lmi-based stability analysis of fuzzy large-scale systems with time delays

Chaos, Solitons and Fractals 25 (2005) 1193–1207

www.elsevier.com/locate/chaos

LMI-based stability analysis of fuzzy large-scale systemswith time delays

Hongbin Zhang *, Juebang Yu

Institute of Electronic Systems, School of Electronic Engineering, University of Electronic Science and Technology of China,

Chengdu, 610054, PR China

Accepted 24 November 2004

Abstract

The stability of fuzzy large-scale systems with time delays both in states and in interconnections are considered in

this paper. The fuzzy large-scale system consists of J interconnected subsystems, which are represented by the

Takagi–Sugeno (T–S) fuzzy models. The stability conditions are derived using Lyapunov–Krasovskii approach, in com-

bined with the linear matrix inequality (LMI) techniques. We also present a stabilization approach for the delayed fuzzy

large-scale systems through fuzzy state feedback controllers. Finally, numerical examples are given to demonstrate the

correctness of the theoretical results.

� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

One of the foremost challenges to system theory brought forth by present-day technological, environmental and

societal process is to overcome the increasing size and complexity of the relevant mathematical models. Since the

amount of computational effort required for analyzing a dynamic process usually grows much faster than the size of

the corresponding systems. It is therefore natural to seek techniques, which can reduce the computational effort. The

methodologies of large-scale system provide such techniques through the manipulation of system structure in some

way. There were considerable interests in the research of large-scale systems in past years [1]. Recently, many methods

have been presented to investigate the stability and stabilization of large-scale systems [2–5].

During the past several years, fuzzy systems of the T–S model [6] have attracted great interests from scientists and

engineers. The method of T–S fuzzy model suggests an efficient method to represent complex nonlinear systems by fuzzy

sets and fuzzy reasoning. Recently, the issue of stability of fuzzy systems has been considered extensively in nonlinear

stability frameworks [7–23]. Specially, the stability analysis and stabilization of fuzzy large-scale systems for both dis-

crete and continuous case were discussed in [22,23].

It is well known that delays appear in many dynamic systems. Generally speaking, the dynamic behaviors of systems

with delays are more complicated than that of systems without any delays. More recently, delayed fuzzy systems were

0960-0779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2004.11.065

* Corresponding author.

E-mail address: [email protected] (H. Zhang).

mailto:[email protected]

1194 H. Zhang, J. Yu / Chaos, Solitons and Fractals 25 (2005) 1193–1207

studied in [24–28]. In [24] and [25], using the Lyapunov–Krasovskii approach and the Lyapunov-Razumikhin method,

the authors studied the stability of delayed fuzzy systems. In [26] output feedback robust H1 control of uncertain fuzzy

dynamic system with time-varying delays was discussed. In [27], the authors studied global exponential stability of fuzzy

systems with bounded uncertain delays by using the method of functional differential inequalities analysis. In [28],

delay-dependent robust stability of uncertain fuzzy system with time-varying delay was studied.

In this paper, we use the Lyapunov–Krasovskii functional approach [29], in combined with the LMI techniques [30]

to study the stability of fuzzy large-scale systems with time delays. Recently, we noticed that in [32,33], the authors also

considered the stability analysis and stabilization of fuzzy time-delay large-scale systems. However, the system consid-

ered in [32,33] contains tine delays only in interconnections, and their analysis was not based on LMIs. The rest of this

paper is organized as follows. In Section 2, the problem to be studied is stated and some preliminaries are presented.

Based on the Lyapunov–Krasovskii stability theory, in combination with the LMI approach, the stability condition for

continuous-time fuzzy large-scale systems with time delays is derived in Section 3, feedback stabilization of continu-

ous delayed fuzzy large-scale systems via the so-called parallel distributed compensation (PDC) scheme is also dis-

cussed in this section. Stability of discrete time delayed fuzzy large-scale system is studied in Section 4. In Section 5,

numerical examples are given to testify the correctness of theoretical results. And finally, we conclude our results in

Section 6.

2. System descriptions and preliminaries

Consider a continuous time fuzzy large-scale system F, with delays both in states and in interconnections, which con-

sists of J interconnected fuzzy subsystems Fj, j = 1,2, . . . ,J. The jth fuzzy subsystem Fj is described by the following

equations

F j :

_xjðtÞ ¼Prji¼1

lijðtÞ½AijxjðtÞ þ Bijxjðt � sjjÞ þ CijujðtÞ� þ /jðtÞ

/jðtÞ ¼PJn¼1n6¼j

½DnjxnðtÞ þ Enjxnðt � snjÞ�

8>>>><>>>>:

ð1Þ

where xj(t) is the state vector, uj(t) is the input vector, Aij, Bij, Cij, Dij, Eij are constant matrices with appropriate dimen-

sions, rj is the number of fuzzy rules, lij are the normalized weights defined in (4), and sij 2 R+ are the time delays.

Each isolated subsystem of F is represented by a Takagi–Sugeno (T–S) fuzzy model composed of a set of fuzzy impli-

cations, and each implication is expressed by a linear system model. The ith rule of this T–S fuzzy model is of the fol-

lowing form.

Rule i:

IF x1j (t) is Mi1j and . . . and xpj(t) is Mipj THEN

_xjðtÞ ¼ AijxjðtÞ þ Bijxjðt � sjjÞ þ CijuðtÞxjðtÞ ¼ x0j ðtÞ; t 2 ½�sjj; 0�; i ¼ 1; 2; . . . ; rj

ð2Þ

where xj(t) = [x1j(t),x2j(t), . . . ,xpj(t)]T, x1j(t) � xpj(t) are the premise variables, and Milj(l = 1,2, . . . ,p) are fuzzy sets.

