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New approach on robust stability for uncertain T–S fuzzy systems with state and input delays
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Chaos, Solitons and Fractals 40 (2009) 2329–2339 www.elsevier.com/locate/chaos
New approach on robust stability for uncertain T–S fuzzy systems with state and input delays Li Li *, Xiaodong Liu Research Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China Accepted 15 October 2007
Abstract This paper proposes an approach for robust delay-dependent stability of a class of uncertain fuzzy systems with state and control input delays. The key features of the approach include a kind of Lyapunov–Krasovskii functional and a descriptor model transformation with a recent result on bounding of cross products of vectors. Unlike existing methodologies, the proposed approach does not involve free weighting matrices. It can, however, lead to much less conservative stability criteria than the existing ones for the systems under consideration. Numerical examples show that the proposed criteria improve the existing results significantly with much less computational effort. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Among various schemes and fuzzy system theory [1] developed for the analysis and synthesis of complex nonlinear systems, fuzzy logic control is an attractive and effective rule-based one. It has emerged as a paradigm of intelligent control capable of dealing with complex and ill-defined systems for which the application of conventional control techniques is not straightforward or feasible. Fuzzy control technique represents a means of collecting human knowledge and expertise. It has been applied to various industrial fields [2,3]. Although the method has been practically successful, it has proved extremely difficult to develop a general analysis and design theory for conventional fuzzy control systems. In many of the model-based fuzzy control approaches, the well-known Takagi–Sugeno (T–S) fuzzy model [4] is a popular and convenient tool in functional approximations. Unlike conventional modeling, it combines some simple local linear dynamic systems with their linguistic description to represent highly nonlinear dynamic systems. Mathematically, the T–S fuzzy model is an interpolation method. The physical complex system is assumed to exhibit explicit linear or nonlinear dynamics around some operating points. These local models are smoothly aggregated via fuzzy inferences, which lead to the construction of complete system dynamics. This method is feasible since in many situations, human experts can provide linguistic descriptions of local systems in terms of IF–THEN rules. The method is quite interesting because it gives a way to smoothly connect local linear systems to form global nonlinear systems by fuzzy membership
*
Corresponding author. E-mail address: [email protected] (L. Li).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.10.026
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L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
functions. So it has been widely and successfully applied to a variety of industrial processes such as batch chemical reactors, cement kilns, etc. Research on the properties, especial stability problem and relaxed stability problem of fuzzy system, has become a very active area, e.g.[5–7], etc. Time-delay and uncertainty are frequently sources of instability and commonly exist in various engineering, biological, and economical systems due to the finite speed of information processing. Many criteria for checking the stability of time-delay and uncertainty systems have been derived, e.g., [6–11] and references therein. For the time-delay system, it is known that delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the size of the delay is small. Recently, delay-dependent stabilization are discussed in [8–10] for T–S fuzzy time-delay systems based on Lyapunov–Krasovksii functional approach. The delay is assumed to be constant and unknown. The state feedback control schemes have been proposed in terms of the feasible solutions to LMIs. In [11], 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 unknown time-varying delay. LMI-based delay-dependent conditions for robust stability and stabilization have been proposed. 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 delay in control system is always inevitable. So far, considerable attention has been paid to the control synthesis for nonlinear systems with both state and input delays. Although the input delay is a technically important issue of frequent occurrence, few related control strategies seem to be available (see [12–15]). It remains yet to be a theoretically challenging issue, and thereby it is more important and more realistic to research the T–S fuzzy model both the state delay and input delay. The existing results [12] based on Lyapunov–Razumikhin functional approaches, the delay-dependent LMI conditions for stability and stabilization have been developed for T–S fuzzy systems with state delay and input delay. But the obtained results 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. And in [12], the authors have studied the delay systems without uncertainty, in which the state delay is equal to the input delay. In [13], the fuzzy systems with input delay is researched without state delay. Lyapunov functional xT Px is constructed, which functional does not relate to information of time-delay. So the obtained results are more conservative. In [14] and [15], the delay-dependent stability conditions are obtained. And the considered uncertainty is norm-bounded uncertainty, which is the special case of linear fractional form uncertainty. In addition, input delay is only considered in [15]. Furthermore, in all the aforementioned results, the free weighting matrix approach is used, that is, a zero equalities is developed. Application of these methods makes the delay-dependent conditions less conservative. But many researchers have realized that too many free variables introduced in the free weighting matrices method will complicate the system synthesis and consequently lead to a significant increase in the computational demand [16]. The work presented in this paper reports our recent progress in the development of a computationally efficient method for significant improvement of delay-dependent stability criteria for uncertain fuzzy systems with state and input delays. Compared with the existing methods, the proposed method does not employ free weighting matrices in the derivation of our results. The main features of our method include: (1) A new Lyapunov–Krasovskii functional is constructed. (2) A combination of integral inequality technique and descriptor system approach [17,18] are used, which method avoids free weighting matrices used. And the results are less conservative and less computational demand. (3) The uncertain structured linear fractional form includes norm-bounded uncertainty as a special case and can describe a class of rational nonlinearities. This paper is organized as follows: In Section 2, the T–S fuzzy models with time-delays are first formulated. The main results are presented in Section 3. Section 4 provides illustrative examples to demonstrate the effectiveness of proposed method. Finally, the conclusions are drawn in Section 5.
