**Robust Decentralized PID Controller Design**

Danica Rosinová and Alena Kozáková

*Slovak University of Technology Slovak Republic* 

## **1. Introduction**

132 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

[1] Multivariable Process control, Pradeep B. Deshpande, ISA, Research Triangle Park, NC,

[4] Multivariable control systems: An Engineering approach, P. Albertos and Sala, Springer

[5] Identification and control of multivariable systems-role of relay feedback, A Ph.D Thesis,

[6] Rames C. Panda, 2008, Synthesis of PID controller using desired closed loop criteria, Ind.

[7] Rames C. Panda, 2009, Synthesis of PID controller for unstable and integrating processes,

Control Engg Lab, Electical Engg Dept., Anna University, Chennai, India, C.

[2] Autotuning of PID controllers, C.C.Yu, Springer Verlag, London, 2nd Edition, 2006 [3] Control configuration selection for multivariable plants, Ali Khaki-Sedigh and Bijan

Moaveni, Springer, Berlin, Heidelberg, 2009.

& Engg. Chem Res., 47(22),1684-92

Chem. Eng. Sci, 64 (12), 2807-16

**9. References** 

U.S.A, 1989

Verlag, London, 2004

Selvakumar, 2009.

Robust stability of uncertain dynamic systems has major importance when real world system models are considered. A realistic approach has to consider uncertainties of various kinds in the system model. Uncertainties due to inherent modelling/identification inaccuracies in any physical plant model specify a certain uncertainty domain, e.g. as a set of linearized models obtained in different working points of the plant considered. Thus, a basic required property of the system is its stability within the whole uncertainty domain denoted as robust stability. Robust control theory provides analysis and synthesis approaches and tools applicable for various kinds of processes, including multi input – multi output (MIMO) dynamic systems. To reduce multivariable control problem complexity, MIMO systems are often considered as interconnection of a finite number of subsystems. This approach enables to employ decentralized control structure with subsystems having their local control loops. Compared with centralized MIMO controller systems, decentralized control structure brings about certain performance deterioration, however weighted against by important benefits, such as design simplicity, hardware, operation and reliability improvement. Robustness is one of attractive qualities of a decentralized control scheme, since such control structure can be inherently resistant to a wide range of uncertainties both in subsystems and interconnections. Considerable effort has been made to enhance robustness in decentralized control structure and decentralized control design schemes and various approaches have been developed in this field both in time and frequency domains (Gyurkovics & Takacs, 2000; Zečevič & Šiljak, 2004; Stankovič et al., 2007).

Recently, the algebraic approach has gained considerable interest in robust control, (Boyd et al., 1994; Crusius & Trofino, 1999; de Oliveira et al., 1999; Ming Ge et al., 2002; Grman et al., 2005; Henrion et al., 2002). Algebraic approach is based on the fact that many different problems in control reduce to an equivalent linear algebra problem (Skelton et al., 1998). By algebraic approach, robust control problem is formulated in algebraic framework and solved as an optimization problem, preferably in the form of Linear Matrix Inequalities (LMI). LMI techniques enable to solve a large set of convex problems in polynomial time (see Boyd et al., 1994). This approach is directly applicable when control problems for linear uncertain systems with a convex uncertainty domain are solved. Still, many important control problems even for linear systems have been proven as NP hard, including structured linear control problems such as decentralized control and simultaneous static output feedback (SOF) designs. In these cases the prescribed structure of control feedback matrix (block diagonal for decentralized control) results in nonconvex problem formulation. There

Robust Decentralized PID Controller Design 135

robust stability conditions. A direct "one-shot" robust DC design methodology based on integration of robust stability conditions in the Equivalent Subsystems Method enables to design local controllers of equivalent subsystems with regard to robust stability of the full system. The frequency domain approach is applicable for both continuous- and discrete-

This section aims at description, and analysis of two input - two output process from literature, which will be later used to demonstrate our proposed methods for decentralized PID controller design. The quadruple-tank process shown in Fig.1 has been introduced in (Johansson et al., 1999; Johansson, 2000) to provide a case study to analyze both minimum and nonminimum phase MIMO systems on the same plant. The aim is to

pump 1 and 2 flows respectively, the controlled outputs y1 and y2 are levels in lower tanks

1 and

(1)

2 are

time PID controller designs.

