Information Sciences 181 (2011) 4287–4299

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Information Sciences

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Delay-range-dependent robust stabilization for uncertain T–S fuzzycontrol systems with interval time-varying delays

Chen Peng a,b, Qing-Long Han b,c,⇑a School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, Jiangsu 210042, PR Chinab Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton, Qld 4702, Australiac School of Information and Communication Technology, Central Queensland University, Rockhampton, Qld 4702, Australia

a r t i c l e i n f o

Article history:Received 20 August 2009Received in revised form 25 November 2010Accepted 30 May 2011Available online 6 June 2011

Keywords:T–S fuzzy control systemsStabilityStabilizationInterval time-varying delays

0020-0255/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.ins.2011.05.025

⇑ Corresponding author at: Centre for Intelligent a49309270; fax: +61 7 49309729.

E-mail addresses: c.peng@njnu.edu.cn (C. Peng),

a b s t r a c t

This paper is concerned with robust stabilization for a class of T–S fuzzy control systemswith interval time-varying delays. An approach is proposed to significantly improve thesystem performance while reducing the number of scalar decision variables in linearmatrix inequalities. The main points of the approach are: (i) two coupling integral inequal-ities are proposed to deal with some integral items in the derivation of the stability criteria;(ii) an appropriate Lyapunov–Krasovskii functional is constructed by including both thelower and upper bounds of the interval time-varying delays; and (iii) neither model trans-formation nor free weighting matrices are employed in the theoretical result derivation. Asa result, some improved sufficient stability criteria are derived, and the maximum allow-able delay bound and controller gains can be obtained simultaneously by solving an opti-mization problem. Numerical examples are given to demonstrate the effectiveness of theproposed approach.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Fuzzy systems in the form of the Takagi–Sugeno (T–S) model [25] have attracted rapidly growing interest in recent years[14]. It has shown that the T–S model-based approach provides a powerful and systematical control methodology to somecomplex nonlinear systems. It combines the flexible fuzzy logic theory and fruitful linear system theory into a unified frame-work to approximate a wide range of complex nonlinear systems, especially those with incomplete information [20,25,29].During the past decades, many industrial applications via T–S model-based control have been reported in the open literature,see, for example [5,23,24] and references therein.

Recently, a new type of time delays, i.e. interval time-varying delays, have been proposed from theoretical and practicalengineering systems and have attracted much attention in the area of time-delay systems, see for examples[9–11,13,14,16,20]. A typical example of dynamical systems with interval time-varying delays is networked control systems[12,18,33,34]. It should be mentioned that compared with the Lyapunov–Razumikhin approach [1,27], the Lyapunov–Krasovskii approach [2,8,31] employed in stability analysis and controller synthesis of T–S fuzzy systems with time-varyingdelays can usually provide less conservative results in particular when time delay is small [14].

Notice that in the existing literature there are some nice results available for stability analysis and controller synthesis ofT–S fuzzy systems with time-varying delays [1,3,4,17,19,29,30,32]. However, there are a few results reported in the literature

. All rights reserved.

nd Networked Systems, Central Queensland University, Rockhampton, Qld 4702, Australia. Tel.: +61 7

q.han@cqu.edu.au (Q.-L. Han).

4288 C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299

on delay-range-dependent stabilization for T–S fuzzy systems with time-varying delays [14,16,26]. In the result mentionedabove, some free weighting matrices were introduced in the theoretical results derivation. It should be pointed out thatoverly using free weighting matrices will lead to significant computational demand, especially when a large number of fuzzyrules are used [12,20,21]; in designing a controller, some free weighting matrices are required to be tuned to obtain subop-timal and parameter-dependent solutions; for example, in [26], three parameters qi (i = 2,3,4) must be pre-determined inorder to guarantee Mi = qiM1, where Mi (i = 2,3,4) are free weighting matrices and M1 must be a nonsingular matrix; this pro-cedure introduces some conservativeness because it is dependent on prescribed constants qi. How to tune qi is still an openquestion since the searching scopes of qi are infinite [12,20,22]. Therefore, how to overcome the disadvantage of the freeweighing matrices mentioned above motivates the current study.

In this paper, we will develop an approach to stability analysis and controller design for uncertain T–S fuzzy control sys-tems with interval time-varying delays. In the approach, we will introduce two coupling integral inequalities to deal withsome integral items, construct an appropriate Lyapunov functional by including both the lower and the upper bounds ofthe interval time-varying delays. Then we will derive some delay-range-dependent stability criteria, which will be less con-servative than some existing results in the existing literature. Based on the stability criteria, we will design feedback control-lers for the system under consideration. The feedback gains can be derived by solving a constrained convex optimizationproblem. We will give two examples to show the effectiveness of the design method.

Notation: N stands for the set of positive integers. Rn denotes the n-dimensional Euclidean space. Rm�n is the set of allm � n real matrices. For symmetric matrices P and Q, the notation P > Q (respectively, P P Q) means that matrix P � Q is po-sitive definite (respectively, positive semi-definite). In is an identity matrix of n � n dimensions. diag(A1,A2, . . . ,An) denotesthe block-diagonal matrix. For an arbitrary matrix W and two symmetric matrices P and Q, the symmetric term in a sym-

metric matrix is denoted by ⁄, i.e. P W� Q

� �¼ P W

WT Q

� �.