Through the use of fuzzy ‘‘blending’’, the final output of the jth isolate fuzzy subsystem is inferred as follows.

_xjðtÞ ¼Prj

i¼1wijðtÞ½AijxjðtÞ þ Bijxjðt � sjjÞ þ CijujðtÞ�Prji¼1wijðtÞ

¼Xrji¼1

lijðtÞ½AijxjðtÞ þ Bijxjðt � sjjÞ þ CijujðtÞ� ð3Þ

with

wijðtÞ ¼Ypl¼1

MiljðxljðtÞÞ; lijðtÞ ¼wijðtÞPrji¼1wijðtÞ

ð4Þ

in which Milj(xlj(t)) is the grade of membership of xlj(t) in Milj. It is assumed that wij(t)P 0, i = 1,2, . . . , rj, andPrji¼1wijðtÞ > 0 for all t. Therefore, lij(t)P 0, and

Prji¼1lijðtÞ ¼ 1 for all t.

The discrete time delayed fuzzy large-scale systems can be defined in similar way. Consider a discrete time delayed

fuzzy large-scale system F, which consists of J interconnected fuzzy subsystems Fj, j = 1,2, . . . ,J. The jth fuzzy subsys-

tem Fj is described by the following equations

H. Zhang, J. Yu / Chaos, Solitons and Fractals 25 (2005) 1193–1207 1195

F j :

xjðt þ 1Þ ¼Prji¼1

lijðtÞ½AijxjðtÞ þ Bijxjðt � hjjÞ þ CijujðtÞ� þ /jðtÞ


½DnjxnðtÞ þ Enjxnðt � hnjÞ�

8>>>><>>>>:

ð5Þ

where xj(t) is the state vector, uj(t) is the input vector, Aij, Bij, Cij, Dij, Eij are constant matrices with appropriate dimen-

sions, rj is the number of fuzzy rules, lij are the normalized weights defined in (8), and hij 2 T+ are time delays.

Each isolated subsystem of F is represented by a discrete time Takagi–Sugeno (T–S) fuzzy model composed of a set

of fuzzy implications, and each implication is expressed by a linear system model. The i th rule of this discrete time T–S

fuzzy model is of the following form.

Rule i:

IF x1j (t) is Mi1j and . . . and xpj(t) is Mipj THEN

xjðt þ 1Þ ¼ AijxjðtÞ þ Bijxjðt � hjjÞ þ CijuðtÞxjðtÞ ¼ x0j ðtÞ; t ¼ �hjj; . . . ;�1; 0; i ¼ 1; 2; . . . ; rj

ð6Þ

where xj(t) = [x1j(t),x2j(t), . . . ,xpj(t)]T, x1j(t) � xpj(t) are the premise variables, and Milj(l = 1,2, . . . ,p) are fuzzy sets.

Through the use of fuzzy ‘‘blending’’, the final output of the jth isolate fuzzy subsystem is inferred as follows.

xjðt þ 1Þ ¼Prj

i¼1wijðtÞ½AijxjðtÞ þ Bijxjðt � hjjÞ þ CijujðtÞ�Prji¼1wijðtÞ

¼Xrji¼1

lijðtÞ½AijxjðtÞ þ Bijxjðt � hjjÞ þ CijujðtÞ�ð7Þ

with

wijðtÞ ¼Ypl¼1

MiljðxljðtÞÞ; lijðtÞ ¼wijðtÞPrji¼1wijðtÞ

ð8Þ

in which Milj(xlj(t)) is the grade of membership of xlj(t) in Milj. It is assumed that wij P 0, i = 1,2, . . . , rj, andPrji¼1wijðtÞ > 0 for all t. Therefore, lij(t) P 0, and

Prji¼1lijðtÞ ¼ 1 for all t.

We will use, WT, W�1 to denote, respectively, the transpose of, and inverse of a square matrix W. We use W > 0

(< 0) to denote a positive (negative) definite matrix W.

Before starting the main results, we need the following lemmas.

Lemma 1 [27]. Let Q be any of a n · n matrix, we will have for any constant k > 0 and any positive definite matrix S > 0

that

2xTQy 6 kxTQS�1QTxþ 1

kyTSy ð9Þ

for all x, y 2 Rn.

Lemma 2 [2]. Consider the (non-singular) matrices A, B, C, which have the same dimension and A = B + C, then for any

positive scalar �, we have

ATA 6 ð1þ eÞBTBþ ð1þ e�1ÞCTC ð10Þ

Lemma 3 [2]. Tchebyshev�s inequality hold for any vector vi 2 Rn

Xp

i¼1

vi

" #T Xp

i¼1

vi

" #6 p

Xp

i¼1

vTi vi ð11Þ

Lemma 4 [25]. Given two matrices A 2 Rm·n, B 2 Rm·n, and two positive definite matrix P 2 Rm·m, Q 2 Rn·n such that

ATPA� Q < 0 and BTPB� Q < 0

then

ATPBþ BTPA� 2Q < 0


3. Stability analysis of continuous time delayed fuzzy large-scale systems

In this section, we first analyze the stability of open-loop continuous time delayed fuzzy large-scale system of the

following form:

F j :

_xjðtÞ ¼Prji¼1

lijðtÞ½AijxjðtÞ þ Bijxjðt � sjjÞ� þ /jðtÞ



8>>>><>>>>:

ð12Þ

Theorem 1. The continuous time delayed fuzzy large-scale system described in (12) is asymptotically stable if there exist

common matrices Pj > 0 (j = 1,2, . . . , J), such that

ATijP j þ P jAij þ ð2J � 1ÞI Mij

MTij �I

" #< 0 i ¼ 1; 2; . . . ; rj ð13Þ

where

Mij ¼ P j½Bij;D1j; . . . ;Dj�1;j;Djþ1;j; . . . ;DJ ;j;E1j; . . . ;Ej�1;j;Ejþ1;j; . . . ;EJ ;j�:

Proof. A proof of this theorem is given in Appendix A. h

Next, we consider the fuzzy controller design problem. The method of the parallel distributed compensation (PDC)

[7,22] is utilized to design fuzzy controllers. The concept of PDC scheme is that each control rule is distributively de-

signed for the corresponding rule of a T–S fuzzy model. The fuzzy controller shares the same fuzzy sets with the fuzzy

model in the premise parts. Since each rule of the fuzzy model is described by a linear state equation, linear control

theory can be used to design the consequent parts of a fuzzy controller. The resulting overall fuzzy controller is achieved

by fuzzy ‘‘blending’’ of each individual linear controller.

The jth fuzzy controller can be described as follows:

Rule i:

IF x1j(t) is Mi1j and . . . and xpj(t) is Mipj THEN

ujðtÞ ¼ �F ijxjðtÞ ð14Þ

for i = 1,2, . . . , rj. The overall state feedback fuzzy control law is represented by

ujðtÞ ¼ �Prj

i¼1wijðtÞF ijxjðtÞPrji¼1wijðtÞ

¼ �Xrji¼1

lijðtÞF ijxjðtÞ ð15Þ

Substituting (15) into (1), we can get the following closed-loop delayed fuzzy large-scale systems. The design of state

feedback fuzzy controller is to determine the local feedback gains Fij such that the following closed-loop fuzzy large-

scale system with time delays is asymptotically stable

F j :

_xjðtÞ ¼Prji¼1

Prjk¼1

lijðtÞlkjðtÞ½ðAij � CijF kjÞxjðtÞ þ Bijxjðt � sjjÞ� þ /jðtÞ



8>>>><>>>>:

ð16Þ

Noting thatPrj

i¼1

Prjk¼1lijlkj ¼ 1, also that the closed-loop system (16) can be looked as a free delayed fuzzy large-scale

system, we can directly use the results of the above free system to derive asymptotical stability condition for (16), and

then get a feedback controller design method.

Theorem 2. If there exist common matrices Pj > 0 (j = 1,2, . . . , J), such that

ðAij � CijF kjÞTP j þ P jðAij � CijF kjÞ þ ð2J � 1ÞI Mij

MTij �I

" #< 0 i ¼ 1; 2; . . . ; rj; k ¼ 1; 2; . . . ; rj ð17Þ


where Mj = Pj[Bij,D1j, . . . ,Dj�1,j,Dj+1,j, . . . ,DJ,j,E1j, . . . ,Ej�1,j,Ej+1,j, . . . ,EJ,j], and I is a identity matrix, then the fuzzy

large-scale system (16) can be asymptotically stabilized by the fuzzy controller (15).

A proof of this theorem can be derived by some slight modifications of the proofs of Theorem 1; therefore details are

omitted.

By using the above theorems, the stability condition and controllers can be obtained directly by solving the LMIs

numerically using, for example, the interior-point algorithms [30].

4. Stability analysis of discrete time delayed fuzzy large-scale systems

In this section, we analyze the stability of the open-loop discrete time delayed fuzzy large-scale systems

F j :

xjðt þ 1Þ ¼Prji¼1

lijðtÞ½AijxjðtÞ þ Bijxjðt � hjjÞ� þ /jðtÞ



8>>>><>>>>:

ð18Þ

Theorem 3. The discrete time delayed fuzzy large-scale system described by (18) is asymptotically stable if there exist

common matrices Pj > 0, Sjn > 0, and positive scalar kj > 0, (j = 1,2, . . . , J), such that the following matrix inequalities are

fulfilled:

Mj11 Mj12

MTj12 Mj22

" #< 0 ð19Þ

where

Mj11 ¼Xrji¼1

½l2ijrj þ lijðJ � 1Þkj�AT

ijP jAij � P j þXJ

n¼1n6¼j

J � 1þ 1

kn

� �DT

jnP nDjn þXJ

n¼1

Sjn ð20Þ

Mj12 ¼Xrji¼1

½l2ijrj þ lijðJ � 1Þkj�AT

ijP jBij þXJ

n¼1n 6¼j

J � 1þ 1

kn

� �DT

jnP nEjn ð21Þ

Mj22 ¼Xrji¼1

½l2ijrj þ lijðJ � 1Þkj�BT

ijP jBij þXJ

n¼1n 6¼j

J � 1þ 1

kn

� �ETjnP nEjn �

XJ

n¼1

Sjn ð22Þ

Proof. A proof of this theorem is given in Appendix B. h

Theorem 3 can be looked as a generalization of the Theorem 1 of [23]. By using Lemmas 2 and 4, we can obtain

another stability condition, which does not utilize the information of the membership functions.

Theorem 4. The discrete time delayed fuzzy large scale system described by (18) is asymptotically stable if there exist

common matrices Pj > 0, Sjn > 0, and positive scalar ej > 0 (j = 1,2, . . . , J), such that the following matrix inequalities are

fulfilled:

Mij11 Mij12

MTij12 Mij22

" #< 0 i ¼ 1; 2; . . . ; rj ð23Þ

where

Mij11 ¼ ð1þ ejÞATijP jAij � P j þ ðJ � 1Þ

XJ

n¼1n 6¼j

ð1þ e�1n ÞDT

jnP nDjn þXJ

n¼1

Sjn ð24Þ


Mij12 ¼ ð1þ ejÞATijP jBij þ ðJ � 1Þ

XJ

n¼1n6¼j

ð1þ e�1n ÞDT

jnP nEjn ð25Þ

Mij22 ¼ ð1þ ejÞBTijP jBij þ ðJ � 1Þ

XJ

n¼1n6¼j

ð1þ e�1n ÞET

jnP nEjn �XJ

n¼1

Sjn ð26Þ

The design of feedback fuzzy controller for this discrete time system via the PDC scheme is similar to that of the con-

tinuous case, and the stability analysis of the closed loop system is similar to that of Theorems 3 and 4, therefore details

are omitted.