1 and 2 respectively.

Fig. 1. Quadruple tank process scheme.

**2. Motivation: Case study - Quadruple tank process** 

control the level in the lower two tanks using two pumps. The inputs

The nonlinear model of the four tanks can be described by state equations

1 1 3 1 1

2 2

*dh a <sup>a</sup> <sup>k</sup> gh gh v dt A A A*

2 2

*dh a a <sup>k</sup> gh gh v dt A A A*

(1 ) <sup>2</sup>

(1 ) <sup>2</sup>

3 3 2 2

*dh a <sup>k</sup> gh <sup>v</sup> dt A A*

*dh a <sup>k</sup> gh <sup>v</sup> dt A A*

3 3 4 4 1 1

4 4

11 1 2 2 4 2 2

22 2

3 2

4 1

1 31

2 42

are basically two approaches to solve the respective nonconvex control problem: 1) to reformulate the problem as LMI using certain convex relaxations (e.g. deOliveira et al., 2000; Rosinová & Veselý, 2003) or, alternatively, adopt an iterative procedure; 2) to formulate and solve the bilinear matrix inequalities (BMI) respective to robust control design problem. A nice review and basic characteristics of LMI and BMI in various control problems can be found in (Van Antwerp & Braatz, 2000).

To reduce the problem size in decentralized control design for large scale systems, the diagonal dominance or block diagonal dominance concept can be adopted. Recently, the so called Equivalent Subsystems Method has been developed for decentralized control in frequency domain, (Kozáková & Veselý, 2009). The main concept of the Equivalent Subsystems Method, originally developed as a Nyquist based frequency domain decentralized controller design technique, is the so called equivalent subsystem; equivalent subsystems are generated by shaping Nyquist plot of each decoupled subsystem using any selected characteristic locus of the matrix of interactions. The point of this approach consists in that local controllers of equivalent subsystems can be independently tuned for stability and required performance specified in terms of a suitable (preferably frequency domain) performance measure (e.g. degree of stability, phase margin, bandwidth), so that the resulting decentralized controller guarantees equivalent performance of the full system.

When designing decentralized control, besides robust stability, performance requirements have to be considered. Performance objectives can be of two basic types: a) achieving required performance in different subsystems; or b) achieving plant-wide desired performance. In this chapter two alternative approaches belonging to the latter group are presented, based on recent research results on robust decentralized PID controller design in the frequency and time domains.

The present chapter further extends the robust decentralized PID controller design techniques from (Kozáková et al., 2009; 2010; 2011; Rosinová et al., 2003; Rosinová & Veselý, 2007; 2011), bringing novel robust control design approaches. The results are illustrated on the case study dealing with robust decentralized controller design for the quadruple tank process. This laboratory process recently presented in (Johansson, 2000; Johansson et al., 1999) is an illustrative two input - two output laboratory plant for studying multivariable dynamic systems for both minimum and nonminimum-phase configurations.

The first presented approach is based on formulation and solution of BMI or LMI for uncertain linear polytopic system to design robust controller in the state space. In the time domain, we introduce the augmented model for closed-loop linear uncertain system with PID controller; this model is in general form, comprising both continuous- and discrete-time cases. For both cases, a general robust stability condition is formulated; the particular design procedures differ only in parameterization of augmented model matrices. A decentralized control design strategy is adopted, where robust PID control design approach is applied for structured - block diagonal controller matrices respective to decentralized controller.

