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New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays
Information Sciences 179 (2009) 1134–1148
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Information Sciences journal homepage: www.elsevier.com/locate/ins
New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays q Li Li a,b,*, Xiaodong Liu a a b
Research Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034, PR China
a r t i c l e
i n f o
Article history: Received 23 July 2006 Received in revised form 19 November 2008 Accepted 22 November 2008
Keywords: Takagi–Sugeno fuzzy control Delay-dependent Linear matrix inequality State and input delays
a b s t r a c t This paper focuses on the delay-dependent stability analysis and stabilization for uncertain T–S fuzzy control systems with state and input delays. The key features of the approach include the introduction of uncorrelated augmented matrix items into the Lyapunov functional and the use of a tighter bounding technology. In fact, these techniques lead to more general and less conservative stability conditions that guarantee a wide stability region. Our delay-dependent stability conditions thus obtained are given in terms of linear matrix inequalities (LMIs). Finally, some examples are included to show that the proposed criteria improve the existing results significantly. Ó 2008 Published by Elsevier Inc.
1. Introduction Recently, Takagi–Sugeno (T–S) [22,23] fuzzy model has attracted considerable attention because it can combine the flexibility of fuzzy logic theory and rigorous mathematical theory of linear or nonlinear system into a unified framework. It gives a way to smoothly connect local linear systems to form global nonlinear systems by fuzzy membership functions. Researches on the properties, especially stability problems of the fuzzy systems, have become a very active area, e.g., [1,2,14–16,20,24– 27,34]. Time-delay phenomena are often encountered in various practical systems, such as AIDS epidemic, aircraft stabilization, chemical engineering systems, distributed networks, inferred grinding model, manual control, microwave oscillator, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines. In practical systems, analysis of a mathematical model is usually an important work for a control engineer as to control a system. However, the mathematical model always contains some uncertain elements. Therefore, under such imperfect knowledge of the mathematical model, seeking to design a robust control such that the system responses can meet desired properties is an important topic in system theory. Hence robust stability and robust stabilization problems for time-delay system have received some attenuation. Many criteria for checking the stability of time delay fuzzy systems have been derived, e.g., [5–7,10,12,33,35] and references therein. The existing results can be classified into two types: delay-independent stability [10,21,33], delay-dependent stability [5–7,12,35], etc.; the former does not include any information on the size of delay while the latter employs such information. It is known that delay-dependent stability conditions are generally less
q This work is supported in part by the National Natural Science Foundation of China under Grant 60575039 60534010 and in part by the National Key Basic Research and Development Program of China under Grant 2002CB312201-06. * Corresponding author. Address: Research Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China. E-mail address: [email protected] (L. Li).
0020-0255/$ - see front matter Ó 2008 Published by Elsevier Inc. doi:10.1016/j.ins.2008.11.039
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
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conservative than delay-independent ones especially when the size of the delay is small. Recently, delay-dependent stabilization is discussed in [5–7,12,32] for T–S fuzzy time-delay systems based on Lyapunov–Krasovksii functional approach. The delay is assumed to be constant. A state feedback control scheme has been proposed in terms of the feasible solutions to LMIs. In [13,30], the delay dependent stability analysis and control synthesis have been carried out by using Lyapunov–Krasovksii functional approach for uncertain T–S fuzzy systems with time-varying delay. In [31], the delay-dependent stability analysis and stabilization problems is researched for continuous-time T–S fuzzy systems with a time-varying delay. A fuzzy Lyapunov–Krasovksii functional is employed to derive the proposed delay-dependent results. However, all the aforementioned results are proposed for T–S fuzzy systems which only contain state delay. In modern industrial systems, sensors, controllers and plants are often connected over a network medium. Since the sampling data and controller signals are transmitted through a network, control input delays in control systems are always inevitable. So, it is important both in theory and in practice to consider the input delay when controllers are designed. Although the input delay is a technically important issue of frequent occurrence, few related control strategies seem to be available except for [8,17–19]. It remains yet to be a theoretically challenging issue, and thereby it is more important and more realistic to study the T–S fuzzy model with state delay and input delay. In [17], the authors have studied the delay systems without uncertainty, in which the state delay is equal to the input delay. In [8,18], the fuzzy systems with input delay is researched without state delay. And the input delay is constant delay in [8]. In [19], the uncertain fuzzy systems with state and input delay is considered. Based on Lyapunov–Razumikhin functional approaches, the delay-dependent LMI conditions for stability and stabilization have been developed in [17,18]. But the obtained results of [17,18] using the Lyapunov–Razumikhin functional approaches are usually more conservative than those using the Lyapunov–Krasovskii functional approaches since the Lyapunov–Krasovskii functional approaches take the advantage of the additional information of the delay. So the obtained results are more conservative. And in [8,19], the delay-dependent stability conditions are obtained with similar methods which are Lyapunov– Krasovskii functional approaches and free weighting matrix method. In this paper, we construct a new Lyapunov–Krasovskii functional, which gives the additional design matrix one more potential relaxation than the Lyapunov–Krasovskii functional methods of [8,19]. Furthermore, a tighter bounding technology for cross terms and the free weighting matrix approach are used to reduce the conservation. Some delay-dependent sufficient conditions are obtained to stabilize the uncertain fuzzy systems with time-varying delay in both state and input. Examples show that our stability conditions are less conservative than other conditions in the existing literatures, e.g. [7,8,19,30,31] etc. This paper is organized as follows: In Section 2, the T–S fuzzy model with time-delays is first formulated. We present some sufficient conditions for state feedback controller design by using PDC in Section 3. Section 4 provides some illustrative examples to demonstrate the effectiveness of proposed methods. Finally, the conclusions are drawn in Section 5. 2. Problem formulation and preliminaries We consider a nonlinear fuzzy system with state and input delays, which is represented by a Takagi–Sugeno (T–S) fuzzy model [22] composed of a set of fuzzy implications. Each implication is expressed by a linear time-delay system and the ith rule of the T–S model is written in the following form: Plant Rule i: IF h1 ðtÞ is Mi1 ; . . ., and hp ðtÞ is Mip , THEN
_ xðtÞ ¼ ðAi þ DAi ÞxðtÞ þ ðAid þ DAid Þxðt s1 ðtÞÞ þ ðBid þ DBid Þuðt s2 ðtÞÞ; 1 ; s 2 Þ; 0; i ¼ 1; 2; . . . ; r; xðtÞ ¼ uðtÞ; t 2 ½ maxðs
ð1Þ
where M ij is the fuzzy set, xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input. Ai ; Aid 2 Rnn ; Bid 2 Rnm ; and h1 ðtÞ; h2 ðtÞ; ; hp ðtÞ are the premise variables. It is assumed that the premise variables do not depend on the input uðtÞ. The time delays in state and input, i.e., s1 ðtÞ and s2 ðtÞ; are time-varying continuous functions satisfying 0 6 si ðtÞ 6 si < 1; s_ i ðtÞ 6 li < 1; ði ¼ 1; 2Þ 8t P 0; where si and li are constants. The matrices DAi ðtÞ; DAid ðtÞ; and DBid ðtÞ denote the uncertainties in system and they are of the forms
DAi ðtÞ ¼ Di FðtÞEia ;
DAid ðtÞ ¼ Di FðtÞEida ;
DBid ðtÞ ¼ Di FðtÞEidb ;
where Di ; Eai ; Eida ; Eidb are known constant matrices and FðtÞ are unknown real time-varying matrices with Lebesgue measurable elements bounded by F T ðtÞFðtÞ 6 I: For the simplicity, let us introduce the following notations:
Ai ¼ Ai þ DAi ðtÞ;
Aid ¼ Aid þ DAid ðtÞ;
Bid ¼ Bid þ DBid ðtÞ:
ð2Þ
By using center-average defuzzifer, product inference and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model:
Pr _ xðtÞ ¼
i¼1 i ðhðtÞÞ½Ai xðtÞ
þ Aid xðt s1 ðtÞÞ þ Bid uðt s2 ðtÞÞ Pr ; i¼1 i ðhðtÞÞ
ð3Þ
1136
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
where hðtÞ ¼ ½h1 ðtÞ; . . . ; hp ðtÞT ; i ðhðtÞÞ : Rp ! ½0; 1 ði ¼ 1; 2; . . . ; rÞ are the membership functions of the system with respect r P i ðhðtÞÞ. In this paper, we assume that i ðhðtÞÞ P 0 for i ¼ 1; 2; . . . ; r and to the ith plant rule, and hi ðhðtÞÞ ¼ i ðhðtÞÞ i¼1 Pr Pr i¼1 i ðhðtÞÞ > 0 for all t. Therefore, hi ðhðtÞÞ P 0 ði ¼ 1; 2; . . . ; rÞ and i¼1 hi ðhðtÞÞ ¼ 1 for all t. Then, (3) can be represented by
_ xðtÞ ¼
r X
hi ðhðtÞÞ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ Bid uðt s2 ðtÞÞ:
ð4Þ
i¼1
Based on the parallel distributed compensation (PDC), the following fuzzy control law is employed to deal with the problem of stabilization via state feedback. Control Rule i: IF h1 ðtÞ is Mi1 ; . . . ; and hp ðtÞ is Mip ; then
uðtÞ ¼ K i xðtÞ;
i ¼ 1; 2; . . . ; r:
ð5Þ
Hence, the overall fuzzy control law is represented by
uðtÞ ¼
r X
hi ðhðtÞÞK i xðtÞ i ¼ 1; 2; . . . ; r;
ð6Þ
i¼1
where K i ði ¼ 1; 2; . . . ; rÞ are the local control gains. When there exists an input delay s2 ðtÞ, we have uðt s2 ðtÞÞ ¼ Pr i¼1 hi ðhðt s2 ðtÞÞÞK i xðt s2 ðtÞÞ: So, it is natural and necessary to make an assumption that the functions 2 ; 0, and satisfy the following properties: hi ðhðt s2 ðtÞÞÞ P 0 hi ðhðtÞÞði ¼ 1; 2; . . . ; rÞ are well defined for all t 2 ½s P ði ¼ 1; 2; . . . ; rÞ; and ri¼1 hi ðhðt s2 ðtÞÞÞ ¼ 1: For the simplicity, let us denote: d:t:
hi ¼ hi ðhðtÞÞ;
s d:t:
hi 2 ¼ hi ðhðt s2 ðtÞÞÞ:
The design of the fuzzy controller is to determine the feedback gains K i ði ¼ 1; 2; . . . ; rÞ such that the resulting closed-loop system is asymptotically stable. With the control law (6), the overall closed-loop system can be written as
8 r r P r P P > s < xðtÞ _ hi hj 2 Bid K j xðt s2 ðtÞÞ; ¼ hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ i¼1 j¼1 j¼1 > : 1 ; s 2 Þ; 0; i; j ¼ 1; 2; . . . ; r: xðtÞ ¼ uðtÞ; t 2 ½ maxðs
ð7Þ
The following lemmas are useful in the proofs of our main results. Lemma 1 [11]. For any positive symmetric constant matrix M 2 Rnn and a scalar c > 0, if there exists a vector function Rc Rc w : ½0; c ! Rn such that the integrals 0 wT ðsÞMwðsÞ ds and 0 wðsÞ ds are well defined, then the following inequality holds:
c
Z
c
wT ðsÞMwðsÞ ds P
Z
0
c
Z wT ðsÞ ds M
0
c
wðsÞ ds :
ð8Þ
0
Applying Lemma 1 and the methods of [9], the following Lemma can be obtained, which is a much tighter bounding technology for cross terms and makes the results derived less conservative. Q 11 Q 12 , and vector P 0, scalar sðtÞ 6 s Lemma 2. For any constant matrices Q 11 ; Q 22 ; Q 12 2 Rnn ; Q 11 P 0; Q 22 P 0; Q 22 ; 0 ! Rn such that the following integration is well defined, then function x_ : ½s
s
Z
t
xT ðsÞ x_ T ðsÞ
Q 11
ts
Q 12 Q 22
2 3T 2 xðtÞ Q 22 xðsÞ 6 7 6 ds 6 4 xðt sðtÞÞ 5 4 Q 22 _ Rt xðsÞ xðsÞ ds Q 12 tsðtÞ
Q 22 Q 22 Q 12
Q T12
32
xðtÞ
3
7 76 Q T12 54 Rxðt sðtÞÞ 5: t xðsÞ ds Q 11 tsðtÞ
Lemma 3 [28]. Given constant matrices Q ; H; and E of appropriate dimensions and with symmetric matrix Q, then
Q þ HFðtÞE þ ET F T ðtÞHT < 0; for all FðtÞ satisfying F T ðtÞFðtÞ 6 I; if and only if there exists a scalar
e > 0 such that
Q þ eHHT þ e1 ET E < 0: 3. Stabilization results In this section, we focus on the problems of developing some delay-dependent stability criteria.