2. Problem statement

Consider the uncertain system with interval time-varying delays described by the following IF-THEN rules

If h1ðtÞ is Mi1 and; . . . ; and hgðtÞ is Mi

g

Then_xðtÞ ¼ AixðtÞ þ Adixðt � sðtÞÞ þ BiuðtÞ; i ¼ 1;2; . . . ; r

xðtÞ ¼ /ðtÞ; t 2 ½�s2;0�

(ð1Þ

where r is the number of If-Then rules; Mij (i = 1,2, . . . ,r; j = 1,2, . . . ,g) are fuzzy sets and hj(t) (j = 1,2, . . . ,g) represent premise

variables; xðtÞ 2 Rn and uðtÞ 2 Rm are state vector and input vector, respectively; s(t) is a time-varying delay satisfying0 6 s1 6 s(t) 6 s2 and /(t) is the initial condition of the system state; Ai ¼ Ai þ DAiðtÞ; Adi ¼ Adi þ DAdiðtÞ andBi ¼ Bi þ DBiðtÞ ði ¼ 1;2; . . . ; rÞ; Ai, Adi and Bi (i = 1,2, . . . ,r) are constant matrices with compatible dimensions; DAi(t), D Adi(t)and DBi(t) (i = 1,2, . . . ,r) are uncertain time-varying matrices satisfying

½DAiðtÞ;DAdiðtÞ;DBiðtÞ� ¼ HiFiðtÞ½Eai; Edi; Ebi�; i ¼ 1;2; . . . ; r ð2Þ

where Hi, Eai, Edi, and Ebi are known constant real matrices with appropriate dimensions that characterize the structures ofuncertainties; Fi(t) is an unknown real time-varying matrix with Lebesgue measurable elements such that

FTi ðtÞFiðtÞ 6 I ð3Þ

Denote h(t) = [h1(t), . . . ,hg(t)]T. In this paper, we assume that h(t) is either given or a function of x(t) and does not depend onu(t).

By using a center-average defuzzifier, product inference and singleton fuzzifier, the global dynamics of T–S fuzzy system(1) can be inferred as

_xðtÞ ¼Pri¼1

liðhðtÞÞ AixðtÞ þ Adixðt � sðtÞÞ þ BiuðtÞh i

xðtÞ ¼ /ðtÞ; t 2 ½�s2; 0�

8<: ð4Þ

where

liðhðtÞÞ ¼hiðhðtÞÞPri¼1hiðhðtÞÞ

; hiðhðtÞÞ ¼ Pgj¼1Mi

jðhjðtÞÞ ð5Þ

with Mijð�Þ being the grade of the membership function of Mi

j and " i 2 {1,2, . . . ,r}, li(h(t)) satisfying

liðhðtÞÞP 0;Xr

i¼1

liðhðtÞÞ ¼ 1: ð6Þ

C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299 4289

In this paper, a T–S fuzzy-model-based controller will be designed via parallel distributed compensation (PDC) to stabilizethe studied system (4). The PDC control structure utilizes a nonlinear state feedback controller which mirrors the structure ofthe associated T–S model. The gains of the controller can be determined by using a linear matrix inequality (LMI) formulation[28].

Assume that the following rules are given

If h1ðtÞ is Mi1 and; . . . ; and hgðtÞ is Mi

g

Then uðtÞ ¼ KixðtÞ; i ¼ 1; . . . ; rð7Þ

The defuzzified output of controller rule (7) is designed as

uðtÞ ¼Xr

j¼1

ljðhðtÞÞKjxðtÞ ð8Þ

where Kj (j = 1,2, . . . ,r) are controller gains to be determined. Notice that the controller only depends on the parameters Ki

and lj(h(t)). So it allows for simplicity, low-cost of implementation and affordability.Substituting (8) into (4) yields the following closed-loop fuzzy system

_xðtÞ ¼ AijxðtÞ þAi

dxðt � sðtÞÞxðtÞ ¼ /ðtÞ; t 2 ½�s2; 0�

(ð9Þ

where

Aij ¼

Xr

i¼1

Xr

j¼1

liðhðtÞÞljðhðtÞÞðAi þ BiKjÞ

Aid ¼

Xr

i¼1

liðhðtÞÞAdi:

In this paper, we will consider the following two scenarios of the interval time-varying delay s(t):

Case 1 s(t) is a differentiable function satisfying

0 6 s1 6 sðtÞ 6 s2; 0 6 _sðtÞ 6 l < 1; 8t P 0 ð10Þ

Case 2 s(t) is a continuous function satisfying

0 6 s1 6 sðtÞ 6 s2; 8t P 0 ð11Þ

where s1, s2 and l are known constants.

Remark 1. Both Cases 1 and 2 consider the upper and lower bounds of the interval time-varying delays. Case 1 is a specialcase of Case 2. If the time-varying delay is differentiable and l < 1, we can obtain less conservative results using Case 1 thanthose using Case 2. However, if the time-varying delay is not differentiable, only Case 2 can be used to handle the situation.

The purpose of this paper is to design a state feedback controller (8) under Cases 1 and 2 such that the closed-loop system(9) is asymptotically stable.

To end this section, we give two improved lemmas, which are useful in deriving our main results. Lemma 1 is derivedbased on Jensen’s integral inequality [7], which presents a tighter bound to deal with integral terms without ignoring anyuseful items. Lemma 2 is used to transfer the time-varying matrix inequality to a set of solvable matrix inequalities.

Lemma 1. For any constant matrix R 2 Rn�n;R ¼ RT > 0, scalars s1 6 s(t) 6 s2, and vector function _x : ½�s2;�s1� ! Rn, such thatthe following integration is well defined, then

�ðs2 � s1ÞZ t�s1

t�s2

_xTðvÞR _xðvÞdv 6X2

i¼1

fTi ðtÞ½1þ pi�XðRÞfiðtÞ ð12Þ

where

f1ðtÞ ¼xðt � s1Þ

xðt � sðtÞÞ

� �; f2ðtÞ ¼

xðt � sðtÞÞxðt � s2Þ

� �; XðRÞ ¼

�R R

R �R

� �; e ¼ s2 � sðtÞ

sðtÞ � s1ð13Þ

and

p1 ¼ �1; p2 ¼ 0; when sðtÞ ¼ s1

p1 ¼ e; p2 ¼ 1e ; when s1 < sðtÞ < s2

p1 ¼ 0; p2 ¼ �1; when sðtÞ ¼ s2

8><>:

4290 C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299

Proof

(1) When s(t) = s1 (respectively, s(t) = s2), it can be seen that the item fT1ðtÞXðRÞf1ðtÞ (respectively, fT

2ðtÞXðRÞf2ðtÞ) in therightmost of Eq. (12) always equals to zero. Then, Eq. (12) will be simplified as the following Jensen’s inequality [7].

