Stability analysis and controller synthesis for discrete-time delayed fuzzy systems via small gain theorem

Information Sciences 226 (2013) 93–104

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Stability analysis and controller synthesis for discrete-time delayed fuzzy systems via small gain theorem Zhicheng Li a, Huijun Gao b,⇑, Ramesh K. Agarwal a a b

Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, 150001, China Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 27 July 2011 Received in revised form 6 November 2012 Accepted 10 November 2012 Available online 28 November 2012 Keywords: Fuzzy systems Stability analysis Stabilization Scaled small gain theorem Time-delay systems

a b s t r a c t This paper investigates the problems of stability analysis and controller synthesis of discrete-time Takagi–Sugeno (T–S) fuzzy systems with time-varying delay via an input–output approach. The original system is reformulated as a feedback subsystem and a forward subsystem with two constant time delays and a delay ‘‘uncertainty’’ satisfying the scaled small gain theorem. The lifting method is employed to convert the forward subsystem to a delay-free system, where the main result concerning the stability analysis of the system is obtained based on the scaled small gain theorem. The merit of the new stability condition lies in its reduced conservatism, which is made possible by a more precise estimation of the time-varying delay. Additionally, a state-feedback controller is designed through the Lyapunov–Krasovskii functional approach combined with the utilization of the scaled small gain theorem. Two illustrative examples are given to demonstrate the less conservatism of the stability condition and the effectiveness of the controller design method. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Majority of real world systems are nonlinear. In recent years, Takagi–Sugeno (T–S) fuzzy model has emerged as a powerful tool for approximating a complex nonlinear system [25]. In T–S model a weakly nonlinear system is locally linearized for the purpose of controller design. Therefore it can be considered as an intermediate model connecting the nonlinear control problem with the linear system theory [5]. During the past decade, T–S fuzzy control of nonlinear systems has been the subject of many investigations dealing with the problems of stability analysis and controller synthesis of these systems [2,26]. Time delays often arise in many dynamical systems and they are often a source of instability. In past several decades, there has been significant emphasis in the literature on the stability analysis and control of time-delay systems. Refs. [11–14,27] investigate the problem of stability analysis for systems with time-varying state delay and complex networks respectively. Several papers address the control of stochastic systems [23,31,33], neural networks [15,16,28] and uncertain systems [22,35]. In resent years, a great deal of attention has been devoted to fuzzy time-delay systems [18,19]. In the continuous-time case, the problem of stability analysis has been studied based on a special class of fuzzy weighting functions [20]. The problem of stabilization of T–S fuzzy systems has been investigated via a delay-partitioning approach in [42], and the problem of guaranteed cost controller design has been investigated in [36]. Additionally, the problems of the observerbased controller design and H1 filtering for continuous-time T–S fuzzy systems with an interval time-varying delay have been investigated in [3,41] respectively. The ever-increasing progress in computer hardware and algorithms has made it possible to treat complex discrete systems of practical relevance. In resent years, several papers have addressed the problem of stability analysis for discrete systems. In ⇑ Corresponding author. E-mail address: [email protected] (H. Gao). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.11.008

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Z. Li et al. / Information Sciences 226 (2013) 93–104

[6], the authors study the problem of stability analysis by defining new Lyapunov functions. The problem of controller design is discussed in [21]. In most industrial applications, there exist both nonlinearity and time delay. Several techniques have been proposed in the literature to study the control problems of nonlinear time-delay systems based on Smith Predictor [17] and PID controller. However, there are well known deficiencies in these control methods. It is now established that the T–S fuzzy model approach can be used as a powerful method for representing the time-delay nonlinear dynamical systems. The discrete-time fuzzy systems with time delay have been successfully applied to model chemical plants and other industrial processes, see for example [24,43] and the references cited therein. The authors in [7] obtain stability analysis results by using the free-weighting matrices technique, which are further improved in [29] by utilizing a delay-partitioning method. In [32], the authors investigate the robust H1 control problem for T–S fuzzy systems, which is further improved in [39] by using a new Lyapunov–Krasovskii functional. These results are obtained by defining different types of Lyapunov functionals and by resorting to different techniques for achieving delay dependence [12,30,34]. It is worth mentioning that these results, although delay dependent, still leave much room for further improvement. It is still an important and challenging problem how to further reduce the conservatism and overdesign for the T–S fuzzy discrete-time systems with timevarying delay; this is the motivation behind the present study. In this paper, we focus on the problems of stability analysis and stabilization for discrete-time T–S fuzzy systems with timevarying delay. An input–output approach is proposed for solving the problems based on the scaled small gain theorem. To begin with, the lower delay bound d1 and the upper delay bound d2 are utilized to estimate the time-varying delay d(k). The original model is transformed into an interconnected form, which consists of a forward subsystem with two constant delays and an uncertain disturbance and a feedback subsystem. Then, the lifting method is used to obtain the corresponding delay-free system, and the bounded real lemma (BRL) is employed to yield the main result for the stability analysis. Additionally, a state-feedback controller is designed such that the closed-loop system is asymptotically stable. Finally, two illustrative examples are employed to demonstrate the less conservatism of the stability condition and the effectiveness of the proposed controller. The remainder of the paper is organized as follows: The problems to be solved are formulated in Section 2. In Section 3, a new criterion guaranteeing the asymptotic stability of T–S fuzzy time-delay systems is presented. In Section 4 the state-feedback controller is designed based on the input–output approach. Illustrative examples are provided in Section 5, and finally the conclusions are given in Section 6. Notations: The notations used in this paper are fairly standard. Rn represents the n-dimensional Euclidean space. The superscript T stands for matrix transpose. The notation P > (P)0 means that matrix P is positive (semi) definite. In denotes an identify matrix of dimension n and 0mn denotes an m  n dimension zero matrix. ⁄denotes the symmetric terms in a block matrix P, sym (A) is used to abbreviate AT + A, and diag{. . .} is used to express a block-diagonal matrix. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi kTk1 denotes P1 T the l2-induced norm of a transfer function matrix or a general operator; for xðkÞ; kxðkÞkl2 ¼ x ðkÞxðkÞ . k¼0 2. Problem statement Many nonlinear systems can be expressed as a set of linear systems in local operating regions, which can be represented by T–S fuzzy systems. The discrete-time T–S fuzzy systems with time-varying delay are expressed by the if-then rules as follows:

F : Rule i : IF h1 ðkÞ is Mi1 and . . . and hp ðkÞ is M ip ; THEN xðk þ 1Þ ¼ Ai xðkÞ þ Adi xðk  dðkÞÞ þ Bi uðkÞ;

ð1Þ

xðkÞ ¼ uðkÞ; k ¼ d2 ; d2 þ 1; . . . ; 0: i ¼ 1; . . . ; r; where xðkÞ 2 Rn is the state vector, uðkÞ 2 Rn is the control input and d(k) is time-varying delay satisfying 1 6 d1 6 d(k) 6 d2. The symbols d1 and d2 are constant positive scalars representing the lower and upper bounds of the delay. u(k) is the given initial condition sequence. Mij is the fuzzy set, r is the number of if-then rules, and h(k) = [h1(k), h2(k), . . . , hp(k)] is the premise variable vector. It is assumed that the premise variables do not depend on the input u(k); Ai, Adi and Bi are known constant matrices with appropriate dimensions. The fuzzy basis functions are given by

li ðhðkÞÞ ; i¼1 li ðhðkÞÞ

hi ðhðkÞÞ ¼ Pr

ð2Þ

li ðhðkÞÞ ¼ Ppj¼1 Mij ðhj ðkÞÞ:

ð3Þ

where

Mij(hj(k)) represents the grade of membership of hj(k) in Mij. Therefore, for all k, we have hi(h(k)) P 0,i = 1, 2, . . . , r and Pr i¼1 hi ðhðkÞÞ ¼ 1. It is assumed that the premise variables do not depend on the input variable u(k) explicitly. Using the center of gravity method for defuzzification, we can express the T–S fuzzy time-delay model in (1) as

xðk þ 1Þ ¼

r X

hi ðhðkÞÞfAi xðkÞ þ Adi xðk  dðkÞÞ þ Bi uðkÞg;

i¼1

yðkÞ ¼

r X hi ðhðkÞÞfC i xðkÞ þ C di xðk  dðkÞÞg; i¼1

xðkÞ ¼ uðkÞ; k ¼ d2 ; d2 þ 1; . . . ; 0: i ¼ 1; . . . ; n:

ð4Þ

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Z. Li et al. / Information Sciences 226 (2013) 93–104

A more compact representation of the system is given by

xðk þ 1Þ ¼ AðkÞxðkÞ þ Ad ðkÞxðk  dðkÞÞ þ BuðkÞ;

ð5Þ

yðkÞ ¼ CðkÞxðkÞ þ C d ðkÞxðk  dðkÞÞ; where

AðkÞ ¼

r X

r r r r X X X X hi ðhðkÞÞAdi ; BðkÞ ¼ hi ðhðkÞÞBi ; CðkÞ ¼ hi ðhðkÞÞC i ; C d ðkÞ ¼ hi ðhðkÞÞC di :

hi ðhðkÞÞAi ; Ad ðkÞ ¼

i¼1

i¼1

i¼1

i¼1

ð6Þ

i¼1

In the next section, the problem of stability analysis for the T–S fuzzy time-delay system is discussed. The model in (5) is transformed into an interconnected form. Then by using the transformed model, the problem of stability analysis for T–S fuzzy system is studied based on the scaled small gain theorem. 3. Stability analysis In this section, we solve the problem of stability analysis for discrete-time T–S fuzzy systems with time-varying delay by an input–output approach. Following the approach described in [10], we first express the time-varying delay by its lower and upper bounds and an ‘‘uncertainty’’ input. Then the lifting method is employed to obtain a delay-free model, which is further used in the stability analysis of the system. 3.1. Approximation of time-varying delay We employ the approximation method to transform the model (5) into an interconnected form as described in [10]. The terms 12 xðk  d1 Þ and 12 xðk  d2 Þ are used to estimate x(k  d(k)). Then the approximation error is obtained as follows:

1 2

1 2

xðkÞ ¼ xðk  dðkÞÞ  xðk  d1 Þ  xðk  d2 Þ:

ð7Þ

Define

dðkÞ ¼ xðk þ 1Þ  xðkÞ;

ð8Þ

and obtain

" # kd1 1 1 X aðiÞdðiÞ ; 2 i¼kd 2  1 when i 6 k  dðkÞ  1

xðkÞ ¼ aðiÞ ¼

1 when i > k  dðkÞ  1

:

Define the mapping

1 DðkÞ : dðkÞ ! xðkÞ ¼ 2

"

kd 1 1 X

#

aðiÞdðiÞ ;

ð9Þ

i¼kd2

and obtain the relationship between d(k) and x(k). We introduce a property of D(k), which has an upper bound of l2 -induced norm. Proposition 1. Operator D(k) in (9) has the property that c0 ðDÞ ¼ supkdk–0 matrix.

kT xkl 2 kTdkl 2

< 12 ðd2  d1 Þ, where T is an invertible positive

Proof. The definition of a(i) implies that ja(i)j = 1. Using the expression in (9) and the Jensen’s inequality, we derive the following inequality under the zero initial condition:

kT xk2l2 ¼

" # id d1 1X 1 1 1 1 X 1X d  d1 X ðd2  d1 Þ a2 ðjÞdT ðjÞT T TdðjÞ ¼ 2 dT ði þ jÞT T Tdði þ jÞ 4 i¼0 4 j¼id j¼id i¼0

1 X

xT ðiÞT T T xðiÞ 6

i¼0

6

d2  d1 4

2

2

 2 1  2 d2  d1 X d2  d1 dT ðiÞT T TdðiÞ ¼ dT ðiÞT T TdðiÞ ¼ kTdk2l2 ; 2 2 i¼0 i¼0

d 1 1 1X X j¼d2

which implies c0 ðDÞ ¼ supkdk–0

kT xkl

2

kTdkl

2

1 6 d2 d . This completes the proof. 2

h

Remark 1. From Proposition 1, we know that if we use the lower and upper bounds (d1 and d2) to approximate the delay d(k), then the error can be controlled within a small range.

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3.2. Scaled small gain theorem and model transformation We first introduce the scaled small gain theorem, then reformulate the T–S fuzzy time-delay system into an interconnected form. Define the following variable and mapping:

mðkÞ ¼

2 xðkÞ; ðd2  d1 Þ

ð10Þ

e ðkÞ : dðkÞ ! mðkÞ: D kT mkl 2 e Þ ¼ sup Then it is easy to prove that c0 ð D kdk–0 kTdkl 6 1 according to Proposition 1. We now transform the system model 2 into two interconnected parts as follows:

GðkÞ : mðkÞ ! dðkÞ; e ðkÞ : dðkÞ ! mðkÞ; D

ð11Þ ð12Þ

where the forward subsystem (11) has an operator G(k) which is a mapping from m(k) to d(k), and the feedback subsystem e ðkÞ. The following lemma provides the scaled small gain theorem with regard to the (12) is described by the mapping D asymptotic stability of the interconnected system [4,37,38]. Lemma 1 (Scaled Small Gain Theorem). Consider an interconnected system composed of a forward subsystem in (11) and a feedback subsystem in (12) and assume that system (11) is internally stable. The interconnected system is asymptotically stable if the following condition holds:

e Þ < 1; c0 ðGÞ  c0 ð D

ð13Þ

where

kTdðkÞkl2 kT mðkÞkl2 e Þ ¼ sup ; c0 ð D ; T ¼ diagf T 1 kT m ðkÞk kmk–0 kdk–0 kTdðkÞkl2 l2

c0 ðGÞ ¼ sup

   T s g > 0;

e is the feedback subsystem. and G is the forward subsystem, while D Then combining (10) and (8), we get the equivalent transformed model as:



xðk þ 1Þ

"

 ¼

dðkÞ

A

1 A 2 d

1 A 2 d

1 ðd2 2

 d1 ÞAd

A  In

1 A 2 d

1 A 2 d

1 ðd2 2

 d1 ÞAd

#

fðkÞ

mðkÞ

 þ

  1 uðkÞ; 0

e ðkÞdðkÞ; mðkÞ ¼ D

xðkÞ ¼ uðkÞ; k ¼ d2 ; d2 þ 1; . . . ; 0;