ð9Þ
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
1137
3.1. Stability of nominal fuzzy system First we consider the control design of a state feedback control law that stabilizes the following nominal fuzzy time-delay system:
8 r r P r P P > s < xðtÞ _ hi hj 2 Bid K j xðt s2 ðtÞÞ; ¼ hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ i¼1 i¼1 j¼1 > : 1 ; s 2 Þ; 0; i; j ¼ 1; 2; . . . ; r: xðtÞ ¼ uðtÞ; t 2 ½ maxðs
ð10Þ
Next, some delay-dependent criteria are proposed for the T–S fuzzy system (10). i , li ði ¼ 1; 2Þ and di ði ¼ 2; 3; 4Þ, the closed-loop system (10) is asymptotically stable for any Theorem 1. Given scalars s i ði ¼ 1; 2Þ via PDC technique if there exist P11 > 0, P 22 > 0; R11 > 0, R22 > 0; Q 11 > 0, Q 22 > 0; S11 > 0, 0 6 si ðtÞ 6 s S22 > 0; R1 > 0, R2 > 0 and real matrices X; P 12 ; R12 ; Q 12 ; S12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
Cii < 0;
i ¼ 1; 2; . . . ; r;
Cij þ Cji 6 0; "
P11
P12
P22
ð11Þ
1 6 i 6 j 6 r;
#
" P 0;
R11
R12
R22
ð12Þ #
" P 0;
Q 11
Q 12
Q 22
#
" P 0;
S11
S12
S22
# P 0:
ð13Þ
where
2 6 6 6 6 Cij ¼ 6 6 6 6 6 4
Hij ð1; 1Þ Hij ð1; 2Þ
PT22 Q T12
Hij ð1; 4Þ
Hij ð2; 2Þ a1 PT22 þ Q T12 Hij ð2; 4Þ
Q 11
0
RT22 ST12 0
3
7 7 7 7 7; Hij ð4; 6Þ 7 7 7 RT12 5
Hij ð2; 6Þ 7 PT12
0 a2 RT22
Hij ð1; 6Þ
þ ST12
Hij ð4; 4Þ
S11
Hij ð6; 6Þ
Hij ð1; 1Þ ¼ P12 þ PT12 þ R12 þ RT12 þ R1 þ R2 þ s21 Q 11 þ s22 S11 Q 22 S22 þ Ai X T þ XATi ; Hij ð1; 2Þ ¼ a1 P 12 þ Q 22 þ Aid X T þ d2 XATi ; Hij ð1; 4Þ ¼ a2 R12 þ S22 þ Bid K j þ d3 XATi ; Hij ð1; 6Þ ¼ P11 þ R11 þ s21 Q 12 þ s22 S12 þ d4 XATi X T ; Hij ð2; 2Þ ¼ a1 R1 Q 22 þ d2 Aid X T þ d2 XATid ; Hij ð2; 4Þ ¼ d2 Bid K j þ d3 XATid ; Hij ð2; 6Þ ¼ d4 XATid d2 X T ; Hij ð4; 4Þ ¼ a2 R2 S22 þ d3 Bid K j þ d3 K Tj BTid ; Hij ð4; 6Þ ¼ d4 K Tj BTid d3 X T ; Hij ð6; 6Þ ¼ s21 Q 22 þ s22 S22 d4 X d4 X T P 11 ¼ XP 11 X T ; P12 ¼ XP 12 X T ; P22 ¼ XP 22 X T ; R11 ¼ XR11 X T ; R12 ¼ XR12 X T ; R22 ¼ XR22 X T ; Q 11 ¼ XQ 11 X T ; Q 12 ¼ XQ 12 X T ; Q 22 ¼ XQ 22 X T ; S11 ¼ XS11 X T ; S12 ¼ XS12 X T ; S22 ¼ XS22 X T ; R1 ¼ XR1 X T ; R2 ¼ XR2 X T ; K j ¼ K j X T ; a1 ¼ ð1 l1 Þ; a2 ¼ ð1 l2 Þ; and ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by
K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð14Þ
Proof. Choose a delay-dependent Lyapunov–Krasovksii functional candidate as
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ þ V 6 ðtÞ
ð15Þ
1138
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
with
V 1 ðtÞ ¼ gT1 ðtÞPg1 ðtÞ; V 2 ðtÞ ¼ gT2 ðtÞRg2 ðtÞ; Z t xT ðsÞR1 xðsÞ ds; V 3 ðtÞ ¼ ts1 ðtÞ t
Z
V 4 ðtÞ ¼
xT ðsÞR2 xðsÞ ds;
ts2 ðtÞ Z t
1 ÞÞnT ðsÞQ nðsÞ ds; ðs ðt s
1 V 5 ðtÞ ¼ s 2 V 6 ðtÞ ¼ s
ts Z t1
2 ÞÞnT ðsÞSnðsÞ ds; ðs ðt s
2 ts
where
g1 ðtÞ ¼ xT ðtÞ R¼
R11
R12
R22
R ;
t ts1 ðtÞ
T T ; xðsÞ ds
P¼
T nðsÞ ¼ xT ðsÞ x_ T ðsÞ ;
P 11
P12
P22
Q¼
;
g2 ðtÞ ¼ xT ðtÞ
Q 11
Q 12
Q 22
S¼
;
R
t ts2 ðtÞ
S11
S12
S22
T T ; xðsÞ ds
:
The derivative of V 1 ðtÞ, V 2 ðtÞ, V 3 ðtÞ, V 4 ðtÞ, V 5 ðtÞ; V 6 ðtÞ along the trajectory of (10) satisfies that
R
T P11 t V_ 1 ðtÞ ¼ 2gT1 ðtÞP g_ 1 ðtÞ ¼ 2 xT ðtÞ xðsÞds ts1 ðtÞ
P12 P22
R
T R11 t V_ 2 ðtÞ ¼ 2gT2 ðtÞP g_ 2 ðtÞ ¼ 2 xT ðtÞ xðsÞ ds ts2 ðtÞ
R12 R22
_ xðtÞ ; xðtÞ ð1 s_1 ðtÞÞxðt s1 ðtÞÞ
_ xðtÞ ; xðtÞ ð1 s_2 ðtÞÞxðt s2 ðtÞÞ
V_ 3 ðtÞ ¼ xT ðtÞR1 xðtÞ ð1 s_ 1 ðtÞÞxT ðt s1 ðtÞÞR1 xðt s1 ðtÞÞð6 xT ðtÞR1 xðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞR1 xðt s1 ðtÞÞ; V_ 4 ðtÞ ¼ xT ðtÞR2 xðtÞ ð1 s_ 2 ðtÞÞxT ðt s2 ðtÞÞR2 xðt s2 ðtÞÞ 6 xT ðtÞR2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞR2 xðt s2 ðtÞÞ; Q 11 21 xT ðtÞ x_ T ðtÞ V_ 5 ðtÞ ¼ s
Q 12
S11 22 xT ðtÞ x_ T ðtÞ V_ 6 ðtÞ ¼ s
S12
Q 22
S22
Z t T Q 11 xðtÞ 1 s x ðsÞ x_ T ðsÞ _ xðtÞ 1 ts
Z t T S11 xðtÞ 2 s x ðsÞ x_ T ðsÞ _ xðtÞ 2 ts
Q 12 Q 22 S12 S22
xðsÞ ds; _ xðsÞ
xðsÞ ds: _ xðsÞ
It follows from (10) that
0 ¼ 2½xT ðtÞT 1 þ xT ðt s1 ðtÞÞT 2 þ xT ðt s2 ðtÞÞT 3 þ x_ T ðtÞT 4 ( ) r r X r X X s _ hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ hi hj 2 Bid K j xðt s2 ðtÞÞ xðtÞ i¼1
i¼1
j¼1
i.e.
2
0¼2
r X r X i¼1
j¼1
T1
3
6T 7 6 27 6 7 607 s2 T 7 hi hj n ðtÞ6 6 T 7 ½ Ai 6 37 6 7 405 2
T4 T 1 Ai
Aid
0 Bid K j
0 I nðtÞ
3
T 1 Aid
0 T 1 Bid K j
0 T 1
6 T 2 Ai 6 6 r r XX 6 0 s2 T hi hj n ðtÞ6 ¼2 6T A 6 3 i i¼1 j¼1 6 4 0
T 2 Aid
0 T 2 Bid K j
0 T 2 7 7 7 0 0 7 7nðtÞ: 0 T 3 7 7 7 0 0 5
T 4 Ai
T 4 Aid
0 T 3 Aid 0
0
0
0 T 3 Bid K j 0
0
0 T 4 Bid K j
0 T 4
ð16Þ
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L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
From Lemma 2 and
1 s
Z
t
Q 11
Q 12 Q 22
xT ðsÞ x_ T ðsÞ
Q 11
2 s
t
xT ðsÞ x_ T ðsÞ
S11
2 ts
S11
Q 12
S12 S22
Q 22
1 ts
Z
P 0,
S12
S22
P 0, we have
2 3T 2 xðtÞ Q 22 xðsÞ 6 xðt s1 ðtÞÞ 7 6 ds 6 4 5 4 Q 22 _ Rt xðsÞ xðsÞ ds Q 12 ts1 ðtÞ
2 3T 2 xðtÞ S22 xðsÞ 6 7 6 ds 6 4 xðt s2 ðtÞÞ 5 4 S22 _ Rt xðsÞ xðsÞ ds S12 ts2 ðtÞ
Q 22 Q 12
ST12
S22 S22 S12
32
3 xðtÞ 7 6 7 Q T12 54 Rxðt s1 ðtÞÞ 5; t xðsÞ ds Q 11 ts1 ðtÞ Q T12
Q 22
32
xðtÞ
3
7 76 ST12 54 Rxðt s2 ðtÞÞ 5: t xðsÞ ds S11 ts2 ðtÞ
By the above inequalities, it is obtained that
T P11 P12 _ xðtÞ xðsÞds 1 ðtÞ P22 xðtÞ ð1 s_1 ðtÞÞxðt s1 ðtÞÞ R
T R11 R12 _ xðtÞ t þ xT ðtÞR1 xðtÞ þ 2 xT ðtÞ xðsÞds ts2 ðtÞ R22 xðtÞ ð1 s_2 ðtÞÞxðt s2 ðtÞÞ
R t _ VðtÞ 6 2 xT ðtÞ ts
ð1 l1 ÞxT ðt s1 ðtÞÞR1 xðt s1 ðtÞÞ þ xT ðtÞR2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞR2 xðt s2 ðtÞÞ Q 11 Q 12 xðtÞ S11 S12 xðtÞ 22 xT ðtÞ x_ T ðtÞ 21 xT ðtÞ x_ T ðtÞ þs þs _ _ Q 22 xðtÞ S22 xðtÞ 2 3T 2 3 T 32 xðtÞ xðtÞ Q 22 Q 22 Q 12 76 6 7 6 7 þ 4 xðt s1 ðtÞÞ 5 4 Q 22 Q 22 Q T12 54 xðt s1 ðtÞÞ 5 Rt Rt xðsÞ ds xðsÞ ds Q 12 Q 12 Q 11 ts1 ðtÞ ts1 ðtÞ 3 2 3T 2 2 3 xðtÞ xðtÞ S22 S22 ST12 76 6 7 6 7 þ 4 xðt s2 ðtÞÞ 5 4 S22 S22 ST12 54 xðt s2 ðtÞÞ 5 þ 2½xT ðtÞT 1 þ xT ðt s1 ðtÞÞT 2 þ xT ðt s2 ðtÞÞT 3 Rt Rt xðsÞ ds xðsÞ ds S12 S12 S11 ts2 ðtÞ ts2 ðtÞ r r r X XX s _ þ x_ T ðtÞT 4 f hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ hi hj 2 Bid K j xðt s2 ðtÞÞ xðtÞg i¼1
¼
r X r X i¼1
i¼1
j¼1
s
hi hj 2 nT ðtÞFij nðtÞ;
ð17Þ
j¼1
where
R
T t nT ðtÞ ¼ xT ðtÞ xT ðt s1 ðtÞÞ xðsÞds ts1 ðtÞ 2 6 6 6 6 6 Fij ¼ 6 6 6 6 6 4
Kij ð1; 1Þ Kij ð1; 2Þ
P T22 Q T12
Kij ð2; 2Þ a1 PT22 þ Q T12
xT ðt s2 ðtÞÞ ð
Rt
xðsÞdsÞT ts2 ðtÞ
Kij ð1; 4Þ
RT22 ST12
T 2 Bid K j þ ATid T T3
0
0
Q 11
Kij ð4; 4Þ
0 a2 RT22
þ ST12
S11
x_ T ðtÞ ;
Kij ð1; 6Þ
7 ATid T T4 T 2 7 7 7 7 PT12 7; 7 T T T K j Bid T 4 T 3 7 7 7 RT12 5 Kij ð6; 6Þ
and
Kij ð1; 1Þ ¼ P12 þ PT12 þ R12 þ RT12 þ R1 þ R2 þ s21 Q 11 þ s22 S11 Q 22 S22 þ T 1 Ai þ ATi T T1 ; Kij ð1; 2Þ ¼ ð1 l1 ÞP12 þ Q 22 þ T 1 Aid þ ATi T T2 ; Kij ð1; 4Þ ¼ ð1 l2 ÞR12 þ S22 þ T 1 Bid K j þ ATi T T3 ; Kij ð1; 6Þ ¼ P11 þ R11 þ s21 Q 12 þ s22 S12 þ ATi T T4 T 1 ; Kij ð2; 2Þ ¼ ð1 l1 ÞR1 Q 22 þ T 2 Aid þ ATid T T2 ; Kij ð4; 4Þ ¼ ð1 l2 ÞR2 S22 þ T 3 Bid K j þ K Tj BTid T T3 ; Kij ð6; 6Þ ¼ s21 Q 22 þ s22 S22 T T4 T 4 ; a1 ¼ ð1 l1 Þ; a2 ¼ ð1 l2 Þ:
3
ð18Þ
1140
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
Denote T 2 ¼ d2 T 1 ; T 3 ¼ d3 T 1 ; T 4 ¼ d4 T 1 . Pre and post-multiplying both side of (18) with diag ½X; X; X; X; X; X and their transT T T T T T pose, defining new variables X ¼ T 1 1 ; P 11 ¼ XP 11 X ; P 12 ¼ XP 12 X ; P 22 ¼ XP 22 X ; R11 ¼ XR11 X ; R12 ¼ XR12 X ; R22 ¼ XR22 X ; T T T T T T T T Q 11 ¼ XQ 11 X ; Q 12 ¼ XQ 12 X ; Q 22 ¼ XQ 22 X ; S11 ¼ XS11 X ; S12 ¼ XS12 X ; S22 ¼ XS22 X ; R1 ¼ XR1 X ; R2 ¼ XR2 X ; K j ¼ K jXT , _ < 0. Therefore the asymptotic staðj ¼ 1; 2; . . . ; rÞ: The obtained result is equivalent to (11) and (12). So, if Fij < 0, then VðtÞ bility of system (10) is established. The proof is completed. Remark 1. It is worth mentioning that a much tighter bounding technology for cross terms is adopted in the proof of Theorem 1. Lemma 2 is a more general and tighter bounding technology for dealing with cross terms than Lemma 1. However, in the previous results, Lemma 1 is often used. Therefore, the stability criteria derived here are less conservative. It can been shown in the examples of Section 4. Rt 2 t Remark 2. For the constructed Lyapunov–Krasovskii functional items (15), the cross terms, e.g., s s2 ðs T 2 ÞÞx ðsÞS12 xðsÞ _ ds etc., are introduced into the Lyapunov–Krasovskii functional. Therefore, the additional design matrix ðt s gives a potential relaxation; and we can expect less conservative results. It can be seen that some existing results ([8,19], etc.) are special cases of our criteria. The methods in [8] are based on (15) with P ¼ 0; R1 ¼ 0; Q ¼ 0; S12 ¼ 0: And the methods in [19] are also based on (15) with P 12 ¼ 0; P22 ¼ 0; R ¼ 0; Q 12 ¼ 0; Q 22 ¼ 0; S12 ¼ 0; S22 ¼ 0. This implies that our Lyapnov– Krasvskii functional is more generalized and includes more weighting matrices than those in [8,19]. These lead to less conservatism as the illustrative examples shown in Section 4. Moreover, a much tighter bounding technology for cross terms is employed here. These make the stability criteria derived here less conservative. Remark 3. Notice that Theorem 1 is not applicable for li ¼ 0 (i=1,2) or li P 1. As for li ¼ 0, the time-delay is time-invariant, i.e. si ðtÞ ¼ hi (hi are constant). When li is unknown, let P 12 ¼ 0; P22 ¼ 0; R12 ¼ 0; R22 ¼ 0; R1 ¼ 0; R2 ¼ 0 and we can directly obtain a rate-independent criterion from Theorem 1. In what follows, Theorem 1 is developed to a more relaxed condition for the systems (10) using the techniques in [29]. i , li ði ¼ 1; 2Þand di ði ¼ 2; 3; 4Þ, the closed-loop system (10) is asymptotically stable for any Theorem 2. Given scalars s i ði ¼ 1; 2Þ via PDC technique if there exist P 11 > 0, P22 > 0; R11 > 0, R22 > 0; Q 11 > 0, Q 22 > 0; S11 > 0, 0 6 si ðtÞ 6 s S22 > 0; R1 > 0, R2 > 0, T ijk P 0; i; j; k ¼ 1; 2; . . . r; i < j and real matrices X; P12 ; R12 ; Q 12 ; S12 ; K j ; Hij ; V ijk ði; j; k ¼ 1; . . . rÞ satisfying (13) and the following LMIs:
2
R1 Z 1k R12k 6 R R2 Z 2k 2rk 6 6 6 4
.. .