2. Problem formulation and preliminaries In this paper, we consider a nonlinear fuzzy system with state and input delays, which is represented by a Takagi– Sugeno (T–S) fuzzy model [4] composed of a set of fuzzy implications. Each implication is expressed by a linear timedelay system and the ith rule of the T–S model is written in the following form: A. Plant Rule i IF h1 ðtÞ is M i1 ; . . . ; and hp ðtÞ is M ip , THEN
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
x_ ðtÞ ¼ ðA1i þ DA1i ðtÞÞxðtÞ þ ðA2i þ DA2i Þxðt s1 ðtÞÞ þ ðBi þ DBi ðtÞÞuðt s2 ðtÞÞ; n
i ¼ 1; 2; . . . ; r;
2331
ð1Þ
m
where M ij is a fuzzy set, xðtÞ 2 R is the state vector, uðtÞ 2 R is the control input. A1i ; A2i ; Bi are constant real matrices with appropriate dimensions 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 real valued functions si ðtÞ are the time-varying delays in state and input and satisfy 0 6 si ðtÞ 6 si < 1 ði ¼ 1; 2Þ. si are real positive constants as the upper bounds of the time-varying delays. It is also assumed that s_ i ðtÞ 6 li < 1 and li ði ¼ 1; 2Þ are known constants. The matrices DA1i ; DA2i ; DBi denote the uncertainties in system and they are of the form ½ DA1i
DA2i
DBi ¼ DD½ E1ai
E2ai
Ebi ;
ð2Þ
where D; E1ai ; E2ai ; Ebi are known constant matrices. The class of parametric uncertainties D that satisfy D ¼ ½I F ðtÞJ 1 F ðtÞ
ð3Þ
is said to be admissible, where J is also a known matrix satisfying I JJ T > 0
ð4Þ
and F ðtÞ is uncertain matrix satisfying F T ðtÞF ðtÞ 6 I:
ð5Þ
Remark 1. The above structured linear fractional form includes norm-bounded uncertainty as a special case and can describe a class of rational nonlinearities. Notice also that conditions (4) and (5) guarantee that I F ðtÞJ is invertible. For the simplicity, let us introduce the following notations: A1i ¼ A1i þ DAi ;
A2i ¼ A2i þ DA2i ;
Bi ¼ Bi þ DBi :
By using center-average defuzzifier, product inference and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model: Pr i ðhðtÞÞ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ Bi uðt s2 ðtÞÞ Pr x_ ðtÞ ¼ i¼1 ; ð6Þ i¼1 i ðhðtÞÞ where hðtÞ ¼ ½h1 ðtÞ; . . . ; hp ðtÞT ; i : Rp ! ½0; 1; i ¼ 1; 2; . . . ; r are the membership functions of the system with respect to . In this paper, we assume that i ðhðtÞÞ P 0 for the ith plant rule, and hi ðhðtÞÞ ¼ Pri ðhðtÞÞ i ðhðtÞÞ i¼1 P Pr i ¼ 1; 2; . . . ; r and i¼1 i ðhðtÞÞ > 0 for all t. Therefore, hi ðhðtÞÞ P 0ði ¼ 1; 2; . . . ; rÞ and ri¼1 hi ðhðtÞÞ ¼ 1. Therefore, (6) can be represented by r X x_ ðtÞ ¼ hi ðhðtÞÞ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ Bi uðt s2 ðtÞÞ: ð7Þ 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 M i1 ; . . . ; and hp ðtÞ is M ip , then uðtÞ ¼ K i xðtÞ;
i ¼ 1; 2; . . . ; r:
Hence, the overall fuzzy control law is represented by r X hi ðhðtÞÞK i xðtÞ; i ¼ 1; 2; . . . ; r; uðtÞ ¼
ð8Þ
ð9Þ
i¼1
¼ 1; 2; . . . ; rÞ are the local control gains. When there exists an input delay s2 ðtÞ, we have that where K i ðiP uðt s2 Þ ¼ ri¼1 hi ðhðt s2 ðtÞÞÞK i xðt s2 ðtÞÞ. So, it is natural and necessary to make an assumption that the functions hi ðhðtÞÞði ¼ 1; 2; . . . ; rÞ are well defined for all t 2 ½s2 ; 0, and satisfy the following properties:
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L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
hi ðhðt s2 ðtÞÞÞ P 0 for i ¼ 1; 2; . . . ; r;
and
r X
hi ðhðt s2 ðtÞÞÞ ¼ 1:
i¼1
For simplicity, let us denote hi ðhðtÞÞ ¼ hi ;
hj ðhðt s2 ðtÞÞÞ ¼ hsj 2 :
The design of the fuzzy controller is to determine the feedback gains K i ði ¼ 1; 2; . . . ; rÞ such that the resulting closedloop system is asymptotically stable. With the control law (9), the overall closed-loop system can be written as r X r X x_ ðtÞ ¼ hi hsj 2 ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ Bi K j xðt s2 ðtÞÞ i¼1
¼
j¼1
r X
hi ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ
i¼1
r X r X i¼1
ð10Þ
hi hsj 2 Bi K j xðt s2 ðtÞÞ;
j¼1
i ¼ 1; 2; . . . ; r: R0 By using the Newton–Lebuniz formula, xðt sÞ ¼ xðtÞ s x_ ðt þ aÞda, an equivalent form of system (10) is that " # " # Z t Z t r r X r X X s2 x_ ðtÞ ¼ x_ ðsÞds þ x_ ðsÞds : ð11Þ hi ðA1i þ A2i ÞxðtÞ A2i hi hj Bi K j xðtÞ Bi K j xðtÞ ¼ uðtÞ;
t 2 ½ maxðs1 ; s2 Þ; 0;
ts1 ðtÞ
i¼1
i¼1
j¼1
ts2 ðtÞ
The following result can be obtained similarly to the derivations in [19], which will be used to deal with the considered uncertain models. Lemma 1 [19]. Suppose that D is given by (3)–(5). Given matrices M ¼ M T , S and N of appropriate dimensions, the inequality M þ SDN þ N T DT S T < 0 holds for all F ðtÞ such that F ðtÞF T ðtÞ 6 I, if and only if, for some d > 0, 3 2 dM S dN T 7 6 T I J T 5 < 0: 4S dN J I Let d1 ¼ q, we will get the following lemma: Lemma 2. Suppose that D is given by (3)–(5). Given matrices M ¼ M T , S and N of appropriate dimensions, the inequality M þ SDN þ N T DT S T < 0 holds for all F ðtÞ such that F ðtÞF T ðtÞ 6 I, if and only if, for some q > 0, 3 2 M qS NT 7 6 T qI qJ T 5 < 0: 4 qS qJ
N
qI
Lemma 3 [20]. Assume that aðÞ 2 Rna , bðÞ 2 Rnb and NðÞ 2 Rna nb are defined on the interval X. Then, for any matrices X 2 Rna nb , and Z 2 Rna nb , the following holds: Z Z aðsÞ T X Y N aðsÞ T 2 a ðsÞNbðsÞds 6 ds; Z Y T NT bðsÞ X X bðsÞ where
X YT
Y Z
P 0:
3. Main result In this section, we focus on the problems of developing some delay-dependent stability criteria which provide the upper bounds of the delays such that the closed-loop system (10) is stable.
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
2333
