaches but also demonstrate its efficacy and potential for wider applications.

Keywords: fuzzy inference systems, fuzzy interpolation, adaptive fuzzy interpolation, sparse rule bases, fuzzy control

## 1. Introduction

Fuzzy logic and fuzzy sets have been used successfully as tools to manage the uncertainty of fuzziness since their introduction in the 1960s, which have been applied to many fields, including [1–6]. The most widely used fuzzy systems are fuzzy rule-based inference systems, each comprising of a rule base and an inference engine. Different inference engines were invented to support different situations, such as the Mamdani inference engine [7] and the TSK inference engine [8]. The rule bases are usually extracted from expert knowledge or learned from data. The TSK model produces crisp outputs due to its polynomial rule consequences in TSK-style rule

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bases, while the Mamdani model is more appealing in handling inferences based on human natural language due to its fuzzy rule consequences. Despite of the wide applications, these conventional fuzzy inference mechanisms are only workable with dense rule bases which fully cover the entire input domain.

Fuzzy interpolation systems (FISs) were proposed to address the above issue [9], and they also help in complexity reduction for fuzzy models with too complex (dense) rule bases. If there is only a spare rule base available and a given input does not overlap with any rule antecedent, conventional fuzzy inference systems will not be applicable. However, FISs are still able to generate a conclusion by means of fuzzy interpolation in such situations, thus enhancing the applicability of conventional fuzzy inference systems. FISs can also improve the efficiency of complex fuzzy inference systems by excluding those rules that can be accurately interpolated or extrapolated using other rules in a complex rule base. Various fuzzy interpolation methods based on Mamdani-style rule bases have been proposed in the literature such as Refs. [9–20], with successful applications in the fields of decision-making support, prediction and control, amongst others.

FISs have also been developed to support TSK-style sparse fuzzy rule bases by extending the traditional TSK fuzzy inference system [21]. This approach was developed based on a modified similarity degree measure that enables the effective utilisation of all rules during inference process to generate a global result. In particular, the modified similarity measure guarantees that the similarity degree between any given input and any rule antecedent is greater than 0 even when they do not overlap at all. Therefore, all the rules in the rule base can be fired to certain degrees such that they all contribute to the final result to some extents and consequently a conclusion still can be generated even when no rule antecedent is overlapped with the given observation. The extended TSK fuzzy model enjoys the advantages of both TSK model and fuzzy interpolation, which is able to obtain crisp inference results from either sparse, dense or unevenly distributed (including dense parts and spare parts) TSK-style fuzzy rule bases.

FISs have been successfully applied to real-world problems. In some real world scenarios, neither complete expert knowledge nor complete data set is available or readily obtainable to generate evenly distributed dense rule bases. FISs therefore have been applied in such situations. For instance, a FIS has been applied to building evaluation in the work of Molnárka et al. [22] in an effort to help estate agencies making decisions for residential building maintenance, when some necessary relevant data have been lost. In Ref. [23], a FIS system was applied successfully to reduce the complexity and improve the efficiency of a fuzzy home heating control system. The work of Bai et al. [24] applied a FIS to calibrate parallel machine tools for industry use. A behaviour-based fuzzy control system is introduced in Ref. [25], which applied a FIS to make decisions when only incomplete knowledge base has been provided or available. Most recently, FISs have also been used to support network quality of service [26] and network intrusion detection [27].

The remainder of this chapter is organised as follows. Section 2 reviews the theoretical underpinnings of conventional fuzzy inference systems, that is, the Mamdani inference system and the TSK inference system. Section 3 discusses different fuzzy interpolation approaches to support sparse Mamdani-style rule bases. Section 4 presents the extension of the conventional TSK inference system in supporting sparse TSK-style rule bases. Section 5 reports two representative examples of fuzzy interpolation systems in the field of system control. Section 6 concludes the chapter and points out the directions for future work.

## 2. Fuzzy inference systems

bases, while the Mamdani model is more appealing in handling inferences based on human natural language due to its fuzzy rule consequences. Despite of the wide applications, these conventional fuzzy inference mechanisms are only workable with dense rule bases which fully

Fuzzy interpolation systems (FISs) were proposed to address the above issue [9], and they also help in complexity reduction for fuzzy models with too complex (dense) rule bases. If there is only a spare rule base available and a given input does not overlap with any rule antecedent, conventional fuzzy inference systems will not be applicable. However, FISs are still able to generate a conclusion by means of fuzzy interpolation in such situations, thus enhancing the applicability of conventional fuzzy inference systems. FISs can also improve the efficiency of complex fuzzy inference systems by excluding those rules that can be accurately interpolated or extrapolated using other rules in a complex rule base. Various fuzzy interpolation methods based on Mamdani-style rule bases have been proposed in the literature such as Refs. [9–20], with successful applications in the fields of decision-making support, prediction and control,

FISs have also been developed to support TSK-style sparse fuzzy rule bases by extending the traditional TSK fuzzy inference system [21]. This approach was developed based on a modified similarity degree measure that enables the effective utilisation of all rules during inference process to generate a global result. In particular, the modified similarity measure guarantees that the similarity degree between any given input and any rule antecedent is greater than 0 even when they do not overlap at all. Therefore, all the rules in the rule base can be fired to certain degrees such that they all contribute to the final result to some extents and consequently a conclusion still can be generated even when no rule antecedent is overlapped with the given observation. The extended TSK fuzzy model enjoys the advantages of both TSK model and fuzzy interpolation, which is able to obtain crisp inference results from either sparse, dense or unevenly distributed (including dense parts and spare parts) TSK-style fuzzy

FISs have been successfully applied to real-world problems. In some real world scenarios, neither complete expert knowledge nor complete data set is available or readily obtainable to generate evenly distributed dense rule bases. FISs therefore have been applied in such situations. For instance, a FIS has been applied to building evaluation in the work of Molnárka et al. [22] in an effort to help estate agencies making decisions for residential building maintenance, when some necessary relevant data have been lost. In Ref. [23], a FIS system was applied successfully to reduce the complexity and improve the efficiency of a fuzzy home heating control system. The work of Bai et al. [24] applied a FIS to calibrate parallel machine tools for industry use. A behaviour-based fuzzy control system is introduced in Ref. [25], which applied a FIS to make decisions when only incomplete knowledge base has been provided or available. Most recently, FISs have also been used to support network quality of service [26] and network

The remainder of this chapter is organised as follows. Section 2 reviews the theoretical underpinnings of conventional fuzzy inference systems, that is, the Mamdani inference system and the TSK inference system. Section 3 discusses different fuzzy interpolation approaches to

cover the entire input domain.

50 Modern Fuzzy Control Systems and Its Applications

amongst others.

rule bases.

intrusion detection [27].

The process of fuzzy inference is basically an iteration of computer paradigm based on fuzzy set theory, fuzzy-if-then-rules and fuzzy reasoning. Each iteration takes an input which can be an observation or a previously inferred result, crisp or fuzzy. Then, these inputs are used to fire the rules in a given rule base, and the output is the aggregation of the inferred results from all the fired rules. There are generally two primary ways to construct a rule base for a given problem. The first way is directly translating expert knowledge to rules, and the fuzzy inference systems with such rule bases are usually called fuzzy expert systems or fuzzy controllers [28]. In this case, rules are fuzzy representations of expert knowledge, and the resultant rule base offers a high semantic level and a good generalisation capability. The difficulty of building rule bases for complex problems has resulted in the development of another approach of rule base construction, which is driven by data, that is, fuzzy rules are obtained from data by employing machine learning techniques rather than expert knowledge [29, 30]. In contrast, the rule bases built in this way lack comprehensibility and transparency. There are two types of rule bases depending on the expression of the consequences of the fuzzy rules composing the rule base. Mamdani-style fuzzy rules consider fuzzy terms or linguistic values in the consequence, while TSK-style fuzzy rules represent the consequences as polynomial functions of crisp inputs.

#### 2.1. Inference with Mamdani-style rule bases

There are a number of fuzzy inference mechanisms that can be utilised to derive a consequence from a given observation using a Mamdani rule base. The two most significant modes are the compositional rule of inference (CRI) [31] and analogy-based reasoning [24, 33], which are introduced below.

### 2.1.1. Compositional rule of inference

The introduction of CRI marks the era of fuzzy inference [31]. Given a rule 'IF x is A, THEN y is B"' and an observation 'x is A\* ', the conclusion B\* can be generated through CRI as:

$$\mu\_{\mathcal{B}^\*} (\upsilon) = \sup\_{\boldsymbol{\mu} \in \mathcal{U}\_x} T \Big( \mu\_{\mathcal{A}^\*} (\boldsymbol{\mu}), \mu\_{\mathcal{R}} (\boldsymbol{\mu}, \upsilon) \Big), \tag{1}$$

where T is a triangular norm, sup represents supremum, and R is the relationship between variables x and y. Essentially, CRI is a fuzzy extension of classical modus ponens which can be viewed from two perspectives. Firstly, classical modus ponens only supports predicates concerning singleton elements, but CRI is able to deal with predicates which concern a set of elements in the variable domain. This is achieved by representing a fuzzy rule as a fuzzy relation over the Cartesian product of the domains of the antecedent and consequent variables. Various fuzzy implication relations have been proposed [7, 32–34], each of which may have its own properties and therefore is suitable for a certain group of applications. Secondly, classical modus ponens only supports Boolean logic, but CRI supports multi-value logic. That is, CRI is able to deal with predicates with partial truth values, which are implemented by a compositional operator sup T, where T represents a t-norm [35].

A number of existing fuzzy reasoning methods based on CRI have been developed [36, 37], including the first successful practical approach, that is, the Mamdani inference [28]. This approach is also the most commonly seen fuzzy methodology in physical control systems thus far. It was originally proposed as an attempt to control a steam engine and boiler combination by synthesising a set of linguistic control rules obtained from experienced human operators. Mamdani inference implements CRI using minimum as the t-norm operator due to its simplicity. In particular, the inferred result from each fired rule is a fuzzy set which is transformed from the rule consequence by restricting the membership of those elements whose memberships are greater than the firing strength. The firing strength is also sometimes termed the satisfaction degree, which is the supremum within the variable domain of the minimum of the rule antecedent and the given observation. A defuzzification process is needed when crisp outputs are required.

#### 2.1.2. Analogy-based fuzzy inference

Despite the success of CRI in various fuzzy system applications, it suffers various criticisms including its complexity and vague underlying semantics [34, 38]. This has led to another group of fuzzy reasoning approaches which are based on similarity degree, usually called analogy-based fuzzy reasoning [38–41]. Similarity considerations play a major role in human cognitive processes [42], so do they in approximate reasoning. It is intuitive that if a given observation is similar to the antecedent of a rule, the conclusion from the observation should also be similar to the consequence of the rule. Different to CRI-based fuzzy reasoning, analogybased fuzzy reasoning does not require the construction of a fuzzy relation. Instead, it is based on the degree of similarity (given a certain similarity metric) between the given observation and the antecedent of a rule. Utilising the computed similarity degree, the consequence of the fired rule can be modified to the consequence of the given observation.

Approximate analogical reasoning schema is a typical analogy-based fuzzy inference approach [34, 38]. In this method, rules are fired according to the similarity degrees between a given observation and the antecedents of rules. If the degree of similarity between the given observation and the antecedent of a rule is greater than a predefined threshold value, the rule will be fired and the consequence of the observation is deduced from the rule consequence by a given modification procedure. Another analogy-based fuzzy inference approach was proposed in Refs. [39, 40], which particularly targets medical diagnostic problems. This approach is based on the cosine angle between the two vectors that represent the actual and the user's specified values of the antecedent variable. Several modification procedures can be found in Refs. [43, 44]. Particularly, a fuzzy reasoning method which employs similarity measures based on the degree of subsethood between the propositions in the antecedent and a given observation is proposed in Ref. [45]. This method has also been extended to consider the weights of the propositions in the antecedent [46]. Analogy-based fuzzy inference approaches usually arrive at solutions with more natural appeal than those introduced in the last section.