.. .
Rr1k
Rr2k
..
R1rk R2rk
.
.. .
3 7 7 7 < 0; 7 5
ð19Þ
Rr Z rk
where for i; j; k ¼ 1; . . . ; r
Ri ¼ Cii ; 8 > < Hjk Z jk ¼ Hkj > : 0 (
Rijk ¼
if j < k; if j > k; if j ¼ k
Cij þ T ijk þ ðV ijk V Tijk Þ þ 12 W ijk if i < j; Cji þ T jik þ ðV jik V Tjik Þ þ 12 W jik if i > j;
W ijk ¼
Hik
if i ¼ k or j ¼ k;
0
if i–k or j–k;
i ði ¼ 1; 2Þ via PDC technique. In this case, the and Cij , i; j ¼ 1; ; r are given by Theorem 1. Then (10) is stable for any 0 6 si ðtÞ 6 s stabilizing control law is given by (14). Proof. The proof of the theorem is similar to that of Theorem 5 in [29] and it is thus omitted. Let
s1 ðtÞ ¼ s2 ðtÞ ¼ sðtÞ; the fuzzy model (10) is reduced to the following model: _ xðtÞ ¼
r X i¼1
hi ½Ai xðtÞ þ Aid xðt sðtÞÞ þ
r X r X i¼1
s
hi hj Bid K j xðt sðtÞÞ;
j¼1
< 1; s_ ðtÞ 6 l < 1 8t P 0: For this fuzzy system, we have the following Corollary of Theorem 1. where 0 6 sðtÞ 6 s
ð20Þ
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
1141
and di ði ¼ 2; 3Þ, the closed-loop system (20) is asymptotically stable for any 0 6 sðtÞ 6 s via PDC Corollary 1. Given scalars s technique if there exist P 11 > 0, P22 > 0; Q 11 > 0, Q 22 > 0; R1 > 0 and real matrices X; P 12 ; Q 12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
Wii < 0; i ¼ 1; 2; . . . ; r; Wij þ Wji 6 0; 1 6 i 6 j 6 r; "
P 11
P12
P22
#
"
P 0;
ð21Þ
Q 11
Q 12
Q 22
ð22Þ
# P 0:
ð23Þ
where
2 6 6 4
ij ð1; 1Þ ij ð1; 2Þ
ij ð2; 2Þ aPT22 þ Q T12
Wij ¼ 6
P T22 Q T12
Q 11
ij ð1; 4Þ
3
ij ð2; 4Þ 7 7 7; 5 PT 12
ij ð4; 4Þ
2 Q 11 Q 22 þ Ai X T þ XATi ; ij ð1; 1Þ ¼ P 12 þ PT12 þ R1 þ s ij ð1; 2Þ ¼ aP12 þ Q 22 þ Aid X T þ Bid K j þ d2 XATi ; 2 Q 12 þ d3 XATi X T ; ij ð1; 4Þ ¼ P 11 þ s ij ð2; 2Þ ¼ aR1 Q 22 þ d2 Aid X T þ d2 Bid K j þ d2 XATid þ d2 K Tj BTid ; ij ð2; 4Þ ¼ d3 XATid þ d3 K Tj BTid d2 X T ; 2 Q 22 d3 X d3 X T ; ij ð4; 4Þ ¼ s P 11 ¼ XP11 X T ; P 12 ¼ XP 12 X T ; P22 ¼ XP 22 X T ; Q 11 ¼ XQ 11 X T ; Q 12 ¼ XQ 12 X T ; Q 22 ¼ XQ 22 X T ; R1 ¼ XR1 X T ; K j ¼ K j X T ,a ¼ ð1 lÞ; and ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by
K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð24Þ
Proof. We Choose a Lyapunov–Krasovksii functional candidate as
Vðxt ; tÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ
ð25Þ
with
V 1 ðtÞ ¼ gT1 ðtÞPg1 ðtÞ; Z t V 2 ðtÞ ¼ xT ðsÞR1 xðsÞ ds; tsðtÞ
V 3 ðtÞ ¼ s
Z
t
ÞÞnT ðsÞQnðsÞ ds; ðs ðt s
ts
and P; Q are the same as the Theorem 1. Because the Corollary 1 is a special case of Theorem 1. From the proof of Theorem 1, we can easily obtain the proof of Corollary 1. The detailed proof is omitted. Remark 4. Yoneyama [32] is not concerned with the input delay, it considers the control design with the delayed feedback. This leads to the closed-loop system having the same structure as that of this paper. However, in [32], input delay is equal to state delay and the delays are constant delays. Therefore, the stability problems of [32] is a special case of this paper. Moreover, let s1 ðtÞ ¼ s2 ðtÞ ¼ s and P12 ¼ 0; P 22 ¼ 0; Q 12 ¼ 0; Q 22 ¼ 0 in (25) of this paper, the Lyapunov–Krasovskii functional of [32] is obtained. This implies that our Lyapunov–Krasovskii functional is more generalized than that in [32]. This leads to less conservative results. If there is no input delay terms (i.e., s2 ðtÞ ¼ 0), the considered fuzzy model (10) is reduced to the same one as those studied in [6,10] etc.
_ xðtÞ ¼
r X r X i¼1
hi ðhÞhj ðhÞ½Ai xðtÞ þ Aid ðt s1 ðtÞÞ þ Bid K j xðtÞ:
j¼1
In this case, the following Corollary is easily obtained from Theorem 1.
ð26Þ
1142
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
1 , l1 and di ði ¼ 2; 3Þ, the closed-loop system (26) is asymptotically stable for any 0 6 s1 ðtÞ 6 s 1 via Corollary 2. Given scalars s PDC technique if there exist P 11 > 0, P 22 > 0; Q 11 > 0, Q 22 > 0; R1 > 0 and real matrices X; P12 ; Q 12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
Xii < 0; i ¼ 1; 2; . . . ; r; Xij þ Xji 6 0; 1 6 i 6 j 6 r; "
P11
P12
P22
#
"
P 0;
ð27Þ
Q 11
Q 12
Q 22
ð28Þ
# P 0;
ð29Þ
where
2 6 6 6 Xij ¼ 6 6 4
Uij ð1; 1Þ Uij ð1; 2Þ
PT22 Q T12
Uij ð1; 4Þ
3 7
Uij ð2; 2Þ a1 PT22 þ Q T12 Uij ð2; 4Þ 7 7
7; 7 5
Q 11
PT12
Uij ð4; 4Þ
Uij ð1; 1Þ ¼ P12 þ P T12 þ R1 þ s21 Q 11 Q 22 þ Ai X T þ Bid K j þ XATi þ K Tj BTid ; Uij ð1; 2Þ ¼ a1 P12 þ Q 22 þ Aid X T þ d2 XATi þ d2 K Tj BTid ; Uij ð1; 4Þ ¼ P11 þ s21 Q 12 þ d3 XATi þ d3 K Tj BTid X T ; Uij ð2; 2Þ ¼ a1 R1 Q 22 þ d2 Aid X T þ d2 XATid ; Uij ð2; 4Þ ¼ d3 XATid d2 X T Uij ð4; 4Þ ¼ s21 Q 22 d3 X d3 X T ; P11 ¼ XP11 X T ; P 12 ¼ XP12 X T ; P 22 ¼ XP 22 X T ; Q 11 ¼ XQ 11 X T ; Q 12 ¼ XQ 12 X T ; Q 22 ¼ XQ 22 X T ; R1 ¼ XR1 X T ; K j ¼ K j X T ,a1 ¼ ð1 l1 Þ; and ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by
K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð30Þ
Remark 5. Chen and coworkers [3,4,6,7,10,12] have considered the robust control for uncertain Takagi–Sugeno (T–S) fuzzy systems with state delays. Some delay-dependent conditions are obtained with a traditional Lyapunov–Krasovskii functional method in [3,4,6,7,10,12]. However, these results can be derived based on (15) with R ¼ 0; R2 ¼ 0; S ¼ 0; P12 ¼ 0; P 22 ¼ 0; Q 12 ¼ 0; Q 22 ¼ 0: Moreover, we employ free weighting matrix method. These lead to the stability criteria derived less conservative than them. 3.2. Robust stability of uncertain fuzzy systems This section considers a problem of robust stablization of the uncertain T–S fuzzy system (7) using state feedback control rule (6), which is a more general stability problem of fuzzy system than the research problem of [7,8,12,30] etc. i , li ði ¼ 1; 2Þ and di ði ¼ 2; 3; 4Þ, the closed-loop system (7) is asymptotically stable for any Theorem 3. Given scalars s i ði ¼ 1; 2Þ via PDC technique if there exist P 11 > 0; P 22 > 0; R11 > 0; R22 > 0; Q 11 > 0, Q 22 > 0; S11 > 0; 0 6 si ðtÞ 6 s S22 > 0; R1 > 0, R2 > 0, eij > 0 ði; j ¼ 1; ; rÞ and real matrices X; P12 ; R12 ; Q 12 ; S12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
2
Cii eii CDi
6 4 2
eii I
Cij eij CDi
6 4
eij I
CTEii
3
7 0 5 < 0; eii I
CTEij
3
2
i ¼ 1; 2; . . . ; r;
Cji eji CDj
7 6 0 5þ4
eij I
eji I
CTEji
ð31Þ 3
7 0 5 6 0;
eji I
1 6 i 6 j 6 r;
ð32Þ
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
"
P11
P12
P22
#
" P 0;
R11
R12
R22
#
" P 0;
Q 11
Q 12
Q 22
#
" P 0;
S11
S12
S22
1143
# P 0;
ð33Þ
where
CTDi ¼ DTi d2 DTi 0 d3 DTi 0 d4 DTi ; CEij ¼ Eai X T Eida X T 0 Eidb K j 0 0 ; Cij is defined in Theorem 1. And ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð34Þ
Proof. Replacing Ai ; Aid and Bid in Cij of Theorem 1 with Ai þ Di FðtÞEai ; Aid þ Di FðtÞEida and Bid þ Di FðtÞEidb , respectively, we obtain (35) corresponding to the system (7):
Cij þ CDi FðtÞCEij þ CTEij F T ðtÞCTDi < 0;
i; j ¼ 1; . . . ; r:
ð35Þ
By Lemma 3, it is obtained T Cij þ CDi FðtÞCEij þ CTEij F T ðtÞCTDi 6 Cij þ eij CDi CTDi þ e1 ij CEij CEij ;
where
CTDi ¼ DTi d2 DTi 0 d3 DTi 0 d4 DTi ; CEij ¼ Eai X T Eida X T 0 Eidb K j 0 0 : By Schur complement, we obtain (31) and (32). The proof is completed.
Though the uncertain T–S fuzzy systems (7) is the same as the fuzzy systems of [19], our proposed stability conditions are less conservative than those of [19] because we use a new Lyapunov–Krasovskii functional and a tighter bounding technology. It can been seen in the example of Section 4. 4. Numerical examples In this section, four examples are presented to illustrate the effectiveness and merits of our results. We compare the previous results ([7,8,19,30,31] etc.) with our results in order to illustrate the effectiveness of our design methods. Example 1. Consider truck trailer system with time delay:
_ xðtÞ ¼
r X
hi ðhðtÞÞ½ðAi þ DAi ÞxðtÞ þ Aid xðt s1 Þ þ ðBid þ DBid Þuðt s2 Þ;
ð36Þ
i¼1
2
a Ltv t0
6 vt A1 ¼ 6 4 a Lt0 2 2 a v2Ltt0 2 A2d
0
0
3
2
0
7 07 5;
6 vt A2 ¼ 6 4 a Lt0 2 2 ad v2Ltt0
vt t0
0
ð1 aÞ Ltv t0
6 vt ¼6 4 ð1 aÞ Lt0 v 2t2 ð1 aÞ d2Lt 0
a Ltv t0
3
0
0
0
7 07 5;
dvt t0
0
3
2 vt 3
7 0 07 5; 0 0
6 07 B1d ¼ B2d ¼ 4 0 5;
0 0
2
ð1 aÞ Ltv t0
6 vt A1d ¼ 6 4 ð1 aÞ Lt0 2 2 ð1 aÞ v2Ltt0
0 0
3
7 0 07 5; 0 0
lt
0
T with v ¼ 1:0; t ¼ 2:0; t 0 ¼ 0:5; L ¼ 5:5; l ¼ 2:8; d ¼ 10 tp0 ; Di ¼ ½ 0:255 0:255 0:255 ; Eai ¼ ½ 0:1 0 0 ; Eidb ¼ 0:15 for i ¼ 1; 2. For simulation purpose, we simply choose h1 ðxÞ ¼ 1=ð1 þ expðx1 þ 0:5ÞÞ and h2 ðxÞ ¼ 1 h1 ðxÞ where xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ x3 ðtÞ T : In the following cases (a), (b), (c), we compare our proposed results with previous results such as [6,8,10,33].
(a) For the system (36), which is more general systems than those of [7,8,19,30,31], let a ¼ 0:7; d2 ¼ 0:4; d3 ¼ 0:1, d4 ¼ 1:3. 2 ¼ 0:55, the control gains are obtained as follows: 1 ¼ 0:1, s By using Theorem 3 with s
K 1 ¼ ½ 4:4456 1:0809 0:0347 ;
K 2 ¼ ½ 4:5117 1:1684 0:0374 :
1144
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148 60
x1 x2 x3
50
State
40 30 20 10 0 −10 0
10
20
Time(Sec)
30
Fig. 1. The responses of the state of the system (36) with
40
s1 ¼ 0:1; s2 ¼ 0:55
30 25 20 15
u(t)
10 5 0 −5 −10 −15 −20
0
10
20
30
40
Time(Sec) Fig. 2. The responses of u(t) of the system (36) with
s1 ¼ 0:1; s2 ¼ 0:55.