Theorem 1. Consider the closed-loop system (10) and scalars s1 ; s2 ; d1 ; d2 . For some q > 0, if there exist X > 0, S 1 > 0, S 2 > 0, Q1 > 0, Q2 > 0; q > 0 and real matrices Y ; Z, W 11 ; W 12 ; W 13 ; W 21 ; W 22 ; W 23 ; K i satisfying the following equations: 3 2 Pij qM N Tij 7 6 ð12Þ 4 qM T qI qJ T 5 < 0; i; j ¼ 1; 2; . . . ; r; N ij
qJ
W 11 6 4
W 12 W 13
2
qI
3 0 7 d1 A2i X 5 P 0; X Q1 1 X
2
W 21 6 4
W 22 W 23
3 0 7 d2 Bi K j 5 P 0; X Q1 2 X
i; j ¼ 1; 2; . . . ; r;
ð13Þ
where 2
U11 6 6 6 6 Pij ¼ 6 6 6 6 4
U12 U22
0 U23
0 U24
s1 Z T s1 Y T
l1 S 1
0 l2 S 2
0 0 s1 Q1
3 s2 Z T s2 Y T 7 7 7 0 7 7; 0 7 7 7 0 5 s2 Q2
T
U11 ¼ Z þ Z þ S 1 þ S 2 þ s1 W 11 þ s2 W 21 ; U12 ¼ Y þ XAT1i Z T þ d1 XAT2i þ d2 K Tj BTi þ s1 W 12 þ s2 W 22 ; U22 ¼ Y Y T þ s1 W 13 þ s2 W 23 ; U23 ¼ ð1 d1 ÞA2i X ; U24 ¼ ð1 d2 ÞBi K j ; M T ¼ ½ 0 DT 0 0 0 0 ; N ij ¼ ½ E1ai X 1
Y ¼
X ¼P ; Q2 ¼
Q1 2 ;
P 1 2 ;
Z¼
1 P 1 ; 2 P 1P
l1 ¼ ð1 l1 Þ;
0 E2ai X
Ki ¼ KiX ;
0
Ebi K j X
S 1 ¼ XT 1 X ;
0 ; S 2 ¼ XT 2 X ;
Q1 ¼ Q1 1 ;
l2 ¼ ð1 l2 Þ;
and ‘‘*’’ denotes the transposed elements in the symmetric positions. Then, system (10) is asymptotically stable for any 0 6 si ðtÞ 6 si ði ¼ 1; 2Þ; moreover, the control gain matrix K i is given by K i ¼ K i X 1 ;
i ¼ 1; 2; . . . ; r:
Proof. Following [17,18], we represent (11) in the following equivalent descriptor system form: yðtÞ ¼ x_ ðtÞ; 0 ¼ yðtÞ þ 0¼
r X
"
r X
" hi ðA1i þ A2i ÞxðtÞ A2i
Z
ts1 ðtÞ
i¼1
hi ðA1i þ A2i ÞxðtÞ A2i
Z
t
Take the Lyapunov functional for system (14) as V ðtÞ ¼
5 X
V i ðtÞ
i¼1
with V 1 ðtÞ ¼ xT ðtÞPxðtÞ; Z t xT ðsÞT 1 xðsÞ ds; V 2 ðtÞ ¼ ts1 ðtÞ
V 3 ðtÞ ¼
Z
0
s1
Z
t
tþh
yðsÞds þ #
yðsÞds yðtÞ þ ts1 ðtÞ
i¼1
#
t
y T ðsÞQ1 yðsÞ ds dh;
r X r X i¼1
j¼1
r X
r X
i¼1
j¼1
"
Z
hi hsj 2
Bi K j xðtÞ Bi K j
"
Z
hi hsj 2 Bi K j xðtÞ Bi K j
#
t
yðsÞds ; ts2 ðtÞ t
#
yðsÞds : ts2 ðtÞ
ð14Þ
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L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
V 4 ðtÞ ¼
Z
t
xT ðsÞT 2 xðsÞ ds;
ts2 ðtÞ
V 5 ðtÞ ¼
Z
0
s2
Z
t
y T ðsÞQ2 yðsÞ ds dh: tþh
Differentiating V i ðtÞ with respect to t and using Eq. (14) yields V_ 1 ðtÞ ¼ 2xT ðtÞP x_ ðtÞ ¼ 2xT ðtÞPyðtÞ ¼ 2gT ðtÞGT ½ y T ðtÞ 0 T ( ( ) Z t r X 0 I 0 T T ¼ hi 2g ðtÞG yðsÞ ds gðtÞ A1i þ A2i I A2i ts1 ðtÞ i¼1 ( ( ) Z t r X r X 0 0 0 s2 T T gðtÞ hi hj 2g ðtÞG yðsÞ ds ; þ Bi K j 0 Bi K j ts2 ðtÞ i¼1 j¼1 P 0 where gT ðtÞ ¼ ½ xT ðtÞ y T ðtÞ ; G ¼ . By Lemma 3, we have P 1 ZP 2 # Z t " t gðtÞ 0 gðtÞ T W 1 M 1 GT ½ 0 AT2i T T T yðsÞds 6 2g ðtÞG ds A2i ts1 ðtÞ yðsÞ Q1 ts1 ðtÞ yðsÞ Z t T ðxðtÞ xðt s1 ðtÞÞÞ; y T ðsÞQ1 yðsÞds þ 2gT ðtÞ M 1 GT 0 AT2i 6 s1 gT ðtÞW 1 gðtÞ þ ts1 ðtÞ W 11 W 12 , where W 1 ¼ W 13
W1
M1 Q1
ð15Þ
P 0:
Similarly, we have # Z t " Z t 0 gðtÞ gðtÞ T W 2 M 2 GT ½ 0 ðBi K j ÞT T T T yðsÞds 6 2g ðtÞG ds Bi K j ts2 ðtÞ yðsÞ Q2 ts2 ðtÞ yðsÞ Z t y T ðsÞQ2 yðsÞds þ 2gT ðtÞðM 2 GT ½ 0 ½Bi K j T T ÞðxðtÞ xðt s2 ðtÞÞÞ; 6 s2 gT ðtÞW 2 gðtÞ þ ts2 ðtÞ W 21 W 22 where W 2 ¼ , W 23 W 2 M2 P 0: Q2 And V_ 2 ðtÞ ¼ xT ðtÞT 1 xðtÞ ð1 s_ 1 ðtÞÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ 6 xT ðtÞT 1 xðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ; Z t V_ 3 ðtÞ 6 s1 y T ðtÞQ1 yðtÞ y T ðsÞQ1 yðsÞds; ts1 ðtÞ
V_ 4 ðtÞ ¼ xT ðtÞT 2 xðtÞ ð1 s_ 2 ðtÞÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ 6 xT ðtÞT 2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ; Z t V_ 5 ðtÞ 6 s2 y T ðtÞQ2 yðtÞ y T ðsÞQ2 yðsÞds: ts2 ðtÞ
Then, we have r X hi 2gT ðtÞGT V_ ðtÞ 6
0 I gðtÞ þ 2gT ðtÞ M 1 GT ½ 0 AT2i T ðxðtÞ xðt s1 ðtÞÞÞ A1i þ A2i I i¼1 r X r X 0 0 s2 T T T T T T gðtÞ þ 2g ðtÞðM 2 G ½ 0 ½Bi K j ÞðxðtÞ xðt s2 ðtÞÞÞ hi hj 2g ðtÞG þ Bi K j 0 i¼1 j¼1 þ s1 gT ðtÞW 1 gðtÞ þ xT ðtÞT 1 xðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ þ s2 gT ðtÞW 2 gðtÞ þ s1 y T ðtÞQ1 yðtÞ
þ s2 y T ðtÞQ2 yðtÞ þ xT ðtÞT 2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ r X 0 I ¼ hi 2gT ðtÞGT gðtÞ þ 2gT ðtÞðM 1 GT ½ 0 AT2i T ÞðxðtÞ xðt s1 ðtÞÞÞ A1i þ A2i I i¼1
ð16Þ
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
þ
r X r X i¼1
j¼1
0 hi hsj 2 2gT ðtÞGT Bi K j
2335
0 gðtÞ þ 2gT ðtÞðM 2 GT ½ 0 ½Bi K j T T ÞðxðtÞ xðt s2 ðtÞÞÞ 0
T
þ s1 g ðtÞW 1 gðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ þ s2 gT ðtÞW 2 gðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ 0 T1 þ T2 gðtÞ þ gT ðtÞ s1 Q1 þ s2 Q2 0 r X r X hi hsj 2 nT ðtÞXij nðtÞ; 6 i¼1
j¼1
where nT ðtÞ ¼ ½ gT ðtÞ xT ðt s1 ðtÞÞ xT ðt s2 ðtÞÞ ; 3 2 Cð1; 1Þ Cð1; 2Þ Cð1; 3Þ 7 6 ð1 l1 ÞT 1 0 Xij ¼ 4 5; ð1 l2 ÞT 2 with
ð17Þ
" # T I 0 I M T1 Cð1; 1Þ ¼ G þ ½ M1 Gþ þ I A1i I 0 0 T1 þ T2 ; þ s1 W 1 þ s2 W 2 þ s1 Q1 þ s2 Q2 0 T
0 A1i
"
M T2 0 þ 0
# þ ½ M2
0
Cð1; 2Þ ¼ GT ½ 0 AT2i T M 1 ; Cð1; 3Þ ¼ GT ½ 0 ½Bi K j T T M 2 : 0 0 , M 2 ¼ d2 GT , where d1 ; d2 are scalars. Multiplying (17) by diagfðG1 ÞT ; X ; Let M 1 ¼ d1 GT Bi K j A2i X g and diagfG1 ; X ; X g on the left and on the right, respectively, by Schur complement the result inequality is obtained as following: 2 6 6 6 6 Kij ¼ 6 6 6 6 4
Kð1; 1Þ
Kð1; 2Þ
0
0
Kð2; 2Þ Kð2; 3Þ ð1 l1 ÞXT 1 X
Kð2; 4Þ 0 ð1 l2 ÞXT 2 X
3 s2 Z T s2 Y T 7 7 7 7 0 7; 7 0 7 7 5 0
s1 Z T s1 Y T 0 0 s1 Q1 1
ð18Þ
s2 Q1 2
where Kð1; 1Þ ¼ Z þ Z T þ X ðT 1 þ T 2 ÞX þ s1 W 11 þ s2 W 21 ; Kð1; 2Þ ¼ Y þ X ðAT1i þ DAT1i Þ Z T þ d1 XAT2i þ d2 XK Tj BTi þ s1 W 12 þ s2 W 22 ; Kð2; 2Þ ¼ Y Y T þ s1 W 13 þ s2 W 23 ; Kð2; 3Þ ¼ ð1 d1 ÞA2i X þ DA2i X ; Kð2; 4Þ ¼ ð1 d2 ÞBi K j X þ DBi K j X ; 1 1 : X ¼ P 1 ; Y ¼ P 1 2 ; Z ¼ P 2 P 1 P
Then (18) can be expressed as follows: 2 Pð1; 1Þ Pð1; 2Þ 0 6 Pð2; 2Þ Pð2; 3Þ 6 6 6 l1 XT 1 X Kij ¼ 6 6 6 6 4
0 Pð2; 4Þ 0
s1 Z T s1 Y T
l2 XT 2 X
0 s1 Q1 1
0
3 s2 Z T s2 Y T 7 7 7 7 0 7 þ MDN ij þ N T DT M T ; ij 7 0 7 7 5 0 s2 Q1 2
ð19Þ
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L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
where Pð1; 1Þ ¼ Z þ Z T þ X ðT 1 þ T 2 ÞX þ s1 W 11 þ s2 W 21 ; Pð1; 2Þ ¼ Y þ XAT1i Z T þ d1 XAT2i þ d2 XK Tj BTi þ s1 W 12 þ s2 W 22 ; Pð2; 2Þ ¼ Y Y T þ s1 W 13 þ s2 W 23 ; Pð2; 3Þ ¼ ð1 d1 ÞA2i X ; Pð2; 4Þ ¼ ð1 d2 ÞBi K j X ; M T ¼ ½ 0 DT 0 0 0 0 ; N ij ¼ ½ E1ai X 0 E2ai X
0 0 : Applying Lemma 2, the resulting inequality (19) is equivalent to (12), that is V_ ðtÞ < 0. It means that the closed-loop system (10) is asymptotically stable. In the following, what we need to do is to consider the constraint of (15) and (16). Multiplying (15) and (16) by diagfðG1 ÞT ; X g and diagfG1 ; X g on the left and on the right, respectively, we have 3 3 2 2 W 11 W 12 0 W 21 W 22 0 7 7 6 6 ð20Þ W 13 d1 A2i X 5 P 0; 4 W 23 d2 Bi K j X 5 P 0: 4
X Q1 1 X
Ebi K j X
X Q1 2 X
The resulting inequality (20) is (13). The proof of the Theorem 1 is completed.