5. Numerical examples

In this section, we give some numerical examples to testify the correctness of the above theoretical analysis.

Example 1. Consider a continuous time delayed fuzzy large-scale system, which consists of three fuzzy subsystems

described by the following equations:

F :

_x1ðtÞ ¼P2i¼1

li1ðtÞ½Ai1x1ðtÞ þ Bi1x1ðt � s11Þ� þ /1ðtÞ

_x2ðtÞ ¼P2i¼1


_x3ðtÞ ¼P2i¼1


/jðtÞ ¼P3n¼1n 6¼j

½DnjxnðtÞ þ Enjxnðt � snjÞ�; j ¼ 1; 2; 3

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð27Þ

in which xT1 ðtÞ ¼ ½x11ðtÞ; x21ðtÞ�, xT2 ðtÞ ¼ ½x12ðtÞ; x22ðtÞ�, xT3 ðtÞ ¼ ½x13ðtÞ; x23ðtÞ�, and

A11 ¼�1 �0:5

1 �4

� �; A21 ¼

�1:5 0:3

1 �2

� �; A12 ¼

�2 0:8

1 �3

� �; A22 ¼

�1 0:3

�1 �4

� �

A13 ¼�3 0:1

0:3 �1

� �; A23 ¼

�2 0:3

�2 �5

� �; B11 ¼

0:1 0

0:2 �0:4

� �; B21 ¼

�0:2 0:3

0:1 0

� �

B12 ¼�0:1 0

0:2 0:5

� �; B22 ¼

0:3 �0:2

0 �0:1

� �; B13 ¼

0:1 0

0 �0:2

� �; B23 ¼

0 0:1

0 0:1

� �

D21 ¼0:1 0

0 0:1

� �; D31 ¼

�0:1 0

0 0

� �; D12 ¼

0 0

0:1 0:1

� �; D32 ¼

0:1 0

0:1 0

� �

D13 ¼0:1 0

�0:1 0:1

� �; D23 ¼

�0:1 0

0 0:1

� �; E21 ¼

0 0

0 0

� �; E31 ¼

0:1 0

0 0

� �

E12 ¼0:1 0

�0:1 0:1

� �; E32 ¼

�0:1 0

0 0:1

� �; E13 ¼

0:1 0

�0:1 0:1

� �; E23 ¼

�0:1 0

0 �0:1

� �

Moreover, the Takagi–Sugeno fuzzy models of the three isolated subsystems are of the following form:

Subsystem 1

Rule 1: IF x11(t) is M111, THEN

_x1ðtÞ ¼ A11x1ðtÞ þ B11x1ðt � s11Þ ð28aÞ



_x1ðtÞ ¼ A21x1ðtÞ þ B21x1ðt � s11Þ ð28bÞ

and the membership function for Rule 1 and Rule 2 are

M111ðx11ðtÞÞ ¼1

1þ expð�2x11ðtÞÞ; M211ðx11ðtÞÞ ¼ 1�M111ðx11ðtÞÞ

Subsystem 2






M112ðx12ðtÞÞ ¼ expð�x212ðtÞÞ; M212ðx12ðtÞÞ ¼ 1�M112ðx12ðtÞÞ

Subsystem 3








Using the MATLAB LMI toolbox to solve (13), we can obtain the following positive definite matrices Pj (j = 1,2,3):

P 1 ¼4:1663 0:1826

0:1826 1:9450

� �; P 2 ¼

4:8945 0:4852

0:4852 2:7085

� �; P 3 ¼

5:2245 0:4568

0:4568 2:7085

� �ð31Þ

Hence, according to Theorem 1, the continuous time delayed fuzzy large-scale system (27) is asymptotically stable. Be-

cause the stability condition is delay-independent, for simplicity, we let all delays snj be 1 in our simulations. Simulation

results of each subsystem with initial conditions x11(t) = x12(t) = x13(t) � 2, x21(t) = x22(t) = x23��2, for t 2 [�1,0] are

shown in Figs. 1–3. From these figures, we can see that the system (27) is indeed asymptotically stable.

If we let

A11 ¼�1 �0:5

1 3

� �

Fig. 1. State response of subsystem 1.




then, it is easy to know that the system (27) is no longer stable. We can apply Theorem 2 to design a controller to sta-

bilize this system. Assuming all Cij be identity matrices, we can find feasible feedback gains Fij by solving (17). For

example, we can testify by using Theorem 2 that the following Fij can stabilize this system:

F 11 ¼ F 21 ¼1 0

0 4

� �; F ij ¼ O; for i ¼ 1; 2; j ¼ 2; 3:

The validation process is similar to that of the open-loop systems, therefore we omit it.