For the system (36), here choose the initial functions uðtÞ ¼ ½4 12T . The simulation results are shown by Figs. 1,2, respectively. It can be seen that the fuzzy controller (6) proposed in this paper is effective for truck trailer system with state and input delays. ¼ 0:75; d3 ¼ 0:1 and d4 ¼ 2:5; the feedback (b) Let a ¼ 1, the systems (36) is the same as the fuzzy system of [8]. When s gain matrices are computed as
K 1 ¼ ½ 3:3219 0:2406 0:0025 ;
K 2 ¼ ½ 3:3272 0:2494 0:0026 :
2 ¼ 0:86. For the same fuzzy systems, in [8], the maximal allowAnd we obtain that the maximal allowable input delay is s ¼ 0:75, and the feedback gain matrices are computed as able input delay is s
K 1 ¼ ½ 3:4227 0:3535 0:0045 ;
K 2 ¼ ½ 3:5215 0:3617 0:0056
(c) Let Bid ¼ DAi ¼ DBid ¼ 0; a ¼ 0:7. Then the system (36) is reduced to the system which is the same as [6,10,33]. Applying Corollary 2, we compare our results with the existing results, and the corresponding results are shown as Table 1.In 1 ¼ 1:9685 in [6]. Then, in the the following, we compare the maximal allowable delay with [6]. The upper delay is s 1 ¼ 18:52 with d2 ¼ 0:1; d3 ¼ 0:5.From the cases (a)– same condition, we obtain that the maximal allowable delay is s (c), it can been seen our proposed stability conditions are always less conservation than the previous results.
Table 1 1 ¼ 0:5; d2 ¼ 0:1; d3 ¼ 0:5. The responding control gains obtained by different methods when s Methods
K1
In [10] In [33] In [6] Corollary 2
½ 62:8420 ½ 47:4174 ½ 17:6283 ½ 13:8540
K2 229:3535 19:6709 216:5310 58:6529 35:0250 4:6282 25:9117 2:9805
½ 61:5791 ½ 45:7796 ½ 17:3474 ½ 14:1456
224:8307 18:9802 206:5476 39:4999 34:2816 4:3444 27:4639 3:0065
1145
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
Example 2. Consider the nonlinear mass-spring-damper mechanical systems in [18]. The state-space is represented as follows:
# " x_ 1 ðtÞ 0:75x31 ðtÞ 0:5x1 ðtÞ þ 0:1ð1 þ 0:11ðtÞÞx2 ðtÞ þ uðt sðtÞÞ ; ¼ x_ 2 ðtÞ x1 ðtÞ
ð37Þ
where j1ðtÞj2 6 1. The system (37) has one nonlinear term 0:75x31 ðtÞ: Now by adopting fuzzy sets, the T–S fuzzy system of (37) can be constructed as follows Rule 1: Rule 2:
_ ¼ ðA11 þ DA11 ÞxðtÞ þ ðB1d þ DB1d Þuðt sðtÞÞ If x1 ðtÞ is about C1 ; then xðtÞ _ ¼ ðA12 þ DA12 ÞxðtÞ þ ðB2d þ DB2d Þuðt sðtÞÞwhere C1 ; C2 are defined in [18] and the If x1 ðtÞ is about C2 ; then xðtÞ associated matrices are given by
A11 ¼
0:5 0:1 1
0
;
" pffiffiffiffiffiffiffi # 0:1 ; D1 ¼ D2 ¼ 0
A12 ¼
1 0:1 1
0
;
DA11 ¼ DA12 ¼
pffiffiffiffiffiffiffi E1a1 ¼ E1a2 ¼ 0 0:1 ;
0 0:11ðtÞ 0
0
;
B1d ¼ B2d ¼
1 0
;
E1db ¼ E2db ¼ ½ 0 0 :
< 2:04 when d3 ¼ 0:03; d4 ¼ 1:7: In order to Applying Theorem 3 to the system (37), we achieve stability for all delay 0 < s show the advantages of our results, a comparison for the upper bound of input delay and feedback gains to guarantee the asymptotic stabilization of fuzzy system is shown in Table 2. We can conclude that Theorem 3 in this paper is less conservative than those of [18,19]. Example 3. Consider the fuzzy system with state delay in [12]
_ xðtÞ ¼
r X
hi ðhðtÞÞ½Ai xðtÞ þ Aid xðt s1 Þ þ Bi uðtÞ;
ð38Þ
i¼1
where
A1 ¼
0
1
0:1 2
A2 ¼
;
B1 ¼ B2 ¼ ½ 0 1 T ;
b¼
0
1
0:1 0:5 1:5b 0:01
p
;
A1d ¼ A2d ¼
0:1
0
0:1 0:2
;
;
and b is used to avoid system matrices being singular. The membership functions are set as follows:
h1 ðhðtÞÞ ¼
1
1 1 ; 1 þ exp½3ððx2 =0:5Þ ðp=2ÞÞ 1 þ exp½3ððx2 =0:5Þ þ ðp=2ÞÞ
h2 ðhðtÞÞ ¼ 1 h1 ðhðtÞÞ:
Here the system (38) is the same form as studied in Example 2 of [30]. Applying Corollary 2, the existing delay-dependent results are compared with our results, and the corresponding results are shown as the following Table 3. Table 2 A comparison for the upper bounds of input delay and feedback gains for (37) with d3 ¼ 0:03; d4 ¼ 1:7. Results
Upper bounds of state delay
Fuzzy control gains
[18] [19] This paper This paper This paper
0.4223(l unknown) 1.75 (l unknown) 1.97 (l unknown) 1:96 (l ¼ 0:1) 2.04 (l ¼ 0:5)
K1 K1 K1 K1 K1
¼ ½ 0:7623 0:5212 ¼ ½ 0:3622 0:299 ¼ ½ 0:4093 0:2844 ¼ ½ 0:4290 0:2840 ¼ ½ 0:3858 0:2604
K2 K2 K2 K2 K2
¼ ½ 0:3690 0:5362 ¼ ½ 0:3625 0:299 ¼ ½ 0:0706 0:0432 ¼ ½ 0:0317 0:1005 ¼ ½ 0:0390 0:0549
Table 3 The control gains and the maximal delay obtained by different methods with d2 ¼ 0:5; d3 ¼ 1:5. Methods
h2
K1
K2
In paper [17] In paper [12] In paper [30] In paper [7] Corollary 2
1.725 3.4385 8.0354 25.7865 27.96
½ 30:6571 10:0442 ½ 0:9390 0:3123 ½ 21:8139 55:0597 ½ 1:2141 0:8750 ½ 1:1916 0:0361
½ 30:6589 11:1803 ½ 0:3277 0:3636 ½ 21:8139 56:1183 ½ 1:2141 0:6202 ½ 1:1916 1:4591
1146
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148 3
x1 x2
2.5
State
1.5
0.5
−0.5
−1.5
0
10
20
30
40
Time(Sec)
50
Fig. 3. The responses of the state of the fuzzy system (38) when
s ¼ 27:96.
0.5 0
u(t)
−1
−2
−3
−4
0
10
20
30
40
Time(Sec)
Fig. 4. Control input response of the fuzzy system (38) when
50
s ¼ 27:96.
The simulation results are depicted in Figs. 3 and 4 for (38). They show the response of the closed-loop system with maximal allowable delay s ¼ 27:56 under initial condition uðtÞ ¼ ½ 3 1 T . From Figs. 3,4, it can be clearly seen that the proposed conditions guarantees the asymptotic stability of the closed-loop system (38). Example 4. Consider the following T–S fuzzy model [31]:
_ xðtÞ ¼
r X
hi ðhðtÞÞ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ Bi uðtÞ;
ð39Þ
i¼1
where
A1 ¼
0 0:6 0
1
;
A2 ¼
1 0 1 0
;
B1 ¼ B2 ¼
1 1
;
A1d ¼
0:5 0:9 0
2
;
A2d ¼
0:9
0
1
1:6
:
The membership functions are as follows:
h1 ðx1 ðtÞÞ ¼
1 ; 1 þ expð2x1 ðtÞÞ
h2 ðx1 ðtÞÞ ¼ 1 h1 ðx1 ðtÞÞ:
In [5,12], the maximum allowed delay is 0.1524 and 0.2302, respectively. In [31], based on Theorem 3 and Corollary 2, the maximum allowed delay is 0.2664 and 0.2574, respectively. However, applying to Corollary 2 of this paper, for d2 ¼ 0:2; d3 ¼ 1:45; the maximum allowed s can be obtained as 1.05. The aforementioned maximum allowed delays are summarized in Table 4. It can be seen from Table 4 that the proposed stabilization results in this paper are clearly less conservative than those in [5,12,31], which are based on a single Lyapunov–Krasovksii functional methods and a fuzzy Lyapunov–Krasovksii functional methods, respectively. Letting sðtÞ ¼ 1:05 and using Corollary 2 with the choice of d2 ¼ 0:2 and d3 ¼ 1:45, we can obtain the following feedback gain matrices:
K 1 ¼ ½ 32:6582 93:9701 ;
K 2 ¼ ½ 34:3946 98:5644 :
1147
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148 Table 4 Comparisons among various delay-dependent stabilization methods. Methods
Maximum allowed
Theorem 2 of [5] Theorem 1 of [12] Corollary 2 of [31] Theorem 3 of [31] Corollary 2 of this paper
0.1524 0.2302 0.2574 0.2664 1.05
5
s
x1 x2
4
State
3 2 1 0 −1 0
10
20
30
Time(Sec)
40
50
Fig. 5. The responses of the state of the fuzzy system (39).
Fig. 5 shows the control results for the system (39) under the initial condition ½2 1T ; t 2 ½1:050. From the simulation re ¼ 1:05. sults, we can see our designed controller is effective for (39) with the maximum allowed delay s From these examples, it is easy to find that our methods give larger delay upper bounds of si ðtÞði ¼ 1; 2Þ and lower feedback gains than the previous results. 5. Conclusion New classes of Lyapunov–Krasovksii functional and a tighter bounding technology are introduced to study the delaydependent stability for more general nonlinear fuzzy system with delays in input and state. The resulting criteria are less conservative than the existing ones. Numerical examples have been given to demonstrate the effectiveness and the merits of the proposed methods. Acknowledgement The authors would like to thank the anonymous referees for their constructive comments and suggestions, which have been very helpful in improving the paper. References [1] X.J. Ban, X.Z. Gao, X.L. Huang, A.V. Vasilakos, Stability analysis of the simplest Takagi–Sugeno fuzzy control system using circle criterion, Information Sciences 177 (20) (2007) 4387–4409. [2] A. Benzaouia, A.E. Hajjaji, M. Naib, Stabilization of a class of constrained fuzzy systems: a positive invariance approach, International Journal of Innovative Computing, Information and Control 2 (4) (2006) 749–760. [3] B. Chen, X.P. Liu, Reliable control design of fuzzy dynamic systems with time-varying delay, Fuzzy Sets and Systems 146 (3) (2004) 49–374. [4] B. Chen, X.P. Liu, Fuzzy guaranteed cost control for nonlinear systems with time-varying delay, IEEE Transactions on Fuzzy Systems 13 (2) (2005) 38– 249. [5] B. Chen, X.P. Liu, Delay-dependent robust H1 control for T–S fuzzy systems with time delay, IEEE Transactions on Fuzzy Systems 13 (2) (2005) 238– 249. [6] B. Chen, X.P. Liu, S.C. Tong, Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay, Fuzzy Sets and Systems 157 (16) (2006) 2224–2240. [7] B. Chen, X.P. Liu, S.C. Tong, New delay-dependent stabilization conditions of T–S fuzzy systems with constant delay, Fuzzy Sets and Systems 158 (20) (2007) 2209–2224. [8] B. Chen, X.P. Liu, S.C. Tong, Robust fuzzy control of nonlinear systems with input delay, Chaos, Solitons & Fractals. doi:10.1016/j.chaos.2006.09.091. [9] P. Chen, Y.C. Tian, Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay, Journal of Computational and Applied Mathematics (2007) doi:10.1016/j.cam.2007.03.009. [10] Y.Y. Cao, P.M. Frank, Analysis and synthesis of nonlinear time-delay system via fuzzy control approach, IEEE Transactions on Fuzzy Systems 8 (2) (2000) 200–211.