#### 2.2. Inference with TSK-style rule bases

elements in the variable domain. This is achieved by representing a fuzzy rule as a fuzzy relation over the Cartesian product of the domains of the antecedent and consequent variables. Various fuzzy implication relations have been proposed [7, 32–34], each of which may have its own properties and therefore is suitable for a certain group of applications. Secondly, classical modus ponens only supports Boolean logic, but CRI supports multi-value logic. That is, CRI is able to deal with predicates with partial truth values, which are implemented by a composi-

A number of existing fuzzy reasoning methods based on CRI have been developed [36, 37], including the first successful practical approach, that is, the Mamdani inference [28]. This approach is also the most commonly seen fuzzy methodology in physical control systems thus far. It was originally proposed as an attempt to control a steam engine and boiler combination by synthesising a set of linguistic control rules obtained from experienced human operators. Mamdani inference implements CRI using minimum as the t-norm operator due to its simplicity. In particular, the inferred result from each fired rule is a fuzzy set which is transformed from the rule consequence by restricting the membership of those elements whose memberships are greater than the firing strength. The firing strength is also sometimes termed the satisfaction degree, which is the supremum within the variable domain of the minimum of the rule antecedent and the given observation. A defuzzification process is needed when crisp

Despite the success of CRI in various fuzzy system applications, it suffers various criticisms including its complexity and vague underlying semantics [34, 38]. This has led to another group of fuzzy reasoning approaches which are based on similarity degree, usually called analogy-based fuzzy reasoning [38–41]. Similarity considerations play a major role in human cognitive processes [42], so do they in approximate reasoning. It is intuitive that if a given observation is similar to the antecedent of a rule, the conclusion from the observation should also be similar to the consequence of the rule. Different to CRI-based fuzzy reasoning, analogybased fuzzy reasoning does not require the construction of a fuzzy relation. Instead, it is based on the degree of similarity (given a certain similarity metric) between the given observation and the antecedent of a rule. Utilising the computed similarity degree, the consequence of the

Approximate analogical reasoning schema is a typical analogy-based fuzzy inference approach [34, 38]. In this method, rules are fired according to the similarity degrees between a given observation and the antecedents of rules. If the degree of similarity between the given observation and the antecedent of a rule is greater than a predefined threshold value, the rule will be fired and the consequence of the observation is deduced from the rule consequence by a given modification procedure. Another analogy-based fuzzy inference approach was proposed in Refs. [39, 40], which particularly targets medical diagnostic problems. This approach is based on the cosine angle between the two vectors that represent the actual and the user's specified values of the antecedent variable. Several modification procedures can be found in Refs. [43, 44]. Particularly, a fuzzy reasoning method which employs similarity measures based

fired rule can be modified to the consequence of the given observation.

tional operator sup T, where T represents a t-norm [35].

52 Modern Fuzzy Control Systems and Its Applications

outputs are required.

2.1.2. Analogy-based fuzzy inference

The TSK fuzzy inference system was proposed for the direct generation of crisp outputs [8]. In difference with the Mamdani-style fuzzy rule bases, TSK-style rule bases are usually generated from data using a clustering algorithm such as K-Means and an algorithm to determine the number of clusters such as Ref. [47]. Also, the consequence of a TSK fuzzy rule is a polynomial function rather than a fuzzy set. A typical TSK fuzzy rule can be defined as:

$$\text{IF } \mathbf{x}\_1 \text{ is } A\_1 \land \dots \land \mathbf{x}\_m \text{ is } A\_m \text{ THEN } z = f(\mathbf{x}\_1, \dots, \mathbf{x}\_m), \tag{2}$$

where A1,…Am are fuzzy values with regard to antecedent variables x1, …, xm respectively, and f(x1,…,xm) is a crisp polynomial function of crisp inputs determining the crisp output value. The rule consequent polynomial functions f(x1,…,xm) are usually zero order or first order. For simplicity, suppose that a TSK-style rule base is formed by two-antecedent rules as follows:

$$\begin{array}{l} R\_i: \text{IF } \mathbf{x} \text{ is } A\_i \land y \text{ is } B\_i \text{ THEN } z = f\_i(\mathbf{x}, y) \\ R\_j: \text{IF } \mathbf{x} \text{ is } A\_j \land y \text{ is } B\_j \text{ THEN } z = f\_j(\mathbf{x}, y). \end{array} \tag{3}$$

Suppose that (x0,y0) is the crisp input pair, then the inference process can be shown in Figure 1. As the input values overlap with both rule antecedents, both rules are fired. Using rules Ri and Rj, the given input then leads to system outputs fi(x0,x0) and fj(x0,x0), respectively. The consequences from both rules are then integrated using weighted average function, where

Figure 1. TSK fuzzy inference [21].

the values of weights represent the matching degrees between the given input and the rule antecedents (often referred to as firing strengths). Assume that μAi ðx0Þ and μBi ðy0Þ are the matching degree between inputs (x<sup>0</sup> and y0) and rule antecedents (Ai and Bi), respectively. The firing strength of rule Ri, denoted as αi, is calculated as:

$$
\mu\_i = \mu\_{A\_i}(\mathbf{x}\_0) \land \mu\_{B\_i}(y\_0),
\tag{4}
$$

where ∧ stands for a t-norm operator. Different implementations can be used for the t-norm operator, with the minimum operator being used most widely. Of course, if another system input (x1, y1) is presented and it is not covered by the rule base, the matching degrees between this new input and rule antecedents of Ri and Rj are equal to 0. In this case, no rule will be fired, and thus traditional TSK is not applicable. In this case, fuzzy interpolation is required, which is introduced in Section 4.

### 3. Fuzzy interpolation with sparse Mamdani-style rule bases

FISs based on Mamdani-style rule bases can be categorised into two classes. One group of approaches were developed based on the decomposition and resolution principle, termed as 'resolution principle-base interpolation'. In particular, the approach represents each fuzzy set as a series of α-cuts (α ∈ (0,1]), and the α-cut of the conclusion is computed from the α-cuts of the observation and the α-cuts of rules. The final fuzzy set is assembled from all the α-cut consequences using the resolution principle [48–50]. The other group of fuzzy interpolation approaches were developed using the analogy reasoning system, thus termed as 'analogybased fuzzy interpolation'. This group of approaches firstly generates an intermediate rule whose antecedent maximally overlaps with the given observation, then the system output is produced from the observation using the intermediate rule. Two representative approaches of the two classes, the KH approach [10] and the scale and move transformation-based approach [9, 51, 52], are discussed in this section based on simple rule bases with two antecedent rules. Despite of the simple examples used herein, both of these approaches have been extended to work with multiple multi-antecedent rules.

#### 3.1. Resolution principle-based interpolation

Single step interpolation approaches are computationally efficient, such as the KH approach proposed in Refs. [9, 10, 53]. Following these approaches, all variables involved in the reasoning process must satisfy a partial ordering, denoted as ≺ [31]. According to the decomposition principle, a normal and convex fuzzy set A can be represented by a series of α-cut intervals, each denoted as Aα, α ∈ (0,1). Given fuzzy sets Ai and Aj which are associated with the same variable, the partial ordering Ai ≺ Aj is defined as:

$$\inf \{ A\_{ia} \} < \inf \{ A\_{ja} \} \text{ and } \sup \{ A\_{ia} \} < \sup \{ A\_{ja} \}, \quad \forall a \in (0, 1]. \tag{5}$$

where inffAiαg and sup{Aiα} denote the infimum and supremum of Aiα, respectively.

Take the KH approach as an example here. For simplicity, suppose there are two fuzzy rules: If x is Ai, then y is Bi, and If x is Aj then y is Bj, shorten as Ai ⟹ Bi and Aj ⟹ Bj, respectively. Also, suppose that these two rules are adjacent, in other words, there is no rule A ⟹ B existing such that Ai ≺ A ≺ Aj or Aj ≺ A ≺ Ai. Given an observation A\* which satisfies Ai ≺ A� ≺ Aj or Aj ≺ A� ≺ Ai, a conclusion B\* can be computed as:

$$\frac{D(A\_{i\alpha} \, A\_{\alpha}^\*)}{D(A\_{\alpha'}^\* \, A\_{\text{ja}})} = \frac{D(B\_{i\alpha'} \, B\_{\alpha}^\*)}{D(B\_{\alpha'}^\* \, B\_{\text{ja}})} \, \tag{6}$$

where given any 0 < α ≤ 1, the distance DðAiα, AjαÞ between the α-cuts Ai<sup>α</sup> and Aj<sup>α</sup> is defined by the interval <sup>½</sup>D<sup>L</sup>ðAiα, AjαÞ, D<sup>U</sup>ðAiα, AjαÞ� with:

$$D^{L}(A\_{\rm ia}, A\_{\rm ja}) = \inf \{ A\_{\rm ja} \} - \inf \{ A\_{\rm ia} \}, \\ D^{L}(A\_{\rm ia}, A\_{\rm ja}) = \sup \{ A\_{\rm ja} \} - \sup \{ A\_{\rm ia} \}. \tag{7}$$

Following Eqs. (4) and (5), the following is resulted:

$$\begin{cases} \min \{ B\_{\alpha}^{\*} \} = \frac{\overline{\inf \{ B\_{\dot{\alpha} \alpha} \}}{\overline{D}^{L} (A\_{\dot{\alpha} \alpha} A\_{\alpha}^{\*})} + \frac{\inf (B\_{\dot{\alpha} \alpha})}{\overline{D}^{L} (A\_{\alpha'}^{\*} A\_{\dot{\alpha} \alpha})} \\ \qquad \overline{D^{L} (A\_{\dot{\alpha} \alpha} A\_{\alpha}^{\*})} + \frac{1}{\overline{D}^{L} (A\_{\alpha'}^{\*} A\_{\dot{\alpha} \alpha})} \\ \max \{ B\_{\alpha}^{\*} \} = \frac{\sup (B\_{\dot{\alpha} \alpha})}{\overline{D}^{L} (A\_{\dot{\alpha} \alpha} A\_{\alpha}^{\*})} + \frac{\sup (B\_{\dot{\beta} \alpha})}{\overline{D}^{L} (A\_{\alpha'}^{\*} A\_{\dot{\beta} \alpha})} \\ \qquad \overline{D^{L} (A\_{\dot{\alpha} \alpha} A\_{\alpha}^{\*})} + \frac{1}{\overline{D}^{L} (A\_{\alpha'}^{\*} A\_{\dot{\beta} \alpha})} \end{cases} (8)$$

For simplicity, let

the values of weights represent the matching degrees between the given input and the rule

matching degree between inputs (x<sup>0</sup> and y0) and rule antecedents (Ai and Bi), respectively.

ðx0Þ ∧ μBi

where ∧ stands for a t-norm operator. Different implementations can be used for the t-norm operator, with the minimum operator being used most widely. Of course, if another system input (x1, y1) is presented and it is not covered by the rule base, the matching degrees between this new input and rule antecedents of Ri and Rj are equal to 0. In this case, no rule will be fired, and thus traditional TSK is not applicable. In this case, fuzzy interpolation is required, which is

FISs based on Mamdani-style rule bases can be categorised into two classes. One group of approaches were developed based on the decomposition and resolution principle, termed as 'resolution principle-base interpolation'. In particular, the approach represents each fuzzy set as a series of α-cuts (α ∈ (0,1]), and the α-cut of the conclusion is computed from the α-cuts of the observation and the α-cuts of rules. The final fuzzy set is assembled from all the α-cut consequences using the resolution principle [48–50]. The other group of fuzzy interpolation approaches were developed using the analogy reasoning system, thus termed as 'analogybased fuzzy interpolation'. This group of approaches firstly generates an intermediate rule whose antecedent maximally overlaps with the given observation, then the system output is produced from the observation using the intermediate rule. Two representative approaches of the two classes, the KH approach [10] and the scale and move transformation-based approach [9, 51, 52], are discussed in this section based on simple rule bases with two antecedent rules. Despite of the simple examples used herein, both of these approaches have been

Single step interpolation approaches are computationally efficient, such as the KH approach proposed in Refs. [9, 10, 53]. Following these approaches, all variables involved in the reasoning process must satisfy a partial ordering, denoted as ≺ [31]. According to the decomposition principle, a normal and convex fuzzy set A can be represented by a series of α-cut intervals, each denoted as Aα, α ∈ (0,1). Given fuzzy sets Ai and Aj which are associated with the same

where inffAiαg and sup{Aiα} denote the infimum and supremum of Aiα, respectively.

inffAiαg < inffAjαg and supfAiα} < sup{Ajαg, ∀α∈ ð0, 1�, ð5Þ

ðx0Þ and μBi

ðy0Þ, ð4Þ

ðy0Þ are the

antecedents (often referred to as firing strengths). Assume that μAi

α<sup>i</sup> ¼ μAi

3. Fuzzy interpolation with sparse Mamdani-style rule bases

The firing strength of rule Ri, denoted as αi, is calculated as:

54 Modern Fuzzy Control Systems and Its Applications

extended to work with multiple multi-antecedent rules.