Example 2. Consider a discrete time delayed fuzzy large-scale system, which consists of three fuzzy subsystems

described by the following equations:

F :

x1ðt þ 1Þ ¼P2i¼1

li1ðtÞ½Ai1x1ðtÞ þ Bi1x1ðt � h11Þ� þ /1ðtÞ





/jðtÞ ¼P3n¼1n 6¼j

½DnjxnðtÞ þ Enjxnðt � hnjÞ�; j ¼ 1; 2; 3

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð32Þ


in which xT1 ðtÞ ¼ ½x11ðtÞ; x21ðtÞ�, xT2 ðtÞ ¼ ½x12ðtÞ; x22ðtÞ�, xT3 ðtÞ ¼ ½x13ðtÞ; x23ðtÞ�, and

A11 ¼0:5 0

0 0:4

� �; A21 ¼

0:2 0:1

0 0:5

� �;A12 ¼

0:2 0

0:1 0:3

� �; A22 ¼

0:3 0

0:1 0:1

� �

A13 ¼0:3 0

0:3 0:1

� �; A23 ¼

0:2 0:3

0 0:1

� �; B11 ¼

0:1 0

0 0:3

� �; B21 ¼

0:2 0

0:1 0

� �

B12 ¼0:1 0

0 0

� �; B22 ¼

0:3 0:1

0 0:1

� �; B13 ¼

0:1 0

0 0:2

� �; B23 ¼

0:2 0:3

0 0:1

� �

D21 ¼0:1 0

0 0:1

� �; D31 ¼

�0:1 0

0 0

� �; D12 ¼

0 0

0:1 0:1

� �; D32 ¼

0:1 0

0:1 0

� �

D13 ¼0:1 0

�0:1 0:1

� �; D23 ¼

�0:1 0

0 0:1

� �; E21 ¼

0 0

0 0

� �; E31 ¼

0:1 0

0 0

� �

E12 ¼0:1 0

0 0:1

� �; E32 ¼

�0:1 0

0 0:1

� �; E13 ¼

0:1 0

�0:1 0:1

� �; E23 ¼

�0:1 0

0 �0:1

� �

Moreover, the discrete time Takagi–Sugeno fuzzy models of the three isolated subsystems are of the following

form:

Subsystem 1


x1ðt þ 1Þ ¼ A11x1ðtÞ þ B11x1ðt � h11Þ ð33aÞ


x1ðt þ 1Þ ¼ A21x1ðtÞ þ B21x1ðt � h11Þ ð33bÞ




Subsystem 2


x2ðt þ 1Þ ¼ A12x2ðtÞ þ B12x2ðt � h22Þ ð34aÞ


x2ðt þ 1Þ ¼ A22x2ðtÞ þ B22x2ðt � h22Þ ð34bÞ


M112ðx12ðtÞÞ ¼ expð�x212ðtÞÞ; M212ðx12ðtÞÞ ¼ 1�M112ðx12ðtÞÞ

Subsystem 3


x3ðtÞ ¼ A13x3ðtÞ þ B13x3ðt � h33Þ ð35aÞ


x3ðtÞ ¼ A23x3ðtÞ þ B23x3ðt � h33Þ ð35bÞ




Let ej = 1 (j = 1,2,3), and using the MATLAB LMI toolbox to solve (23), we can obtain and the following positive def-

inite matrices Pj, Sjn (j = 1,2,3):


P 1 ¼96:7464 0� 11:6880

�11:6880 146:5433

� �; P 2 ¼

27:3720 2:3010

2:3010 10:5988

� �; P 3 ¼

9:6591 �0:3956

�0:3956 5:2997

� �

S1n ¼14:5573 �1:4654

�1:4654 20:2415

� �; S2n ¼

3:4579 0:9173

0:9173 0:7581

� �; S3n ¼

3:4579 0:9173

0:9173 0:7581

� �; n ¼ 1; 2; 3

ð36Þ

Hence, according to Theorem 4, the discrete time delayed fuzzy large-scale system (32) is asymptotically stable. Because

the stability condition is delay-independent, for simplicity, we let all delays hnj be 1 in our simulations. Simulation re-

sults of each subsystem with random initial conditions are shown in Fig. 4–6. From these figures, we can see that the

system (32) is indeed asymptotically stable.





6. Conclusions

In this paper, the stability and stabilization of fuzzy large-scale systems with time delay both in states and in inter-

connections has been studied. Stability conditions for both continuous time and discrete time delayed fuzzy large-scale

systems were derived using the Lyapunov–Krasovskii approach, in combined with the linear matrix inequality (LMI)

techniques. Based on these stability conditions, a set of fuzzy controllers was designed via the concept of PDC. Finally,

numerical examples were provided to demonstrate the correctness of the theoretical results.

Appendix A. Proof of Theorem 1

Using the concept of vector Lyapunov function [31], we select a Lyapunov functional as

V ðtÞ ¼XJ

j¼1

V jðtÞ ¼XJ

j¼1

xTj ðtÞP jxjðtÞ þXJ

n¼1

Z t

t�snj

xTn ðsÞxnðsÞds" #

ðA:1Þ

where Pj > 0 is to be selected. In the following, we will prove the asymptotic stability of the time delay system (12) based

on the Lyapunov–Krasovskii theorem [28]. It is obviously that there exist r1 and r2 such that

r1kxjðtÞk2 6 V jðxjðtÞÞ 6 r2kxjðtÞk2 ðA:2Þ

The derivative of the Lyapunov functional along the trajectories of (12) is" #
_V ðtÞ ¼
XJ

i¼1

_V jðtÞ ¼XJ

j¼1

_xTj ðtÞP jxjðtÞ þ xTj ðtÞPj _xjðtÞ þXJ

n¼1

xTn ðtÞxnðtÞ �XJ

n¼1

xTn ðt � snjÞxnðt � snjÞ

¼XJ

j¼1

Xrji¼1

lijðtÞ xTj ðtÞATijP jxjðtÞ þ xTj ðt � sjjÞBT

ijP jxjðtÞ þ xTj ðtÞP jAijxjðtÞ þ xTj ðtÞP jBijxjðt � sjjÞ� �"