�ðs2 � s1ÞZ t�s1

t�s2

_xTðvÞR _xðvÞdv 6xðt � s1Þxðt � s2Þ

� �T

XðRÞxðt � s1Þxðt � s2Þ

� �ð14Þ

(2) When s1 < s(t) < s2, from Jensen’s inequality [7] and Leibnitz–Newton formula, it follows that

�ðs2 � s1ÞZ t�s1

t�s2

_xTðvÞR _xðvÞdv ¼ �ðs2 � s1ÞZ t�s1

t�sðtÞ_xTðvÞR _xðvÞdv þ

Z t�sðtÞ

t�s2

_xTðvÞR _xðvÞdv" #

¼ �½1þ e�ðsðtÞ � s1ÞZ t�s1

t�sðtÞ_xTðvÞR _xðvÞdv � 1þ 1

e

� �ðs2 � sðtÞÞ

�Z t�sðtÞ

t�s2

_xTðvÞR _xðvÞdv

6 �½1þ e�Z t�s1

t�sðtÞ_xTðvÞdvR

Z t�s1

t�sðtÞ_xðvÞdv � 1þ 1

e

� � Z t�s1

t�sðtÞ_xTðvÞdvR

Z t�s1

t�sðtÞ_xðvÞdv

¼ �½1þ e�½xðt � s1Þ � xðt � sðtÞÞ�T R½xðt � s1Þ � xðt � sðtÞÞ� � 1þ 1e

� �½xðt � sðtÞÞ

� xðt � s2Þ�T R½xðt � sðtÞÞ � xðt � s2Þ� ð15Þ

Re-arranging term of (15) yields the result (12). This completes the proof. h

Lemma 2. For given matrices P 2 Rm�m; nðtÞ 2 Rm�1; X 2 Rm�m and F 2 Rm�m;X 6 0and F 6 0, scalars s1 6 s(t) 6 s2, if

nTðtÞ½Pþ 3Xþ F�nðtÞ 6 0 ð16ÞnTðtÞ½PþXþ 3F�nðtÞ 6 0 ð17Þ

then

nTðtÞfPþ ½1þ p1�Xþ ½1þ p2�FgnðtÞ 6 0 ð18Þ

where p1 and p2 are defined in Lemma 1, and when s(t) = s1, nT(t)Xn(t) � 0, when sðtÞ ¼ s2; nTðtÞFnðtÞ � 0.

Proof. Define uðtÞ ¼ nTðtÞ½PþXþ F�nðtÞ. First, when s(t) = s1(respectively, s(t) = s2), note that nT(t)Xn(t) � 0 and F 6 0(respectively, nT(t)Fn(t) � 0 and X 6 0), then (18) is readily derived by (16) (respectively, (17)). Secondly, whens1 < s(t) < s2, we consider u(t) > 0 and u(t) 6 0, respectively.

� If u(t) > 0, multiplying (16) and (17) by 12 e and 1

2e, respectively, and summing them up, we have

uðtÞ2

eþ enTðtÞXnðtÞ þuðtÞ2eþ nTðtÞFnðtÞ

e6 0 ð19Þ

Then based on uðtÞ2 eþ uðtÞ

2e P uðtÞ, we have (18).� If u(t) 6 0, due to X 6 0 and F 6 0, then (18) holds obviously.

This completes the proof. h

Remark 2. It is worth mentioning that Lemmas 1 and 2 include the constant delay as a special case, i.e. when s1 = s(t) = s2,both the rightmost and leftmost terms in (16) equals to zero, and (16)–(18) are fully equivalent due tonT(t)Xn(t) = nT(t)Fn(t) � 0.

Remark 3. There are some remarkable features in above Lemma 1, which can be highlighted as (i) compared with someexisting results [14,30], in which free weighting matrices are introduced to deal with integral terms, the approach to bedeveloped in this paper based on Lemma 1 does not employ any free weighting matrices and thus can be expected to givesimpler but improved results; (ii) without ignoring any useful terms, e.g.

R t�sðtÞt�s2

_xTðvÞR _xðvÞdv in (15); and (iii) without arbi-trarily magnifying �(s2 � s1) as �(s(t) � s1) or �(s2 � s(t)) in (15). Clearly, magnifying �(s2 � s1) as �(s(t) � s1) or�(s2 � s(t)) in (15), Eq. (16) will be simplified as

C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299 4291

�ðs2 � s1ÞZ t�s1

t�s2

_xTðvÞR _xðvÞdv 6 fT1ðtÞXðRÞf1ðtÞ þ fT

2ðtÞXðRÞf2ðtÞ ð20Þ

However, one can expect to obtain less conservative results based on (16) than those based on (20). Therefore, Lemma 1plays an important role in the derivation of less conservative criteria for robust delay-dependent stability analysis in thispaper.

Remark 4. Combining Lemma 1 with Lemma 2 in the derivation, less conservative results can be expected, due to the factthat the additional information eXþ 1

e F in (18) is effectively expressed by (16) and (17). This will be demonstrated laterthrough numerical examples.

3. Robust stability analysis

In this section, we analyze robust stability of the closed-loop system (9). First, we consider the stability of the nominalclosed-loop system (9). We now state and establish the following result.