3.1. Resolution principle-based interpolation

variable, the partial ordering Ai ≺ Aj is defined as:

introduced in Section 4.

$$\begin{cases} A\_{\alpha}^{L} = \frac{\inf\{A\_{\alpha}^{\*}\} - \inf\{A\_{i\alpha}\}}{\inf\{A\_{j\alpha}\} - \inf\{A\_{i\alpha}\}} \\\\ A\_{\alpha}^{U} = \frac{\sup\{A\_{\alpha}^{\*}\} - \sup\{A\_{i\alpha}\}}{\sup\{A\_{j\alpha}\} - \sup\{A\_{i\alpha}\}} \end{cases} \tag{9}$$

Also, denote <sup>Λ</sup> ¼ ½Λ<sup>L</sup> <sup>α</sup>, Λ<sup>U</sup> <sup>α</sup> � hereafter. From this, Eq. (8) can be re-written as:

$$\begin{cases} \min\{B\_{\alpha}^{\*}\} = (1 - \Lambda\_{\alpha}^{L})\inf\{B\_{i\alpha}\} + \Lambda\_{\alpha}^{L}\inf\{B\_{j\alpha}\}\\ \max\{B\_{\alpha}^{\*}\} = (1 - \Lambda\_{\alpha}^{\mathrm{II}})\sup\{B\_{i\alpha}\} + \Lambda\_{\alpha}^{\mathrm{II}}\sup\{B\_{j\alpha}\} \end{cases} \tag{10}$$

This means B� <sup>α</sup> ¼ ½minfB� αg, maxfB� <sup>α</sup>g] is generated. The final consequence B\* is then reassembled as:

$$B^\* = \mathcal{U}\_{\mathfrak{a} \in (0,1]} \mathfrak{a} B\_{\mathfrak{a}}^\*. \tag{11}$$

The KH approach may generate invalid interpolated results [54], which is usually called 'the abnormal problem'. To eliminate this deficiency, a number of modifications or improvements have been proposed, including Refs. [9, 10, 13, 14, 18, 53, 55–60]. Approaches such as Refs. [15, 16, 61–63] also belong to this group.

#### 3.2. Analogy-based interpolation

The scale and move transformation-based fuzzy interpolation [51, 52, 64] is a representative approach in the analogy-based interpolation group. For simplicity, following the same assumption of a simple rule base containing two rules with two antecedents, the transformation-based approach is shown in Figure 2 and outlined as follows.

Given neighbouring rules If x is Ai, then y is Bi, and If x is Aj then y is Bj and observation A\*, this method first maps fuzzy sets Ai, Aj and A\* to real numbers ai, aj and a\* (named as representative values) respectively, using real function f1. Then, the location relationship between A\* and rule antecedents (Ai and Aj) is computed. This is achieved by another mapping function f2, which results in the relative placement factor λ. In contrast to the resolution-based interpolation approaches, the generated relative placement factor in analogy-based fuzzy interpolation approach is a crisp real number. Finally, linear interpolation is implemented using mapping function f<sup>3</sup> of λ, which leads to the intermediate rule A�0 ) <sup>B</sup>�0 .

Note that the representative value of intermediate rule antecedent A�<sup>0</sup> equals to that of A\* (the given observation), although A�0 and A\* are not identical for most of the situations. In the scale and move transformation-based fuzzy interpolation approach, the similarity degree between two fuzzy sets A\* and A\*' with the same representative value is expressed as the scale rate s, scale ratio S and move rate M, which is obtained by real function f4. From this, the consequence B\* is calculated from B�0 using a transformation function f<sup>5</sup> which imposes the similarity degree between A\* and A\*'. Different approaches have been developed for intermediate rule generation and final conclusion production from the intermediate rule [17, 55, 63, 65].

#### 3.3. Adaptive fuzzy interpolation

Fuzzy interpolation strengthens the power of fuzzy inference by enhancing the robustness of fuzzy systems and reducing the systems' complexity. Common to both classes of fuzzy

Figure 2. Transformation-based fuzzy interpolation [12].

interpolation approaches discussed above is the fact that interpolation is carried out in a linear manner. This may conflict with the nature of some realistic problems and consequently this may lead to inconsistencies during rule interpolation processes. Adaptive fuzzy interpolation was proposed to address this [12, 66–68]. It was developed upon FIS approaches, which detects inconsistencies, locates possible fault candidates and modifies the candidates in order to remove all the inconsistencies.

Each pair of neighbouring rules is defined as a fuzzy reasoning component in adaptive fuzzy interpolation. Each fuzzy reasoning component takes a fuzzy value as input and produces another as output. The process of adaptive interpolation is summarised in Figure 3. Firstly, the interpolator carries out interpolation and passes the interpolated results to the truth maintenance system (ATMS) [69, 70], which records the dependencies between an interpolated value (including any contradiction) and its proceeding interpolation components. Then, the ATMS relays any β0-contradictions (i.e. inconsistency between two different values for a common variable at least to the degree of a given threshold β<sup>0</sup> (0 ≤ β<sup>0</sup> ≤ 1)) as well as their dependent fuzzy reasoning components to the general diagnostic engine (GDE) [71] which diagnoses the problem and generates all possible component candidates. After that, a modification process takes place to correct a certain candidate to restore consistency by modifying the original linear interpolation to become first-order piecewise linear.

The adaptive approach has been further generalised [11, 72, 73], which allows the identification and modification of observations and rules, in addition to that of interpolation procedures that were addressed in Ref. [12]. This is supported by introducing extra information of certainty degrees associated with such basic elements of FIS. The work also allows for all candidates for modification to be prioritised, based on the extent to which a candidate is likely to lead to all the detected contradictions, by extending the classic ATMS and GDE. This study has significantly improved the efficiency of the work in Ref. [12] by exploiting more information during both the diagnosis and modification processes. Another alternative implementation of the adaptive approach has also been reported in Ref. [74].

#### 3.4. Sparse rule base generation

have been proposed, including Refs. [9, 10, 13, 14, 18, 53, 55–60]. Approaches such as Refs.

The scale and move transformation-based fuzzy interpolation [51, 52, 64] is a representative approach in the analogy-based interpolation group. For simplicity, following the same assumption of a simple rule base containing two rules with two antecedents, the

Given neighbouring rules If x is Ai, then y is Bi, and If x is Aj then y is Bj and observation A\*, this method first maps fuzzy sets Ai, Aj and A\* to real numbers ai, aj and a\* (named as representative values) respectively, using real function f1. Then, the location relationship between A\* and rule antecedents (Ai and Aj) is computed. This is achieved by another mapping function f2, which results in the relative placement factor λ. In contrast to the resolution-based interpolation approaches, the generated relative placement factor in analogy-based fuzzy interpolation approach is a crisp real number. Finally, linear interpolation is implemented using mapping

and move transformation-based fuzzy interpolation approach, the similarity degree between two fuzzy sets A\* and A\*' with the same representative value is expressed as the scale rate s, scale ratio S and move rate M, which is obtained by real function f4. From this, the consequence

between A\* and A\*'. Different approaches have been developed for intermediate rule genera-

Fuzzy interpolation strengthens the power of fuzzy inference by enhancing the robustness of fuzzy systems and reducing the systems' complexity. Common to both classes of fuzzy

tion and final conclusion production from the intermediate rule [17, 55, 63, 65].

) <sup>B</sup>�0 .

using a transformation function f<sup>5</sup> which imposes the similarity degree

and A\* are not identical for most of the situations. In the scale

equals to that of A\* (the

transformation-based approach is shown in Figure 2 and outlined as follows.

function f<sup>3</sup> of λ, which leads to the intermediate rule A�0

Note that the representative value of intermediate rule antecedent A�<sup>0</sup>

[15, 16, 61–63] also belong to this group.

56 Modern Fuzzy Control Systems and Its Applications

3.2. Analogy-based interpolation

given observation), although A�0

3.3. Adaptive fuzzy interpolation

Figure 2. Transformation-based fuzzy interpolation [12].

B\* is calculated from B�0

A Mamdani-style fuzzy rule base is usually implemented through either a data-driven approach [75] or a knowledge-driven approach [76]. The data-driven approach using artificial intelligence approach extracts rules from data sets, while the knowledge-driven approach generates rules by human expert. Due to the limited availability of expert knowledge, datadriven approaches have been increasingly widely applied. However, the application of such

Figure 3. Adaptive fuzzy interpolation [12].

approaches usually requires a large amount of training data, and it often leads to dense rule bases to support conventional fuzzy inference systems, despite of the availability of rule simplification approaches such as Refs. [77, 78].

A recent development or rule base generation has been reported with compact sparse rule bases targeted [79]. This approach firstly partitions the problem domain into a number of subregions and each sub-region is expressed as a fuzzy rule. Then, the importance of each subregion is analysed using curvature value by artificially treating the problem space as a geography object (and high-dimensional problem space is represented as a collection of sub-threedimensional spaces). Briefly, the profile curvature of a surface expresses the extent to which the geometric object deviates from being 'flat' or 'straight', the curvature values of the sub-regions are then calculated to represent how important they are in terms of linear interpolation. Given a predefined threshold, important sub-regions can be identified, and their corresponding rules are selected to generate a raw sparse rule base. The generated raw rule base can then be optimised by fine-tuning the membership functions using an optimisation algorithm. Generic algorithm has been widely used for various optimisation problems, such as Ref. [80], which has also been used in the work of Ref. [79].

Compared to most of the existing rule base generation approaches, the above approach differs in its utilisation of the curvature value in rule selection. Mathematically, curvature is the second derivate of a surface or the slope of slope. The profile curvature [81] is traditionally used in geography to represent the rate at which a surface slope changes whilst moving in the direction, which represents the steepest downward gradient for the given direction. Given a sub-region f(x, y) and a certain direction, the curvature value is calculated as the directional derivative which refers to the rate at which any given scalar field is changing. The overall linearity of a sub-region can thus be accurately represented as the maximum profile curvature value on all directions. From this, those rules corresponding to sub-regions with higher profile curvature values (with respect to a given threshold) are selected, which jointly form the sparse rule base to support fuzzy rule interpolation.

FISs relax the requirement of complete expert knowledge or large data sets covering the entire input domain from the conventional fuzzy inference systems. However, it is still difficult for some real-world applications to obtain sufficient data or expert knowledge for rule base generation to support FISs. In addition, the generated rule resulted from most of the existing rule base generation approaches are fixed and cannot support changing situations. An experience-based rule base generation and adaptation approach for FISs has therefore been proposed for control problems [82]. Briefly, the approach initialises the rule base with very limited rules first. Then, the initialised rule base is revised by adding accurate interpolated rules and removing out-of-date rules guided by the performance index from a feedback mechanism and the performance experiences of rules.

### 4. Fuzzy interpolation with sparse TSK-style rule base

The traditional TSK inference system has been extended to work with sparse TSK fuzzy rule base [21]. This approach, in the same time, also enjoys the benefit from its original version, which directly generates crisp outputs. The extended TSK inference approach is built upon a modified similarity measure which always generates greater than zero similarity degrees between observations and rule antecedents even when they do not overlap at all. Thanks to this property, a global consequence can always be generated by integrating the results from all rules in the rule base.