The second approach is based on the Nyquist-type decentralized control design technique for uncertain MIMO systems described by a transfer function matrix. The decentralized controller is designed on subsystem level using the recently developed Equivalent Subsystem Method (Kozáková et al., 2009). Application of this method in the design for robust stability and nominal performance can be found e.g. in (Kozáková & Veselý, 2009) within a two-stage design scheme: 1. design of decentralized controller for nominal performance; 2. controller redesign with modified performance requirements to meet the

are basically two approaches to solve the respective nonconvex control problem: 1) to reformulate the problem as LMI using certain convex relaxations (e.g. deOliveira et al., 2000; Rosinová & Veselý, 2003) or, alternatively, adopt an iterative procedure; 2) to formulate and solve the bilinear matrix inequalities (BMI) respective to robust control design problem. A nice review and basic characteristics of LMI and BMI in various control problems can be

To reduce the problem size in decentralized control design for large scale systems, the diagonal dominance or block diagonal dominance concept can be adopted. Recently, the so called Equivalent Subsystems Method has been developed for decentralized control in frequency domain, (Kozáková & Veselý, 2009). The main concept of the Equivalent Subsystems Method, originally developed as a Nyquist based frequency domain decentralized controller design technique, is the so called equivalent subsystem; equivalent subsystems are generated by shaping Nyquist plot of each decoupled subsystem using any selected characteristic locus of the matrix of interactions. The point of this approach consists in that local controllers of equivalent subsystems can be independently tuned for stability and required performance specified in terms of a suitable (preferably frequency domain) performance measure (e.g. degree of stability, phase margin, bandwidth), so that the resulting decentralized controller guarantees equivalent performance of the full system. When designing decentralized control, besides robust stability, performance requirements have to be considered. Performance objectives can be of two basic types: a) achieving required performance in different subsystems; or b) achieving plant-wide desired performance. In this chapter two alternative approaches belonging to the latter group are presented, based on recent research results on robust decentralized PID controller design in

The present chapter further extends the robust decentralized PID controller design techniques from (Kozáková et al., 2009; 2010; 2011; Rosinová et al., 2003; Rosinová & Veselý, 2007; 2011), bringing novel robust control design approaches. The results are illustrated on the case study dealing with robust decentralized controller design for the quadruple tank process. This laboratory process recently presented in (Johansson, 2000; Johansson et al., 1999) is an illustrative two input - two output laboratory plant for studying multivariable

The first presented approach is based on formulation and solution of BMI or LMI for uncertain linear polytopic system to design robust controller in the state space. In the time domain, we introduce the augmented model for closed-loop linear uncertain system with PID controller; this model is in general form, comprising both continuous- and discrete-time cases. For both cases, a general robust stability condition is formulated; the particular design procedures differ only in parameterization of augmented model matrices. A decentralized control design strategy is adopted, where robust PID control design approach is applied for

structured - block diagonal controller matrices respective to decentralized controller.

The second approach is based on the Nyquist-type decentralized control design technique for uncertain MIMO systems described by a transfer function matrix. The decentralized controller is designed on subsystem level using the recently developed Equivalent Subsystem Method (Kozáková et al., 2009). Application of this method in the design for robust stability and nominal performance can be found e.g. in (Kozáková & Veselý, 2009) within a two-stage design scheme: 1. design of decentralized controller for nominal performance; 2. controller redesign with modified performance requirements to meet the

dynamic systems for both minimum and nonminimum-phase configurations.

found in (Van Antwerp & Braatz, 2000).

the frequency and time domains.

robust stability conditions. A direct "one-shot" robust DC design methodology based on integration of robust stability conditions in the Equivalent Subsystems Method enables to design local controllers of equivalent subsystems with regard to robust stability of the full system. The frequency domain approach is applicable for both continuous- and discretetime PID controller designs.

#### **2. Motivation: Case study - Quadruple tank process**

This section aims at description, and analysis of two input - two output process from literature, which will be later used to demonstrate our proposed methods for decentralized PID controller design. The quadruple-tank process shown in Fig.1 has been introduced in (Johansson et al., 1999; Johansson, 2000) to provide a case study to analyze both minimum and nonminimum phase MIMO systems on the same plant. The aim is to control the level in the lower two tanks using two pumps. The inputs 1 and 2 are pump 1 and 2 flows respectively, the controlled outputs y1 and y2 are levels in lower tanks 1 and 2 respectively.

Fig. 1. Quadruple tank process scheme.