h
If D of (2) is F(t), that is J ¼ 0, the uncertainties in system are norm-bounded parameter uncertainties. For D ¼ F ðtÞ, we get the following theorem: Theorem 2. Consider the closed-loop system (10) and scalars s1 , s2 ; d1 ; d2 . If there exist X > 0, S 1 > 0, S 2 > 0, Q1 > 0, Q2 > 0, e > 0 and real matrices Y ; Z, W 11 ; W 12 ; W 13 ; W 21 ; W 22 ; W 23 ; K i satisfying the following equations: 3 2 s2 Z T s1 Z T U12 0 0 XET1ai U11 6 U þ eDDT s1 Y T s2 Y T U23 U24 0 7 7 6 22 7 6 T 7 6 l S 0 0 0 XE 1 1 2ai 7 6 6 l2 S 2 0 0 K Tj ETbi 7 ð21Þ 7 < 0; i; j ¼ 1; 2; . . . ; r; 6 7 6 7 6 s1 Q1 0 0 7 6 7 6 4 s2 Q2 0 5 2
W 11 6 4
W 12 W 13
3
0 7 d1 A2i X 5 P 0; X Q1 1 X
2 W 21 6 4
W 22 W 23
3
eI
0 7 d2 Bi K j 5 P 0; X Q1 2 X
i; j ¼ 1; 2; . . . ; r:
ð22Þ
where e ¼ e1 , U11 ; U12 ; U22 ; U23 ; U24 are same as Theorem 1. And ‘‘*’’ denotes the transposed elements in the symmetric positions. Then, system (10) is asymptotically stable for any 0 6 si ðtÞ 6 si ði ¼ 1; 2Þ; moreover, the control gain matrix K i is given by K i ¼ K i X 1 ;
i ¼ 1; 2; . . . ; r:
It is noted that the resulting conditions for robust stabilization in Theorems 1 and 2 are no longer LMIs conditions 1 because of the nonlinear term X Q1 1 X ; X Q2 X in (13) and (22). Because the way to solve Theorem 1 is similar to the way to solve Theorem 2, we mainly analysis the way to solve Theorem 1. An easy way to solve (12) and (13) is simply to set Q1 ¼ a1 X , Q2 ¼ a2 X , where a1 ; a2 > 0 is a tuning parameter. Then by applying a simple one-dimensional search over the positive variable a1 ; a2 , we can find the feasible solution of the resulting LMI conditions. However, this method may increase the conservativeness. Similar to [21], another less conservative approach can be developed using an iterative algorithm next. For the sake of simplicity, we only consider how to apply the algorithm to solve matrix inequalities (12) and (13). First, we need to introduce a new variable L1 ; L2 > 0 such that 1 X Q1 1 X P L1 ; X Q2 X P L2 ;
and replace the condition (13) with 3 2 W 11 W 12 0 7 6 W 13 d1 A2i X 5 P 0; 4
L1
ð23Þ (23) and 2 W 21 W 22 6 W 23 4
3 0 7 d2 Bi K j 5 P 0: L2
ð24Þ
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
2337
1 1 1 1 1 Since (23) is equivalent to X 1 Qi X 1 6 L1 , and i ði ¼ 1; 2Þ, by setting L1 ¼ L1 ; L2 ¼ L2 ; Q1 ¼ Q1 ; Q2 ¼ Q2 ; P ¼ X using Schur complement, condition (13) can be replaced with (24) and " # " # L1 P L2 P P 0; P 0; ð25Þ P Q1 P Q2
L1 L1 ¼ I;
L2 L2 ¼ I;
XP ¼ I;
Q1 Q1 ¼ I;
Q2 Q2 ¼ I:
ð26Þ
Note that condition (26) still includes nonlinear conditions. However, using the idea in a cone complementary linearization algorithm of [22], the problem of finding a feasible solution of the non-convex matrix inequalities (12), (24), (25), (26) can be considered as a cone complementary problem involving LMIs conditions Minimize TraceðL1 L1 þ L2 L2 þ XP þ Q1 Q1 þ Q2 Q2 Þ; Subject to ð12Þ; ð24Þ; ð25Þ and L1 I L2 I X I P 0; P 0; P 0; I L1 I L2 I P " # " # Q1 I Q2 I P 0; P 0: I Q1 I Q2
ð27Þ
The above cone complementary problem (27) can be solved using the following algorithm. Algorithm 1. (i) Find a feasible solution fq0 ; Y 0 ; Z 0 ; K i0 ; W 110 ; W 120 ; W 130 ; W 210 ; W 220 ; W 230 ; S 10 ; S 20 ; X 0 ; P 0 ; L10 ; L10 ; L20 ; L20 ; Q10 ; Q10 ; Q20 ; Q20 g of the LMIs in (27). If there are none, then exit. Set k ¼ 0. (ii) Solve the following LMI optimization problem for the variables fq; Y ; Z; K i ; W 11 ; W 12 ; W 13 ; W 21 ; W 22 ; W 23 ; S 1 ; S 2 ; X ; P ; L1 ; L1 ; L2 ; L2 ; Q1 ; Q1 ; Q2 ; Q2 g: Minimize
Trace Nk ;
subject to LMIs in ð27Þ; where Nk ¼ ðL1k L1 þ L1k L1 þ L2k L2 þ L2k L2 þ X k P þ P k X þ Q1k Q1 þ Q1k Q1 þ Q2k Q2 þ Q2k Q2 Þ. Then set L1kþ1 ¼ L1 ; L1kþ1 ¼ L1 ; L2kþ1 ¼ L2 ; L2kþ1 ¼ L2 ; X kþ1 ¼ X ; P kþ1 ¼ P ; Q1kþ1 ¼ Q1 ; Q1kþ1 ¼ Q1 ; Q2kþ1 ¼ Q2 ; Q2kþ1 ¼ Q2 . (iii) If the above LMI optimization problem leads to positive definite matrices X ; Q such that the resulting LMIs (12) and (13) are feasible, then exit. Otherwise, set k ¼ k þ 1, and go to step (ii). Remark 2. Theorems 1 and 2 provide some new sufficient conditions of the delay-dependent fuzzy systems with state and input delays. The methods are based on the descriptor model transformation [17,18] and a recent result on bounding of cross products of vectors. We would like to emphasize that, to the best of our knowledge, so far there has been no result appeared in the literature for the fuzzy controller design by using the descriptor model transformation. Our results given here are less conservative for the case. Remark 3. By Algorithm 1, we can get optimal delay upper bounds with the cone complementary linearization algorithm [21,22] for solving the stabilization problems of time-varying fuzzy systems.