Theorem 1. For some given constants 0 6 s1 6 s2, l, and matrices Kj, the equilibrium of the nominal closed-loop system (9) isasymptotically stable in the large for any s(t) satisfying (10), if there exist real matrices P > 0, R1 > 0, R2 > 0 and Qi > 0 (i = 1,2,3)with appropriate dimensions such that the following linear matrix inequalities (LMIs) hold for i, j = 1,2, . . . , r, 1 6 i < j 6 r ande = 0,1

Pii11ðeÞ Pii

12

� P22

" #< 0 ð21Þ

Pij11ðeÞ Pij

12 Pji12

� P22 0� � P22

264375 < 0 ð22Þ

where

Pii11ðeÞ ¼ Pii

11 þ 2ð1� eÞXþ 2eFþXþ F

Pij11ðeÞ ¼ Pij

11 þPji11 þ 4ð1� eÞXþ 4eFþ 2Xþ 2F

Pij11 ¼

Cij11 C12 Cij

13 0� C22 0 0� � C33 0� � � C44

266664377775

P22 ¼ diagf�R1;�R2g

Pij12 ¼

s1ðAi þ BiKjÞT R1 ðs2 � s1ÞðAi þ BiKjÞT R2

0 0s1AT

diR1 ðs2 � s1ÞATdiR2

0 0

266664377775

and

X ¼

0 0 0 00 �R2 R2 00 R2 �R2 00 0 0 0

2666437775; F ¼

0 0 0 00 0 0 00 0 �R2 R2

0 0 R2 �R2

2666437775

Cij11 ¼ PðAi þ BiKjÞ þ ðAi þ BiKjÞT P þ Q 1 þ Q 2 þ Q 3 � R1

C12 ¼ R1; Cij13 ¼ PAdi; C22 ¼ �Q 1 � R1

C33 ¼ �ð1� lÞQ 2; C44 ¼ �Q 3

4292 C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299

Proof. Construct a Lyapunov–Krasovskii functional candidate as

VðxtÞ ¼ xTðtÞPxðtÞ þZ t

t�s1

xTðtÞQ 1xðtÞdt þZ t

t�sðtÞxTðtÞQ 2xðtÞdt þ

Z t

t�s2

xTðtÞQ 3xðtÞdt

þ s1

Z 0

�s1

Z t

tþs

_xTðvÞR1 _xðvÞdv dsþ ðs2 � s1ÞZ �s1

�s2

Z t

tþs

_xTðvÞR2 _xðvÞdv ds ð23Þ

where P > 0, R1 > 0, R2 > 0 and Qi > 0 (i = 1,2,3). Taking the time derivative of V(xt) with respect to t along the trajectory of Eq.(9) yields

_VðxtÞ ¼ 2xTðtÞP _xðtÞ þ xTðtÞ½Q1 þ Q 2 þ Q3�xðtÞ � xTðt � s1ÞQ 1xðt � s1Þ � ð1� lÞxTðt � sðtÞÞQ 2xðt � sðtÞÞ

� xTðt � s2ÞQ3xðt � s2Þ þ _xTðtÞ½s21R1 þ ðs2 � s1Þ2R2� _xðtÞ � s1

Z t

t�s1

_xTðvÞR1 _xðvÞdv � ðs2 � s1ÞZ t�s1

t�s2

_xTðvÞR2 _xðvÞdv

ð24Þ

Applying Jensen’s inequality [7] and Lemma 1 in this paper, we have

�s1

Z t

t�s1

_xTðvÞR1 _xðvÞdv 6xðtÞ

xðt � s1Þ

� �T

XðR1ÞxðtÞ

xðt � s1Þ

� �ð25Þ

�ðs2 � s1ÞZ t�s1

t�s2

_xTðvÞR2 _xðvÞdv 6X2

i¼1

fTi ðtÞ½1þ pi�XðR2ÞfiðtÞ ð26Þ

Considering (24)–(26) together, we have

_VðxtÞ 6Xr

i¼1

Xr

j¼1

liðhðtÞÞljðhðtÞÞnTðtÞ Pij

11 þ ½1þ p1�Xþ ½1þ p2�F�Pij12P

�122 Pij

12

� �T� �

nðtÞ

¼Xr

i¼1

liðhðtÞÞ2nTðtÞ Pii

11 þ � �Pii12P

�122 Pii

12

� �T� �

nðtÞ þXr�1

i¼1

Xr

j>i

liðhðtÞÞljðhðtÞÞnTðtÞ Pij

11 þPji11 þ 2�

h�Pij

12P�122 Pij

12

� �T�Pji

12P�122 Pji

12

� �T�nðtÞ ð27Þ

where nTðtÞ ¼ ½xTðtÞ; xTðt � s1Þ; xTðt � sðtÞÞ; xTðt � s2Þ�; � ¼ ½1þ p1�Xþ ½1þ p2�F; X; F and Pij11 ði; j ¼ 1; . . . ; rÞ are defined in

Theorem 1.Using Lemma 2 and combining Eqs. (21) and (22) together, we derive that

Pii11 �Pii

12P�122 ðP

ii12Þ

T þ � < 0

Pij11 þPji

11 �Pij12P

�122 Pij

12

� �T�Pji

12P�122 Pji

12

� �Tþ 2� < 0

8<: ð28Þ

From (27) and (28), we have _VðxtÞ < 0. This means _VðxtÞ < �qkxðtÞk2 for a sufficiently small q > 0, and ensures the equilib-rium of system (9) is asymptotically stable for any delay satisfying s1 6 s(t) 6 s2. This completes the proof. h

From the proof of Theorem 1, one can seen that if (21) and (22) are satisfied, then the equilibrium of the nominal closed-loop system (9) is asymptotically stable in the large for any membership function-dependent li(h(t)) and lj(h(t)), that is, thesystem has robustness to any fuzzy membership functions described in (5).