The nonlinear model of the four tanks can be described by state equations

$$\begin{aligned} \frac{dh\_1}{dt} &= -\frac{a\_1}{A\_1}\sqrt{2gh\_1} + \frac{a\_3}{A\_1}\sqrt{2gh\_3} + \frac{\gamma\_1 k\_1}{A\_1}v\_1\\ \frac{dh\_2}{dt} &= -\frac{a\_2}{A\_2}\sqrt{2gh\_2} + \frac{a\_4}{A\_2}\sqrt{2gh\_4} + \frac{\gamma\_2 k\_2}{A\_2}v\_2\\ \frac{dh\_3}{dt} &= -\frac{a\_3}{A\_3}\sqrt{2gh\_3} + \frac{(1-\gamma\_2)k\_2}{A\_3}v\_2\\ \frac{dh\_4}{dt} &= -\frac{a\_4}{A\_4}\sqrt{2gh\_4} + \frac{(1-\gamma\_1)k\_1}{A\_4}v\_1 \end{aligned} \tag{1}$$

Robust Decentralized PID Controller Design 137

pumps. To achieve this aim, the decentralized control structure is employed, with two

Decentralized control design consists of several steps, the crucial ones for controller design

We consider the standard approach for the former two steps presented below; in Sections 3

Frequently used index to assess input-output pairing is the Relative Gain Array (RGA) index, see e.g. (Ogunnaike & Ray, 1994), (Skogestad & Postletwhaite, 2009), computed

Individual subsystems are then specified by the chosen pairing and their transfer functions are placed in the diagonal of the transfer function matrix. To check structural stabilizability

> det (0) ( ( (0)) *G*

If 0 *NI* , the system cannot be stabilized using the chosen pairing and the pairing must be

In our case study, the steady state RGA(0) is considered to choose appropriate pairing with

*RGA G G*

For quadruple tank system (1), we consider the uncertainty to be a change of valve position,

In minimum phase region: In nonminimum phase region:

= 0.4 WP1: 1

= 0.8 (7) WP3: 1

, uncertainty domain is specified by three working points.

 = 0.1, 2 

<sup>1</sup> <sup>1</sup>

*T*

 and 2 

<sup>1</sup> ( ) ( ). \* [ ( ) ] *<sup>T</sup> RGA s G s G s* (4)

*diag G* (5)

(6)

exclusively. The diagonal

(minimum phase system) and the respective

(nonminimm phase system), the opposite

= 0.3; WP2: 1

 = 0.1, 2 

 = 0.3, 2 = 0.1

= 0.1 (8)

and 4 we concentrate on the last step – robust decentralized control design.

where G(s) is a square transfer function matrix of the linearized system.

using the chosen control configuration, the Niederlinski index is applied:

the respective RGA elements positive and closest possible to 1.

*NI*

(0) 0 . \* 0 <sup>1</sup>

depends on valve parameters 1

pairing 1 22 1 *v yv y* , is indicated. This result is approved by Niederlinski index.

 

> 

control loops respective to output values y1 and y2.


*Pairing and structural stability* 


are

as

modified.

where 1 2

 

i.e. change of 1

 = 0.4, 2 

> = 0.8, 2

WP1: <sup>1</sup> 

WP3: 1 

1 2 1 

elements λ are positive for 1 2 1 2

 and 2 

= 0.8; WP2: 1

pairing is 1 12 2 *v yv y* , . For 1 2 0 1

**2.2 Quadruple tank process – uncertainty domain** 

 = 0.8, 2 

where Ai is cross-section of tank i, ai is cross-section of the outlet hole of tank i, hi is water level in tank i, g is acceleration of gravity, the flow corresponding to pump i is kivi*.* Parameter 1 denotes position of the valve dividing the pump 1 flow into the lower tank 1 and related upper tank 4 and similarly 2 divides flow from pump 2 to the tanks 2 and 3. The flow to tank 1 is 111 *k v* and to tank 4 it is 1 11 (1 ) *k v* , analogically for the tanks 2 and 3. The nonlinear model (1) can be linearized around the working point given by the water levels in tanks 10 20 30 40 *hhhh* ,,, . The deviation state space model was considered with *iii*<sup>0</sup> *xhh* and the respective control variables *uvv iii* <sup>0</sup> . The linearized state space model for quadruple tank (1) is then