4. Numerical examples In this section, we present several numerical examples to illustrate the usefulness of the proposed theoretical results. Example 1. Consider the nonlinear mass-spring-damper mechanical systems in [13]. The state-space is represented as following: x_ 1 ðtÞ 0:75x31 ðtÞ 0:5x1 ðtÞ þ 0:1ð1 þ 0:11ðtÞÞx2 ðtÞ þ uðt sðtÞÞ ; ð28Þ ¼ x_ 2 ðtÞ x1 ðtÞ where j1ðtÞj2 6 1. The system (28) has one nonlinear term 0:75x31 ðtÞ. Now by adopting fuzzy sets, the T–S fuzzy system of (28) can be constructed as follows:
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Table 1 A comparison for allowable input delay Methods
In [13]
In [14]
In this paper
s
0.4223
1.75
3.06
Rule 1: If x1 ðtÞ is about C1 ; then x_ ðtÞ ¼ ðA11 þ DA11 ÞxðtÞ þ ðB1 þ DB1 Þuðt sðtÞÞ, Rule 2: If x1 ðtÞ is about C2 , then x_ ðtÞ ¼ ðA12 þ DA12 ÞxðtÞ þ ðB2 þ DB2 Þuðt sðtÞÞ, where the associated matrices are given by 0:5 0:1 1 0:1 0 0:11ðtÞ 1 ; A12 ¼ ; DA11 ¼ DA12 ¼ ; B1 ¼ B2 ¼ ; A11 ¼ 1 0 1 0 0 0 0 " pffiffiffiffiffiffiffi # pffiffiffiffiffiffiffi 0:1 ; E1a1 ¼ E1a2 ¼ ½ 0 D1 ¼ D2 ¼ 0:1 ; Eb1 ¼ Eb2 ¼ ½ 0 0 ; J ¼ 0: 0 Applying our algorithm to this paper, system (28) achieves stability for all delay 0 < s < 3:06 when d2 ¼ 0:95. Table 1 lists the maximum upper bound of input delay for the system (28) in [13,14] and this paper. When s < 3:06, the feedback gain is K 1 ¼ ½ 0:4669 0:2856 ; K 2 ¼ ½ 0:0501 0:0679 . Example 2. In the following, we will consider the truck-trailer system [15], which can be described by the following T–S fuzzy system: vt x1 ðtÞ is about 0, THEN Rule 1: IF hðtÞ ¼ x2 ðtÞ þ 2L x_ ðtÞ ¼ ðA1 þ DA1 ÞxðtÞ þ ðB1 þ DB1 Þuðt sÞ: vt Rule 2: IF hðtÞ ¼ x2 ðtÞ þ 2L x1 ðtÞ is about p or p, THEN x_ ðtÞ ¼ ðA2 þ DA2 ÞxðtÞ þ ðB2 þ DB2 Þuðt sÞ,
where 2 6 A1 ¼ 6 4
Ltvt0 vt Lt0 2 2
v t 2Lt 0
3
2
7 0 07 5; vt 0 t0
6 A2 ¼ 6 4
0 0
3
Ltvt0
0
0
vt Lt0
0
7 07 5;
2 2
v t d 2Lt 0
dvt t0
2
vt lt0
3
6 7 B1 ¼ B2 ¼ 4 0 5;
0
0
with a ¼ 0:7; v ¼ 1:0; t ¼ 2:0; t0 ¼ 0:5; L ¼ 5:5; l ¼ 2:8; d ¼ 10 tp0 . D1 ¼ D2 ¼ ½ 0:255 0:255
0:255 T ;
E1a1 ¼ E1a2 ¼ ½ 0:1 0 0 ;
Eb1 ¼ Eb2 ¼ ½0:15;
The following fuzzy rules are employed 1 1 h1 ðxÞ ¼ 1 and 1 expð3ðhðtÞ 0:5pÞÞ 1 expð3ðhðtÞ þ 0:5pÞÞ
J ¼ 0:
h2 ðxÞ ¼ 1 h1 ðxÞ:
Applying our algorithm to this paper, the maximal allowable input delay is 5.92 for the proposed fuzzy systems with input delay when d2 ¼ 1:9. But in [15], the maximal allowable input delay is s ¼ 0:75 for the same fuzzy systems. In order to show our result less conservative, we compare the feedback gain matrices and the maximal allowable input delay in Table 2.
Table 2 Comparison of the feedback gain for different input delay Methods
s
K1
K2
Chen et al. [15] Our result
0.75 0.75 5.92
½ 3:4227 0:3535 0:0045 ½ 0:2007 0:0859 1:1886 ½ 0:2282 0:0674 19:8924
½ 3:5215 0:3617 0:0056 ½ 0:2040 0:0990 1:4765 ½ 0:2262 0:1140 20:0603
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
2339
5. Conclusion By using a type of new Lyapunov function and the descriptor system method, we establish some new delay-dependent criteria for robust stabilization of uncertain fuzzy systems with state and input delays. The numerical algorithm underlying our controller design methodology involves solving a series of parameterized LMI problems. Numerical examples are given to demonstrate the superiority of the proposed method. Furthermore, our controller design methodology leads to a state feedback controller which can stabilize a larger class of uncertain systems than the existing methods.
Acknowledgements This work is supported in part by the National Natural Science Foundation of China under Grant 60575039 and in part by the National Key Basic Research and Development Program of China under Grant 2002CB312201-06.