ð14Þ

ð15Þ

where

fðkÞ ¼ colf xðkÞ xðk  d1 Þ xðk  d2 Þ g: The model in (14) and (15) has the same form as the interconnected systems in (11) and (12). Thus, we can employ the scaled small gain theorem to this model. e Þ 6 1 in (10) by Proposition 1, and thus, the scaled small gain condition becomes Remark 2. It is easy to show that c0 ð D c0(G) < 1 in the forward subsystem in (14) with the internal stability of the subsystem in (14) by Lemma 1. Note that there are two constant state delays in the forward subsystem in (14), and thus the scaled small gain condition in Lemma 1 cannot be employed to solve this system directly by the bounded real lemma. Thus, one possible way is to transform the forward subsystem in (14) with u(k) = 0 into a delay-free system so that the bounded real lemma can be used. To this end, we use state lifting method to complete the transformation. Let

xðkÞ ¼ colf xðkÞ xðk  1Þ . . . xðk  d2 Þ g; xð0Þ ¼ colf uð0Þ . . . uðd1 Þ . . . uðd2 Þ g; " # " # d12 1 1 Ad 2 b d ¼ A 0ðd1 1Þnn 2 Ad 0ðd12 1Þnn 2 Ad ; B bd ¼ ; A Id2 n 0d2 nn 0d2 nn h i bd ¼ A  I 0 1 1 b d ¼ d12 Ad ; C ; D A 0 A ðd1 1Þnn d ðd 1Þnn d 12 2 2 2

ð16Þ

where u(k), k = d2, d2 + 1, . . . , 0 is defined in the system in (1). Then the forward subsystem in (14) with u(k) = 0 can be rewritten as



xðk þ 1Þ dðkÞ

"

 ¼

bd A bd C

# b d  xðkÞ  B : b d mðkÞ D

ð17Þ

Z. Li et al. / Information Sciences 226 (2013) 93–104

97

It can be seen that the time-delay forward subsystem in (14) with u(k) = 0 has been transformed into the forward subsystem without delay in (17). Now the bounded real lemma combined with the scaled small gain theorem can be used directly in the interconnected system described by (15) and (17), and we obtain the following theorem. Theorem 1. The system in (1) with u(k) = 0 is asymptotically stable for all d(k) satisfying 1 6 d1 6 d(k) 6 d2, if there exist matrices 0 < Pi ¼ PTi 2 Rðd2 þ1Þnðd2 þ1Þn ; 0 < S ¼ ST ¼ T T T 2 Rnn , such that

Pii < 0; ði ¼ 1; . . . ; rÞ; Pij þ Pji < 0; ð0 6 i < j 6 rÞ;

ð18Þ

where

2

ej Pi A

Pi

ej Pi B

0

3

7 6 6 eTS 7 7 6  Pi 0 C j 7; Pij ¼ 6 7 6 6  eTS 7  S D 4 j 5    S " Aj 0ðd1 1Þnn 12 Adj 0ðd12 1Þnn ej ¼ A Id2 n 0d2 nn  e j ¼ Aj  I C

0ðd1 1Þnn

1 A 2 dj

1 A 2 dj

#

"d

ej ¼ ; B

0ðd12 1Þnn

1 A 2 dj

12

2

Adj

0d2 nn

# ;

e j ¼ d12 Adj ; ; D 2

1Þ d12 ¼ ðd2 d , and Aj, Adj are defined in the system in (1). 2

Proof. We will use the bounded real lemma and the scaled small gain theorem to prove this theorem. From the inequalities in (18), we get the following inequality: r r1 X r X X hi ðhðkÞÞPii þ hi ðhðkÞÞhj ðhðkÞÞðPij þ Pji Þ < 0: i¼1

ð19Þ

i¼1 j¼iþ1

bd; B b d; D bd; C b d , the inequality in (19) is equivalent to the following inequality: According to the definitions of A

2

P

6 6  6 6 4  

bd PA

bd PB

P

0



S





3 0 bTS 7 7 C d 7 < 0: bTS 7 5 D d

ð20Þ

S

Moreover, the inequality in (20) implies that

"

P

bd PA



P

# < 0;

ð21Þ

which guarantees the internal stability of the forward system in (17). According to Lemma 3.2 in [1] we set P ¼ X and S = cL, then the inequality in (20) guarantees c0(G) < 1. Thus, combining Lemma 1 and Remark 2, it is clear that the interconnected system described by (15) and (17) is asymptotically stable. The system in(1) with u(k) = 0 and the interconnected system in (15) and (17) describe the same system. Thus the system in (1) with u(k) = 0 is asymptotically stable. This completes the proof. h Note that Theorem 1 can be cast into a convex optimization problem subject to LMI constraints, which can be readily solved using the standard numerical software. In Section 5, Example 1 is given to show the less conservatism of Theorem 1. 4. State-feedback controller design In this section, we design a state-feedback controller based on the scaled small gain theorem such that the closed-loop discrete-time T–S fuzzy system with time-varying delay in (1) is asymptotically stable. We assume that all the states are available for the state-feedback controller, which is represented by the following rules:

Rule i : IF h1 ðkÞ is M i1 and . . . and hp ðkÞ is M ip ; THEN uðkÞ ¼ K i xðkÞ; i 2 R; where K i 2 Rmn ; i 2 R is the local control gain. Thus, the controller in (22) can be represented by the following form:

ð22Þ

98

Z. Li et al. / Information Sciences 226 (2013) 93–104 r X hi ðhðkÞÞK i xðkÞ ¼ KðkÞxðkÞ:

uðkÞ ¼

ð23Þ

i¼1

Substituting the controller form in (23) into the forward subsystem in (14), we obtain the whole closed-loop fuzzy interconnected system

"

"

#

xðk þ 1Þ

¼

dðkÞ

A þ BðkÞKðkÞ

1 A 2 d

1 A 2 d

1 ðd2 2

 d1 ÞAd

A  In

1 A 2 d

1 A 2 d

1 ðd2 2

 d1 ÞAd

#"

fðkÞ

mðkÞ

# ;

e ðkÞdðkÞ: mðkÞ ¼ D

ð24Þ

ð25Þ

The design of the state-feedback fuzzy controller is to determine the feedback gains Ki such that the closed-loop fuzzy interconnected system described by the forward subsystem in (24) and the feedback subsystem in (25) is asymptotically stable. The closed-loop fuzzy interconnected system in (24) and (25) is equivalent to the system in (1). Thus the closed-loop controller is still effective for the system in (1). Introduce the following matrices: r r r X X X hi ðkÞPi > 0; Q 1 ðkÞ ¼ hi ðkÞQ 1i > 0; Q 2 ðkÞ ¼ hi ðkÞQ 2i > 0; R1 > 0; R2 > 0:

PðkÞ ¼

i¼1

i¼1

i¼1

We construct the Lyapunov–Krasovskii functional as:

VðkÞ ¼ V 1 ðkÞ þ V 2 ðkÞ þ V 3 ðkÞ;

ð26Þ

where

V 1 ðkÞ ¼ xT ðkÞPðkÞxðkÞ; 2 X k1 X

V 2 ðkÞ ¼

xT ðiÞQ l ðiÞxðiÞ;

l¼1 i¼kdl 2 X 1 X k1 X

V 3 ðkÞ ¼

dl dT ðjÞRl dðjÞ;

l¼1 i¼dl j¼kþi

dðiÞ ¼ xði þ 1Þ  xðiÞ: The problem of stabilization for the system in (1) is solved in the following theorem by utilizing the Lyapunov–Krasovskii functional in (26) and the scaled small gain theorem. Theorem 2. Given a time-varying delay d(k) satisfying 1 6 d1 6 d(k) 6 d2, there exists a stabilizing state-feedback controller in the form (23), such that the closed-loop fuzzy system in (1) is asymptotically stable if there exist matrices e 1i > 0; Q e 2i > 0; R e 1 > 0; R e 2 > 0, Z; e and a constant scalar  > 0, satisfying e i > 0; Q P

Cstlii < 0; ðs; t; l; i ¼ 1; . . . ; rÞ; Cstlij þ Cstlji < 0; ðs; t; l ¼ 1; . . . ; rÞ; ð1 6 i < j 6 rÞ;

ð27Þ

where

"

Cstlij ¼ 2

Ptli 

Xi

#

H1ij d1 H2ij d2 H2ij H2ij ; Ns

e1 Re 1 e1 Re 2 0

ð28Þ 3

e 1i þ Q e 2i  e2 R e 1  e2 R e2; ei þ Q o 5:Xi ¼  P e 2l  R e2 Z e e2 Q   Rxi ¼ Ai 12 eAdi 12 eAdi 12 eðh2  h1 Þ Adi ; Ryi ¼ Ai  In 12 eAdi 12 eAdi 12 eðh2  h1 ÞAdi ; H1ij ¼ LT RTxi þ RT1 Y Tj BTi ; n o e 1  e symðLÞ R e 2  e symðLÞ Z e  e symðLÞ ; R1 ¼ ½ In 0n3n ; H2ij ¼ LT RTyi þ RT1 Y Tj BTi ; Ns ¼ diag Pe s  symðLÞ R R2 ¼ ½ 0nn In 0n2n ; ; R3 ¼ ½ 0n2n In 0nn ; R4 ¼ ½ 0n3n In :

Ptli ¼ 4



n e 1t  R e1 diag e2 Q

The state-feedback controller gains are

K i ¼ Y i L1 :

Z. Li et al. / Information Sciences 226 (2013) 93–104

99

Proof. We assume that all the inequalities in (27) hold. Define the following matrices:

F ¼ eL;

H ¼ diag L1

F

1

F

1

F

1

I

I

I

ð29Þ

 I :