$$
\begin{bmatrix}
\dot{\boldsymbol{x}}\_{1} \\
\dot{\boldsymbol{x}}\_{3} \\
\dot{\boldsymbol{x}}\_{2} \\
\dot{\boldsymbol{x}}\_{4}
\end{bmatrix} = \begin{bmatrix}
\frac{-1}{T\_{1}} & \frac{A\_{3}}{T\_{3}A\_{1}} & 0 & 0 \\
0 & \frac{-1}{T\_{3}} & 0 & 0 \\
0 & 0 & \frac{-1}{T\_{2}} & \frac{A\_{4}}{T\_{4}A\_{2}} \\
0 & 0 & 0 & \frac{-1}{T\_{4}}
\end{bmatrix} \cdot \begin{bmatrix}
\boldsymbol{x}\_{1} \\
\boldsymbol{x}\_{1} \\
\boldsymbol{x}\_{3} \\
\boldsymbol{x}\_{2} \\
\boldsymbol{x}\_{4}
\end{bmatrix} + \begin{bmatrix}
\frac{\boldsymbol{\gamma}\_{1}k\_{1}}{A\_{1}} & 0 \\
0 & \frac{(1-\gamma\_{2})k\_{2}}{A\_{3}} \\
0 & \frac{\boldsymbol{\gamma}\_{2}k\_{2}}{A\_{1}} \\
\frac{(1-\gamma\_{1})k\_{1}}{A\_{4}} & 0
\end{bmatrix} \cdot \begin{bmatrix}
\boldsymbol{u}\_{1} \\
\boldsymbol{u}\_{2}
\end{bmatrix} \tag{2}
$$
  $\frac{A\_{i}}{\sigma} \sqrt{\frac{2I\_{i0}}{\sigma}}$ ,  $i = 1, \ldots, 4$ .

where <sup>0</sup> <sup>2</sup> , 1,...,4 *i i i i A h T i a g* .

The argument t has been omitted; the state variables corresponding to levels in tanks 2 and 3 have been interchanged in state vector so that subsystems respective to input u1 from pump 1 (tanks 1 and 3) and u2 from pump 2 (tanks 2 and 4) are more apparent. This decomposition into two subsystems is used for decentralized control design.

The respective transfer function matrix having inputs v1 and v2 and outputs y1 and y2 is

$$G(s) = \begin{vmatrix} \frac{c\_1 \mathcal{V}\_1}{T\_1 s + 1} & \frac{c\_1 (1 - \mathcal{V}\_2)}{(T\_3 s + 1)(T\_1 s + 1)}\\ \frac{c\_2 (1 - \mathcal{V}\_1)}{(T\_4 s + 1)(T\_2 s + 1)} & \frac{c\_2 \mathcal{V}\_2}{T\_2 s + 1} \end{vmatrix} \tag{3}$$

where <sup>0</sup> <sup>2</sup> , 1,2 *ii i i i Tk h c i A g* .

The plant can be shifted from minimum to nonminimum phase configuration and vice versa simply by changing a valve controlling the flow ratios 1 and 2 between lower and upper tanks. The minimum-phase configuration corresponds to 1 2 1 2 and the nonminimum-phase one to 1 2 0 1 .

#### **2.1 Decentralized control of quadruple tank – problem formulation and pairing selection**

The basic control aim for quadruple tank is to reach the given level in the lower two tanks, i.e. prescribed values of y1 and y2 by controlling input flows v1 and v2 delivered by two

where Ai is cross-section of tank i, ai is cross-section of the outlet hole of tank i, hi is water level in tank i, g is acceleration of gravity, the flow corresponding to pump i is kivi*.*

The nonlinear model (1) can be linearized around the working point given by the water levels in tanks 10 20 30 40 *hhhh* ,,, . The deviation state space model was considered with *iii*<sup>0</sup> *xhh* and the respective control variables *uvv iii* <sup>0</sup> . The linearized state space