#### 4.1. Rule firing strength

approaches usually requires a large amount of training data, and it often leads to dense rule bases to support conventional fuzzy inference systems, despite of the availability of rule

A recent development or rule base generation has been reported with compact sparse rule bases targeted [79]. This approach firstly partitions the problem domain into a number of subregions and each sub-region is expressed as a fuzzy rule. Then, the importance of each subregion is analysed using curvature value by artificially treating the problem space as a geography object (and high-dimensional problem space is represented as a collection of sub-threedimensional spaces). Briefly, the profile curvature of a surface expresses the extent to which the geometric object deviates from being 'flat' or 'straight', the curvature values of the sub-regions are then calculated to represent how important they are in terms of linear interpolation. Given a predefined threshold, important sub-regions can be identified, and their corresponding rules are selected to generate a raw sparse rule base. The generated raw rule base can then be optimised by fine-tuning the membership functions using an optimisation algorithm. Generic algorithm has been widely used for various optimisation problems, such as Ref. [80], which

Compared to most of the existing rule base generation approaches, the above approach differs in its utilisation of the curvature value in rule selection. Mathematically, curvature is the second derivate of a surface or the slope of slope. The profile curvature [81] is traditionally used in geography to represent the rate at which a surface slope changes whilst moving in the direction, which represents the steepest downward gradient for the given direction. Given a sub-region f(x, y) and a certain direction, the curvature value is calculated as the directional derivative which refers to the rate at which any given scalar field is changing. The overall linearity of a sub-region can thus be accurately represented as the maximum profile curvature value on all directions. From this, those rules corresponding to sub-regions with higher profile curvature values (with respect to a given threshold) are selected, which jointly form the sparse

FISs relax the requirement of complete expert knowledge or large data sets covering the entire input domain from the conventional fuzzy inference systems. However, it is still difficult for some real-world applications to obtain sufficient data or expert knowledge for rule base generation to support FISs. In addition, the generated rule resulted from most of the existing rule base generation approaches are fixed and cannot support changing situations. An experience-based rule base generation and adaptation approach for FISs has therefore been proposed for control problems [82]. Briefly, the approach initialises the rule base with very limited rules first. Then, the initialised rule base is revised by adding accurate interpolated rules and removing out-of-date rules guided by the performance index from a feedback

The traditional TSK inference system has been extended to work with sparse TSK fuzzy rule base [21]. This approach, in the same time, also enjoys the benefit from its original version,

simplification approaches such as Refs. [77, 78].

58 Modern Fuzzy Control Systems and Its Applications

has also been used in the work of Ref. [79].

rule base to support fuzzy rule interpolation.

mechanism and the performance experiences of rules.

4. Fuzzy interpolation with sparse TSK-style rule base

The modified similarity measure is developed from the work described in Ref. [83]. Suppose there are two fuzzy sets A and A<sup>0</sup> in a normalised variable domain. Without loss generality, a fuzzy set with any membership can be approximated by a polygonal fuzzy membership function with n odd points. Therefore, A and A<sup>0</sup> can be represented as A ¼ ða1, a2, …anÞ and A<sup>0</sup> ¼ ða<sup>0</sup> 1, a<sup>0</sup> <sup>2</sup>, …a<sup>0</sup> <sup>n</sup>0Þ, as shown in Figure 4. The similarity degree S(A,A<sup>0</sup> ) between A and A<sup>0</sup> is computed as:

$$S(A, A') = \left(1 - \frac{\sum\_{i=1}^{n} |a\_i - a'\_i|}{n}\right) (DF)^{\tilde{B}(sup\_{A'} supp\_{A'})} \frac{\min\left(\mu(\mathsf{C}\_A), \mu(\mathsf{C}\_{A'})\right)}{\max\left(\mu(\mathsf{C}\_A), \mu(\mathsf{C}\_{A'})\right)},\tag{12}$$

where cA is the centre of gravity of fuzzy sets A, and μ(cA) is the membership of the centre of gravity of fuzzy set A; DF represents a distance factor which is a function of the distance between two concerned fuzzy sets, and BðsuppA, suppA0Þ is defined as follows:

$$B(\operatorname{supp}\_{A'} \operatorname{supp}\_{A'}) = \begin{cases} 1, & \operatorname{if} \operatorname{supp}\_A + \operatorname{supp}\_{A'} \neq 0, \\ 0, & \operatorname{if} \operatorname{supp}\_A + \operatorname{supp}\_{A'} = 0, \end{cases} \tag{13}$$

where suppA and suppA<sup>0</sup> are the supports of A and A<sup>0</sup> , respectively.

In Eq. (13), BðsuppA, suppA0Þ is used to determine whether distance factor is considered. That is, if both A and A<sup>0</sup> are of crisp values, the distance factor DF will not take into consideration during the calculation of the similarity degree; otherwise, DF will be considered. The centre of gravity of a fuzzy set is commonly approximated as the average of its odd points. That is:

Figure 4. An arbitrary fuzzy set with n odd points.

$$c\_A = \frac{a\_1 + a\_2 + \dots + a\_n}{n},\tag{14}$$

$$
\mu(c\_A) = \frac{\mu(a\_1) + \mu(a\_2) + \dots + \mu(a\_n)}{n}.\tag{15}
$$

The distance factor DF is represented as:

$$DF = 1 - \frac{1}{1 + e^{-hd + 5}} \tag{16}$$

where d is the distance between the two fuzzy sets, and h(h > 0) is a sensitivity factor. The smaller the value of h is, the more sensitive the similarity degree to their distance is. The value of h is usually within the range of (20, 60), but the exact value is problem specific.

#### 4.2. Fuzzy interpolation

Using the modified similarity measure as traduced above, the similarity between any given observation and a rule antecedent is always greater than zero. This means that all the rules in the rule base are fired for inference. Therefore, if only a sparse rule base is available and a given observation is not covered by the sparse rule base, a consequence still can be generated by firing all the rules in the rule base. The inference process is summarised as below:


$$
\alpha\_i = \mathbb{S}(A^\*, A\_i) \land \mathbb{S}(B^\*, B\_i). \tag{17}
$$

3. Compute the consequence of each rule in line with the given input and the polynomial function in rule consequent:

$$f\_i(A^\*, A\_i) = \alpha\_i \cdot c\_{A^\*} + b\_i \cdot c\_{B^\*} + c\_i. \tag{18}$$

4. Obtain the final result z by integrating the sub-consequences from all m rules in the rule base:

$$z = \frac{\sum\_{i=1}^{n} \alpha\_{i} f\_{i}(A^\*, B^\*)}{\sum\_{i=1}^{n} \alpha\_{i}}.\tag{19}$$

#### 5. Applications of fuzzy interpolation

Fuzzy interpolation systems have been successfully applied to a number of real-world problems including Refs. [23, 22, 25, 52, 57], two of which are reviewed in the section below.

#### 5.1. Truck backer-upper control

cA <sup>¼</sup> <sup>a</sup><sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> an

<sup>μ</sup>ðcAÞ ¼ <sup>μ</sup>ða1Þ þ <sup>μ</sup>ða2Þ þ … <sup>þ</sup> <sup>μ</sup>ðan<sup>Þ</sup>

DF <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

of h is usually within the range of (20, 60), but the exact value is problem specific.

firing all the rules in the rule base. The inference process is summarised as below:

α<sup>i</sup> ¼ SðA�

z ¼

X<sup>n</sup> i¼1 αif <sup>i</sup> ðA� , B� Þ

(Ai, Bi) and the input values (A\*, B\*) based on Eq. (12).

f i ðA�

the input items and rule antecedents as calculated in Step 1:

where d is the distance between the two fuzzy sets, and h(h > 0) is a sensitivity factor. The smaller the value of h is, the more sensitive the similarity degree to their distance is. The value

Using the modified similarity measure as traduced above, the similarity between any given observation and a rule antecedent is always greater than zero. This means that all the rules in the rule base are fired for inference. Therefore, if only a sparse rule base is available and a given observation is not covered by the sparse rule base, a consequence still can be generated by

, AiÞ and SðB�

2. Determine the firing strength of each rule by integrating the matching degrees between

3. Compute the consequence of each rule in line with the given input and the polynomial

4. Obtain the final result z by integrating the sub-consequences from all m rules in the rule base:

X<sup>n</sup> i¼1 αi

Fuzzy interpolation systems have been successfully applied to a number of real-world problems including Refs. [23, 22, 25, 52, 57], two of which are reviewed in the section below.

, AiÞ ∧ SðB�

The distance factor DF is represented as:

60 Modern Fuzzy Control Systems and Its Applications

1. Calculate the matching degree SðA�

function in rule consequent:

5. Applications of fuzzy interpolation

4.2. Fuzzy interpolation

<sup>n</sup> , <sup>ð</sup>14<sup>Þ</sup>

<sup>n</sup> : <sup>ð</sup>15<sup>Þ</sup>

<sup>1</sup> <sup>þ</sup> <sup>e</sup>�hdþ<sup>5</sup> <sup>ð</sup>16<sup>Þ</sup>

, BiÞ between each pair of rule antecedent

, BiÞ: ð17Þ

: ð19Þ

, AiÞ ¼ α<sup>i</sup> � cA� þ bi � cB� þ ci: ð18Þ

Backing a trailer truck to a loading dock is a challenging task for all yet the most skilled truck drivers. Due to the difficulties, this challenge has been used as a control benchmark problem with various solutions proposed [75, 84, 85]. For instance, an artificial neural network has been applied to this problem, but a large amount of training data is required [84]. An adaptive fuzzy control system was also proposed for this problem, but the generation of the rule base is computationally expensive. Another solution combines empirical knowledge and data [85]. That is, a combined fuzzy rule base is generated by joining the previously generated rules (data-driven) and linguistic rules (expert knowledge-driven). More recently, a supervisory control system was proposed with fewer number of state variables required due to its capability to the decomposition of the control task, thus relieving the curse of dimensionality [86].

Fuzzy interpolation system has also been applied to the trailer truck backer-upper problem [52] to further reduce the system complexity. The problem can be formally formulated as θ ¼ fðx, y, ∅Þ. Variables x and y represent the coordinate values corresponding to horizontal and vertical axes; ∅ refers to the azimuth angle between the truck's onward direction and the horizontal axis; and θ is the steering angle of the truck. Given that enough clearance is present between the truck and loading lock in most cases, variable y can be safely omitted and hence results in a simplified formula to θ ¼ fðx, ∅Þ. By evenly partitioning each variable domain into three fuzzy sets, nine (i.e. 3\*3) fuzzy rules were generated using FISMAT [87] and each of which is denoted as IF x is A AND ∅ is B THEN θ is C, where A, B and C are three linguistic values. Noting that domain partitions appear to be symmetrical in some sense, the three rules which are flanked by other rule pairs were removed from the rule base resulting a more compact rule base with only six fuzzy rules.

If the traditional fuzzy inference system were applied, the sparse rule base would cause a sudden break of the truck for some situations as no rule would be fired when the truck is in the position that can be represented by the omitted rules. In this case, fuzzy interpolation is naturally applied and the sudden break problem can be avoided. In addition, thanks to the great generalisation ability of the fuzzy interpolation systems, smooth performance is also demonstrated compared to the conventional fuzzy inference approaches. This study clearly demonstrates that fuzzy interpolation systems are able to simplify rule bases and support inferences with sparse rule bases.

#### 5.2. Heating system control

The demotic energy waste contributes a large part of CO<sup>2</sup> emissions in the UK, and about 60% of the household energy has been used for space heating. Various heating controllers have been developed to reduce the waste of energy on heating unoccupied properties, which are usually programmable and developed using a number of sensors. These systems are able to successfully switch off heating systems when a property is unoccupied [88–92], but they cannot intelligently preheat the properties by warming the property before users return home without manual inputs or leaving the heating systems on unnecessarily for longer time. A smart home heating controller has been developed using a FIS, which allows efficient home heating by accurately predicting the users' home time using users' historic and current location data obtained from portable devices [23].