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[11] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhauser, Basel, 2003. [12] X.P. Guan, C.L. Chen, Delay-dependent guaranteed cost control for T–S fuzzy systems with time delays, IEEE Transactions on Fuzzy Systems 12 (2) (2004) 236–249. [13] X.F. Jiang, Q.L. Han, Robust H1 control for uncertain Takagi–Sugeno fuzzy systems with interval time-varying delay, IEEE Transactions on Fuzzy Systems 15 (2) (2007) 321–331. [14] J. Li, H.O. Wang, D. Niemann, K. Tanaka, Dynamic parallel distributed compensation for Takagi–Sugeno fuzzy systems: an LMI approach, Information Sciences 123 (3-4) (2000) 201–221. [15] X.D. Liu, Q.L. Zhang, New approaches to H1 controller designs based on fuzzy observers for T–S fuzzy systems via LMI, Automatica 39 (2003) 1571– 1582. [16] X.D. Liu, Q.L. Zhang, Approaches to quadratic stability conditions and H1 control designs for T–S fuzzy systems, IEEE Transactions on Fuzzy System 11 (6) (2003) 830–839. [17] C. Lin, Q.G. Wang, T.H. Lee, Delay-dependent LMI conditions for stability and stabilization of T–S fuzzy systems with bounded time-delay, Fuzzy Sets and Systems 157 (9) (2006) 1229–1247. [18] H.J. Lee, J.B. Park, Y.H. Joo, Robust control for uncertain Takagi–Sugeno fuzzy systems with time-varying input delay, ASME Journal of Dynamic Systems Measurement and Control 127 (2005) 302–306. [19] C.H. Lien, K.W. Yu, Robust control for Takagi–Sugeno fuzzy systems with time-varying state and input delays, Chaos, Solitons & Fractals 35 (5) (2008) 1003–1008. [20] S.K. Nguang, P. Shi, Robust output feedback control design for fuzzy dynamic systems with quadratic stability constraints: an LMI approach, Information Sciences 176 (15) (2006) 161–2191. [21] X.M. Sun, J. Zhao, W. Wang, Two design schemes for robust adaptive control of a class of linear uncertain neutral delay systems, International Journal of Innovative Computing, Information and Control 3 (2) (2007) 85–396. [22] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems Man and Cybernetics 15 (1) (1985) 116–132. [23] K. Tanaka, M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45 (1992) 135–156. [24] C.S. Ting, Stability analysis and design of Takagi–Sugeno fuzzy systems, Information Sciences 176 (19) (2006) 2817–2845. [25] S.C. Tong, W. Wang, L.J. Qu, Decentralized robust control for uncertain T–S fuzzy large-scale systems with time-delay, International Journal of Innovative Computing, Information and Control 3 (3) (2007) 657–672. [26] M. Wang, B. Chen, K.F. Liu, Adaptive fuzzy tracking control of nonlinear time-delay systems with unknown virtual control coefficients, Information Sciences 178 (2008) 4326C4340. [27] J. Yoneyama, Robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays, Information Sciences 178 (2008) 1935C1947. [28] L. Xie, Output feedback H1 control of systems with parameter uncertainty, International Journal of Control 63 (4) (1996) 741–750. [29] M.C.M. Teixeira, E. Assuncao, R.G. Avellar, On relaxed LMI-based design for fuzzy regulators and fuzzy observers, IEEE Transactions on Fuzzy Systems 11 (2003) 613–623. [30] E.G. Tian, C. Peng, Delay-dependent stability analysis and synthesis of uncertain T–S fuzzy systems with time-varying delay, Fuzzy Sets and Systems 157 (2006) 544–559. [31] H.N. Wu, H.X. Li, New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with time-varying delay, IEEE Transactions on Fuzzy Systems 15 (3) (2007) 482–493. [32] J. Yoneyama, Robust stability and stabilization for uncertain Takagi–Sugeno fuzzy time-delay systems, Fuzzy Sets and Systems 158 (2) (2007) 115– 134. [33] Y. Zhang, A.H. Pheng, Stability of fuzzy systems with bounded uncertain delays, IEEE Transactions on Fuzzy Systems 10 (1) (2002) 92–97. [34] B.Y. Zhang, S.S. Zhou, T. Li, A new approach to robust and non-fragile H1 control for uncertain fuzzy systems, Information Sciences 177 (22) (2007) 5118–5133. [35] Z.Q. Zuo, Y.J. Wang, Robust stability criteria of uncertain fuzzy systems with time-varying delays, IEEE International Conference on Systems, Man and Cybernetics (2005) 1303–1307.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays q Li Li a,b,*, Xiaodong Liu a a b
Research Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034, PR China
a r t i c l e
i n f o
Article history: Received 23 July 2006 Received in revised form 19 November 2008 Accepted 22 November 2008
Keywords: Takagi–Sugeno fuzzy control Delay-dependent Linear matrix inequality State and input delays
a b s t r a c t This paper focuses on the delay-dependent stability analysis and stabilization for uncertain T–S fuzzy control systems with state and input delays. The key features of the approach include the introduction of uncorrelated augmented matrix items into the Lyapunov functional and the use of a tighter bounding technology. In fact, these techniques lead to more general and less conservative stability conditions that guarantee a wide stability region. Our delay-dependent stability conditions thus obtained are given in terms of linear matrix inequalities (LMIs). Finally, some examples are included to show that the proposed criteria improve the existing results significantly. Ó 2008 Published by Elsevier Inc.
1. Introduction Recently, Takagi–Sugeno (T–S) [22,23] fuzzy model has attracted considerable attention because it can combine the flexibility of fuzzy logic theory and rigorous mathematical theory of linear or nonlinear system into a unified framework. It gives a way to smoothly connect local linear systems to form global nonlinear systems by fuzzy membership functions. Researches on the properties, especially stability problems of the fuzzy systems, have become a very active area, e.g., [1,2,14–16,20,24– 27,34]. Time-delay phenomena are often encountered in various practical systems, such as AIDS epidemic, aircraft stabilization, chemical engineering systems, distributed networks, inferred grinding model, manual control, microwave oscillator, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines. In practical systems, analysis of a mathematical model is usually an important work for a control engineer as to control a system. However, the mathematical model always contains some uncertain elements. Therefore, under such imperfect knowledge of the mathematical model, seeking to design a robust control such that the system responses can meet desired properties is an important topic in system theory. Hence robust stability and robust stabilization problems for time-delay system have received some attenuation. Many criteria for checking the stability of time delay fuzzy systems have been derived, e.g., [5–7,10,12,33,35] and references therein. The existing results can be classified into two types: delay-independent stability [10,21,33], delay-dependent stability [5–7,12,35], etc.; the former does not include any information on the size of delay while the latter employs such information. It is known that delay-dependent stability conditions are generally less
q This work is supported in part by the National Natural Science Foundation of China under Grant 60575039 60534010 and in part by the National Key Basic Research and Development Program of China under Grant 2002CB312201-06. * Corresponding author. Address: Research Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China. E-mail address: [email protected] (L. Li).
0020-0255/$ - see front matter Ó 2008 Published by Elsevier Inc. doi:10.1016/j.ins.2008.11.039
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
1135
conservative than delay-independent ones especially when the size of the delay is small. Recently, delay-dependent stabilization is discussed in [5–7,12,32] for T–S fuzzy time-delay systems based on Lyapunov–Krasovksii functional approach. The delay is assumed to be constant. A state feedback control scheme has been proposed in terms of the feasible solutions to LMIs. In [13,30], the delay dependent stability analysis and control synthesis have been carried out by using Lyapunov–Krasovksii functional approach for uncertain T–S fuzzy systems with time-varying delay. In [31], the delay-dependent stability analysis and stabilization problems is researched for continuous-time T–S fuzzy systems with a time-varying delay. A fuzzy Lyapunov–Krasovksii functional is employed to derive the proposed delay-dependent results. However, all the aforementioned results are proposed for T–S fuzzy systems which only contain state delay. In modern industrial systems, sensors, controllers and plants are often connected over a network medium. Since the sampling data and controller signals are transmitted through a network, control input delays in control systems are always inevitable. So, it is important both in theory and in practice to consider the input delay when controllers are designed. Although the input delay is a technically important issue of frequent occurrence, few related control strategies seem to be available except for [8,17–19]. It remains yet to be a theoretically challenging issue, and thereby it is more important and more realistic to study the T–S fuzzy model with state delay and input delay. In [17], the authors have studied the delay systems without uncertainty, in which the state delay is equal to the input delay. In [8,18], the fuzzy systems with input delay is researched without state delay. And the input delay is constant delay in [8]. In [19], the uncertain fuzzy systems with state and input delay is considered. Based on Lyapunov–Razumikhin functional approaches, the delay-dependent LMI conditions for stability and stabilization have been developed in [17,18]. But the obtained results of [17,18] using the Lyapunov–Razumikhin functional approaches are usually more conservative than those using the Lyapunov–Krasovskii functional approaches since the Lyapunov–Krasovskii functional approaches take the advantage of the additional information of the delay. So the obtained results are more conservative. And in [8,19], the delay-dependent stability conditions are obtained with similar methods which are Lyapunov– Krasovskii functional approaches and free weighting matrix method. In this paper, we construct a new Lyapunov–Krasovskii functional, which gives the additional design matrix one more potential relaxation than the Lyapunov–Krasovskii functional methods of [8,19]. Furthermore, a tighter bounding technology for cross terms and the free weighting matrix approach are used to reduce the conservation. Some delay-dependent sufficient conditions are obtained to stabilize the uncertain fuzzy systems with time-varying delay in both state and input. Examples show that our stability conditions are less conservative than other conditions in the existing literatures, e.g. [7,8,19,30,31] etc. This paper is organized as follows: In Section 2, the T–S fuzzy model with time-delays is first formulated. We present some sufficient conditions for state feedback controller design by using PDC in Section 3. Section 4 provides some illustrative examples to demonstrate the effectiveness of proposed methods. Finally, the conclusions are drawn in Section 5. 2. Problem formulation and preliminaries We consider a nonlinear fuzzy system with state and input delays, which is represented by a Takagi–Sugeno (T–S) fuzzy model [22] composed of a set of fuzzy implications. Each implication is expressed by a linear time-delay system and the ith rule of the T–S model is written in the following form: Plant Rule i: IF h1 ðtÞ is Mi1 ; . . ., and hp ðtÞ is Mip , THEN
_ xðtÞ ¼ ðAi þ DAi ÞxðtÞ þ ðAid þ DAid Þxðt s1 ðtÞÞ þ ðBid þ DBid Þuðt s2 ðtÞÞ; 1 ; s 2 Þ; 0; i ¼ 1; 2; . . . ; r; xðtÞ ¼ uðtÞ; t 2 ½ maxðs
ð1Þ
where M ij is the fuzzy set, xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input. Ai ; Aid 2 Rnn ; Bid 2 Rnm ; and h1 ðtÞ; h2 ðtÞ; ; hp ðtÞ are the premise variables. It is assumed that the premise variables do not depend on the input uðtÞ. The time delays in state and input, i.e., s1 ðtÞ and s2 ðtÞ; are time-varying continuous functions satisfying 0 6 si ðtÞ 6 si < 1; s_ i ðtÞ 6 li < 1; ði ¼ 1; 2Þ 8t P 0; where si and li are constants. The matrices DAi ðtÞ; DAid ðtÞ; and DBid ðtÞ denote the uncertainties in system and they are of the forms
DAi ðtÞ ¼ Di FðtÞEia ;
DAid ðtÞ ¼ Di FðtÞEida ;
DBid ðtÞ ¼ Di FðtÞEidb ;
where Di ; Eai ; Eida ; Eidb are known constant matrices and FðtÞ are unknown real time-varying matrices with Lebesgue measurable elements bounded by F T ðtÞFðtÞ 6 I: For the simplicity, let us introduce the following notations:
Ai ¼ Ai þ DAi ðtÞ;
Aid ¼ Aid þ DAid ðtÞ;
Bid ¼ Bid þ DBid ðtÞ:
ð2Þ
By using center-average defuzzifer, product inference and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model:
Pr _ xðtÞ ¼
i¼1 i ðhðtÞÞ½Ai xðtÞ
þ Aid xðt s1 ðtÞÞ þ Bid uðt s2 ðtÞÞ Pr ; i¼1 i ðhðtÞÞ
ð3Þ
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L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
where hðtÞ ¼ ½h1 ðtÞ; . . . ; hp ðtÞT ; i ðhðtÞÞ : Rp ! ½0; 1 ði ¼ 1; 2; . . . ; rÞ are the membership functions of the system with respect r P i ðhðtÞÞ. In this paper, we assume that i ðhðtÞÞ P 0 for i ¼ 1; 2; . . . ; r and to the ith plant rule, and hi ðhðtÞÞ ¼ i ðhðtÞÞ i¼1 Pr Pr i¼1 i ðhðtÞÞ > 0 for all t. Therefore, hi ðhðtÞÞ P 0 ði ¼ 1; 2; . . . ; rÞ and i¼1 hi ðhðtÞÞ ¼ 1 for all t. Then, (3) can be represented by
_ xðtÞ ¼
r X
hi ðhðtÞÞ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ Bid uðt s2 ðtÞÞ:
ð4Þ
i¼1
Based on the parallel distributed compensation (PDC), the following fuzzy control law is employed to deal with the problem of stabilization via state feedback. Control Rule i: IF h1 ðtÞ is Mi1 ; . . . ; and hp ðtÞ is Mip ; then
uðtÞ ¼ K i xðtÞ;
i ¼ 1; 2; . . . ; r:
ð5Þ
Hence, the overall fuzzy control law is represented by
uðtÞ ¼
r X
hi ðhðtÞÞK i xðtÞ i ¼ 1; 2; . . . ; r;
ð6Þ
i¼1
where K i ði ¼ 1; 2; . . . ; rÞ are the local control gains. When there exists an input delay s2 ðtÞ, we have uðt s2 ðtÞÞ ¼ Pr i¼1 hi ðhðt s2 ðtÞÞÞK i xðt s2 ðtÞÞ: So, it is natural and necessary to make an assumption that the functions 2 ; 0, and satisfy the following properties: hi ðhðt s2 ðtÞÞÞ P 0 hi ðhðtÞÞði ¼ 1; 2; . . . ; rÞ are well defined for all t 2 ½s P ði ¼ 1; 2; . . . ; rÞ; and ri¼1 hi ðhðt s2 ðtÞÞÞ ¼ 1: For the simplicity, let us denote: d:t:
hi ¼ hi ðhðtÞÞ;
s d:t:
hi 2 ¼ hi ðhðt s2 ðtÞÞÞ:
The design of the fuzzy controller is to determine the feedback gains K i ði ¼ 1; 2; . . . ; rÞ such that the resulting closed-loop system is asymptotically stable. With the control law (6), the overall closed-loop system can be written as
8 r r P r P P > s < xðtÞ _ hi hj 2 Bid K j xðt s2 ðtÞÞ; ¼ hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ i¼1 j¼1 j¼1 > : 1 ; s 2 Þ; 0; i; j ¼ 1; 2; . . . ; r: xðtÞ ¼ uðtÞ; t 2 ½ maxðs
ð7Þ
The following lemmas are useful in the proofs of our main results. Lemma 1 [11]. For any positive symmetric constant matrix M 2 Rnn and a scalar c > 0, if there exists a vector function Rc Rc w : ½0; c ! Rn such that the integrals 0 wT ðsÞMwðsÞ ds and 0 wðsÞ ds are well defined, then the following inequality holds:
c
Z
c
wT ðsÞMwðsÞ ds P
Z
0
c
Z wT ðsÞ ds M
0
c
wðsÞ ds :
ð8Þ
0
Applying Lemma 1 and the methods of [9], the following Lemma can be obtained, which is a much tighter bounding technology for cross terms and makes the results derived less conservative. Q 11 Q 12 , and vector P 0, scalar sðtÞ 6 s Lemma 2. For any constant matrices Q 11 ; Q 22 ; Q 12 2 Rnn ; Q 11 P 0; Q 22 P 0; Q 22 ; 0 ! Rn such that the following integration is well defined, then function x_ : ½s
s
Z
t
xT ðsÞ x_ T ðsÞ
Q 11
ts
Q 12 Q 22
2 3T 2 xðtÞ Q 22 xðsÞ 6 7 6 ds 6 4 xðt sðtÞÞ 5 4 Q 22 _ Rt xðsÞ xðsÞ ds Q 12 tsðtÞ
Q 22 Q 22 Q 12
Q T12
32
xðtÞ
3
7 76 Q T12 54 Rxðt sðtÞÞ 5: t xðsÞ ds Q 11 tsðtÞ
Lemma 3 [28]. Given constant matrices Q ; H; and E of appropriate dimensions and with symmetric matrix Q, then
Q þ HFðtÞE þ ET F T ðtÞHT < 0; for all FðtÞ satisfying F T ðtÞFðtÞ 6 I; if and only if there exists a scalar
e > 0 such that
Q þ eHHT þ e1 ET E < 0: 3. Stabilization results In this section, we focus on the problems of developing some delay-dependent stability criteria.