þ/Tj ðtÞP jxjðtÞ þ xTj ðtÞPj/jðtÞ þ

XJ

n¼1

xTn ðtÞxnðtÞ �XJ

n¼1

xTn ðt � snjÞxnðt � snjÞ#

ðA:3Þ

Using Lemma 1, and noting thatPrj

i¼1lijðtÞ ¼ 1, we have

_V ðtÞ 6XJ

j¼1

Xrji¼1

lijðtÞxTj ðtÞ ATijP j þ PjAij þ P jBijBT

ijP j

� �xjðtÞ þ

XJ

n¼1

xTn ðtÞxnðtÞ"

þXJ

n¼1n6¼j

xTn ðtÞDTnjP jxjðtÞ þ xTj ðtÞP j

XJ

n¼1n 6¼j

DnjxnðtÞ þXJ

n¼1n6¼j

xTn ðt � snjÞETnjP jxjðtÞ

þxTj ðtÞP j

XJ

n¼1n6¼j

Enjxnðt � snjÞ �XJ

n¼1n6¼j

xTn ðt � snjÞxnðt � snjÞ#

ðA:4Þ

Completing the quadratic form for the last three terms by adding and subtracting xTj ðtÞPjPJ

n¼1;n6¼jEnjETnjP jxjðtÞ to the

above equation, and also using Lemma 1, results in

_V ðtÞ 6XJ

j¼1

Xrji¼1

lijðtÞxTj ðtÞðATijP j þ PjAij þ P jBijBT

ijP jÞxjðtÞ þXJ

n¼1

xTn ðtÞxnðtÞ"

þXJ

n¼1n6¼j

xTn ðtÞxnðtÞ þ xTj ðtÞP j

XJ

n¼1n6¼j

DnjDTnjP jxjðtÞ þ xTj ðtÞPj

XJ

n¼1n6¼j

EnjETnjP jxjðtÞ

#

¼XJ

j¼1

Xrji¼1

lijðtÞxTj ðtÞ ATijP j þ PjAij þ P jBijBT

ijP j þ P j

XJ

n¼1n6¼j

DnjDTnjP j þ P j

XJ

n¼1n 6¼j

EnjETnjP j

0BB@

1CCAxjðtÞ

2664

3775

þXJ

j¼1

XJ

n¼1

xTn ðtÞxnðtÞ þXJ

n¼1n6¼j

xTn ðtÞxnðtÞ

2664

3775 ðA:5Þ


The last term of (A.5) can be recast as

XJ

j¼1

XJ

n¼1

xTn ðtÞxnðtÞ þXJ

n¼1n6¼j

xTn ðtÞxnðtÞ

2664

3775 ¼

XJ

j¼1

XJ

n¼1

xTj ðtÞxjðtÞ þXJ

n¼1n 6¼j

xTj ðtÞxjðtÞ

2664

3775 ¼

XJ

j¼1

xTj ðtÞ½ð2J � 1ÞI �xjðtÞ ðA:6Þ

Thus, we have

_V ðtÞ 6XJ

j¼1

Xrji¼1

lijðtÞxTj ðtÞ ATijP j þ P jAij þ PjBijBT

ijP j þ Pj

XJ

n¼1n6¼j

ðDnjDTnj þ EnjET

njÞP j þ ð2J � 1ÞI

0BB@

1CCAxjðtÞ

2664

3775 ðA:7Þ

From Schur complement, we know that the LMI (13) is equivalent to

ATijP j þ P jAij þ PjBijBT

ijP j þ P j

XJ

n¼1n 6¼j

ðDnjDTnj þ EnjET

njÞP j þ ð2J � 1ÞI < 0 ðA:8Þ

So, we have _V ðtÞ < 0 if xj 6¼ 0ð _V ðtÞ ¼ 0 if and only if xjðtÞ ¼ 0Þ. From Lyapunov–Krasovskii theorem, the whole con-

tinuous time delayed fuzzy large-scale system (12) is asymptotically stable.

Appendix B. Proof of Theorem 3

Let the Lyapunov functional be

V ðtÞ ¼XJ

j¼1

V jðtÞ ¼XJ

j¼1

xTj ðtÞP jxjðtÞ þXJ

n¼1

Xhnjl¼1

xTn ðt � lÞSnjxnðt � lÞ" #

ðB:1Þ

where Pj > 0 and Snj > 0 are to be selected.