Remark 5. Notice that Theorem 1 includes the constant delay, that is s(t) = s1 = s2, as a special case. However, in this case, asmentioned in Remark 2, the proposed method is not effective to reduce conservativeness, and results in the superfluousvariables, such as Q2, Q3 and R2,included in constructed Lyapunov–Krasovskii functional (23). Therefore, in the following, weonly pay attention on time varying delay case.

Remark 6. A key feature of the proposed approach is that neither model transformation nor free weighting matrices areemployed. Particularly, none of the useful items are being ignored in the derivation of our stability criteria. Therefore, theresults derived in this paper are expected to be less conservative than some existing results. Moreover, the approach pro-posed in this paper employs a smaller number of scalar decision variables in LMIs. This will be demonstrated later throughnumerical examples.

C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299 4293

Remark 7. From the proof of Theorem 1, the basic procedure of our approach can be generalized as follows: (i) construct anew type of Lyapunov functional candidate. Generally speaking, under the framework of Lyapunov’s stability theory, con-struct an appropriate Lyapunov functional candidate is a key step. It is our belief that one can always obtain improved resultsif the Lyapunov functional candidate is tweaked a little. For example, the following items: (1) simple integrating factors; (2)quadratic integrating factors related with the lower and upper delay bounds; (3) augmented integrating items included inLyapunov functional candidate will help to obtain the less conservative results; (ii) construct an uncorrelated augmentedmatrix n(t), in which none of the terms are a linear combination of other terms in n(t) and thus free-weighting matricesbecome unnecessary; (iii) employ Lemma 1 and Jensen’s inequality [7], e.g. (25) and (26); and (iv) introduce Lemma 2 toensure the feasibility of Eq. ( 28) based on Eqs. (21) and (22).

Theorem 1 provides a stability criterion for the nominal closed-loop system (9). When taking parameter uncertaintiesinto account, by Theorem 1, we can derive the following result for robust stability of the closed-loop system (9).

Theorem 2. For some given constants 0 6 s1 < s2, l, and matrices Kj, the equilibrium of the closed-loop system (9) is robustlyasymptotically stable in the large for any s(t) satisfying (10), if there exist real matrices P > 0, R1 > 0, R2 > 0 and Qi > 0 (i = 1,2,3)with appropriate dimensions, scalars ki > 0, such that the following LMIs hold for i, j = 1,2, . . . , r, 1 6 i < j 6 r and e = 0,1

Pii11ðeÞ Pii

12 Pii13

� P22 P23

� � Pi33

264375 < 0 ð29Þ

Pij11ðeÞ Pij

12 Pji12 Pij

13 Pji13

� P22 0 P23 P23

� � P22 0 0� � � Pi

33 0

� � � � Pi33

26666664

37777775 < 0 ð30Þ

where

Pij13 ¼

PM kiðEai þ EbiKjÞT

0 00 kiE

Tdi

0 0

266664377775; P23 ¼

s1R1M 0ðs2 � s1ÞR2M 0

� �Pi

33 ¼ diagf�kiI;�kiIg

and Pij11ðeÞ; Pij

12ðeÞ; P22; X and F are defined in Theorem 1.

Proof. Decomposing the resulting matrix inequalities (21) into nominal and uncertain parts leads to

R1ii þ DFiðtÞEii þ ET

iiFTi ðtÞD

T < 0

R2ii þ DFiðtÞEii þ ET

iiFTi ðtÞD

T < 0

(ð31Þ

where

D ¼

PM

000

s1R1Mðs2 � s1ÞR2M

2666666664

3777777775; ET

ij ¼

ðEai þ EbiKjÞT

0ET

di

000

26666666664

37777777775

R1

ii and R2ii are defined in (22). It is clear that there exist scalars ki > 0, thus the following inequalities always hold

R1ij þ DFiðtÞEii þ ET

iiFTi ðtÞD

T6 R1

ij þ Dik�1i DT

i þ ETiikiEii

R2ij þ DFiðtÞEii þ ET

iiFTi ðtÞD

T6 R2

ij þ Dik�1i DT

i þ ETiikiEii

(ð32Þ

Using Schur’s complement, one can obtain (29). Similarly, one can also obtain (30) from matrix inequalities (22). This com-pletes the proof. h

When there is no information about the derivative of the interval time-varying delays, condition (10) cannot be directlyused. Instead, condition (11) has to be employed for stability analysis. Setting Q2 , 0 in Theorem 2, we have the followingresult.

4294 C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299

Corollary 1. For some given constants 0 6 s1 < s2, and matrices Kj, the equilibrium of the closed-loop system (9) is robustlyasymptotically stable in the large for any s(t) satisfying (11), if there exist real matrices P > 0, R1 > 0, R2 > 0, Q1 > 0, and Q3 > 0 withappropriate dimensions and scalars ki > 0 such that LMIs (29) and (30) hold for Q2 , 0, i, j = 1,2, . . . , r, 1 6 i < j 6 r and e = 0,1.

Neglecting the p1X and p2F in the proof of Theorem 2, we can obtain the following Corollary 2. The proof is very similar tothe proof of Theorem 2, hence, it is omitted here.

Corollary 2. For some given constants 0 6 s1 < s2, l, and matrices Kj, the equilibrium of the closed-loop system (9) is robustlyasymptotically stable in the large for any s(t) satisfying (10), if there exist real matrices P > 0, R1 > 0, R2 > 0 and Qi > 0 (i = 1,2,3)with appropriate dimensions, scalars ki > 0, such that the following LMIs hold for i, j = 1,2, . . . , r and 1 6 i < j 6 r

Pii11 þXþ F Pii

12 Pii13

� P22 P23

� � Pi33

264375 < 0 ð33Þ

Pij11 þPji

11 þXþ F Pij12 Pji

12 Pij13 Pji

13

� P22 0 P23 P23

� � P22 0 0� � � Pi

33 0

� � � � Pi33

26666664

37777775 < 0 ð34Þ

where all symbols are defined in Theorem 2.