ð9Þ
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
1137
3.1. Stability of nominal fuzzy system First we consider the control design of a state feedback control law that stabilizes the following nominal fuzzy time-delay system:
8 r r P r P P > s < xðtÞ _ hi hj 2 Bid K j xðt s2 ðtÞÞ; ¼ hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ i¼1 i¼1 j¼1 > : 1 ; s 2 Þ; 0; i; j ¼ 1; 2; . . . ; r: xðtÞ ¼ uðtÞ; t 2 ½ maxðs
ð10Þ
Next, some delay-dependent criteria are proposed for the T–S fuzzy system (10). i , li ði ¼ 1; 2Þ and di ði ¼ 2; 3; 4Þ, the closed-loop system (10) is asymptotically stable for any Theorem 1. Given scalars s i ði ¼ 1; 2Þ via PDC technique if there exist P11 > 0, P 22 > 0; R11 > 0, R22 > 0; Q 11 > 0, Q 22 > 0; S11 > 0, 0 6 si ðtÞ 6 s S22 > 0; R1 > 0, R2 > 0 and real matrices X; P 12 ; R12 ; Q 12 ; S12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
Cii < 0;
i ¼ 1; 2; . . . ; r;
Cij þ Cji 6 0; "
P11
P12
P22
ð11Þ
1 6 i 6 j 6 r;
#
" P 0;
R11
R12
R22
ð12Þ #
" P 0;
Q 11
Q 12
Q 22
#
" P 0;
S11
S12
S22
# P 0:
ð13Þ
where
2 6 6 6 6 Cij ¼ 6 6 6 6 6 4
Hij ð1; 1Þ Hij ð1; 2Þ
PT22 Q T12
Hij ð1; 4Þ
Hij ð2; 2Þ a1 PT22 þ Q T12 Hij ð2; 4Þ
Q 11
0
RT22 ST12 0
3
7 7 7 7 7; Hij ð4; 6Þ 7 7 7 RT12 5
Hij ð2; 6Þ 7 PT12
0 a2 RT22
Hij ð1; 6Þ
þ ST12
Hij ð4; 4Þ
S11
Hij ð6; 6Þ
Hij ð1; 1Þ ¼ P12 þ PT12 þ R12 þ RT12 þ R1 þ R2 þ s21 Q 11 þ s22 S11 Q 22 S22 þ Ai X T þ XATi ; Hij ð1; 2Þ ¼ a1 P 12 þ Q 22 þ Aid X T þ d2 XATi ; Hij ð1; 4Þ ¼ a2 R12 þ S22 þ Bid K j þ d3 XATi ; Hij ð1; 6Þ ¼ P11 þ R11 þ s21 Q 12 þ s22 S12 þ d4 XATi X T ; Hij ð2; 2Þ ¼ a1 R1 Q 22 þ d2 Aid X T þ d2 XATid ; Hij ð2; 4Þ ¼ d2 Bid K j þ d3 XATid ; Hij ð2; 6Þ ¼ d4 XATid d2 X T ; Hij ð4; 4Þ ¼ a2 R2 S22 þ d3 Bid K j þ d3 K Tj BTid ; Hij ð4; 6Þ ¼ d4 K Tj BTid d3 X T ; Hij ð6; 6Þ ¼ s21 Q 22 þ s22 S22 d4 X d4 X T P 11 ¼ XP 11 X T ; P12 ¼ XP 12 X T ; P22 ¼ XP 22 X T ; R11 ¼ XR11 X T ; R12 ¼ XR12 X T ; R22 ¼ XR22 X T ; Q 11 ¼ XQ 11 X T ; Q 12 ¼ XQ 12 X T ; Q 22 ¼ XQ 22 X T ; S11 ¼ XS11 X T ; S12 ¼ XS12 X T ; S22 ¼ XS22 X T ; R1 ¼ XR1 X T ; R2 ¼ XR2 X T ; K j ¼ K j X T ; a1 ¼ ð1 l1 Þ; a2 ¼ ð1 l2 Þ; and ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by
K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð14Þ
Proof. Choose a delay-dependent Lyapunov–Krasovksii functional candidate as
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ þ V 6 ðtÞ
ð15Þ
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L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
with
V 1 ðtÞ ¼ gT1 ðtÞPg1 ðtÞ; V 2 ðtÞ ¼ gT2 ðtÞRg2 ðtÞ; Z t xT ðsÞR1 xðsÞ ds; V 3 ðtÞ ¼ ts1 ðtÞ t
Z
V 4 ðtÞ ¼
xT ðsÞR2 xðsÞ ds;
ts2 ðtÞ Z t
1 ÞÞnT ðsÞQ nðsÞ ds; ðs ðt s
1 V 5 ðtÞ ¼ s 2 V 6 ðtÞ ¼ s
ts Z t1
2 ÞÞnT ðsÞSnðsÞ ds; ðs ðt s
2 ts
where
g1 ðtÞ ¼ xT ðtÞ R¼
R11
R12
R22
R ;
t ts1 ðtÞ
T T ; xðsÞ ds
P¼
T nðsÞ ¼ xT ðsÞ x_ T ðsÞ ;
P 11
P12
P22
Q¼
;
g2 ðtÞ ¼ xT ðtÞ
Q 11
Q 12
Q 22
S¼
;
R
t ts2 ðtÞ
S11
S12
S22
T T ; xðsÞ ds
:
The derivative of V 1 ðtÞ, V 2 ðtÞ, V 3 ðtÞ, V 4 ðtÞ, V 5 ðtÞ; V 6 ðtÞ along the trajectory of (10) satisfies that
R
T P11 t V_ 1 ðtÞ ¼ 2gT1 ðtÞP g_ 1 ðtÞ ¼ 2 xT ðtÞ xðsÞds ts1 ðtÞ
P12 P22
R
T R11 t V_ 2 ðtÞ ¼ 2gT2 ðtÞP g_ 2 ðtÞ ¼ 2 xT ðtÞ xðsÞ ds ts2 ðtÞ
R12 R22
_ xðtÞ ; xðtÞ ð1 s_1 ðtÞÞxðt s1 ðtÞÞ
_ xðtÞ ; xðtÞ ð1 s_2 ðtÞÞxðt s2 ðtÞÞ
V_ 3 ðtÞ ¼ xT ðtÞR1 xðtÞ ð1 s_ 1 ðtÞÞxT ðt s1 ðtÞÞR1 xðt s1 ðtÞÞð6 xT ðtÞR1 xðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞR1 xðt s1 ðtÞÞ; V_ 4 ðtÞ ¼ xT ðtÞR2 xðtÞ ð1 s_ 2 ðtÞÞxT ðt s2 ðtÞÞR2 xðt s2 ðtÞÞ 6 xT ðtÞR2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞR2 xðt s2 ðtÞÞ; Q 11 21 xT ðtÞ x_ T ðtÞ V_ 5 ðtÞ ¼ s
Q 12
S11 22 xT ðtÞ x_ T ðtÞ V_ 6 ðtÞ ¼ s
S12
Q 22
S22
Z t T Q 11 xðtÞ 1 s x ðsÞ x_ T ðsÞ _ xðtÞ 1 ts
Z t T S11 xðtÞ 2 s x ðsÞ x_ T ðsÞ _ xðtÞ 2 ts
Q 12 Q 22 S12 S22
xðsÞ ds; _ xðsÞ
xðsÞ ds: _ xðsÞ
It follows from (10) that
0 ¼ 2½xT ðtÞT 1 þ xT ðt s1 ðtÞÞT 2 þ xT ðt s2 ðtÞÞT 3 þ x_ T ðtÞT 4 ( ) r r X r X X s _ hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ hi hj 2 Bid K j xðt s2 ðtÞÞ xðtÞ i¼1
i¼1
j¼1
i.e.
2
0¼2
r X r X i¼1
j¼1
T1
3
6T 7 6 27 6 7 607 s2 T 7 hi hj n ðtÞ6 6 T 7 ½ Ai 6 37 6 7 405 2
T4 T 1 Ai
Aid
0 Bid K j
0 I nðtÞ
3
T 1 Aid
0 T 1 Bid K j
0 T 1
6 T 2 Ai 6 6 r r XX 6 0 s2 T hi hj n ðtÞ6 ¼2 6T A 6 3 i i¼1 j¼1 6 4 0
T 2 Aid
0 T 2 Bid K j
0 T 2 7 7 7 0 0 7 7nðtÞ: 0 T 3 7 7 7 0 0 5
T 4 Ai
T 4 Aid
0 T 3 Aid 0
0
0
0 T 3 Bid K j 0
0
0 T 4 Bid K j
0 T 4
ð16Þ
1139
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
From Lemma 2 and
1 s
Z
t
Q 11
Q 12 Q 22
xT ðsÞ x_ T ðsÞ
Q 11
2 s
t
xT ðsÞ x_ T ðsÞ
S11
2 ts
S11
Q 12
S12 S22
Q 22
1 ts
Z
P 0,
S12
S22
P 0, we have
2 3T 2 xðtÞ Q 22 xðsÞ 6 xðt s1 ðtÞÞ 7 6 ds 6 4 5 4 Q 22 _ Rt xðsÞ xðsÞ ds Q 12 ts1 ðtÞ
2 3T 2 xðtÞ S22 xðsÞ 6 7 6 ds 6 4 xðt s2 ðtÞÞ 5 4 S22 _ Rt xðsÞ xðsÞ ds S12 ts2 ðtÞ
Q 22 Q 12
ST12
S22 S22 S12
32
3 xðtÞ 7 6 7 Q T12 54 Rxðt s1 ðtÞÞ 5; t xðsÞ ds Q 11 ts1 ðtÞ Q T12
Q 22
32
xðtÞ
3
7 76 ST12 54 Rxðt s2 ðtÞÞ 5: t xðsÞ ds S11 ts2 ðtÞ
By the above inequalities, it is obtained that
T P11 P12 _ xðtÞ xðsÞds 1 ðtÞ P22 xðtÞ ð1 s_1 ðtÞÞxðt s1 ðtÞÞ R
T R11 R12 _ xðtÞ t þ xT ðtÞR1 xðtÞ þ 2 xT ðtÞ xðsÞds ts2 ðtÞ R22 xðtÞ ð1 s_2 ðtÞÞxðt s2 ðtÞÞ
R t _ VðtÞ 6 2 xT ðtÞ ts
ð1 l1 ÞxT ðt s1 ðtÞÞR1 xðt s1 ðtÞÞ þ xT ðtÞR2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞR2 xðt s2 ðtÞÞ Q 11 Q 12 xðtÞ S11 S12 xðtÞ 22 xT ðtÞ x_ T ðtÞ 21 xT ðtÞ x_ T ðtÞ þs þs _ _ Q 22 xðtÞ S22 xðtÞ 2 3T 2 3 T 32 xðtÞ xðtÞ Q 22 Q 22 Q 12 76 6 7 6 7 þ 4 xðt s1 ðtÞÞ 5 4 Q 22 Q 22 Q T12 54 xðt s1 ðtÞÞ 5 Rt Rt xðsÞ ds xðsÞ ds Q 12 Q 12 Q 11 ts1 ðtÞ ts1 ðtÞ 3 2 3T 2 2 3 xðtÞ xðtÞ S22 S22 ST12 76 6 7 6 7 þ 4 xðt s2 ðtÞÞ 5 4 S22 S22 ST12 54 xðt s2 ðtÞÞ 5 þ 2½xT ðtÞT 1 þ xT ðt s1 ðtÞÞT 2 þ xT ðt s2 ðtÞÞT 3 Rt Rt xðsÞ ds xðsÞ ds S12 S12 S11 ts2 ðtÞ ts2 ðtÞ r r r X XX s _ þ x_ T ðtÞT 4 f hi ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ hi hj 2 Bid K j xðt s2 ðtÞÞ xðtÞg i¼1
¼
r X r X i¼1
i¼1
j¼1
s
hi hj 2 nT ðtÞFij nðtÞ;
ð17Þ
j¼1
where
R
T t nT ðtÞ ¼ xT ðtÞ xT ðt s1 ðtÞÞ xðsÞds ts1 ðtÞ 2 6 6 6 6 6 Fij ¼ 6 6 6 6 6 4
Kij ð1; 1Þ Kij ð1; 2Þ
P T22 Q T12
Kij ð2; 2Þ a1 PT22 þ Q T12
xT ðt s2 ðtÞÞ ð
Rt
xðsÞdsÞT ts2 ðtÞ
Kij ð1; 4Þ
RT22 ST12
T 2 Bid K j þ ATid T T3
0
0
Q 11
Kij ð4; 4Þ
0 a2 RT22
þ ST12
S11
x_ T ðtÞ ;
Kij ð1; 6Þ
7 ATid T T4 T 2 7 7 7 7 PT12 7; 7 T T T K j Bid T 4 T 3 7 7 7 RT12 5 Kij ð6; 6Þ
and
Kij ð1; 1Þ ¼ P12 þ PT12 þ R12 þ RT12 þ R1 þ R2 þ s21 Q 11 þ s22 S11 Q 22 S22 þ T 1 Ai þ ATi T T1 ; Kij ð1; 2Þ ¼ ð1 l1 ÞP12 þ Q 22 þ T 1 Aid þ ATi T T2 ; Kij ð1; 4Þ ¼ ð1 l2 ÞR12 þ S22 þ T 1 Bid K j þ ATi T T3 ; Kij ð1; 6Þ ¼ P11 þ R11 þ s21 Q 12 þ s22 S12 þ ATi T T4 T 1 ; Kij ð2; 2Þ ¼ ð1 l1 ÞR1 Q 22 þ T 2 Aid þ ATid T T2 ; Kij ð4; 4Þ ¼ ð1 l2 ÞR2 S22 þ T 3 Bid K j þ K Tj BTid T T3 ; Kij ð6; 6Þ ¼ s21 Q 22 þ s22 S22 T T4 T 4 ; a1 ¼ ð1 l1 Þ; a2 ¼ ð1 l2 Þ:
3
ð18Þ
1140
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
Denote T 2 ¼ d2 T 1 ; T 3 ¼ d3 T 1 ; T 4 ¼ d4 T 1 . Pre and post-multiplying both side of (18) with diag ½X; X; X; X; X; X and their transT T T T T T pose, defining new variables X ¼ T 1 1 ; P 11 ¼ XP 11 X ; P 12 ¼ XP 12 X ; P 22 ¼ XP 22 X ; R11 ¼ XR11 X ; R12 ¼ XR12 X ; R22 ¼ XR22 X ; T T T T T T T T Q 11 ¼ XQ 11 X ; Q 12 ¼ XQ 12 X ; Q 22 ¼ XQ 22 X ; S11 ¼ XS11 X ; S12 ¼ XS12 X ; S22 ¼ XS22 X ; R1 ¼ XR1 X ; R2 ¼ XR2 X ; K j ¼ K jXT , _ < 0. Therefore the asymptotic staðj ¼ 1; 2; . . . ; rÞ: The obtained result is equivalent to (11) and (12). So, if Fij < 0, then VðtÞ bility of system (10) is established. The proof is completed. Remark 1. It is worth mentioning that a much tighter bounding technology for cross terms is adopted in the proof of Theorem 1. Lemma 2 is a more general and tighter bounding technology for dealing with cross terms than Lemma 1. However, in the previous results, Lemma 1 is often used. Therefore, the stability criteria derived here are less conservative. It can been shown in the examples of Section 4. Rt 2 t Remark 2. For the constructed Lyapunov–Krasovskii functional items (15), the cross terms, e.g., s s2 ðs T 2 ÞÞx ðsÞS12 xðsÞ _ ds etc., are introduced into the Lyapunov–Krasovskii functional. Therefore, the additional design matrix ðt s gives a potential relaxation; and we can expect less conservative results. It can be seen that some existing results ([8,19], etc.) are special cases of our criteria. The methods in [8] are based on (15) with P ¼ 0; R1 ¼ 0; Q ¼ 0; S12 ¼ 0: And the methods in [19] are also based on (15) with P 12 ¼ 0; P22 ¼ 0; R ¼ 0; Q 12 ¼ 0; Q 22 ¼ 0; S12 ¼ 0; S22 ¼ 0. This implies that our Lyapnov– Krasvskii functional is more generalized and includes more weighting matrices than those in [8,19]. These lead to less conservatism as the illustrative examples shown in Section 4. Moreover, a much tighter bounding technology for cross terms is employed here. These make the stability criteria derived here less conservative. Remark 3. Notice that Theorem 1 is not applicable for li ¼ 0 (i=1,2) or li P 1. As for li ¼ 0, the time-delay is time-invariant, i.e. si ðtÞ ¼ hi (hi are constant). When li is unknown, let P 12 ¼ 0; P22 ¼ 0; R12 ¼ 0; R22 ¼ 0; R1 ¼ 0; R2 ¼ 0 and we can directly obtain a rate-independent criterion from Theorem 1. In what follows, Theorem 1 is developed to a more relaxed condition for the systems (10) using the techniques in [29]. i , li ði ¼ 1; 2Þand di ði ¼ 2; 3; 4Þ, the closed-loop system (10) is asymptotically stable for any Theorem 2. Given scalars s i ði ¼ 1; 2Þ via PDC technique if there exist P 11 > 0, P22 > 0; R11 > 0, R22 > 0; Q 11 > 0, Q 22 > 0; S11 > 0, 0 6 si ðtÞ 6 s S22 > 0; R1 > 0, R2 > 0, T ijk P 0; i; j; k ¼ 1; 2; . . . r; i < j and real matrices X; P12 ; R12 ; Q 12 ; S12 ; K j ; Hij ; V ijk ði; j; k ¼ 1; . . . rÞ satisfying (13) and the following LMIs:
2
R1 Z 1k R12k 6 R R2 Z 2k 2rk 6 6 6 4
.. .