Taking the backward difference of V(t), we have

DV ðtÞ ¼ V ðt þ 1Þ � V ðtÞ

¼XJ

j¼1

xTj ðt þ 1ÞP jxjðt þ 1Þ þXJ

n¼1

Xhnj�1

l¼0


�XJ

j¼1

xTj ðtÞPjxjðtÞ þXJ

n¼1

Xhnjl¼1


¼XJ

j¼1

Xrji¼1

lij½AijxjðtÞ þ Bijxjðt � hjjÞ� þXJ

n¼1n6¼j


2664

3775

T

P j

�Xrji¼1

lij½AijxjðtÞ þ Bijxjðt � hjjÞ� þXJ

n¼1n6¼j


2664

3775

þXJ

j¼1

XJ

n¼1

Xhnj�1

l¼0

xTn ðt � lÞSnjxnðt � lÞ � xTj ðtÞP jxjðtÞ �XJ

n¼1

Xhnjl¼1


¼XJ

j¼1

Xrji¼1

lijAij�xjðtÞ þXJ

n¼1n6¼j

Dnj�xnðtÞ

2664

3775

T

P j

Xrji¼1

lijAij�xjðtÞ þXJ

n¼1n6¼j

Dnj�xnðtÞ

2664

3775

þXJ

j¼1

��xTj ðtÞPj�xjðtÞ þXJ

n¼1

�xTn ðtÞSnj�xnðtÞ" #

¼ T 1 þ T 2 ðB:2Þ


where

�xnðtÞ ¼xnðtÞ

xnðt � hnjÞ

" #; Aij ¼ ½AijBij�; Dij ¼ ½DijEij�; P j ¼

P j 0

0 0

" #; Snj ¼

Snj 0

0 �Snj

" #ðB:3Þ

T 1 ¼XJ

j¼1

Xrji¼1

lijAij�xjðtÞ" #T

P j

Xrji¼1

lijAij�xjðtÞ" #

þXJ

n¼1n 6¼j

Dnj�xnðtÞ

2664

3775

T

P j

XJ

n¼1n6¼j

Dnj�xnðtÞ

2664

3775

8>><>>:

þXJ

n¼1n6¼j

Xrji¼1

lij½�xTj ðtÞAT

ijP jDnj�xnðtÞ þ �xTn ðtÞDT

njP jAij�xjðtÞ�

9>>=>>; ðB:4Þ

Using Lemmas 1 and 3, we have

T 1 6

XJ

j¼1

Xrji¼1

l2ijrj�x

Tj ðtÞA

T

ijP jAij�xjðtÞ þXJ

j¼1

ðJ � 1ÞXJ

n¼1n 6¼j

�xTn ðtÞDT

njP jDnj�xnðtÞ

þXJ

j¼1

XJ

n¼1n6¼j

Xrji¼1

lij kj�xTj ðtÞAT

ijP jAij�xjðtÞ þ1

kj�xTn ðtÞD

T

njP jDnj�xnðtÞ� �

¼XJ

j¼1

Xrji¼1

l2ijrj�x

Tj ðtÞA

T


j¼1

ðJ � 1ÞXJ

n¼1n6¼j

�xTj ðtÞDT

jnP nDjn�xjðtÞ

þXJ

j¼1

Xrji¼1

lijðJ � 1Þkj�xTj ðtÞAT


j¼1

XJ

n¼1n6¼j

1

kn�xTj ðtÞD

T

jnP nDjn�xjðtÞ

¼XJ

j¼1

�xTjXrji¼1

½l2ijrj þ lijðJ � 1Þkj�A

T

ijP jAij þXJ

n¼1n6¼j

J � 1þ 1

kn

� �D

T

jnP nDjn

2664

3775�xjðtÞ ðB:5Þ

and

T 2 ¼XJ

j¼1

�xTj ðtÞ �P j þXJ

n¼1

Sjn

" #�xjðtÞ ðB:6Þ

So

DV ðtÞ ¼ T 1 þ T 2

6

XJ

j¼1

�xTjXrji¼1


T

ijP jAij þXJ

n¼1n6¼j

J � 1þ 1

kn

� �D

T

jnP nDjn � P j þXJ

n¼1

Sjn

2664

3775�xjðtÞ ðB:7Þ

Using (B.3), it is easy to check that (19)–(22) is equivalent to the following inequality

Xrji¼1


T

ijP jAij þXJ

n¼1n 6¼j

J � 1þ 1

kn

� �D

T

jnP nDjn � P j þXJ

n¼1

Sjn < 0 ðB:8Þ

Thus, DV(t) < 0 if xj (t)5 0 (DV(t) = 0 if and only if xj(t) = 0). So the discrete time delayed fuzzy large-scale system (18)

is asymptotically stable.


Appendix C. Proof of Theorem 4

The definitions of V(t), T1, T2 and DV(t) are the same as that in Appendix B.

By using Lemma 2, we have

T 1 6

XJ

j¼1

Xrji¼1

Xrjk¼1

lijlkjð1þ ejÞ�xTj ðtÞAT

ijP jAkj�xjðtÞ þXJ

j¼1

ð1þ e�1j ÞðJ � 1Þ

XJ

n¼1n6¼j

�xTn ðtÞDT

njP jDnj�xnðtÞ

¼XJ

j¼1

Xrji¼1

Xrjk¼1

lijlkjð1þ ejÞ�xTj ðtÞAT

ijP jAkj�xjðtÞ þXJ

j¼1

ðJ � 1ÞXJ

n¼1n6¼j

ð1þ e�1n Þ�xTj ðtÞD

T

jnP nDjn�xjðtÞ ðC:1Þ

DV ðtÞ ¼ T 1 þ T 2

6

XJ

j¼1

Xrji¼1

Xrjk¼1

lijlkj�xTj ðtÞ ð1þ ejÞA

T

ijP jAkj � P j þXJ

n¼1

Sjn þ ðJ � 1ÞXJ

n¼1n 6¼j

ð1þ e�1n ÞDT

jnP nDjn

2664

3775�xjðtÞ

¼XJ

j¼1

Xrji¼1

l2ij�x

Tj ðtÞ ð1þ ejÞA

T

ijP jAij � P j þXJ

n¼1

Sjn þ ðJ � 1ÞXJ

n¼1n 6¼j

ð1þ e�1n ÞDT

jnP nDjn

2664

3775�xjðtÞ

þXJ

j¼1

Xrji¼1

Xrjk¼1k<i

lijlkj�xTj ðtÞ½ð1þ ejÞðA

T

ijP jAkj þ AT

kjP jAijÞ � 2P j þ 2XJ

n¼1

Sjn

þ 2ðJ � 1ÞXJ

n¼1n6¼j

ð1þ e�1n ÞDT

jnP nDjn��xjðtÞ ðC:2Þ

From Lemma 4, if the matrix inequalities in (23)–(26) hold, then DV(t) < 0 if xj(t)50 (DV(t) = 0 if and only if xj(t) = 0),

therefore (18) is asymptotically stable.