3 1 1

*A k T TA A*

<sup>1</sup> 0 0 <sup>0</sup>

<sup>1</sup> (1 ) 0 00 <sup>0</sup>

1 1 2 2 3 31 3 3 2 4 2 2 2 2

*x x k x xu T A x xk A u*

4 4 2 42 1

*x x T TA A*

<sup>1</sup> (1 ) <sup>000</sup> <sup>0</sup>

4 4

*T A*

The argument t has been omitted; the state variables corresponding to levels in tanks 2 and 3 have been interchanged in state vector so that subsystems respective to input u1 from pump 1 (tanks 1 and 3) and u2 from pump 2 (tanks 2 and 4) are more apparent. This

1 ( 1)( 1) ( ) (1 )

*c c Ts Ts Ts*

*c c Ts Ts Ts*

( 1)( 1) 1

The plant can be shifted from minimum to nonminimum phase configuration and vice versa

The basic control aim for quadruple tank is to reach the given level in the lower two tanks, i.e. prescribed values of y1 and y2 by controlling input flows v1 and v2 delivered by two

1 1 1 2 1 31 2 1 2 2 42 2

The respective transfer function matrix having inputs v1 and v2 and outputs y1 and y2 is

 

1 31 1

<sup>1</sup> 0 0 <sup>0</sup>

decomposition into two subsystems is used for decentralized control design.

tanks. The minimum-phase configuration corresponds to 1 2 1 2

**2.1 Decentralized control of quadruple tank – problem formulation and pairing** 

*k v* and to tank 4 it is 1 11 (1 )

denotes position of the valve dividing the pump 1 flow into the lower tank 1

1 1

(1 )

 

 and 2 

 

*k*

divides flow from pump 2 to the tanks 2 and 3.

(2)

(3)

between lower and upper

and the

 

*k v* , analogically for the tanks 2 and 3.

Parameter 1

The flow to tank 1 is 111

 

where <sup>0</sup> <sup>2</sup> , 1,...,4 *i i*

where <sup>0</sup> <sup>2</sup> , 1,2 *ii i*

*A g* .

nonminimum-phase one to 1 2 0 1

*i Tk h c i*

*i*

**selection** 

*a g* .

*G s*

simply by changing a valve controlling the flow ratios 1

 .

*i A h T i*

*i*

and related upper tank 4 and similarly 2

model for quadruple tank (1) is then

pumps. To achieve this aim, the decentralized control structure is employed, with two control loops respective to output values y1 and y2.

Decentralized control design consists of several steps, the crucial ones for controller design are


We consider the standard approach for the former two steps presented below; in Sections 3 and 4 we concentrate on the last step – robust decentralized control design.

*Pairing and structural stability* 

Frequently used index to assess input-output pairing is the Relative Gain Array (RGA) index, see e.g. (Ogunnaike & Ray, 1994), (Skogestad & Postletwhaite, 2009), computed as

$$RGA(s) = G(s).\* \left[G(s)^T\right]^{-1} \tag{4}$$

where G(s) is a square transfer function matrix of the linearized system.

Individual subsystems are then specified by the chosen pairing and their transfer functions are placed in the diagonal of the transfer function matrix. To check structural stabilizability using the chosen control configuration, the Niederlinski index is applied:

$$NI = \frac{\det(G(0))}{II \, (diag(G(0)))} \tag{5}$$

If 0 *NI* , the system cannot be stabilized using the chosen pairing and the pairing must be modified.

In our case study, the steady state RGA(0) is considered to choose appropriate pairing with the respective RGA elements positive and closest possible to 1.

$$RGA(0) = G(0). \, ^\ast \left[ G(0)^{-1} \right]^T = \begin{bmatrix} \mathcal{\lambda} & 1 - \mathcal{\lambda} \\ 1 - \mathcal{\lambda} & \mathcal{\lambda} \end{bmatrix} \tag{6}$$

where 1 2 1 2 1 depends on valve parameters 1 and 2 exclusively. The diagonal

elements λ are positive for 1 2 1 2 (minimum phase system) and the respective pairing is 1 12 2 *v yv y* , . For 1 2 0 1 (nonminimm phase system), the opposite pairing 1 22 1 *v yv y* , is indicated. This result is approved by Niederlinski index.