.. .
Rr1k
Rr2k
..
R1rk R2rk
.
.. .
3 7 7 7 < 0; 7 5
ð19Þ
Rr Z rk
where for i; j; k ¼ 1; . . . ; r
Ri ¼ Cii ; 8 > < Hjk Z jk ¼ Hkj > : 0 (
Rijk ¼
if j < k; if j > k; if j ¼ k
Cij þ T ijk þ ðV ijk V Tijk Þ þ 12 W ijk if i < j; Cji þ T jik þ ðV jik V Tjik Þ þ 12 W jik if i > j;
W ijk ¼
Hik
if i ¼ k or j ¼ k;
0
if i–k or j–k;
i ði ¼ 1; 2Þ via PDC technique. In this case, the and Cij , i; j ¼ 1; ; r are given by Theorem 1. Then (10) is stable for any 0 6 si ðtÞ 6 s stabilizing control law is given by (14). Proof. The proof of the theorem is similar to that of Theorem 5 in [29] and it is thus omitted. Let
s1 ðtÞ ¼ s2 ðtÞ ¼ sðtÞ; the fuzzy model (10) is reduced to the following model: _ xðtÞ ¼
r X i¼1
hi ½Ai xðtÞ þ Aid xðt sðtÞÞ þ
r X r X i¼1
s
hi hj Bid K j xðt sðtÞÞ;
j¼1
< 1; s_ ðtÞ 6 l < 1 8t P 0: For this fuzzy system, we have the following Corollary of Theorem 1. where 0 6 sðtÞ 6 s
ð20Þ
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
1141
and di ði ¼ 2; 3Þ, the closed-loop system (20) is asymptotically stable for any 0 6 sðtÞ 6 s via PDC Corollary 1. Given scalars s technique if there exist P 11 > 0, P22 > 0; Q 11 > 0, Q 22 > 0; R1 > 0 and real matrices X; P 12 ; Q 12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
Wii < 0; i ¼ 1; 2; . . . ; r; Wij þ Wji 6 0; 1 6 i 6 j 6 r; "
P 11
P12
P22
#
"
P 0;
ð21Þ
Q 11
Q 12
Q 22
ð22Þ
# P 0:
ð23Þ
where
2 6 6 4
ij ð1; 1Þ ij ð1; 2Þ
ij ð2; 2Þ aPT22 þ Q T12
Wij ¼ 6
P T22 Q T12
Q 11
ij ð1; 4Þ
3
ij ð2; 4Þ 7 7 7; 5 PT 12
ij ð4; 4Þ
2 Q 11 Q 22 þ Ai X T þ XATi ; ij ð1; 1Þ ¼ P 12 þ PT12 þ R1 þ s ij ð1; 2Þ ¼ aP12 þ Q 22 þ Aid X T þ Bid K j þ d2 XATi ; 2 Q 12 þ d3 XATi X T ; ij ð1; 4Þ ¼ P 11 þ s ij ð2; 2Þ ¼ aR1 Q 22 þ d2 Aid X T þ d2 Bid K j þ d2 XATid þ d2 K Tj BTid ; ij ð2; 4Þ ¼ d3 XATid þ d3 K Tj BTid d2 X T ; 2 Q 22 d3 X d3 X T ; ij ð4; 4Þ ¼ s P 11 ¼ XP11 X T ; P 12 ¼ XP 12 X T ; P22 ¼ XP 22 X T ; Q 11 ¼ XQ 11 X T ; Q 12 ¼ XQ 12 X T ; Q 22 ¼ XQ 22 X T ; R1 ¼ XR1 X T ; K j ¼ K j X T ,a ¼ ð1 lÞ; and ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by
K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð24Þ
Proof. We Choose a Lyapunov–Krasovksii functional candidate as
Vðxt ; tÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ
ð25Þ
with
V 1 ðtÞ ¼ gT1 ðtÞPg1 ðtÞ; Z t V 2 ðtÞ ¼ xT ðsÞR1 xðsÞ ds; tsðtÞ
V 3 ðtÞ ¼ s
Z
t
ÞÞnT ðsÞQnðsÞ ds; ðs ðt s
ts
and P; Q are the same as the Theorem 1. Because the Corollary 1 is a special case of Theorem 1. From the proof of Theorem 1, we can easily obtain the proof of Corollary 1. The detailed proof is omitted. Remark 4. Yoneyama [32] is not concerned with the input delay, it considers the control design with the delayed feedback. This leads to the closed-loop system having the same structure as that of this paper. However, in [32], input delay is equal to state delay and the delays are constant delays. Therefore, the stability problems of [32] is a special case of this paper. Moreover, let s1 ðtÞ ¼ s2 ðtÞ ¼ s and P12 ¼ 0; P 22 ¼ 0; Q 12 ¼ 0; Q 22 ¼ 0 in (25) of this paper, the Lyapunov–Krasovskii functional of [32] is obtained. This implies that our Lyapunov–Krasovskii functional is more generalized than that in [32]. This leads to less conservative results. If there is no input delay terms (i.e., s2 ðtÞ ¼ 0), the considered fuzzy model (10) is reduced to the same one as those studied in [6,10] etc.
_ xðtÞ ¼
r X r X i¼1
hi ðhÞhj ðhÞ½Ai xðtÞ þ Aid ðt s1 ðtÞÞ þ Bid K j xðtÞ:
j¼1
In this case, the following Corollary is easily obtained from Theorem 1.
ð26Þ
1142
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
1 , l1 and di ði ¼ 2; 3Þ, the closed-loop system (26) is asymptotically stable for any 0 6 s1 ðtÞ 6 s 1 via Corollary 2. Given scalars s PDC technique if there exist P 11 > 0, P 22 > 0; Q 11 > 0, Q 22 > 0; R1 > 0 and real matrices X; P12 ; Q 12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
Xii < 0; i ¼ 1; 2; . . . ; r; Xij þ Xji 6 0; 1 6 i 6 j 6 r; "
P11
P12
P22
#
"
P 0;
ð27Þ
Q 11
Q 12
Q 22
ð28Þ
# P 0;
ð29Þ
where
2 6 6 6 Xij ¼ 6 6 4
Uij ð1; 1Þ Uij ð1; 2Þ
PT22 Q T12
Uij ð1; 4Þ
3 7
Uij ð2; 2Þ a1 PT22 þ Q T12 Uij ð2; 4Þ 7 7
7; 7 5
Q 11
PT12
Uij ð4; 4Þ
Uij ð1; 1Þ ¼ P12 þ P T12 þ R1 þ s21 Q 11 Q 22 þ Ai X T þ Bid K j þ XATi þ K Tj BTid ; Uij ð1; 2Þ ¼ a1 P12 þ Q 22 þ Aid X T þ d2 XATi þ d2 K Tj BTid ; Uij ð1; 4Þ ¼ P11 þ s21 Q 12 þ d3 XATi þ d3 K Tj BTid X T ; Uij ð2; 2Þ ¼ a1 R1 Q 22 þ d2 Aid X T þ d2 XATid ; Uij ð2; 4Þ ¼ d3 XATid d2 X T Uij ð4; 4Þ ¼ s21 Q 22 d3 X d3 X T ; P11 ¼ XP11 X T ; P 12 ¼ XP12 X T ; P 22 ¼ XP 22 X T ; Q 11 ¼ XQ 11 X T ; Q 12 ¼ XQ 12 X T ; Q 22 ¼ XQ 22 X T ; R1 ¼ XR1 X T ; K j ¼ K j X T ,a1 ¼ ð1 l1 Þ; and ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by
K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð30Þ
Remark 5. Chen and coworkers [3,4,6,7,10,12] have considered the robust control for uncertain Takagi–Sugeno (T–S) fuzzy systems with state delays. Some delay-dependent conditions are obtained with a traditional Lyapunov–Krasovskii functional method in [3,4,6,7,10,12]. However, these results can be derived based on (15) with R ¼ 0; R2 ¼ 0; S ¼ 0; P12 ¼ 0; P 22 ¼ 0; Q 12 ¼ 0; Q 22 ¼ 0: Moreover, we employ free weighting matrix method. These lead to the stability criteria derived less conservative than them. 3.2. Robust stability of uncertain fuzzy systems This section considers a problem of robust stablization of the uncertain T–S fuzzy system (7) using state feedback control rule (6), which is a more general stability problem of fuzzy system than the research problem of [7,8,12,30] etc. i , li ði ¼ 1; 2Þ and di ði ¼ 2; 3; 4Þ, the closed-loop system (7) is asymptotically stable for any Theorem 3. Given scalars s i ði ¼ 1; 2Þ via PDC technique if there exist P 11 > 0; P 22 > 0; R11 > 0; R22 > 0; Q 11 > 0, Q 22 > 0; S11 > 0; 0 6 si ðtÞ 6 s S22 > 0; R1 > 0, R2 > 0, eij > 0 ði; j ¼ 1; ; rÞ and real matrices X; P12 ; R12 ; Q 12 ; S12 ; K j ðj ¼ 1; 2; . . . ; rÞ satisfying the following LMIs:
2
Cii eii CDi
6 4 2
eii I
Cij eij CDi
6 4
eij I
CTEii
3
7 0 5 < 0; eii I
CTEij
3
2
i ¼ 1; 2; . . . ; r;
Cji eji CDj
7 6 0 5þ4
eij I
eji I
CTEji
ð31Þ 3
7 0 5 6 0;
eji I
1 6 i 6 j 6 r;
ð32Þ
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
"
P11
P12
P22
#
" P 0;
R11
R12
R22
#
" P 0;
Q 11
Q 12
Q 22
#
" P 0;
S11
S12
S22
1143
# P 0;
ð33Þ
where
CTDi ¼ DTi d2 DTi 0 d3 DTi 0 d4 DTi ; CEij ¼ Eai X T Eida X T 0 Eidb K j 0 0 ; Cij is defined in Theorem 1. And ‘‘” denotes the transposed elements in the symmetric positions. Moreover, the control gain matrix K j is given by K j ¼ K j ðX T Þ1 ;
j ¼ 1; 2; . . . ; r:
ð34Þ
Proof. Replacing Ai ; Aid and Bid in Cij of Theorem 1 with Ai þ Di FðtÞEai ; Aid þ Di FðtÞEida and Bid þ Di FðtÞEidb , respectively, we obtain (35) corresponding to the system (7):
Cij þ CDi FðtÞCEij þ CTEij F T ðtÞCTDi < 0;
i; j ¼ 1; . . . ; r:
ð35Þ
By Lemma 3, it is obtained T Cij þ CDi FðtÞCEij þ CTEij F T ðtÞCTDi 6 Cij þ eij CDi CTDi þ e1 ij CEij CEij ;
where
CTDi ¼ DTi d2 DTi 0 d3 DTi 0 d4 DTi ; CEij ¼ Eai X T Eida X T 0 Eidb K j 0 0 : By Schur complement, we obtain (31) and (32). The proof is completed.