The overall flow chart of the smart home heating system is shown in Figure 5. The controller first extracts the resident's location and moving information. There are four types of residents' location and moving information that need to be considered: At Home, Way Back Home, Leaving Home and Static (i.e. at Special Location). The user's current location and moving states are obtained effectively using the GPS information provided by user's portable devices. From this, if the resident's current state is At Home, the algorithm terminates; and if the residents' current state is Leaving Home, that is the residents are moving away from home,

Figure 5. The flow chart of the heating controller [23].

the boiler is off and the system will check the resident's location and moving information again in a certain period of time. Otherwise, the time to arriving home (denoted as TAH) is predicted and the time to preheat the home to a comfortable temperature (denoted as TPH) is also calculated, based on the resident's current situation and the current environment around home.

The user's current travel modes (i.e. driving, walking or bicycling) can be detected by employing a naïve Bayes classifier [93] using the GPS information. Then the travel distance and time between the current location and home can be estimated using Google Distance Matrix API. Note that the time spent on different locations may vary significantly, and also different residents usually spend different amount of times at the same special location as people have their own living styles. The time that the residents spent at the current location is therefore estimated using fuzzy interpolation systems, thanks to the complexity of the problem. In particular, the fuzzy interpolation engine takes five fuzzy inputs and produces one fuzzy output which is the estimate of the time to getting home. The five inputs are the current location, the day of the week, the time of the day, the time already spent at the current location and the estimated travel between the current location and home.

If each input domain is fuzzy partitioned by 5 to 13 fuzzy, tens of thousands of rules will be resulted which requires significant resources during inferences. The proposed system, however, has selected the most important 72 rules forming a sparse rule base to support fuzzy rule interpolation, which significantly improve the system performance. Once the home time is calculated, the home can then be accurately preheated based on a heating gain table developed based on the particular situation and environment of a concerned property [91]. This work has been applied to a four-bedroom detached house with a total hearing space of 100 m<sup>2</sup>ðf loor areaÞ � <sup>2</sup>:4 mðhightÞ. The house is heated by a 15 kW heating boiler. The study has shown that the controller developed using fuzzy inference has successfully reduced the burning time of the boiler for heating and more accurately preheat the home.

Despite of the success of the applications introduced above, there is a potential for FISs to be applied to more and larger scales real-world problems, especially in the field of system control. Note that robotics has taken the centre in the control field to perform tasks from basic robot calligraphy system [94] to complex tasks which require hand-eye (camera) coordination [95]. FISs can also be applied to such advanced areas in the field of robotics, which require further investigation.

## 6. Conclusions

heating by accurately predicting the users' home time using users' historic and current location

The overall flow chart of the smart home heating system is shown in Figure 5. The controller first extracts the resident's location and moving information. There are four types of residents' location and moving information that need to be considered: At Home, Way Back Home, Leaving Home and Static (i.e. at Special Location). The user's current location and moving states are obtained effectively using the GPS information provided by user's portable devices. From this, if the resident's current state is At Home, the algorithm terminates; and if the residents' current state is Leaving Home, that is the residents are moving away from home,

Begin

Check User Location State

Away from home

TETAgreater

Calculate TWARM FRI Reasoning

TWARM TETA

N

Compare TWARM, TETA

Heating System ON

Home Yet ?

END

Y

Special Location Meet ?

Y

Similar Or Twarm greater

Way Back, Special Location

At Home

N

Calculate Travel Time

TETA

data obtained from portable devices [23].

62 Modern Fuzzy Control Systems and Its Applications

Heating System OFF

Heating System OFF

Figure 5. The flow chart of the heating controller [23].

This chapter reviewed fuzzy interpolation systems and their applications in the field of control. There are basically two groups of fuzzy interpolation approaches using the two most common types of fuzzy rule bases (i.e. Mamdani-style rule bases and TSK-style rule bases) to supplement the two groups of widely used fuzzy inference approaches (i.e. the Mamdani inference and the TSK inference). The applications of fuzzy interpolation systems have also been discussed in the chapter which demonstrate the power of the approaches. FISs can be further improved despite of its promising performance. Firstly, type-2 FISs have already been proposed in the literature, but how type-2 FISs can be applied in real-world applications requires further investigation. Also, more theoretical analysis for FISs is needed to mathematically prove the convergence property of the approaches. In addition, most of the existing fuzzy interpolation approaches are proposed as a supplementary of the existing fuzzy inference models. It is interesting to investigate the development of a united platform which integrates both the existing fuzzy models and fuzzy inference systems such that the new system can benefit from both approaches.

## Acknowledgements

This work was jointly supported by the National Natural Science Foundation of China (No. 61502068) and the China Postdoctoral Science Foundation (Nos. 2013M541213 and 2015T80239).

## Author details

Longzhi Yang<sup>1</sup> \*, Zheming Zuo<sup>1</sup> , Fei Chao2 and Yanpeng Qu3


## References


[5] Yang L, Neagu D. Towards the integration of heterogeneous uncertain data. In: 2012 IEEE 13th International Conference on Information Reuse & Integration (IRI). 2012. pp. 295– 302

proposed in the literature, but how type-2 FISs can be applied in real-world applications requires further investigation. Also, more theoretical analysis for FISs is needed to mathematically prove the convergence property of the approaches. In addition, most of the existing fuzzy interpolation approaches are proposed as a supplementary of the existing fuzzy inference models. It is interesting to investigate the development of a united platform which integrates both the existing fuzzy models and fuzzy inference systems such that the new

This work was jointly supported by the National Natural Science Foundation of China (No. 61502068) and the China Postdoctoral Science Foundation (Nos. 2013M541213 and

, Fei Chao2 and Yanpeng Qu3

1 Department of Computer and Information Sciences, Northumbria University, Newcastle, UK

3 Information Science and Technology College, Dalian Maritime University, Dalian, PR China

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2 School of Information Science and Engineering, Xiamen University, Xiamen, PR China

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\*, Zheming Zuo<sup>1</sup>

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\*Address all correspondence to: longzhi.yang@northumbria.ac.uk

Acknowledgements

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Longzhi Yang<sup>1</sup>

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67974

#### Abstract

The chapter deals with implementing fuzzy logic for transition of descriptions in natural language to formal fuzzy and stochastic models and their further optimization in terms of effectiveness and efficiency of information modeling and prediction systems. The theoretical methods are implemented in lifelong learning business for developmentspecific virtual trainings for adult students.

Keywords: fuzzy, fuzzy set, fuzzy risk, lifelong learning, virtual learning, training method prediction, fuzzy analysis, complex definition area

## 1. Introduction

In the first part of the chapter the author examines challenges in transforming subject description in a natural language to a formal model. Establishing collaboration between specialists in different knowledge areas (technology, information, business, etc.) is usually a very complicated problem. In the process of common work there is a strong need to estimate model clearness, effectiveness, and efficiency for specialists in different knowledge areas. We consider effectiveness as correct interpretation of a subject in a real world by a model. And efficiency is calculated based on the share of the service states of the model.

In the second part, the author shows the implementation of the approach in virtual training design. Developing specific training methods for adult lifelong students is a very complicated task. We implemented fuzzy-based analysis to determine the best learning methods for different student groups and course types.

The special section is devoted to adapting and implementing virtual training for hard of hearing people.

## 2. Extension for fuzzy model definition area

#### 2.1. Introduction

Information systems are widely spread in our daily life: offices, industry, and home. Each system represents or controls an extra object, such as household appliances or industrial control devices. Efficiency of those information and control systems depends on a wide range of factors. Wide-known methods for control systems' synthesis demand the formal model represented in analytical format [1]. The main problem of the discrete automate models is the strong necessity of formalization at the beginning of the process. It is a rather difficult task for complicated technological and information processes. Moreover, it is more appropriate to set a task of the adaptive control system with parametric adaptation features synthesis.

Let us consider the control system for an informational object as a "black box" with two sets of inputs and outputs [3]. Two main inputs are a math (or analytical) model, and data. One of the important features is a number and percentage of internal states of an information system/ control device. Internal states are always necessary to control stability of work, prevent user's mistakes, etc. These internal states and their properties require additional computing resources. The author examines an approach aimed at balancing advantages and resource usage of internal states [2].

Meanwhile these both inputs are (or can be) cross-dependent. Creating an analytical model as the first step of the system synthesis is based on intrinsic and semantic analysis of input data. Establishment of collaboration between specialists in different knowledge areas (technology, information, business, or others) is usually one of the main problems and challenges almost in all practice-based projects [3]. Factor analysis or lingual models may provide partial decision of the problem. Lingual models combine a system description in both formal and descriptive terms. A restriction of linguistic models is in their insufficient formalization and a high level of dependence on subjective expert appraisals.

Fuzzy models implementation ensures an analysis of processes in technical and informationbased systems with nonlinear and/or multifactor-based behavior [1, 5]. The author presents the approach that combines model analysis based on natural language with strict formal systems by implementing fuzzy logic approaches.

#### 2.2. General approach based on lingual models

We consider a process of analytical model synthesis as a first phase of a control model synthesis. The first important step at this phase is transition of a description based on natural language to a formal model. We should consider an interaction between specialists in different knowledge areas while creating such models as they may use different terms. Another important task is establishing back coupling between input data and information/control system. It is quite necessary for creating information and control systems with adaptive features. We implement fuzzy logic to solve the task of information system synthesis for complicated technological objects.

Implementing fuzzy logic ensures the opportunity of a semantic analysis of the object description made on natural-based language due to partial entrance of an element in one or several sets. The conceptual structure of the research area is represented as a set of abstract entities relying on concepts and terms of both a natural language and fuzzy sets. Next, we need to extend the existing fuzzy sets' model as an analytical model contains features, appropriate to elements of analyzable object, and own properties of the model, that provide its integrity.

Second step should be to deal with input data analysis to adapt the model and control system to environment changes. We do not need to change our model but make it adaptive. We need to establish clear borders for research object and its analytical model. To resolve this task, we need also to extend current fuzzy sets and models' methods and features. In this work, we present an extension range of definition of fuzzy sets and its elements' logical division to two sets: corresponding to object properties and internal properties of a system itself.

#### 2.3. Model analysis in a partially defined environment based on fuzzy sets

Any analytical model has several internal features that ensure its internal integrity and reliability of a control system to controlled object. One may implement internal model states for internal data integrity control, exchange, additional logic control, etc.

Let's name a set of objects to be synthesized as S ¼ {a[i]}. Each noun in a lingual model compares to an object a[i] with value (weight) a(i) and functionality H(a[i]). Lets' define an expansion of a definition range values for fuzzy attributes to imaginary area:

$$\begin{cases} \text{ } \text{\(\ast\)} \text{\(\)}, & \text{if an object exists only in a formal model} \\ \text{ } a \text{\(i\)} = a \text{ (i)} \quad \text{if an object compares to a modelling subject} \end{cases} \tag{1}$$

Based on the above, it is obvious that model S always consists of objects with both rational and imaginary features and functionality weights. Hþ and H�[4, 6, 7]. In this case, total number of object a[i] features is h(n).


2. Extension for fuzzy model definition area

72 Modern Fuzzy Control Systems and Its Applications

Information systems are widely spread in our daily life: offices, industry, and home. Each system represents or controls an extra object, such as household appliances or industrial control devices. Efficiency of those information and control systems depends on a wide range of factors. Wide-known methods for control systems' synthesis demand the formal model represented in analytical format [1]. The main problem of the discrete automate models is the strong necessity of formalization at the beginning of the process. It is a rather difficult task for complicated technological and information processes. Moreover, it is more appropriate to set a

Let us consider the control system for an informational object as a "black box" with two sets of inputs and outputs [3]. Two main inputs are a math (or analytical) model, and data. One of the important features is a number and percentage of internal states of an information system/ control device. Internal states are always necessary to control stability of work, prevent user's mistakes, etc. These internal states and their properties require additional computing resources. The author examines an approach aimed at balancing advantages and resource

Meanwhile these both inputs are (or can be) cross-dependent. Creating an analytical model as the first step of the system synthesis is based on intrinsic and semantic analysis of input data. Establishment of collaboration between specialists in different knowledge areas (technology, information, business, or others) is usually one of the main problems and challenges almost in all practice-based projects [3]. Factor analysis or lingual models may provide partial decision of the problem. Lingual models combine a system description in both formal and descriptive terms. A restriction of linguistic models is in their insufficient formalization and a high level of

Fuzzy models implementation ensures an analysis of processes in technical and informationbased systems with nonlinear and/or multifactor-based behavior [1, 5]. The author presents the approach that combines model analysis based on natural language with strict formal systems

We consider a process of analytical model synthesis as a first phase of a control model synthesis. The first important step at this phase is transition of a description based on natural language to a formal model. We should consider an interaction between specialists in different knowledge areas while creating such models as they may use different terms. Another important task is establishing back coupling between input data and information/control system. It is quite necessary for creating information and control systems with adaptive features. We implement fuzzy logic to solve the task of information system synthesis for complicated

task of the adaptive control system with parametric adaptation features synthesis.