Remark 8. Although the results based on Corollary 2 is less conservative than some existing results in [16,26], comparedwith the results based on Theorem 2 without ignoring any useful items in the proof, which is still conservative. This alsoshows the effectiveness of the proposed approach in this paper, and will be demonstrated later through numerical examples.

4. State feedback controller design

In this section, we design the state feedback controller for the closed-loop system (9). When designing the state feedbackcontroller, the control objective is guaranteeing robust asymptotic stability of the equilibrium of the closed-loop system (9).

Theorem 3. For some given constants 0 6 s1 < s2 and l, the equilibrium of the system (9) is robustly asymptotically stable in thelarge with feedback gains Kj = YjX

�T (j = 1,2, . . . , r) for any s(t) satisfying (10), if there exist real matrices X > 0; eR1 > 0; eR2 > 0and eQ i > 0 (i = 1,2,3) with appropriate dimensions, scalars ki > 0, such that the following matrix inequalities hold for i, j = 1,2, . . . , r,1 6 i < j 6 r and e = 0,1

Uii11ðeÞ Uii

12 Uii13

� U22 U23

� � Ui33

264375 < 0 ð35Þ

Uij11ðeÞ Uij

12 Uji12 Uij

13 Uji13

� U22 0 U23 U23

� � U22 0 0� � � Ui

33 0

� � � � Ui33

26666664

37777775 < 0 ð36Þ

where

Uii11ðeÞ ¼ Uii

11 þ 2ð1� eÞXþ 2e�FþXþ �F

Uij11ðeÞ ¼ Uij

11 þUji11 þ 4ð1� eÞXþ 4e�Fþ 2Xþ 2�F

X ¼

0 0 0 00 �eR2

eR2 00 eR2 �eR2 00 0 0 0

2666437775; �F ¼

0 0 0 00 0 0 00 0 �eR2

eR2

0 0 eR2 �eR2

2666437775

C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299 4295

Uij11 ¼

Kij11 K12 Kij

13 0

� K22 0 0

� � K33 0

� � � K44

2666664

3777775; Uij13 ¼

M ðXETai þ YT

j ETdiÞ

0 0

0 XETbi

0 0

2666664

3777775

Uij12 ¼

s1ðXATi þ YT

j BTi Þ ðs2 � s1ÞðXAT

i þ YTj BT

i Þ0 0

s1XATdi ðs2 � s1ÞXAT

di

0 0

2666664

3777775U22 ¼ diagf�XeR�1

1 X;�XeR�12 Xg; Ui

33 ¼ diagf�kiI;�k�1i Ig

U23 ¼s1M 0

ðs2 � s1ÞM 0

� �

and

Kij11 ¼ Ai þ BiYj þ AT

i þ YTj BT

i þ eQ 1 þ eQ 2 þ eQ 3 � eR1; K12 ¼ eR1

Kij13 ¼ AdiX

T ;K33 ¼ �ð1� lÞeQ 2; K22 ¼ �eQ 1 � eR1; K44 ¼ � eQ 3

Proof. Pre- and post-multiply both sides of Eq. (29) with diag(X, X, X, X, I, I, I, I) and its transpose, and Eq. (30 ) with diag(X, X,X, X, I, I, I, I, I, I, I, I) and its transpose, respectively. Define X ¼ P�1; XR1XT ¼ ~R1; XR2XT ¼ eR2; XQ iX

T ¼ eQ i ði ¼ 1;2;3) andYj = KjX

T. Then using Schur’s complement, we arrive at (35) and (36). This completes the proof. h

Similar to the derivation of Corollary 2 from Theorem 2, the derivation of the following corollary is carried out from The-orem 3 by setting 3X and 3�F as X and �F, respectively, and all the symbols in (37) and ( 38) are defined in Theorem 3.

Corollary 3. For some given constants 0 6 s1 < s2 and l, the equilibrium of the system (9) is robustly asymptotically stable in thelarge with feedback gains Kj = YjX

�T (j = 1,2, . . . , r) for any s(t) satisfying (10), if there exist real matrices X > 0; eR1 > 0; eR2 > 0and eQ i > 0 (i = 1,2,3) with appropriate dimensions, scalars ki > 0, such that the following matrix inequalities hold for i, j = 1,2, . . . , rand 1 6 i < j 6 r

Uii11 þXþ �F Uii

12 Uii13

� U22 U23

� � U33

26643775 < 0 ð37Þ

Uij11 þUji

11 þXþ �F Uij12 Uji

12 Uij13 Uji

13

� U22 0 U23 U23

� � U22 0 0

� � � U33 0

� � � � U33

2666666664

3777777775< 0 ð38Þ

It is worth mentioning that although there are nonlinear items U22, U33 in Theorem 3 and Corollary 3, definingeU22 ¼ diagf�M1;�M2g, eU33 ¼ diagf�kiI;�Nig, they are easily to be transferred to the following nonlinear minimizationproblem instead of the original non-convex feasibility problem

Minimize Trace XP þP2i¼1ðNiki þ SiMi þ Ui

eRi�

subject to ð35Þ�; ð36Þ�

P I

� X

" #> 0;

Mi I

� Si

" #> 0;

eRi I

� Ui

" #> 0;

Ni I

� ki

" #> 0

8>>>>>>><>>>>>>>:ð39Þ

>where U22 and U33 in (35) and (36) are replaced by eU22 and eU33, respectively.

4296 C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299

For this optimization problem, similar to those in [6,19], the cone complementary linearization algorithm can be adoptedto solve this non-convex problem. Detailed discussion of this algorithm is omitted in this paper.