Let

e 1i ¼ LT Q L; Q e 2i ¼ LT Q L; R e i ¼ LT Pi L; Q e 1 ¼ F T R1 F; P 1i 2i e ¼ F T ZF: e 2 ¼ F T R2 F; Z R Using the known inequalities

e 1 ð P e s  LÞT P e s  LÞ P 0; ðP s e 1 ð R e 1  FÞ P 0; e 1  FÞT R ðR 1

e 1 ð R e 2  FÞ P 0; e 2  FÞT R ðR 2 T e 1 e e ð Z  FÞ Z ð Z  FÞ P 0; we get

e s  symðLÞ P LT P e 1 L; P s T e 1 F; e 1  symðFÞ P F R R 1

ð30Þ

e 1 F; e 2  symðFÞ P F T R R 2 e 1 F: e  symðFÞ P F T Z Z

By performing a congruence transformation H to Cstlij and combining the inequalities in (30), we obtain the following inequalities:

ustlii 6 HT Cstlii H < 0; ðs; t; l; i ¼ 1; . . . ; rÞ; ustlij þ ustlji 6 HT ðCstlij þ Cstlji ÞH < 0; ðs; t; l ¼ 1; . . . ; rÞ; ð1 6 i < j 6 rÞ; where

"

ustlij ¼

b tli P 

b T d1 H b T d2 H bT H 1ij 2ij 2ij

diag P1 s

R1 1

bT H 2ij

R1 2

Z 1

ð31Þ

#  ;

RT1 ðP i þ Q 1i þ Q 2i ÞR1  RT2 Q 1t R2  RT3 Q 2l R3

b tli ¼ P

RTR1 R1 RR1  RTR2 R2 RR2  RT4 Z R4 ; b xi ¼ Ai 1 Adi 1 Adi 1 ðd2  d1 ÞAdi ; R 2 2 2  b yi ¼ Ai  In 1 Adi 1 Adi 1 ðd2  d1 ÞAdi ; R 2 2 2 b T þ RT K T BT ; H b 2ij ¼ R b T þ RT K T BT ; b 1ij ¼ R H 

xi

1

j

i

yi

1

j

i

RR1 ¼ ½ In In 0nn 0nn ; RR2 ¼ ½ In 0nn In 0nn : Then combining the definitions of PðkÞ; Q 1 ðkÞ; Q 2 ðkÞ; R1 ; R2 ; Z, and the inequalities in (31), we obtain the following inequalities:

UðkÞ ¼

8 > > > > > > r X r X r > < X s¼1 t¼1

9 > > 9> > > > > > > > >= = < 0: > > > r1 X r l¼1 > > > X > > > > > >> > >þ > >> hi ðhðkÞÞhj ðhðkÞÞðustlij þ ustlji Þ > > > : ; : > ; hs ðhðk þ 1ÞÞht ðhðk  d1 ÞÞhl ðhðk  d2 ÞÞ 8 r X > > hi ðhðkÞÞustlii > > < i¼1

ð32Þ

i¼1 j¼iþ1

Taking the forward difference of V(k), we get

DV 1 ðkÞ ¼ xT ðk þ 1ÞPðk þ 1Þxðk þ 1Þ  xT ðkÞPðkÞxðkÞ; 2 X DV 2 ðkÞ ¼ fxT ðkÞQ l ðkÞxðkÞ  xT ðk  dl ÞQ l ðk  dl Þxðk  dl Þg; l¼1 8 9 2 < k1 = X X 2 T T DV 3 ðkÞ ¼ dl d ðkÞRl dðkÞ  dl d ðiÞRl dðiÞ : : ; l¼1 i¼kd l

ð33Þ

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Using the Jensen’s inequality, we get

 d1

k1 X

dT ðiÞR1 dðiÞ 6 ½xðkÞ  xðk  d1 ÞR1 ½xðkÞ  xðk  d1 Þ;

i¼kd1

 d2

k1 X

ð34Þ T

d ðiÞR2 dðiÞ 6 ½xðkÞ  xðk  d2 ÞR2 ½xðkÞ  xðk  d2 Þ:

i¼kd2

Combining (33) and (34), we obtain the following inequalities:

DV 6 DV 1 ðkÞ þ DV 2 ðkÞ þ DV 3 ðkÞ

ð35Þ

T

6 n ðkÞKðkÞnðkÞ; where

 T nðkÞ ¼ xT ðkÞ xT ðk  d1 Þ xT ðk  d2 Þ : According to Sylvester’s criterion, the formula in (32) can guarantee K(k) < 0. Thus, the forward subsystem in (24) is internally stable. Let S > 0 and introduce the following equation:

J,

1 X ðdT ðkÞSdðkÞ  mT ðkÞSmðkÞÞ;

ð36Þ

k¼0

where S = TTT. It is clear that

J 6 J þ Vð1Þ  Vð0Þ ¼

1 1 X X fJðkÞ þ DVðkÞg < gT ðkÞUðkÞgðkÞ; k¼0



ð37Þ

k¼0



gðkÞ ¼ nT ðkÞ mT ðkÞ T : Thus the inequality in (32) can guarantee J 6 0, which means that if we let T > 0 and S = TTT > 0, then the scaled small gain of G satisfies

c0 ðGÞ ¼ sup

kTdðkÞkl2 < 1: mðkÞkl2

kmk–0 kT

ð38Þ

e Þ 6 1. Now, considering Lemma 1, we have established According to Proposition 1 and the formula in (10), we have c0 ð D that the closed-loop interconnected system in (24) and (25) is asymptotically stable. The interconnected system in (24) and (25) and the system in (1) describe the same system, and therefore the closed-loop system in (1) is asymptotically stable. This completes the proof. h Remark 3. In [42], delay-partitioning idea was incorporated in Lyapunov–Krasovskii functional method to derive the stability conditions. This idea can be easily applied to Theorem 2 to further reduce the conservatism of the result presented in this paper. Remark 4. In Theorem 2, we define a scalar e in (29), with the aim to obtain a tractable matrix condition without increasing the conservatism of the results. By choosing the scalar appropriately, the conservatism of the result can be further reduced. Remark 5. The small gain theorem is important in nonlinear control system theory; it has recently been employed to solve the problem of stability analysis for linear time-delay systems. Its basic theory is given in [9] and a direct Lyapunov method has been presented in [10]. Using this theorem, we have shown that the asymptotic stability of the equivalent interconnected system given by (14) and (15) is equivalent to the asymptotic stability of the system given by (1). The input–output approach is different from the traditional ones; it transforms the problem of asymptotic stability analysis for the system (1) into the problem of asymptotic stability analysis for the equivalent interconnected system given by (14) and (15). Additionally, the conservatism of the result is reduced by the method given in this paper as shown in the next section by the illustrative examples.

5. Numerical examples In this section, two model examples are considered to illustrate the power of the approach described in Sections 3 and 4 above. The first example shows the less conservatism of stability analysis based on the scaled small gain theorem. The second example demonstrates the effectiveness of the controller design method.

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Example 1. Consider the fuzzy system in (1) with

 A1 ¼

0:291

1

0

0:95



 ;

A2 ¼

0:1

0

1

0:2

 ;

 Ad1 ¼

0:012 0:014 0

0:015



 ;

Ad2 ¼

0:01

0

0:01 0:015

 :

ð39Þ

For a given d1, we are interested in finding the upper bound d2 such that the system is asymptotically stable. For d1 = 3, it can be found that d2 = 14 by [7] and d2 = 24 by delay-partitioning approach with the case m = 3 in [29]. However, even for d2 = 38, we can still get feasible results by Theorem 1 of this paper, which shows less conservatism of Theorem 1. Additional calculated results are listed in Table 1. From Table 1, we can conclude that the method proposed in this paper is less conservative compared to the results given in [7,29]. Table 2 shows the comparison of results between Theorem 1 in [7] and Theorem 2 in this paper. From Table 2, we can conclude that Theorem 2 is less conservative than the results given in [7]. This example demonstrates that the conservatism of the results is reduced by the method proposed in this paper. Example 2. This example is used to show the effectiveness of the proposed controller design method. Consider the following Henon system [8], which is a discrete-time dynamical system and is one of the most studied examples of dynamical systems that exhibits chaotic behavior:

x1 ðk þ 1Þ ¼ fcx1 ðkÞ þ ð1  cÞx1 ðk  dðkÞÞg2 þ 0:3x2 ðkÞ þ 1:4 þ uðkÞ; ð40Þ

x2 ðk þ 1Þ ¼ cx1 ðkÞ þ ð1  cÞx1 ðk  dðkÞÞ; yðkÞ ¼ cx1 ðkÞ þ ð1  cÞx1 ðk  dðkÞÞ;

where the constant c 2 [0, 1] is the retarded coefficient. Let h(k) = cx1(k) + (1  c)x1(k  d(k)). Assume that h(k) 2 [m, m], m > 0. By using the same conformation of system as in [7], the nonlinear term h2(k) can be exactly represented as:

h2 ðkÞ ¼ h1 ðhðkÞÞðmÞhðkÞ þ h2 ðhðkÞÞmhðkÞ;

ð41Þ

where h1(h(k)), and h2(h(k)) 2 [0, 1], and h1(h(k)) + h2(h(k)) = 1. By solving the equations in (41), the membership functions h1(h(k)) and h2(h(k)) are obtained as

h1 ðhðkÞÞ ¼

  1 hðkÞ 1 ; 2 m

h2 ðhðkÞÞ ¼

  1 hðkÞ 1þ : 2 m

It can be seen from the above expressions that h1(h(k)) = 1, and h2(h(k)) = 0 when h(k) is m and that h1(h(k)) = 0, and h2(h(k)) = 1 when h(k) is m. Then, the nonlinear system in (40) can be approximately represented by the following T–S fuzzy time-delay model:

Plant rule 1 : IF hðkÞ is  m; THEN xðk þ 1Þ ¼ A1 ðkÞ þ Ad1 xðk  dðkÞÞ þ B1 u ðkÞ yðkÞ ¼ C 1 xðkÞ þ C d1 xðk  dðkÞÞ Plant rule 2 : IF hðkÞ is m; THEN xðk þ 1Þ ¼ A2 ðkÞ þ Ad2 xðk  dðkÞÞ þ B2 u ðkÞ yðkÞ ¼ C 2 xðkÞ þ C d2 xðk  dðkÞÞ ⁄

where u (k) = 1.4 + u(k) and

Table 1 Calculated upper delay bound d2 for different d1. d1

3

5

10

12

Theorem 1 of [7] Theorem 1 of [29] Theorem 1

14 24 38

16 26 40

20 31 45

21 33 47

Table 2 Calculated upper delay bound d2 for different d1. d1

8

10

12

14

18

22

Theorem 1 of [7] Theorem 2

18 19

20 21

21 23

23 25

27 29

31 33

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Z. Li et al. / Information Sciences 226 (2013) 93–104

4.5 4

Time−varying delay

3.5 3 2.5 2 1.5 1 0.5

0

10

20

30

40

50

60

70

time, k Fig. 1. Time-varying delay.

2 1 0 −1 −2

0

10

20

30

0

10

20

30

40

50

60

70

40

50

60

70

2 1 0 −1 −2

Time, k Fig. 2. State response of open-loop system.



   cm 0:3 cm 0:3 A1 ¼ ; A2 ¼ ; c 0 c 0     ð1  cÞm 0 ð1  cÞm 0 ; Ad2 ¼ ; Ad1 ¼ 1c 0 1c 0 B1 ¼ B2 ¼ ½ 1 0 T ; C 1 ¼ C 2 ¼ ½ c 

In this example, we set xðkÞ ¼ xT1 ðkÞ

0 ; C d1 ¼ C d2 ¼ ½ 1  c xT2 ðkÞ

T

0 :

; c ¼ 0:8; m ¼ 0:6 and d(k) represents a time-varying state delay. For simh i k ulation, the initial condition is assumed to be uðkÞ ¼ ed2 0 for all k = d2, d2 + 1, . . . , 0. Our goal is to design a statefeedback controller in the form of (40) such that the resulting closed-loop system is asymptotically stable. Let d2 = 4, d1 = 1 and by setting e = 50, we obtain the fuzzy controller gains by Theorem 2 as

K 1 ¼ ½ 0:2463 0:2820 ; K 2 ¼ ½ 0:8205 0:3159 :

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Z. Li et al. / Information Sciences 226 (2013) 93–104

1 0.5 0 −0.5 −1 −1.5

0

10

20

30

0

10

20

30

40

50

60

70

40

50

60

70

1.5 1 0.5 0 −0.5 −1

Time, k Fig. 3. State response of closed-loop system.

Table 3 Calculated upper delay bound d2 for different e(d1 = 1).

e

0.1

1

2

3

6

10

Theorem 2

–

1

2

3

4

5

In the simulation, let the delay d(k) change randomly between d2 = 4 and d1 = 1. Fig. 1 shows the change of time-varying delay. Fig. 2 plots the states of the open-loop system, while the states of the closed-loop system are shown in Fig. 3. As time goes on, two states of the closed-loop system approach zero in Fig. 3 showing the effectiveness of the controller design. In Remark 4, we have discussed the influence of e in Theorem 2. From Table 3, it can be seen that as e become bigger and bigger, the results become less and less conservative. Example 2 demonstrates that our method can control complex systems with chaotic behavior. The method can also be employed to control practical complex systems, for example, the problem of crane control described in [40]. 6. Conclusions This paper has dealt with the problems of stability analysis and controller synthesis for T–S discrete-time fuzzy systems with time-varying delay via an input–output approach. The system model is transformed into an interconnected form with a forward delayed subsystem and a feedback subsystem by approximation approach. Using lifting method and the bounded real lemma combined with the scaled small gain theorem, a new less conservatism stability criterion has been established. Additionally, a state-feedback controller existence criterion is proposed using the Lyapunov–Krasovskii functional method and the scaled small gain theorem, which guarantees the asymptotic stability of the closed-loop system. Two illustrative examples are employed to demonstrate the effectiveness and merit of the proposed approach for stability analysis and control synthesis of discrete-time delayed fuzzy systems. Acknowledgement This work was supported by NSFC (NO.61021002 and NO.61273201). References [1] P. Apkarian, P. Gahinet, A convex characterization of gain-scheduled H1 controllers, IEEE Transactions on Automatic Control 40 (5) (1995) 853–864. [2] X.H. Chang, G.H. Yang, Relaxed stabilization conditions for continuous-time Takagi–Sugeno fuzzy control systems, Information Sciences 180 (17) (2010) 3273–3287.

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