2.1. Introduction

usage of internal states [2].

dependence on subjective expert appraisals.

by implementing fuzzy logic approaches.

technological objects.

2.2. General approach based on lingual models

4. A functional as a summary (vector) weight of features divided into rational and imaginary parts a[i] -> H.

A model functional parameter may be represented as the following complex number:

$$H = H + + \text{j}H - \tag{2}$$

#### 2.4. Extending fuzzy logic procedures for analysis and synthesis of information systems

Below, we show the extension is correct and does not break the fuzzy logic postulates. The definition area would be the following:

$$H\left(A|B\right) \in \left[0; \approx[+j([0, \approx])\right)\tag{3}$$

Lemmas:

If A is the element of a real object and B is the internal element:

$$H\left(A|B\right) \in \left[0; \ast\right] \tag{4}$$

If both A, B are internal elements:

$$H \ (A|B) \in \mathfrak{j}([0; \curvearrowright]) \tag{5}$$

$$H(\mathcal{U}) = \approx \tag{6}$$

U is a universal set.

We provided a transformation of the traditional fuzzy sets axioms and features to a specific fuzzy model for using complex-based definition area to specify features of models for synthesis of control automation systems.

#### 2.4.1. Fuzzy model synthesis based on lingual model analysis for technological and information objects

First, we need to transform a lingual model based on subject area terms and definitions to a metalanguage-based model. This model still may include some terms from its predecessor the certain lingual model. Let us mark sets of keywords, and consider them as fuzzy model objects. At the first step, we may consider all nouns as fuzzy model objects [1–3].

Next let's define a transition between fuzzy objects:

$$\begin{cases} \text{fi} \\ a[i] \to a[i+1], \text{fi is an } a[i] \text{ object's method} \end{cases} \tag{7}$$

If weight a(i) is an imaginary one, then the object a[i] does not match any object in a lingual model, and only meta model contains it.

Then, based on δ-operation, as it is defined in common fuzzy logic, we provide the following transformation between logically tied objects:

$$\begin{cases} a[i] \ \delta \dot{a}[i+1] \ \Rightarrow \ a[i+1] \\ \qquad \text{fi}+1 \\ a[i] \ \delta i+1 \ [i+1] \ \Rightarrow \ a[i] \ \rightarrow \ a[i+2] \end{cases} \tag{8}$$

So, we can formalize an object's method based on a 2-step δ-operation.

Let's introduce a weight of a fuzzy object H (M)¼A. We consider ∑H(a[i])>¼H(P)), as the model contains both objects and their methods f(i), that define a consequence of transitions between objects in a model. f(i) methods are described by γ and δ operations. δ -operation defines a consequence of operations, events, and nodes of a model, and γ-operation defines weights (for

example, possibility, availability, resilience, etc.) for nodes, events, operations. If we take into consideration both types and an optimization factor H(M)¼A!max, then the following equality is true:

HðPÞ ¼ Hða½1�γHða½2�γ γHða½i�γ γHða½n� ð9Þ

The following set of conditions provides model and based on it control system integrity:

$$\begin{cases} \mathbf{H}(\mathbf{M}) \to \max; \\ \mathbf{H}(\mathbf{M}) = \Gamma a[i] \,\Delta a[i]; \\ \nabla a[i], \mathbf{H}(a[i] \neq 0; \\ \nabla a[i], \mathbf{H}(a[i]) \to \max; \\ \gamma \mathbf{i} \in a[i], \ \mathbf{\delta i} \in \, a[i]; \\ \mathbf{f}(\mathbf{i}) = \gamma \mathbf{i} \cup \delta \mathbf{i} \end{cases} \tag{10}$$

The following operations provide structuration of fuzzy sets in A:

1. Establish relations between sets:

1.1. Relation of entrance Ai ∈ Aj

H ðAjBÞ ∈ ½0; ∞½þjð½0;∞½Þ ð3Þ

H ðAjBÞ ∈ ½0;∞½ ð4Þ

H ðAjBÞ∈ jð½0;∞½Þ ð5Þ

HðUÞ ¼ ∞ ð6Þ

ð7Þ

ð8Þ

Lemmas:

If both A, B are internal elements:

74 Modern Fuzzy Control Systems and Its Applications

sis of control automation systems.

Next let's define a transition between fuzzy objects:

�

model, and only meta model contains it.

transformation between logically tied objects:

8 < :

So, we can formalize an object's method based on a 2-step δ-operation.

fi

U is a universal set.

If A is the element of a real object and B is the internal element:

We provided a transformation of the traditional fuzzy sets axioms and features to a specific fuzzy model for using complex-based definition area to specify features of models for synthe-

2.4.1. Fuzzy model synthesis based on lingual model analysis for technological and information objects First, we need to transform a lingual model based on subject area terms and definitions to a metalanguage-based model. This model still may include some terms from its predecessor the certain lingual model. Let us mark sets of keywords, and consider them as fuzzy model

a½i� ! a½iþ1�; fi is an a½i� object's method

If weight a(i) is an imaginary one, then the object a[i] does not match any object in a lingual

Then, based on δ-operation, as it is defined in common fuzzy logic, we provide the following

a½i� δi þ1 ½iþ1� ) a½i� ! a½iþ2�

Let's introduce a weight of a fuzzy object H (M)¼A. We consider ∑H(a[i])>¼H(P)), as the model contains both objects and their methods f(i), that define a consequence of transitions between objects in a model. f(i) methods are described by γ and δ operations. δ -operation defines a consequence of operations, events, and nodes of a model, and γ-operation defines weights (for

a½i� δia½iþ1� ) a½iþ1� fi þ 1

objects. At the first step, we may consider all nouns as fuzzy model objects [1–3].

1.2. Relation of inheritance. Ai inherits Aj if a(i) ∈ Ai has the fixed set of values, and methods no less than a(j) ∈ A(j).

2. Let's introduce meta-set M, which describes a set of sets:

$$\text{Ai}: \ M = \mathsf{U}(\mathsf{n}(\mathsf{i}), \mathsf{u}\mathsf{h}(\mathsf{i}, \mathsf{j})), \tag{11}$$

And a system of their internal relationships, where n(i), Ai's unique identifier (usually string name); h(i,j), a weight of relation between Ai and Aj sets.

An optimization of model parameters provides by consequent iterations. A rate, which weights reliability between fuzzy meta-model and preceding lingual model, is one of the most important optimization criteria. It is based on a feature ∇a½i�∈ M, that indicates a belongingness of a set element to a real or imaginary areas.

2.4.2. Fuzzy model synthesis algorithm based on a preceding lingual model


$$\mathbf{h} \rhd \mathbf{n} \rhd = \mathbf{1}, \nabla \mathbf{h}(\mathbf{j})(A\mathbf{i}) = \mathbf{f}(\mathbf{h}(\mathbf{j}), \mathbf{A}(\mathbf{i}), \mathbf{H}\mathbf{j}, [A\mathbf{i}] \triangle \mathbf{i}).$$

8. Establish fuzzy model M ðAi; H½aðjÞ�.

Synthesis of a control system for technical and/or information object is based on the defined fuzzy set M.

### 2.5. Scope definition for fuzzy sets usage in control and simulation systems for technical and information systems

While designing formal model (including fuzzy one), it is necessary to estimate the following items:


$$\mathbf{M} = \mathbf{A} \cup \mathbf{B} \cup \mathbf{C} \cup \cup \mathbf{N} \tag{12}$$

#### 2.5.1. Internal consistency criteria

1. Redundancy property:

$$\forall \mathsf{H}(a[i]\exists a[j] > 0$$

2. Compliance property:

$$\forall \mathsf{H}(a[i]|a[j] \neq \infty)$$

3. Efficiency and resilience balance:

$$\begin{cases} \sum a[i](\mathbf{h}+) > 0\\ \sum a[i](\mathbf{h}-) > 0\\ \sum a[i](\mathbf{h}) > 0 \end{cases} \tag{13}$$

#### 2.5.2. Compliance to modeling object criteria

(1) Compliance between M - model transformations results, and an object fact features: implementing of any possible track of operations of M model cannot lead an object to a prohibited state.

$$\forall \text{H}(\text{Xi}(\text{Ai})) > 0,\tag{14}$$

or

5. Create subsets S(i) as based on the S set. S(i) elements are united into a certain subset

6. Create sets H(j) containing experts', knowledgebase articles and other data sources'

n >¼ 1;∇hðjÞðAiÞ ¼ fðhðjÞ; AðiÞ; Hj; ½Ai�\CiÞ:

Synthesis of a control system for technical and/or information object is based on the defined

2.5. Scope definition for fuzzy sets usage in control and simulation systems for technical

While designing formal model (including fuzzy one), it is necessary to estimate the following


∀Hða½i�βa½j� > 0

∀Hða½i�ja½j� 6¼ ∞

<sup>X</sup>a½i�ð<sup>h</sup> þ Þ <sup>&</sup>gt; <sup>0</sup> <sup>X</sup>a½i�ð<sup>h</sup> �Þ <sup>&</sup>gt; <sup>0</sup>

8 >>><

>>>:

<sup>X</sup>a½i�ðh<sup>Þ</sup> <sup>&</sup>gt; <sup>0</sup>

M ¼ A ∪ B ∪ C ∪ ∪ N ð12Þ

ð13Þ


7. Create sets A(i) including attributes a(i), as Ai ∇ S,t(i)A1, An, where

according to C(i) criteria.

76 Modern Fuzzy Control Systems and Its Applications

weights: Ci ! HjðCiÞ.

fuzzy set M.

items:

and information systems

8. Establish fuzzy model M ðAi; H½aðjÞ�.

behavior if used in common [2, 3]:

2.5.1. Internal consistency criteria

1. Redundancy property:

2. Compliance property:

3. Efficiency and resilience balance:

Implementing of any possible track in M, model can transform the controlled system or object R into a possible state or common null state

$$\forall H(\text{Xi}(\text{Ai})) >>= 0 \tag{15}$$

Thus, we consider a fuzzy model is applicable in case that its real functional is positive and there is at least one set of allowed methods that transform a control system from its initial to final state.

#### 2.6. Conclusion

Finally, we found that the synthesis of control systems based on descriptive models of natural language may be adequately implemented based on fuzzy sets. Logical separation of elements of fuzzy sets, in which the real domain includes the attributes and functional elements that describe the state of an object, but to the imaginary one—own internal state of the model and the management system that are required to make it operational. Based on this logical separation, we may estimate effectiveness and resilience of control system.

Finally, the authors resume that automated systems' synthesis is appropriately presented and formalized by fuzzy sets' models. Fuzzy logic definition area has been extended to an imaginary area. We established the logical division of model components to real and imaginary areas per their role. Internal objects of a model are presented in the imaginary area, and objects that describe the modeling system to the real area. We introduced necessary functional extensions for fuzzy logic to operate with logical extension.

Transformation algorithm is developed, and we recommend the certain implementation area for it.