Remark 9. Notice that within the framework of T–S fuzzy models and PDC control design, conditions for the stability andperformance of a nonlinear system can be stated in terms of the feasibility of a set of LMIs. The main advantage of LMIs isthat while most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations, theyoften bring a numerical resolution to control problems in the LMI framework. This makes LMI-based design a valuablealternative to classical ‘‘analytical’’ methods. Due to the fact that there exist very efficient numerical algorithms fordetermining the feasibility of LMIs, such as Matlab/LMI toolbox, even large scale analysis and design problems arecomputationally tractable, so the employed method of this work is practical to real world applications [5,23,24,28]. However,simple and affordable of an LMI approach to practical on-line control engineering is still changeling.

5. Numerical examples

Generally, there are two methods to obtain the T–S fuzzy models in the literature. One common method is based on thefuzzy identification and reasoning by using the input–output data [25]. If there are sufficient number of noiseless output datafor the identification, then simple local linear dynamic systems with corresponding linguistic description could be identifiedto approximately express original nonlinear systems. However, when the input–output data are difficult to be obtained, thenthis method is not available again. Another method to derive concerned models is based on the concept of linearization [14].It is based on the replacement of the original nonlinear systems by a set of local linear dynamic systems, whose coefficientsare obtained by setting the nonlinear items to linear items in some local subspaces. This method leads to different solutionson the basis of the particular criteria of linearization adopted. Moreover, the prerequisite condition of this method is that thenonlinear system models, that is, the mechanism model must be known a priori. In fact, the mechanism model is difficult tobe derived for most of practical systems. The choice between these two methods based on the usability of input–output dataand the availability of nonlinear model for the system under study.

Assume that the nonlinear model and the mechanism model have been known, this section gives two numerical exam-ples to demonstrate the effectiveness of the proposed approach. The T–S fuzzy model of Example 1 is derived from a time-delayed nonlinear system [16,26], and the modeling of Example 2 is obtained from a practical backing-up control of a com-puter simulated truck-trailer [1].

Example 1. Consider the following time-delayed nonlinear system:

Table 1s2 unde

Meth

WuTianLien

CoroCoro

Over

_x1ðtÞ ¼ 0:5ð1� sin2ðhðtÞÞÞx2ðtÞ � x1ðt � sðtÞÞ � ð1þ sin2ðhðtÞÞÞx1ðtÞ_x2ðtÞ ¼ sgnðjhðtÞj � p=2Þð0:9 cos2ðhðtÞÞ � 1Þx1ðt � sðtÞÞ

� x2ðt � sðtÞÞ � ð0:9þ 0:1 cos2ðhðtÞÞÞx2ðtÞð40Þ

which can be exactly expressed as a T–S delayed system (9) with the following rules [16,26].

R1 : If hðtÞ is p=2Then _xðtÞ ¼ A1xðtÞ þ Ad1xðt � sðtÞÞ

ð41Þ

R2 : If hðtÞ is 0Then _xðtÞ ¼ A2xðtÞ þ Ad2xðt � sðtÞÞ

ð42Þ

The membership functions for above rules 1 and 2 are

l1ðx1ðtÞÞ ¼1

1þ expð�2x1ðtÞÞ; l2ðx1ðtÞÞ ¼ 1� l1ðx1ðtÞÞ ð43Þ

where Ai and Adi (i = 1,2) are given as

r different values of s1 and unknown l.

od s(t) = s s1 = 0 s1 = 0.4 s 1 = 0.8 s1 = 1.2 Num. var

and Li [30] 1.5974 – – – – 9and Peng [26] 1.5974 0.7210 0.8839 1.0935 1.3362 8 + 8ret al. [16] 1.5974 0.8310 0.8829 1.0677 1.3181 7

llary 2 1.5974 0.9822 1.0437 1.1699 1.3648 6llary 1 1.5974 0.9822 1.1818 1.3130 1.4333 6

[16] Nil >18% >33% >22% >8% –

Table 2s2 unde

Meth

Lien

CoroTheo

Over

C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299 4297

A1 ¼�2 00 �0:9

� �; A2 ¼

�1 0:50 �1

� �; Ad1 ¼

�1 0�1 �1

� �; Ad2 ¼

�1 00:1 �1

� �

The calculated maximum allowable delay bounds are tabulated in Tables 1 and 2 under different lower delay bounds s1 andwith known or unknown time delay differential l, respectively. The second column of Table 1 shows the case of constantdelay, that is, s1 = s2 = s. It is seen that most recent methods including those proposed in this paper give the same results = 1.5974.

Considering the interval time-varying delay case, the methods developed in Li et al. [15], Wu and Li [30], and Chen et al.[3] fail to make any conclusion as they are proposed for constant delay or time-varying delay. Without any information aboutthe derivative of the time delay, the third through sixth columns of Table 1 list the upper delay bounds derived from Tian andPeng [26], Lien et al. [16] and the method proposed in this paper. It is seen from Table 1 that the results obtained from bothCorollaries 1 and 2 in this paper are better than those obtained from the other methods. However, the results obtained fromCorollary 1 are much better than Corollary 2 in this paper, since none of useful items are ignored in the proof of Theorem 1.

The last column of Table 1 also shows that the method proposed in this paper utilizes the least number of scalar decisionvariables in LMIs for stability computation and thus is more computationally efficient than other methods (Num. var meansnumber of variables in Tables). Wu and Li [30] requires 9 variables; Lien et al. [16] needs 7 variables; and Tian and Peng [26]uses 8 + 8r variables, where r is the number of fuzzy rules. However, our results are dependent on 6 unknown variables,which also show the effectiveness of our proposed method. When the delay differential l is known, our results are listedin Table 2, it is also seen from which that the similar improved results can be obtained.

Example 2. Consider the following modified truck-trailer model with a time delay [30].