References [1] El Naschie MS. On two new fuzzy Ka¨hler manifolds, Klein modular space and ‘t Hooft holographic principles. Chaos, Solitons & Fractals 2006;29:876–81. [2] Tanaka K, Wang HO. Fuzzy control systems design and analysis. New York: Wiley; 2001. [3] Wang LX. A course in fuzzy systems and control. London, UK: Prentice-Hall; 1997. [4] Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst, Man, Cybern 1985;15(1):116–32. [5] Liu XD, Zhang QL. New approaches to H 1 controller designs based on fuzzy observers for T–S fuzzy systems via LMI. Automatica 2003;39:1571–82. [6] Cao YY, Frank PM. Analysis and synthesis of nonlinear time-delay system via fuzzy control approach. IEEE Trans Fuzzy Syst 2000;8(2):200–11. [7] Zhang Y, Pheng AH. Stability of fuzzy systems with bounded uncertain delays. IEEE Trans Fuzzy Syst 2002;10(1):92–7. [8] Guan XP, Chen CL. Delay-dependent guaranteed cost control for T–S fuzzy systems with time delays. IEEE Trans Fuzzy Syst 2004;12(2):236–49. [9] Chen B, Liu XP, Tong SC. Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay. Fuzzy Set Syst 2006;157(16):2224–40. [10] Chen B, Liu XP, Tong SC. New delay-dependent stabilization conditions of T–S fuzzy systems with constant delay. Fuzzy Set Syst; in press. doi:10.1016/j.fss.2007.02.018. [11] Tian EG, Peng C. Delay-dependent stability analysis and synthesis of uncertain T–S fuzzy systems with time-varying delay. Fuzzy Set Syst 2006;157:544–59. [12] Lin C, Wang QG, Lee TH. Delay-dependent LMI conditions for stability and stabilization of T–S fuzzy systems with bounded time-delay. Fuzzy Set Syst 2006;157:1229–47. [13] Lee HJ, Park JB, Joo YH. Robust control for uncertain Takagi–Sugeno fuzzy systems with time-varying input delay. ASME J Dyn Syst Meas Contr 2005;127:302–6. [14] Lien CH, Yu KW. Robust control for Takagi–Sugeno fuzzy systems with time-varying state and input delays. Chaos Solitons & Fractals 2008;35(5):1003–8. [15] Chen B et al., Robust fuzzy control of nonlinear systems with input delay. Chaos, Solitons & Fractals; in press. doi:10.1016/ j.chaos.2006.09.091. [16] Chen WH, Zheng WX. Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica 2007;43:95–104. [17] Fridman E. A descriptor systems approach to H 1 control of linear time-delay systems. IEEE Trans Automat Contr 2002;47:253–70. [18] Fridman E, Shaked U. An improved stabilization method for linear time-delay systems. IEEE Trans Automat Contr 2002;47:1931–7. [19] Zhou SS, Lam J. Robust stabilization of delayed singular systems with linear fractional parametric uncertainties. Circuits Syst Signal Process 2003;22(6):579–88. [20] Moon YS. Delay-dependent robust stabilization of uncertain state delayed systems. Int J Contr 2001;74:1447–55. [21] Gao H, Wang C. Comments and further results on ‘‘A descriptor system approach to H 1 control of linear time-delay systems’’. IEEE Trans Automat Contr 2003;48:520–5. [22] Ghaoui LE, Oustry F, AitRami M. A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans Automat Contr 1997;42:1171–6.
Chaos, Solitons and Fractals 40 (2009) 2329–2339 www.elsevier.com/locate/chaos
New approach on robust stability for uncertain T–S fuzzy systems with state and input delays Li Li *, Xiaodong Liu Research Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China Accepted 15 October 2007
Abstract This paper proposes an approach for robust delay-dependent stability of a class of uncertain fuzzy systems with state and control input delays. The key features of the approach include a kind of Lyapunov–Krasovskii functional and a descriptor model transformation with a recent result on bounding of cross products of vectors. Unlike existing methodologies, the proposed approach does not involve free weighting matrices. It can, however, lead to much less conservative stability criteria than the existing ones for the systems under consideration. Numerical examples show that the proposed criteria improve the existing results significantly with much less computational effort. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Among various schemes and fuzzy system theory [1] developed for the analysis and synthesis of complex nonlinear systems, fuzzy logic control is an attractive and effective rule-based one. It has emerged as a paradigm of intelligent control capable of dealing with complex and ill-defined systems for which the application of conventional control techniques is not straightforward or feasible. Fuzzy control technique represents a means of collecting human knowledge and expertise. It has been applied to various industrial fields [2,3]. Although the method has been practically successful, it has proved extremely difficult to develop a general analysis and design theory for conventional fuzzy control systems. In many of the model-based fuzzy control approaches, the well-known Takagi–Sugeno (T–S) fuzzy model [4] is a popular and convenient tool in functional approximations. Unlike conventional modeling, it combines some simple local linear dynamic systems with their linguistic description to represent highly nonlinear dynamic systems. Mathematically, the T–S fuzzy model is an interpolation method. The physical complex system is assumed to exhibit explicit linear or nonlinear dynamics around some operating points. These local models are smoothly aggregated via fuzzy inferences, which lead to the construction of complete system dynamics. This method is feasible since in many situations, human experts can provide linguistic descriptions of local systems in terms of IF–THEN rules. The method is quite interesting because it gives a way to smoothly connect local linear systems to form global nonlinear systems by fuzzy membership
*
Corresponding author. E-mail address: [email protected] (L. Li).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.10.026
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L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
functions. So it has been widely and successfully applied to a variety of industrial processes such as batch chemical reactors, cement kilns, etc. Research on the properties, especial stability problem and relaxed stability problem of fuzzy system, has become a very active area, e.g.[5–7], etc. Time-delay and uncertainty are frequently sources of instability and commonly exist in various engineering, biological, and economical systems due to the finite speed of information processing. Many criteria for checking the stability of time-delay and uncertainty systems have been derived, e.g., [6–11] and references therein. For the time-delay system, it is known that delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the size of the delay is small. Recently, delay-dependent stabilization are discussed in [8–10] for T–S fuzzy time-delay systems based on Lyapunov–Krasovksii functional approach. The delay is assumed to be constant and unknown. The state feedback control schemes have been proposed in terms of the feasible solutions to LMIs. In [11], 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 unknown time-varying delay. LMI-based delay-dependent conditions for robust stability and stabilization have been proposed. 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 delay in control system is always inevitable. So far, considerable attention has been paid to the control synthesis for nonlinear systems with both state and input delays. Although the input delay is a technically important issue of frequent occurrence, few related control strategies seem to be available (see [12–15]). It remains yet to be a theoretically challenging issue, and thereby it is more important and more realistic to research the T–S fuzzy model both the state delay and input delay. The existing results [12] based on Lyapunov–Razumikhin functional approaches, the delay-dependent LMI conditions for stability and stabilization have been developed for T–S fuzzy systems with state delay and input delay. But the obtained results 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. And in [12], the authors have studied the delay systems without uncertainty, in which the state delay is equal to the input delay. In [13], the fuzzy systems with input delay is researched without state delay. Lyapunov functional xT Px is constructed, which functional does not relate to information of time-delay. So the obtained results are more conservative. In [14] and [15], the delay-dependent stability conditions are obtained. And the considered uncertainty is norm-bounded uncertainty, which is the special case of linear fractional form uncertainty. In addition, input delay is only considered in [15]. Furthermore, in all the aforementioned results, the free weighting matrix approach is used, that is, a zero equalities is developed. Application of these methods makes the delay-dependent conditions less conservative. But many researchers have realized that too many free variables introduced in the free weighting matrices method will complicate the system synthesis and consequently lead to a significant increase in the computational demand [16]. The work presented in this paper reports our recent progress in the development of a computationally efficient method for significant improvement of delay-dependent stability criteria for uncertain fuzzy systems with state and input delays. Compared with the existing methods, the proposed method does not employ free weighting matrices in the derivation of our results. The main features of our method include: (1) A new Lyapunov–Krasovskii functional is constructed. (2) A combination of integral inequality technique and descriptor system approach [17,18] are used, which method avoids free weighting matrices used. And the results are less conservative and less computational demand. (3) The uncertain structured linear fractional form includes norm-bounded uncertainty as a special case and can describe a class of rational nonlinearities. This paper is organized as follows: In Section 2, the T–S fuzzy models with time-delays are first formulated. The main results are presented in Section 3. Section 4 provides illustrative examples to demonstrate the effectiveness of proposed method. Finally, the conclusions are drawn in Section 5.