## 3. Defining appropriate training methods for lifelong learning organization

#### 3.1. Introduction

The author has developed a method and algorithms of fuzzy analysis for lingual models with complex digits' implementation. The author used the approach that differentiates native data, attributes of an object and internal model data, and attributes [2, 3]. Dividing these classes into real and imaginary leads to the decrease of dimensions in a model, and in this way to the decrease of computing capacity. This fact allows to decrease the risk of incorrect interpretation of results, and it provides also an opportunity to estimate costs of efficiency.

This article is devoted to implementing fuzzy analysis to define and implement various virtual training methods in a lifelong learning educational organization and reaches the highest possible satisfaction level by different categories of adult students as defined in Ref. [4].

#### 3.2. Big challenges in lifelong learning

The lifelong professional learning training center offers short-term trainings and postdiploma programs to upgrade professional skills or gain a new specialization for adult professionals. A lot of students take multiple courses as bundles or periodically in accordance to new versions of software, technological equipment, or professional standards. It's of great importance for the training organization to analyze big data interdependencies to find out trends, develop new courses, make targeted offers for students, and create specific training methods for certain client groups. Since 2009, the author has been deeply involved into developing and implementing various virtual—online, and blended—training methods. During this work, the author carried out a regular analysis of data from different sources to determine customer requirements, demands for courses, and ways of their representation, technical, and mythological opportunities [4, 5]. The goal of those continual research efforts was the development of strongly targeted training options for certain student groups and courses. The fuzzy-based modeling is considered as the most appropriate approach to the task, because students', customers', trainers', and other staff's feedback, requirements, as well as demand estimations, are mostly represented as a nonformal or mixed way. For example, rating A in a feedback means "more than I can expect." It is obvious that the level of expectations differs among students, and customer representatives.

#### 3.3. Opportunities and threats in lifelong learning

Based on M\_o\_R™ and Total Risk Management® concepts fundamental characteristics of any risk define organization behavior for it, for example: tolerance level, impact, mitigation and contingency strategies, management level, as well as level of financial reserves. While examining risk nature we often consider that a single risk belongs to different characteristic sets. For example, a risk of incorrect professional behavior can belong to human and organizational, and technical sets simultaneously. Therefore, we can create a fuzzy description of a risk:

$$r(i) \in O, r(i) \in T, \mathbf{r(i)} \, H \tag{16}$$

where O, T, H, fuzzy sets (organizational, technical, human features).

Implementing risk analysis in fuzzy terms ensures complex analysis for risk source, impact, mitigation, and contingency. The author examined complex risk analysis for portfolio (both projects and operational activities) of virtual learning methods in a lifelong adult training center. As a service-based and private user (a student) oriented business, its success depends dramatically on a subjective personalized opinion of students and partially of corporate HR managers. Their feedbacks are represented both in partially formalized manner, and comments in a natural language.

Another challenge concerns representing risk dependencies, or so-called domino effect. As it's a rather complicated task to formalize risks interdependencies, we can implement an approach, starting with an informal description in a natural language with further formalizing it by means of fuzzy-based algorithms.

The fuzzy analysis is the very appropriate tool to transfer statements in a natural language into a formal model, and explore threats and opportunities. The fuzzy analysis is implemented as described by the author in Ref. [5]. Identifying and analyzing risks, and their interdependencies, we include both negative (threat) and positive (opportunity) parts of risk analysis with the primary aim for finding new opportunities for development and quality improvement.

Main threats for an adult professional training organization are in customer dissatisfaction, and on the opposite main opportunities are based on reaching continuous education of students personally, and corporate customers. Let's examine a simple example about modern technology-based virtual learning implementation, and consider online learning process. There are several main opportunities of online learning for an educational organization, which are:


On the other side, there are threats:


real and imaginary leads to the decrease of dimensions in a model, and in this way to the decrease of computing capacity. This fact allows to decrease the risk of incorrect interpretation

This article is devoted to implementing fuzzy analysis to define and implement various virtual training methods in a lifelong learning educational organization and reaches the highest possible satisfaction level by different categories of adult students as defined in Ref. [4].

The lifelong professional learning training center offers short-term trainings and postdiploma programs to upgrade professional skills or gain a new specialization for adult professionals. A lot of students take multiple courses as bundles or periodically in accordance to new versions of software, technological equipment, or professional standards. It's of great importance for the training organization to analyze big data interdependencies to find out trends, develop new courses, make targeted offers for students, and create specific training methods for certain client groups. Since 2009, the author has been deeply involved into developing and implementing various virtual—online, and blended—training methods. During this work, the author carried out a regular analysis of data from different sources to determine customer requirements, demands for courses, and ways of their representation, technical, and mythological opportunities [4, 5]. The goal of those continual research efforts was the development of strongly targeted training options for certain student groups and courses. The fuzzy-based modeling is considered as the most appropriate approach to the task, because students', customers', trainers', and other staff's feedback, requirements, as well as demand estimations, are mostly represented as a nonformal or mixed way. For example, rating A in a feedback means "more than I can expect." It is obvious that the level of expectations differs among

Based on M\_o\_R™ and Total Risk Management® concepts fundamental characteristics of any risk define organization behavior for it, for example: tolerance level, impact, mitigation and contingency strategies, management level, as well as level of financial reserves. While examining risk nature we often consider that a single risk belongs to different characteristic sets. For example, a risk of incorrect professional behavior can belong to human and organizational, and technical sets simultaneously. Therefore, we can create a fuzzy description of a risk:

Implementing risk analysis in fuzzy terms ensures complex analysis for risk source, impact, mitigation, and contingency. The author examined complex risk analysis for portfolio (both projects and operational activities) of virtual learning methods in a lifelong adult training center. As a service-based and private user (a student) oriented business, its success depends dramatically on a subjective personalized opinion of students and partially of corporate HR

rðiÞ ∈ O;rðiÞ ∈ T;rðiÞ H ð16Þ

of results, and it provides also an opportunity to estimate costs of efficiency.

3.2. Big challenges in lifelong learning

78 Modern Fuzzy Control Systems and Its Applications

students, and customer representatives.

3.3. Opportunities and threats in lifelong learning

where O, T, H, fuzzy sets (organizational, technical, human features).

Those risks—both positive and negative—are well-known when we talk about them in a natural language, but training organization's decision-making process requires qualitative and quantitative estimates. As shown in reference [1] we can implement fuzzy analysis to transform natural language to a weak formalized fuzzy model, by placing model-internal risks into an imaginary area, and objective risks into a real area of the model.

#### 3.4. Building current data analysis with fuzzy logic

We investigated our students, trainers, corporate clients, and internal administrative staff feedbacks to discover additional training opportunities.

In 2009 we started online webinar trainings, which are held as simultaneous trainings in groups consisting of online (webinar), and class-based students (named as "webinar-in-class™"). The example of feedbacks is given in a Table 1. Total number of feedbacks: 10,000þ student feedbacks, 700þ by trainers, 1000þ by training center administrative staff, and 500þ by corporate customers'representatives.

We compared and analyzed feedbacks of webinars with the excerpt feedbacks of traditional classbased trainings. As the "specialist computer training center (CCT)" has been operated since 1991, we extracted feedbacks for class trainings for previous 5 years, e.g., we included 25,000 students', 5000 trainers', 5000 administrative staff's, and 2000 customer representatives' feedbacks into comparative analysis against "webinar-in-class" feedbacks, which are presented in Table 2.

The "trapeze" form of a fuzzy interpretation, as shown at Figure 1, is used to represent fuzzy component, because rating values are subjective and personal oriented. For example, we use ratings from "1" or "E" (minimum value, means that a client is completely dissatisfied) to "5" or "A" (maximum value, indicates that a service exceeds customer's expectations). Rate "3" or "C" indicates customer's general satisfaction. These estimations are personally based and depend on a lot of factors, such as professional specialization, job function and rank, individual specialties. Only ratings "E" and "A" strongly indicate satisfactory level. For example, rating "3" or "C" at design or HR trainings is mainly considered as more dissatisfied than satisfied. On the other side a "C" rating at business or project management trainings is mainly considered as satisfied' and rating A" is very rare, because business managers are mostly not as emotional.


Table 1. An excerpt from webinar-N-class studies.


Table 2. An excerpt from traditional class-based studies.

Figure 1. Trapeze interpretation of satisfaction level.

example of feedbacks is given in a Table 1. Total number of feedbacks: 10,000þ student feedbacks, 700þ by trainers, 1000þ by training center administrative staff, and 500þ by corporate

We compared and analyzed feedbacks of webinars with the excerpt feedbacks of traditional classbased trainings. As the "specialist computer training center (CCT)" has been operated since 1991, we extracted feedbacks for class trainings for previous 5 years, e.g., we included 25,000 students', 5000 trainers', 5000 administrative staff's, and 2000 customer representatives' feedbacks into comparative analysis against "webinar-in-class" feedbacks, which are presented in Table 2.

The "trapeze" form of a fuzzy interpretation, as shown at Figure 1, is used to represent fuzzy component, because rating values are subjective and personal oriented. For example, we use ratings from "1" or "E" (minimum value, means that a client is completely dissatisfied) to "5" or "A" (maximum value, indicates that a service exceeds customer's expectations). Rate "3" or "C" indicates customer's general satisfaction. These estimations are personally based and depend on a lot of factors, such as professional specialization, job function and rank, individual specialties. Only ratings "E" and "A" strongly indicate satisfactory level. For example, rating "3" or "C" at design or HR trainings is mainly considered as more dissatisfied than satisfied. On the other side a "C" rating at business or project management trainings is mainly considered as satisfied' and

Trainer Technical facilities Course Willing for further training

Trainer Technical facilities Course Willing for further training

rating A" is very rare, because business managers are mostly not as emotional.

A 32 27 45 42 B 48 51 42 47 C 12 14 10 7 D5 6 2 2 E3 2 1 1

A 39 42 41 42 B 46 41 44 47 C 11 13 13 7 D3 4 2 2 E1 2 1 1

customers'representatives.

80 Modern Fuzzy Control Systems and Its Applications

Rate Parameter (%)

Rate Parameter (%)

Table 1. An excerpt from webinar-N-class studies.

Table 2. An excerpt from traditional class-based studies.

In fact, the total number of analyzed attributes is more than 100, and it changes regularly to follow customer, market demands, technical, and methodological facilities.

Below is a partial list of main attributes in the information system, which contain basic data and we consider them as a real area attributes in an analytical model:

• Student, Client/Customer, Learning format, Country/region, Course, Year/season/month, Vendor, Product Trainer, Trainer rating, Course rating, Training method rating, Number of courses taken by a student afterwards

The total number of real area attributes is more than 50.

Next I show an excerpt from a list of more than 25 additional (information model based) attributes. We consider these attributes in an imaginary area of our fuzzy model:

• Total rating of a training method, comparative rating to class-based training, comparative rating of webinars against class and self-learning combined method, views and filters across client types, regions, time, season, etc.

To build an integral customer satisfaction rating we use multidimensional fuzzy analysis of different partial (single-parameter) rankings as shown at Figure 2. Also, different filters and constraints are implemented to localize problems, challenges and find grow-points.

#### 3.5. Investigating students' satisfaction against educational organization efficiency

We can investigate an integral satisfaction/dissatisfaction level of a training group based on the following attributes: trainer ratings, course ratings, willingness to continue training at a next course, and ratings of technical facilities. Each set is fuzzy and contains ranking values (ratings) by each student.

Figure 2. Determining the satisfaction area on an example of four parameter analysis.

In each case (for a student/group/trainer/course, etc.) we form a fuzzy area, in which we may consider that a course/study/trainer/class, etc. are satisfactory. According to company goals and statistical data we can also implement additional weights for attributes. For example, a trainer rank has coefficient "1, 5" and class ranking—"0, 8". Attribute "willing for further training" will have the maximum weight coefficient "2", as it's the most important factor for commercial adult learning company. This model shows a subjective satisfaction level as a set (family) of polygons. Each polygon draws by connecting points, which reflect partial ratings. For example, an area of subjective satisfaction is defined by the following partial ratings:

Trainer rating ¼ 2 AND

Course rating ¼ 4 AND

Technical facilities ¼ 3 AND

Willing to continue education ¼ 5.