_x1ðtÞ ¼ �av�t

ðLþ DLðtÞÞt0x1ðtÞ � ð1� aÞ v�t

ðLþ DLðtÞÞt0x1ðt � sðtÞÞ þ v�t

ðlþ DlðtÞÞt0uðtÞ

_x2ðtÞ ¼ av�t

ðLþ DLðtÞÞt0x1ðtÞ þ ð1� aÞ v�t

ðLþ DLðtÞÞt0x1ðt � sðtÞÞ

_x3ðtÞ ¼v�tt0

sin x2ðtÞ þ av�t

2ðLþ DLðtÞÞ x1ðtÞ þ ð1� aÞ v�t2ðLþ DLðtÞÞ x1ðt � sÞ

� � ð44Þ

where a ¼ 0:7; v ¼ �1:0; �t ¼ 2:0; t0 ¼ 0:5; L ¼ 5:5; l ¼ 2:8; �0:2619 6 DLðtÞ 6 0:2895; �0:1333 6 DlðtÞ 6 0:1474.Consider the special cases of sin(h(t)) in above _x3ðtÞ, that is, h(t) equals to 0 or ±p, respectively. Then the following T–S

fuzzy model is used for fuzzy controller design:

R1 : If hðtÞ ¼ x2ðtÞ þ a v�t2L x1ðtÞ þ ð1� aÞ v�t

2L x1ðt � sðtÞÞ is 0

Then _xðtÞ ¼ A1x1ðtÞ þ Ad1x1ðt � sðtÞÞ þ B1uðtÞð45Þ

R2 : If hðtÞ ¼ x2ðtÞ þ a v�t2L x1ðtÞ þ ð1� aÞ v�t

2L x1ðt � sðtÞÞ is pThen _xðtÞ ¼ A2x1ðtÞ þ Ad2x1ðt � sðtÞÞ þ B2uðtÞ

ð46Þ

where

A1 ¼

�a v�tLt0

0 0

a v�tLt0

0 0

�a v2�t2

2Lt0

v�tt0

0

26643775; Ad1 ¼

�ð1� aÞ v�tLt0

0 0

ð1� aÞ v�tLt0

0 0

ð1� aÞ v2�t2

2Lt00 0

26643775

A2 ¼

�a v�tLt0

0 0

a v�tLt0

0 0

ad v2�t2

2Lt0

dv�tt0

0

26643775; Ad2 ¼

�ð1� aÞ v�tLt0

0 0

ð1� aÞ v�tLt0

0 0

ð1� aÞ dv2�t2

2Lt00 0

26643775

B1 ¼ B2 ¼v�tlt0

; 0; 0� �

; DAi ¼ DAdi ¼ ½0:255; 0:255; 0:255� sinðtÞ½0:1;0;0�:

r different values of s1 and l = 0.1.

od s1 = 0 s1 = 0.4 s1 = 0.8 s1 = 1.2 Num. var

et al. [16] 1.4840 1.4841 1.4841 1.4841 7

llary 2 1.4847 1.4849 1.4855 1.4870 6rem 1 1.4847 1.6539 1.8069 1.6940 6

[16] >29% >11% >21% >14% –

Table 3s2 and controller feedback gains under different values of s1(Example 2).

s1 s2 K1, K2

0 4.64 K1K2

� �¼ 2:7479 �2:6517 0:0429

2:7894 �2:7607 0:0444

� �1 5.07 K1

K2

� �¼ 2:3876 �2:1257 0:0245

2:4079 �2:2064 0:0251

� �2 5.98 K1

K2

� �¼ 2:2025 �1:8865 0:0178

2:2156 �1:9397 0:0181

� �

0 10 20 30 40 50 60−5

0

5

x 1(t)

0 10 20 30 40 50 60−5

0

5

x 2(t)

0 10 20 30 40 50 60−20

−10

0

x 3(t)

Simulation Time (Sec.)

Fig. 1. State responses of the studied system (Example 2).

4298 C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299

As in [1], set d = 10t0/pand employ the following membership functions

l1ðhðtÞÞ ¼ 1� 11þ expð�3ðhðtÞ � 0:5pÞÞ

� �1

1þ expð�3ðhðtÞ þ 0:5pÞÞ

� �l2ðhðtÞÞ ¼ 1� l1ðhðtÞÞ: ð47Þ

Under different levels of lower delay bound s1, Table 3 lists the results of some feasible solutions K1 and K2 and the corre-sponding maximum allowable delay bound s2 that guarantee asymptotic stability of the studied system.

With initial state conditions [0.5p �0.75p �2]T, Fig. 1 depicts the state responses of the nonlinear system under the abovedesigned fuzzy controller with 0 6 s(t) 6 4.64. It shows that the system with interval time-varying delays has been effec-tively controlled using the designed fuzzy controller.

6. Conclusion

An approach has been proposed for stability analysis and controller design of T–S fuzzy control systems with intervaltime-varying delays. By employing two coupling integral inequalities to deal with some integral items and constructingan appropriate Lyapunov–Krasovskii functional including the interval delay characteristic, some delay-range-dependent cri-teria have been derived for stability analysis and controller design, respectively. Because neither model transformation norfree weighting matrices have been employed to deal with integral terms in the derivation of our stability criteria, the resultsobtained in this paper are naturally less conservative and more computationally efficient than those derived from someexisting free weighting methods. Moreover, by solving a constrained convex optimization problem based on a standard conecomplementary algorithm, the maximum allowable upper delay bound and feedback controller gains can be obtained simul-taneously. Numerical examples have also been given to demonstrate the effectiveness of the proposed approach.

Acknowledgments

This work was in part jointly supported by the National Science Foundation of China under Grants 61074024 and60904013, the Natural Science Foundation of Jiangsu Province of China under Grant BK2010543, the Australian Research

C. Peng, Q.-L. Han / Information Sciences 181 (2011) 4287–4299 4299

Council Discovery Projects under Grants DP1096780 and DP0986376, and the Research Advancement Awards Scheme Pro-gram (January 2010–December 2012) at Central Queensland University, Australia.

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