2. Problem formulation and preliminaries In this paper, we consider a nonlinear fuzzy system with state and input delays, which is represented by a Takagi– Sugeno (T–S) fuzzy model [4] composed of a set of fuzzy implications. Each implication is expressed by a linear timedelay system and the ith rule of the T–S model is written in the following form: A. Plant Rule i IF h1 ðtÞ is M i1 ; . . . ; and hp ðtÞ is M ip , THEN
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
x_ ðtÞ ¼ ðA1i þ DA1i ðtÞÞxðtÞ þ ðA2i þ DA2i Þxðt s1 ðtÞÞ þ ðBi þ DBi ðtÞÞuðt s2 ðtÞÞ; n
i ¼ 1; 2; . . . ; r;
2331
ð1Þ
m
where M ij is a fuzzy set, xðtÞ 2 R is the state vector, uðtÞ 2 R is the control input. A1i ; A2i ; Bi are constant real matrices with appropriate dimensions 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 real valued functions si ðtÞ are the time-varying delays in state and input and satisfy 0 6 si ðtÞ 6 si < 1 ði ¼ 1; 2Þ. si are real positive constants as the upper bounds of the time-varying delays. It is also assumed that s_ i ðtÞ 6 li < 1 and li ði ¼ 1; 2Þ are known constants. The matrices DA1i ; DA2i ; DBi denote the uncertainties in system and they are of the form ½ DA1i
DA2i
DBi ¼ DD½ E1ai
E2ai
Ebi ;
ð2Þ
where D; E1ai ; E2ai ; Ebi are known constant matrices. The class of parametric uncertainties D that satisfy D ¼ ½I F ðtÞJ 1 F ðtÞ
ð3Þ
is said to be admissible, where J is also a known matrix satisfying I JJ T > 0
ð4Þ
and F ðtÞ is uncertain matrix satisfying F T ðtÞF ðtÞ 6 I:
ð5Þ
Remark 1. The above structured linear fractional form includes norm-bounded uncertainty as a special case and can describe a class of rational nonlinearities. Notice also that conditions (4) and (5) guarantee that I F ðtÞJ is invertible. For the simplicity, let us introduce the following notations: A1i ¼ A1i þ DAi ;
A2i ¼ A2i þ DA2i ;
Bi ¼ Bi þ DBi :
By using center-average defuzzifier, product inference and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model: Pr i ðhðtÞÞ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ Bi uðt s2 ðtÞÞ Pr x_ ðtÞ ¼ i¼1 ; ð6Þ i¼1 i ðhðtÞÞ where hðtÞ ¼ ½h1 ðtÞ; . . . ; hp ðtÞT ; i : Rp ! ½0; 1; i ¼ 1; 2; . . . ; r are the membership functions of the system with respect to . In this paper, we assume that i ðhðtÞÞ P 0 for the ith plant rule, and hi ðhðtÞÞ ¼ Pri ðhðtÞÞ i ðhðtÞÞ i¼1 P Pr i ¼ 1; 2; . . . ; r and i¼1 i ðhðtÞÞ > 0 for all t. Therefore, hi ðhðtÞÞ P 0ði ¼ 1; 2; . . . ; rÞ and ri¼1 hi ðhðtÞÞ ¼ 1. Therefore, (6) can be represented by r X x_ ðtÞ ¼ hi ðhðtÞÞ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ Bi uðt s2 ðtÞÞ: ð7Þ 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 M i1 ; . . . ; and hp ðtÞ is M ip , then uðtÞ ¼ K i xðtÞ;
i ¼ 1; 2; . . . ; r:
Hence, the overall fuzzy control law is represented by r X hi ðhðtÞÞK i xðtÞ; i ¼ 1; 2; . . . ; r; uðtÞ ¼
ð8Þ
ð9Þ
i¼1
¼ 1; 2; . . . ; rÞ are the local control gains. When there exists an input delay s2 ðtÞ, we have that where K i ðiP uðt s2 Þ ¼ ri¼1 hi ðhðt s2 ðtÞÞÞK i xðt s2 ðtÞÞ. So, it is natural and necessary to make an assumption that the functions hi ðhðtÞÞði ¼ 1; 2; . . . ; rÞ are well defined for all t 2 ½s2 ; 0, and satisfy the following properties:
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L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
hi ðhðt s2 ðtÞÞÞ P 0 for i ¼ 1; 2; . . . ; r;
and
r X
hi ðhðt s2 ðtÞÞÞ ¼ 1:
i¼1
For simplicity, let us denote hi ðhðtÞÞ ¼ hi ;
hj ðhðt s2 ðtÞÞÞ ¼ hsj 2 :
The design of the fuzzy controller is to determine the feedback gains K i ði ¼ 1; 2; . . . ; rÞ such that the resulting closedloop system is asymptotically stable. With the control law (9), the overall closed-loop system can be written as r X r X x_ ðtÞ ¼ hi hsj 2 ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ Bi K j xðt s2 ðtÞÞ i¼1
¼
j¼1
r X
hi ½A1i xðtÞ þ A2i xðt s1 ðtÞÞ þ
i¼1
r X r X i¼1
ð10Þ
hi hsj 2 Bi K j xðt s2 ðtÞÞ;
j¼1
i ¼ 1; 2; . . . ; r: R0 By using the Newton–Lebuniz formula, xðt sÞ ¼ xðtÞ s x_ ðt þ aÞda, an equivalent form of system (10) is that " # " # Z t Z t r r X r X X s2 x_ ðtÞ ¼ x_ ðsÞds þ x_ ðsÞds : ð11Þ hi ðA1i þ A2i ÞxðtÞ A2i hi hj Bi K j xðtÞ Bi K j xðtÞ ¼ uðtÞ;
t 2 ½ maxðs1 ; s2 Þ; 0;
ts1 ðtÞ
i¼1
i¼1
j¼1
ts2 ðtÞ
The following result can be obtained similarly to the derivations in [19], which will be used to deal with the considered uncertain models. Lemma 1 [19]. Suppose that D is given by (3)–(5). Given matrices M ¼ M T , S and N of appropriate dimensions, the inequality M þ SDN þ N T DT S T < 0 holds for all F ðtÞ such that F ðtÞF T ðtÞ 6 I, if and only if, for some d > 0, 3 2 dM S dN T 7 6 T I J T 5 < 0: 4S dN J I Let d1 ¼ q, we will get the following lemma: Lemma 2. Suppose that D is given by (3)–(5). Given matrices M ¼ M T , S and N of appropriate dimensions, the inequality M þ SDN þ N T DT S T < 0 holds for all F ðtÞ such that F ðtÞF T ðtÞ 6 I, if and only if, for some q > 0, 3 2 M qS NT 7 6 T qI qJ T 5 < 0: 4 qS qJ
N
qI
Lemma 3 [20]. Assume that aðÞ 2 Rna , bðÞ 2 Rnb and NðÞ 2 Rna nb are defined on the interval X. Then, for any matrices X 2 Rna nb , and Z 2 Rna nb , the following holds: Z Z aðsÞ T X Y N aðsÞ T 2 a ðsÞNbðsÞds 6 ds; Z Y T NT bðsÞ X X bðsÞ where
X YT
Y Z
P 0:
3. Main result In this section, we focus on the problems of developing some delay-dependent stability criteria which provide the upper bounds of the delays such that the closed-loop system (10) is stable.