Though the uncertain T–S fuzzy systems (7) is the same as the fuzzy systems of [19], our proposed stability conditions are less conservative than those of [19] because we use a new Lyapunov–Krasovskii functional and a tighter bounding technology. It can been seen in the example of Section 4. 4. Numerical examples In this section, four examples are presented to illustrate the effectiveness and merits of our results. We compare the previous results ([7,8,19,30,31] etc.) with our results in order to illustrate the effectiveness of our design methods. Example 1. Consider truck trailer system with time delay:
_ xðtÞ ¼
r X
hi ðhðtÞÞ½ðAi þ DAi ÞxðtÞ þ Aid xðt s1 Þ þ ðBid þ DBid Þuðt s2 Þ;
ð36Þ
i¼1
2
a Ltv t0
6 vt A1 ¼ 6 4 a Lt0 2 2 a v2Ltt0 2 A2d
0
0
3
2
0
7 07 5;
6 vt A2 ¼ 6 4 a Lt0 2 2 ad v2Ltt0
vt t0
0
ð1 aÞ Ltv t0
6 vt ¼6 4 ð1 aÞ Lt0 v 2t2 ð1 aÞ d2Lt 0
a Ltv t0
3
0
0
0
7 07 5;
dvt t0
0
3
2 vt 3
7 0 07 5; 0 0
6 07 B1d ¼ B2d ¼ 4 0 5;
0 0
2
ð1 aÞ Ltv t0
6 vt A1d ¼ 6 4 ð1 aÞ Lt0 2 2 ð1 aÞ v2Ltt0
0 0
3
7 0 07 5; 0 0
lt
0
T with v ¼ 1:0; t ¼ 2:0; t 0 ¼ 0:5; L ¼ 5:5; l ¼ 2:8; d ¼ 10 tp0 ; Di ¼ ½ 0:255 0:255 0:255 ; Eai ¼ ½ 0:1 0 0 ; Eidb ¼ 0:15 for i ¼ 1; 2. For simulation purpose, we simply choose h1 ðxÞ ¼ 1=ð1 þ expðx1 þ 0:5ÞÞ and h2 ðxÞ ¼ 1 h1 ðxÞ where xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ x3 ðtÞ T : In the following cases (a), (b), (c), we compare our proposed results with previous results such as [6,8,10,33].
(a) For the system (36), which is more general systems than those of [7,8,19,30,31], let a ¼ 0:7; d2 ¼ 0:4; d3 ¼ 0:1, d4 ¼ 1:3. 2 ¼ 0:55, the control gains are obtained as follows: 1 ¼ 0:1, s By using Theorem 3 with s
K 1 ¼ ½ 4:4456 1:0809 0:0347 ;
K 2 ¼ ½ 4:5117 1:1684 0:0374 :
1144
L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148 60
x1 x2 x3
50
State
40 30 20 10 0 −10 0
10
20
Time(Sec)
30
Fig. 1. The responses of the state of the system (36) with
40
s1 ¼ 0:1; s2 ¼ 0:55
30 25 20 15
u(t)
10 5 0 −5 −10 −15 −20
0
10
20
30
40
Time(Sec) Fig. 2. The responses of u(t) of the system (36) with
s1 ¼ 0:1; s2 ¼ 0:55.
For the system (36), here choose the initial functions uðtÞ ¼ ½4 12T . The simulation results are shown by Figs. 1,2, respectively. It can be seen that the fuzzy controller (6) proposed in this paper is effective for truck trailer system with state and input delays. ¼ 0:75; d3 ¼ 0:1 and d4 ¼ 2:5; the feedback (b) Let a ¼ 1, the systems (36) is the same as the fuzzy system of [8]. When s gain matrices are computed as
K 1 ¼ ½ 3:3219 0:2406 0:0025 ;
K 2 ¼ ½ 3:3272 0:2494 0:0026 :
2 ¼ 0:86. For the same fuzzy systems, in [8], the maximal allowAnd we obtain that the maximal allowable input delay is s ¼ 0:75, and the feedback gain matrices are computed as able input delay is s
K 1 ¼ ½ 3:4227 0:3535 0:0045 ;
K 2 ¼ ½ 3:5215 0:3617 0:0056
(c) Let Bid ¼ DAi ¼ DBid ¼ 0; a ¼ 0:7. Then the system (36) is reduced to the system which is the same as [6,10,33]. Applying Corollary 2, we compare our results with the existing results, and the corresponding results are shown as Table 1.In 1 ¼ 1:9685 in [6]. Then, in the the following, we compare the maximal allowable delay with [6]. The upper delay is s 1 ¼ 18:52 with d2 ¼ 0:1; d3 ¼ 0:5.From the cases (a)– same condition, we obtain that the maximal allowable delay is s (c), it can been seen our proposed stability conditions are always less conservation than the previous results.
Table 1 1 ¼ 0:5; d2 ¼ 0:1; d3 ¼ 0:5. The responding control gains obtained by different methods when s Methods
K1
In [10] In [33] In [6] Corollary 2
½ 62:8420 ½ 47:4174 ½ 17:6283 ½ 13:8540
K2 229:3535 19:6709 216:5310 58:6529 35:0250 4:6282 25:9117 2:9805
½ 61:5791 ½ 45:7796 ½ 17:3474 ½ 14:1456
224:8307 18:9802 206:5476 39:4999 34:2816 4:3444 27:4639 3:0065
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L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148
Example 2. Consider the nonlinear mass-spring-damper mechanical systems in [18]. The state-space is represented as follows:
# " x_ 1 ðtÞ 0:75x31 ðtÞ 0:5x1 ðtÞ þ 0:1ð1 þ 0:11ðtÞÞx2 ðtÞ þ uðt sðtÞÞ ; ¼ x_ 2 ðtÞ x1 ðtÞ
ð37Þ
where j1ðtÞj2 6 1. The system (37) has one nonlinear term 0:75x31 ðtÞ: Now by adopting fuzzy sets, the T–S fuzzy system of (37) can be constructed as follows Rule 1: Rule 2:
_ ¼ ðA11 þ DA11 ÞxðtÞ þ ðB1d þ DB1d Þuðt sðtÞÞ If x1 ðtÞ is about C1 ; then xðtÞ _ ¼ ðA12 þ DA12 ÞxðtÞ þ ðB2d þ DB2d Þuðt sðtÞÞwhere C1 ; C2 are defined in [18] and the If x1 ðtÞ is about C2 ; then xðtÞ associated matrices are given by
A11 ¼
0:5 0:1 1
0
;
" pffiffiffiffiffiffiffi # 0:1 ; D1 ¼ D2 ¼ 0
A12 ¼
1 0:1 1
0
;
DA11 ¼ DA12 ¼
pffiffiffiffiffiffiffi E1a1 ¼ E1a2 ¼ 0 0:1 ;
0 0:11ðtÞ 0
0
;
B1d ¼ B2d ¼
1 0
;
E1db ¼ E2db ¼ ½ 0 0 :
< 2:04 when d3 ¼ 0:03; d4 ¼ 1:7: In order to Applying Theorem 3 to the system (37), we achieve stability for all delay 0 < s show the advantages of our results, a comparison for the upper bound of input delay and feedback gains to guarantee the asymptotic stabilization of fuzzy system is shown in Table 2. We can conclude that Theorem 3 in this paper is less conservative than those of [18,19]. Example 3. Consider the fuzzy system with state delay in [12]
_ xðtÞ ¼
r X
hi ðhðtÞÞ½Ai xðtÞ þ Aid xðt s1 Þ þ Bi uðtÞ;
ð38Þ
i¼1
where
A1 ¼
0
1
0:1 2
A2 ¼
;
B1 ¼ B2 ¼ ½ 0 1 T ;
b¼
0
1
0:1 0:5 1:5b 0:01
p
;
A1d ¼ A2d ¼
0:1
0
0:1 0:2
;
;
and b is used to avoid system matrices being singular. The membership functions are set as follows:
h1 ðhðtÞÞ ¼
1
1 1 ; 1 þ exp½3ððx2 =0:5Þ ðp=2ÞÞ 1 þ exp½3ððx2 =0:5Þ þ ðp=2ÞÞ
h2 ðhðtÞÞ ¼ 1 h1 ðhðtÞÞ:
Here the system (38) is the same form as studied in Example 2 of [30]. Applying Corollary 2, the existing delay-dependent results are compared with our results, and the corresponding results are shown as the following Table 3. Table 2 A comparison for the upper bounds of input delay and feedback gains for (37) with d3 ¼ 0:03; d4 ¼ 1:7. Results
Upper bounds of state delay
Fuzzy control gains
[18] [19] This paper This paper This paper
0.4223(l unknown) 1.75 (l unknown) 1.97 (l unknown) 1:96 (l ¼ 0:1) 2.04 (l ¼ 0:5)
K1 K1 K1 K1 K1
¼ ½ 0:7623 0:5212 ¼ ½ 0:3622 0:299 ¼ ½ 0:4093 0:2844 ¼ ½ 0:4290 0:2840 ¼ ½ 0:3858 0:2604
K2 K2 K2 K2 K2
¼ ½ 0:3690 0:5362 ¼ ½ 0:3625 0:299 ¼ ½ 0:0706 0:0432 ¼ ½ 0:0317 0:1005 ¼ ½ 0:0390 0:0549
Table 3 The control gains and the maximal delay obtained by different methods with d2 ¼ 0:5; d3 ¼ 1:5. Methods
h2
K1
K2
In paper [17] In paper [12] In paper [30] In paper [7] Corollary 2
1.725 3.4385 8.0354 25.7865 27.96
½ 30:6571 10:0442 ½ 0:9390 0:3123 ½ 21:8139 55:0597 ½ 1:2141 0:8750 ½ 1:1916 0:0361
½ 30:6589 11:1803 ½ 0:3277 0:3636 ½ 21:8139 56:1183 ½ 1:2141 0:6202 ½ 1:1916 1:4591
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L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148 3
x1 x2
2.5
State
1.5
0.5
−0.5
−1.5
0
10
20
30
40
Time(Sec)
50
Fig. 3. The responses of the state of the fuzzy system (38) when
s ¼ 27:96.
0.5 0
u(t)
−1
−2
−3
−4
0
10
20
30
40
Time(Sec)
Fig. 4. Control input response of the fuzzy system (38) when
50
s ¼ 27:96.
The simulation results are depicted in Figs. 3 and 4 for (38). They show the response of the closed-loop system with maximal allowable delay s ¼ 27:56 under initial condition uðtÞ ¼ ½ 3 1 T . From Figs. 3,4, it can be clearly seen that the proposed conditions guarantees the asymptotic stability of the closed-loop system (38). Example 4. Consider the following T–S fuzzy model [31]:
_ xðtÞ ¼
r X
hi ðhðtÞÞ½Ai xðtÞ þ Aid xðt s1 ðtÞÞ þ Bi uðtÞ;
ð39Þ
i¼1
where
A1 ¼
0 0:6 0
1
;
A2 ¼
1 0 1 0
;
B1 ¼ B2 ¼
1 1
;
A1d ¼
0:5 0:9 0
2
;
A2d ¼
0:9
0
1
1:6
:
The membership functions are as follows:
h1 ðx1 ðtÞÞ ¼
1 ; 1 þ expð2x1 ðtÞÞ
h2 ðx1 ðtÞÞ ¼ 1 h1 ðx1 ðtÞÞ:
In [5,12], the maximum allowed delay is 0.1524 and 0.2302, respectively. In [31], based on Theorem 3 and Corollary 2, the maximum allowed delay is 0.2664 and 0.2574, respectively. However, applying to Corollary 2 of this paper, for d2 ¼ 0:2; d3 ¼ 1:45; the maximum allowed s can be obtained as 1.05. The aforementioned maximum allowed delays are summarized in Table 4. It can be seen from Table 4 that the proposed stabilization results in this paper are clearly less conservative than those in [5,12,31], which are based on a single Lyapunov–Krasovksii functional methods and a fuzzy Lyapunov–Krasovksii functional methods, respectively. Letting sðtÞ ¼ 1:05 and using Corollary 2 with the choice of d2 ¼ 0:2 and d3 ¼ 1:45, we can obtain the following feedback gain matrices:
K 1 ¼ ½ 32:6582 93:9701 ;
K 2 ¼ ½ 34:3946 98:5644 :
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L. Li, X. Liu / Information Sciences 179 (2009) 1134–1148 Table 4 Comparisons among various delay-dependent stabilization methods. Methods
Maximum allowed
Theorem 2 of [5] Theorem 1 of [12] Corollary 2 of [31] Theorem 3 of [31] Corollary 2 of this paper
0.1524 0.2302 0.2574 0.2664 1.05
5
s
x1 x2
4
State
3 2 1 0 −1 0
10
20
30
Time(Sec)
40
50
Fig. 5. The responses of the state of the fuzzy system (39).
Fig. 5 shows the control results for the system (39) under the initial condition ½2 1T ; t 2 ½1:050. From the simulation re ¼ 1:05. sults, we can see our designed controller is effective for (39) with the maximum allowed delay s From these examples, it is easy to find that our methods give larger delay upper bounds of si ðtÞði ¼ 1; 2Þ and lower feedback gains than the previous results. 5. Conclusion New classes of Lyapunov–Krasovksii functional and a tighter bounding technology are introduced to study the delaydependent stability for more general nonlinear fuzzy system with delays in input and state. The resulting criteria are less conservative than the existing ones. Numerical examples have been given to demonstrate the effectiveness and the merits of the proposed methods. Acknowledgement The authors would like to thank the anonymous referees for their constructive comments and suggestions, which have been very helpful in improving the paper. References [1] X.J. Ban, X.Z. Gao, X.L. Huang, A.V. Vasilakos, Stability analysis of the simplest Takagi–Sugeno fuzzy control system using circle criterion, Information Sciences 177 (20) (2007) 4387–4409. [2] A. Benzaouia, A.E. Hajjaji, M. Naib, Stabilization of a class of constrained fuzzy systems: a positive invariance approach, International Journal of Innovative Computing, Information and Control 2 (4) (2006) 749–760. [3] B. Chen, X.P. Liu, Reliable control design of fuzzy dynamic systems with time-varying delay, Fuzzy Sets and Systems 146 (3) (2004) 49–374. [4] B. Chen, X.P. Liu, Fuzzy guaranteed cost control for nonlinear systems with time-varying delay, IEEE Transactions on Fuzzy Systems 13 (2) (2005) 38– 249. [5] B. Chen, X.P. Liu, Delay-dependent robust H1 control for T–S fuzzy systems with time delay, IEEE Transactions on Fuzzy Systems 13 (2) (2005) 238– 249. [6] B. Chen, X.P. Liu, S.C. Tong, Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay, Fuzzy Sets and Systems 157 (16) (2006) 2224–2240. [7] B. Chen, X.P. Liu, S.C. Tong, New delay-dependent stabilization conditions of T–S fuzzy systems with constant delay, Fuzzy Sets and Systems 158 (20) (2007) 2209–2224. [8] B. Chen, X.P. Liu, S.C. Tong, Robust fuzzy control of nonlinear systems with input delay, Chaos, Solitons & Fractals. doi:10.1016/j.chaos.2006.09.091. [9] P. Chen, Y.C. Tian, Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay, Journal of Computational and Applied Mathematics (2007) doi:10.1016/j.cam.2007.03.009. [10] Y.Y. Cao, P.M. Frank, Analysis and synthesis of nonlinear time-delay system via fuzzy control approach, IEEE Transactions on Fuzzy Systems 8 (2) (2000) 200–211.
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