#### **2.2 Quadruple tank process – uncertainty domain**

For quadruple tank system (1), we consider the uncertainty to be a change of valve position, i.e. change of 1 and 2 , uncertainty domain is specified by three working points.

In minimum phase region: In nonminimum phase region: WP1: <sup>1</sup> = 0.4, 2 = 0.8; WP2: 1 = 0.8, 2 = 0.4 WP1: 1 = 0.1, 2 = 0.3; WP2: 1 = 0.3, 2 = 0.1 WP3: 1 = 0.8, 2 = 0.8 (7) WP3: 1 = 0.1, 2 = 0.1 (8)

a) minimum phase configuration b) nonminimum phase configuration

Robust Decentralized PID Controller Design 139

1

 *xt A xt B ut* () ( )() ( )() 

> 

1 2 1 2 1 2 ( ) ( ... ), ( ) ( ... ), ( ) ( ... ) *TT T TT T TT T xt x x x ut u u u yt y y y NNN* are state, control and output vectors

system matrices of corresponding dimensions respective to the subsystems, matrices

A closed loop system performance is assessed considering the guaranteed cost notion; the

for a continuous-time and

<sup>1</sup> ( ) { ( ),..., ( )}, *B diag B B d N*

 , ( ) *Bi* , ( ) *Aij* , ( ) *Bij* 

, 1 1

*p p*

*N i i*

*<sup>i</sup> xt R* , ( ) *mi ut R <sup>i</sup>* , ( ) *<sup>i</sup> <sup>p</sup> <sup>i</sup> <sup>y</sup> t R* are the subsystem state, control and output

*B B* 

*B B* 

> 

 , *BB B* ( ) 

*B B* 

> 

for a discrete-time systems (13)

 *d m* 

1 1 ( ) , 1, 0 *K K*

and off-diagonal blocks *Aijk* , *<sup>B</sup>*( ) *<sup>k</sup>* has diagonal blocks

*k k*

 

 

> 

1 1 ( ) , 1, 0 *K K ij k ijk k k k k*

*xt xt* ( ) ( 1) for discrete-time

are from polytopic

; *Ci* are matrices with corresponding

( ) , 1, 0 *K K i k ik k k k k*

 

 

() () *<sup>d</sup> y t C xt* (11)

 ( ) and

(12)

<sup>1</sup> { ,..., } *C diag C C d N* are overall

*kk k k*

 

. (10)

,

*xt xt* () () for continuous-time system model;

1

*m m*

,

*i*

 

 

The whole interconnected system model in the compact form is

 S: 

*kk k k*

 

 *d m* 

[ ( ) ( ) ( ) ( )] *T T cJ x t Qx t u t Ru t dt*

[ ( ) ( ) ( ) ( )] *T T*

*J x t Qx t u t Ru t*

, ( )

*N i*

1

1 1 ( ) , 1, 0 *K K i k ik k k k k*

1 1 ( ) , 1, 0 *K K ij k ijk k k k k*

uncertain system matrix ( ) ( ) *AA A* 

> ( ) 1 1 ( ) , 1, 0 *K K*

*Bik* and off-diagonal blocks *Bijk* respective to (10); and

 

quadratic cost function known from LQ theory is used.

0

0

*d k* correspond to interconnections.

*k k*

 

*A A* 

where *A*( ) *<sup>k</sup>* has diagonal blocks *Aik*

of the overall system S; <sup>1</sup> ( ) { ( ),..., ( )}, *A diag A A d N*

*Am*( ) , *Bm*( ) 

 ,

dimensions. Uncertain model matrices ( ) *Ai*

 

 

*A A* 

*A A* 

*n n*

*N i i*

where

system model; ( ) *ni*

vectors respectively,

uncertainty domains

Fig. 2. Uncertainty domain specified by working points