In this case, we define that a group is partially satisfied/partially dissatisfied with a trainer, but if most of students of a certain group are ready to continue education at next courses, we can mark this group as "satisfied."

For complex estimation of job effectiveness of an adult life-long training organization we developed more complicated approaches, which include statistical data based on more than 70 attributes (both in real and imaginary areas) and collected them in multidimensional databases—OLAP cubes. This cube has the appropriate number (more than 70) of dimensions, and we need to build a set of fuzzy models, and optimize them for daily calculations and analysis. A real part of a model includes basic facts, and on the other side an imaginary part of a model includes filters, views, and additional states of a model or database. Due to this approach, we decrease number of dimensions to 50 in total, which leads to decrease in computing capacity requirements. Practical result: we have an opportunity to process analytical reports in a real-time mode, and postpone few complicated reports for nonworking hours (night time and Sundays).

Let us examine a comparatively simple set of fuzzy sets, which describes an integral satisfactory factor and training organization efficiency:

• By course, a certain trainer and/or trainer group, a company—customer, learning location, a certain period, a training method, a training branch (e.g., Management, ITSM, software development, HR, etc.).

The developed model contains both real area attributes, which reflect basic states, and imaginary area attributes, which reflect temporary, service states, filter conditions, identifiers, etc.

Below, I show an excerpt from a model. Attributes named in a lingual model terms to simplify understanding

$$\begin{cases} \text{M1} = (\text{CN} + \text{St} + \text{Tr} + \text{TM}) + \text{j}(\text{V} + \text{TP} + \text{ST}),\\ \text{M2} = (\text{CN} + \text{Tr} + \text{CCF} + \text{TCF} + \text{WFT}) + \text{j}(\text{TP}(\text{i}) + \text{ACR}(\text{i}) + \text{ATR}(\text{i}) + \text{V}),\\ \text{M3} = (\text{CN} + \text{TL} + \text{C} + \text{TM}) + \text{j}(\text{TP}(\text{i}) + \text{ACR}(\text{i}) + \text{MAR}(\text{i}) + \text{V}), \end{cases} (17)$$

where CN, Course name; St, Student identifier; Tr, trainer identifier; V, view name; TP(i), selected time period; St, threshold level of students' satisfaction for the certain model; TM, training method; CCF, course cash flow; TCF, cash flow on courses by a certain trainer; WFT, student's willing for continuous education; ACR, average course ranking for selected period; ATR, average trainer ranking for selected period; TL, training location; C, corporate customer name; and AMR, average ranking of a certain training method for selected period.

Set M1 reflects mean level of students' satisfaction for a certain course and a certain trainer for selected time period.

Set M2 reflects current level of economic efficiency of a certain trainer, based on dynamic trend of students' satisfaction across a number of time periods (for example, month to month or quarter to quarter).

Set M3 reflects dynamics of corporate customers' satisfaction for a certain training location, course, and a training method. This set gives a control how a certain training location provides quality for a certain course and a training method, for example, webinar, or blended, or selfpaced, etc.

#### 3.6. Defining training methods and models

In each case (for a student/group/trainer/course, etc.) we form a fuzzy area, in which we may consider that a course/study/trainer/class, etc. are satisfactory. According to company goals and statistical data we can also implement additional weights for attributes. For example, a trainer rank has coefficient "1, 5" and class ranking—"0, 8". Attribute "willing for further training" will have the maximum weight coefficient "2", as it's the most important factor for commercial adult learning company. This model shows a subjective satisfaction level as a set (family) of polygons. Each polygon draws by connecting points, which reflect partial ratings. For example, an area of subjective satisfaction is defined by the following partial ratings:

Figure 2. Determining the satisfaction area on an example of four parameter analysis.

In this case, we define that a group is partially satisfied/partially dissatisfied with a trainer, but if most of students of a certain group are ready to continue education at next courses, we can

For complex estimation of job effectiveness of an adult life-long training organization we developed more complicated approaches, which include statistical data based on more than 70 attributes (both in real and imaginary areas) and collected them in multidimensional databases—OLAP cubes. This cube has the appropriate number (more than 70) of dimensions, and we need to build a set of fuzzy models, and optimize them for daily calculations and analysis. A real part of a model includes basic facts, and on the other side an imaginary part of a model includes filters, views, and additional states of a model or database. Due to this approach, we decrease number of dimensions to 50 in total, which leads to decrease in computing capacity requirements. Practical result: we have an opportunity to process analytical reports in a real-time mode, and postpone few complicated reports for nonworking hours

Trainer rating ¼ 2 AND Course rating ¼ 4 AND

Technical facilities ¼ 3 AND

Willing to continue education ¼ 5.

82 Modern Fuzzy Control Systems and Its Applications

mark this group as "satisfied."

(night time and Sundays).

Adult learning training organization should offer various training opportunities for its students, such as long- and short-term trainings, class-based, virtual, blended, synchronous, and asynchronous, etc. Based on the analysis model at our Specialist CCT we develop balanced cost-effective vs. "student satisfactory" training methods for precisely defined customer audience and course bundles.

While analyzing results of modeling, we find several maximums. Each of the maximums is characterized by a certain set of parameters, as shown in Figure 3.

Figure 3. Multidimensional estimation, where I – Individual approach level; II – Trainer level; III – Technical facilities; IV – Training center economical effectiveness.

For example, the maximum satisfaction level which is found in an analytical cube is defined by the following attributes:


The "webinar-in-class" training option is the best for mentioned attributes, because a webinarbased student has an opportunity to study anywhere, and has no travel or accommodation expenses. Simultaneously, he/she has an access to the best trainers, and can collaborate with classmates in a class and other webinar students, using a very simple software. The webinar students have an access to the same technical facilities such as labs. Thus, the webinar-in-class training method becomes very popular solution for students studying technical (Microsoft, Oracle, CISCO, etc.) courses, as well as project, and IT service management courses.

We worked further to analyze maximums, and another point is based on the following attributes:




In this example, we see more attributes, and it was a bit more difficult to create an appropriate training method. The result was a kind of blended learning, which we named an "open learning."


To resume I want to stress that the estimating process is everlasting, as well as optimization of the research model. While preparing this article, one more training method was developed, which proves the efficiency of the described approach. The described methods won several professional awards [6, 7].

## 4. Identifying special tools for virtual training of hard of hearing people

#### 4.1. Introduction

For example, the maximum satisfaction level which is found in an analytical cube is defined by

Figure 3. Multidimensional estimation, where I – Individual approach level; II – Trainer level; III – Technical facilities;

The "webinar-in-class" training option is the best for mentioned attributes, because a webinarbased student has an opportunity to study anywhere, and has no travel or accommodation expenses. Simultaneously, he/she has an access to the best trainers, and can collaborate with classmates in a class and other webinar students, using a very simple software. The webinar students have an access to the same technical facilities such as labs. Thus, the webinar-in-class training method becomes very popular solution for students studying technical (Microsoft,

We worked further to analyze maximums, and another point is based on the following attri-

Oracle, CISCO, etc.) courses, as well as project, and IT service management courses.

the following attributes:


butes:




IV – Training center economical effectiveness.

84 Modern Fuzzy Control Systems and Its Applications







More than 10% of people on the Earth suffers from different hearing impairs as the World Health Organization data shows. Many of them are young people, or employees, which are involved into lifelong learning. They have difficulties with taking both class and virtual trainings, if they do not have or use special hearing aids. Meanwhile, many young people do not use special devices due to medical recommendations or having scruples.

Based on our research of students' with hearing impairs demands our research team deliver a special computer-based technology—named as Petralex©. It is implemented in mobile Apps and Windows driver, which works as a personal hearing assistant [6]. A student passes an "in cito" hearing test and the application creates a personal hearing profile for each place or environment (for example, public transport, café, car, room at home, classroom, and workplace). Mobile App acts as a hearing aid in a smartphone, so a student can easily attend classes. The Windows-based driver creates a Virtual audio device (VAD), which adapts streaming audio for both online and asynchronous virtual learning to an appropriate user's hearing profile.

Different virtual training methods—synchronous, asynchronous, blended—which are defined in Ref. [4], include online and/or off-line listening in videos, podcasts, as well as online training delivery, including real-time discussion with a trainer and classmates. So, students with hearing disabilities should have opportunities to be involved into the entire training process.

#### 4.2. Synchronous learning methods

A student with partial hearing losses can feel uncomfortable while studying in class, or at online webinar. If a student studies in a class, he or she can implement the Petralex® mobile app as a hearing assistant, and involve in-depth into a learning process. In a synchronous learning training content is delivered in an online mode as shown in Figure 4. A student accesses it using the special audio driver, which ensures audio stream adjustment to personal hearing profile. As a result, a student can attend studies for a long time—up to 8 hours per day —due to improved hearing tolerance, reducing fatigue for long listening sessions, and attenuation of excessive sound pressure [6, 7].

#### 4.2.1. Typical learning cases for online trainings


#### 4.3. Asynchronous learning methods

In this section let us consider different scenarios for asynchronous learning of students with partial hearing losses. The most popular tools for asynchronous learning are: learning management systems (LMS), stream and offline video, and audio services.

As shown in Figure 5, an external audio signal from a learning tool goes through the virtual audio driver, which transforms it according to an activated user hearing profile. Thanks to it, a student can study anywhere. At our training center, and with our partners, the following scenarios, as shown in Figure 5, were tested:

In a special class,

At home,

At a workplace, and

On a beach.

Figure 4. The learning schema for synchronous virtual learning.

Implementing Complex Fuzzy Analysis for Business Planning Systems http://dx.doi.org/10.5772/67974 87

Figure 5. Asynchronous learning scenarios.

One of the most impressive cases in our training practices is short-term learning for adult busy people. We defined lifelong business students as a separate category in our model. Lifelong learners often study during their vacations or weekends. Usually they are strongly motivated, so they can easily combine their rest and studies. As an example, a student creates a "beach" profile and can listen learning records on a comfortable manner for his/her hearing.

#### 5. Resume

app as a hearing assistant, and involve in-depth into a learning process. In a synchronous learning training content is delivered in an online mode as shown in Figure 4. A student accesses it using the special audio driver, which ensures audio stream adjustment to personal hearing profile. As a result, a student can attend studies for a long time—up to 8 hours per day —due to improved hearing tolerance, reducing fatigue for long listening sessions, and attenu-

1. A trainer explains learning materials: a student studies at home. S/he connects to an online tool (for example, Skype®, Citrix GoTo®, WebEx, Adobe® Connect®, or any other), activates "My room" profile. Next, our driver transforms audio stream in a real-time mode with only 10–50 ms delay, so a student can hear a teacher clearly, have concentration on studying, and ask or answer questions; present his/her work, and discuss with other

2. A business game: business games and other forms of group studies are very popular according to our model (Eq. (16)). An online student usually plays a role of a virtual team member or help desk agent. Implementing audio-driver provides both parties an oppor-

In this section let us consider different scenarios for asynchronous learning of students with partial hearing losses. The most popular tools for asynchronous learning are: learning man-

As shown in Figure 5, an external audio signal from a learning tool goes through the virtual audio driver, which transforms it according to an activated user hearing profile. Thanks to it, a student can study anywhere. At our training center, and with our partners, the following

ation of excessive sound pressure [6, 7].

86 Modern Fuzzy Control Systems and Its Applications

students in a real-time mode.

4.3. Asynchronous learning methods

scenarios, as shown in Figure 5, were tested:

Figure 4. The learning schema for synchronous virtual learning.

In a special class,

At a workplace, and

At home,

On a beach.

tunity to collaborate in a clear mode without delays.

agement systems (LMS), stream and offline video, and audio services.

4.2.1. Typical learning cases for online trainings

Implementing extended definition area for fuzzy set analysis provides vast opportunities for representation of control objects by information systems, their analysis and optimization. Based on implementation of fuzzy analysis the author succeeded in creating and launching various virtual training models for lifelong learning, including people suffering with partial hearing losses.

#### Author details

Danil Dintsis

Address all correspondence to: consult@dintsis.org

Educational Private organization "Specialist", Moscow, Russian Federation

#### References