References

[1] Siljak DD. Large-Scale Dynamic Systems: Stability and Structure. New York: Elsevier North-Hollad; 1978.

[2] Lee TN, Radovic UL. General decentralized stabilization of large-scale linear continuous and discrete time-delay systems. Int J

Contr 1987;46(6):2127–40.

[3] Hu Z. Decentralized stabilization of large-scale interconnection systems with delays. IEEE Trans Auto Contr 1994;39(1):180–2.

[4] Wang WJ, Mau LG. Stabilization and estimation for perturbed discrete time-delay large-scale systems. IEEE Trans Auto Contr

1997;42(9):1277–82.

[5] Tsay JC, Liu PL, Su TJ. Robust stability for perturbed large-scale time-delay systems. IEE Proc Contr Theory Appl

1996;143(3):233–6.

[6] Takagi T, Sugeno M. Fuzzy identification of systtems and its application to modeling and control. IEEE Trans Syst Man Cybern

1985;15:116–32.

[7] Tanaka K, Sugeno M. Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst 1992;45:135–56.

[8] Tanaka K, Sano M. Fuzzy stability criterion of a class of nonlinear systems. Inform Sci 1993;70(1):3–26.

[9] Tanaka K, Sano M. A robust stabilization problem of fuzzy control systems its application to backing up control of a truck-

trailer. IEEE Trans Fuzzy Syst 1994;2(2):119–34.

[10] Tanaka K. Stabilization and stability of fuzzy-neural-linear control systems. IEEE Trans Fuzzy Syst 1995;3(4):438–47.

[11] Cao SG, Rees NW, Feng G. Stability analysis and design for a class of continuous-time fuzzy control systems. Int J Contr

1996;64(6):1069–87.

[12] Kim E, Lee H. New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Trans Fuzzy Syst

2000;8(5):523–4.

[13] Akar M, Ozguner U. Decentralized techniques for the analysis and control of takagi-sugeno fuzzy systems. IEEE Trans Fuzzy

Syst 2000;8(6):691–704.

[14] Tanaka K, Ikeda T, Wang HO. Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE

Trans Fuzzy Syst 1998;6(2):250–65.


[15] French M, Rogers E. Input/output stability theory for direct neuro-fuzzy controllers. IEEE Trans Fuzzy Syst 1998;6(3):331–45.

[16] Calcev G. Some remarks on the stability of mamdani fuzzy control systems. IEEE Trans Fuzzy Syst 1998;6(3):436–42.

[17] Feng M, Harris CJ. Piecewise lyapunov stability conditions of fuzzy systems. IEEE Trans SMC-B 2001;31(2):259–62.

[18] Kim E, Kim D. Stability analysis and synthesis for an affine fuzzy systems via LMI and ILMI: discrete case. IEEE Trans SMC-B

2001;31(1):132–40.

[19] Kim E, Kim S. Stability analysis and synthesis for an affine fuzzy systems via LMI and ILMI: continuous case. IEEE Trans Fuzzy

Syst 2002;10(3):391–400.

[20] Feng G. Approaches to quadratic stabilization of uncertain fuzzy dynamic systems. IEEE Trans Circuit Syst 2001;48(6):760–9.

[21] Feng G, Ma J. Quadratic stabilization of uncertain discrete-time fuzzy dynamic systems. IEEE Trans Circuit Syst

2001;48(11):1337–44.

[22] Hsiao FH, Hwang JD. Decentralized stabilization of fuzzy large-scale systems. In: Proc IEEE Conf Decision and Control, Sydney,

2000:3447–52.

[23] Hsiao FH, Hwang JD. Stability analysis of fuzzy large-scale systems. IEEE Trans SMC-B 2002;32(1):122–6.

[24] Cao YY, Frank PM. Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi–Sugeno fuzzy models.

Fuzzy Sets and Systems 2001;124:213–29.

[25] Cao YY, Frank PM. Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach. IEEE Trans Fuzzy

Systems 2000;8(2):200–11.

[26] Lee KR, Kim JH, Jeung ET, Park HB. Output feedback H1 control of uncertain fuzzy dynamic systems with time-varying delay.

IEEE Trans Fuzzy Systems 2000;8(6):657–64.

[27] Zhang Y, Heng PA. Stability of fuzzy control systems with bounded uncertain delays. IEEE Trans Fuzzy Systems

2002;10(1):92–7.

[28] Li C, Wang H, Liao X. Delay-dependent robust stability of uncertain fuzzy systems with time-varying delays. IEE Proc Contr

Theory Appl 2004;151(5):417–21.

[29] Hale JK. Theory of functional differential equations. NY: Springer-verlag; 1977.

[30] Boyd S, Ghaoui L, Feron E, Balakrishnan V. Linear Matrix inequalities in system and control theory. Philadephia: SIAM; 1994.

[31] Lakshmikantham V, et al. Vector Lyapunov functions and stability analysis of nonlinear systems. Kluwer Academic Publishers;

1991.

[32] Li C, Liao X, Wang H. Stability analysis of fuzzy large-scale systems with time delays in interconnections. Int J Gen Syst

2004;33(6):621–33.

[33] Wang RJ, Wang WJ. Fuzzy control for perturbed fuzzy time-delay large-scale systems. IEEE Int Conf Fuzzy Syst 2003:537–42,

May 25–28, St. Louis, Missouri, USA.

lmi-based stability analysis of fuzzy large-scale systems with time delays

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