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
2333
Theorem 1. Consider the closed-loop system (10) and scalars s1 ; s2 ; d1 ; d2 . For some q > 0, if there exist X > 0, S 1 > 0, S 2 > 0, Q1 > 0, Q2 > 0; q > 0 and real matrices Y ; Z, W 11 ; W 12 ; W 13 ; W 21 ; W 22 ; W 23 ; K i satisfying the following equations: 3 2 Pij qM N Tij 7 6 ð12Þ 4 qM T qI qJ T 5 < 0; i; j ¼ 1; 2; . . . ; r; N ij
qJ
W 11 6 4
W 12 W 13
2
qI
3 0 7 d1 A2i X 5 P 0; X Q1 1 X
2
W 21 6 4
W 22 W 23
3 0 7 d2 Bi K j 5 P 0; X Q1 2 X
i; j ¼ 1; 2; . . . ; r;
ð13Þ
where 2
U11 6 6 6 6 Pij ¼ 6 6 6 6 4
U12 U22
0 U23
0 U24
s1 Z T s1 Y T
l1 S 1
0 l2 S 2
0 0 s1 Q1
3 s2 Z T s2 Y T 7 7 7 0 7 7; 0 7 7 7 0 5 s2 Q2
T
U11 ¼ Z þ Z þ S 1 þ S 2 þ s1 W 11 þ s2 W 21 ; U12 ¼ Y þ XAT1i Z T þ d1 XAT2i þ d2 K Tj BTi þ s1 W 12 þ s2 W 22 ; U22 ¼ Y Y T þ s1 W 13 þ s2 W 23 ; U23 ¼ ð1 d1 ÞA2i X ; U24 ¼ ð1 d2 ÞBi K j ; M T ¼ ½ 0 DT 0 0 0 0 ; N ij ¼ ½ E1ai X 1
Y ¼
X ¼P ; Q2 ¼
Q1 2 ;
P 1 2 ;
Z¼
1 P 1 ; 2 P 1P
l1 ¼ ð1 l1 Þ;
0 E2ai X
Ki ¼ KiX ;
0
Ebi K j X
S 1 ¼ XT 1 X ;
0 ; S 2 ¼ XT 2 X ;
Q1 ¼ Q1 1 ;
l2 ¼ ð1 l2 Þ;
and ‘‘*’’ denotes the transposed elements in the symmetric positions. Then, system (10) is asymptotically stable for any 0 6 si ðtÞ 6 si ði ¼ 1; 2Þ; moreover, the control gain matrix K i is given by K i ¼ K i X 1 ;
i ¼ 1; 2; . . . ; r:
Proof. Following [17,18], we represent (11) in the following equivalent descriptor system form: yðtÞ ¼ x_ ðtÞ; 0 ¼ yðtÞ þ 0¼
r X
"
r X
" hi ðA1i þ A2i ÞxðtÞ A2i
Z
ts1 ðtÞ
i¼1
hi ðA1i þ A2i ÞxðtÞ A2i
Z
t
Take the Lyapunov functional for system (14) as V ðtÞ ¼
5 X
V i ðtÞ
i¼1
with V 1 ðtÞ ¼ xT ðtÞPxðtÞ; Z t xT ðsÞT 1 xðsÞ ds; V 2 ðtÞ ¼ ts1 ðtÞ
V 3 ðtÞ ¼
Z
0
s1
Z
t
tþh
yðsÞds þ #
yðsÞds yðtÞ þ ts1 ðtÞ
i¼1
#
t
y T ðsÞQ1 yðsÞ ds dh;
r X r X i¼1
j¼1
r X
r X
i¼1
j¼1
"
Z
hi hsj 2
Bi K j xðtÞ Bi K j
"
Z
hi hsj 2 Bi K j xðtÞ Bi K j
#
t
yðsÞds ; ts2 ðtÞ t
#
yðsÞds : ts2 ðtÞ
ð14Þ
2334
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
V 4 ðtÞ ¼
Z
t
xT ðsÞT 2 xðsÞ ds;
ts2 ðtÞ
V 5 ðtÞ ¼
Z
0
s2
Z
t
y T ðsÞQ2 yðsÞ ds dh: tþh
Differentiating V i ðtÞ with respect to t and using Eq. (14) yields V_ 1 ðtÞ ¼ 2xT ðtÞP x_ ðtÞ ¼ 2xT ðtÞPyðtÞ ¼ 2gT ðtÞGT ½ y T ðtÞ 0 T ( ( ) Z t r X 0 I 0 T T ¼ hi 2g ðtÞG yðsÞ ds gðtÞ A1i þ A2i I A2i ts1 ðtÞ i¼1 ( ( ) Z t r X r X 0 0 0 s2 T T gðtÞ hi hj 2g ðtÞG yðsÞ ds ; þ Bi K j 0 Bi K j ts2 ðtÞ i¼1 j¼1 P 0 where gT ðtÞ ¼ ½ xT ðtÞ y T ðtÞ ; G ¼ . By Lemma 3, we have P 1 ZP 2 # Z t " t gðtÞ 0 gðtÞ T W 1 M 1 GT ½ 0 AT2i T T T yðsÞds 6 2g ðtÞG ds A2i ts1 ðtÞ yðsÞ Q1 ts1 ðtÞ yðsÞ Z t T ðxðtÞ xðt s1 ðtÞÞÞ; y T ðsÞQ1 yðsÞds þ 2gT ðtÞ M 1 GT 0 AT2i 6 s1 gT ðtÞW 1 gðtÞ þ ts1 ðtÞ W 11 W 12 , where W 1 ¼ W 13
W1
M1 Q1
ð15Þ
P 0:
Similarly, we have # Z t " Z t 0 gðtÞ gðtÞ T W 2 M 2 GT ½ 0 ðBi K j ÞT T T T yðsÞds 6 2g ðtÞG ds Bi K j ts2 ðtÞ yðsÞ Q2 ts2 ðtÞ yðsÞ Z t y T ðsÞQ2 yðsÞds þ 2gT ðtÞðM 2 GT ½ 0 ½Bi K j T T ÞðxðtÞ xðt s2 ðtÞÞÞ; 6 s2 gT ðtÞW 2 gðtÞ þ ts2 ðtÞ W 21 W 22 where W 2 ¼ , W 23 W 2 M2 P 0: Q2 And V_ 2 ðtÞ ¼ xT ðtÞT 1 xðtÞ ð1 s_ 1 ðtÞÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ 6 xT ðtÞT 1 xðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ; Z t V_ 3 ðtÞ 6 s1 y T ðtÞQ1 yðtÞ y T ðsÞQ1 yðsÞds; ts1 ðtÞ
V_ 4 ðtÞ ¼ xT ðtÞT 2 xðtÞ ð1 s_ 2 ðtÞÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ 6 xT ðtÞT 2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ; Z t V_ 5 ðtÞ 6 s2 y T ðtÞQ2 yðtÞ y T ðsÞQ2 yðsÞds: ts2 ðtÞ
Then, we have r X hi 2gT ðtÞGT V_ ðtÞ 6
0 I gðtÞ þ 2gT ðtÞ M 1 GT ½ 0 AT2i T ðxðtÞ xðt s1 ðtÞÞÞ A1i þ A2i I i¼1 r X r X 0 0 s2 T T T T T T gðtÞ þ 2g ðtÞðM 2 G ½ 0 ½Bi K j ÞðxðtÞ xðt s2 ðtÞÞÞ hi hj 2g ðtÞG þ Bi K j 0 i¼1 j¼1 þ s1 gT ðtÞW 1 gðtÞ þ xT ðtÞT 1 xðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ þ s2 gT ðtÞW 2 gðtÞ þ s1 y T ðtÞQ1 yðtÞ
þ s2 y T ðtÞQ2 yðtÞ þ xT ðtÞT 2 xðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ r X 0 I ¼ hi 2gT ðtÞGT gðtÞ þ 2gT ðtÞðM 1 GT ½ 0 AT2i T ÞðxðtÞ xðt s1 ðtÞÞÞ A1i þ A2i I i¼1
ð16Þ
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
þ
r X r X i¼1
j¼1
0 hi hsj 2 2gT ðtÞGT Bi K j
2335
0 gðtÞ þ 2gT ðtÞðM 2 GT ½ 0 ½Bi K j T T ÞðxðtÞ xðt s2 ðtÞÞÞ 0
T
þ s1 g ðtÞW 1 gðtÞ ð1 l1 ÞxT ðt s1 ðtÞÞT 1 xðt s1 ðtÞÞ þ s2 gT ðtÞW 2 gðtÞ ð1 l2 ÞxT ðt s2 ðtÞÞT 2 xðt s2 ðtÞÞ 0 T1 þ T2 gðtÞ þ gT ðtÞ s1 Q1 þ s2 Q2 0 r X r X hi hsj 2 nT ðtÞXij nðtÞ; 6 i¼1
j¼1
where nT ðtÞ ¼ ½ gT ðtÞ xT ðt s1 ðtÞÞ xT ðt s2 ðtÞÞ ; 3 2 Cð1; 1Þ Cð1; 2Þ Cð1; 3Þ 7 6 ð1 l1 ÞT 1 0 Xij ¼ 4 5; ð1 l2 ÞT 2 with
ð17Þ
" # T I 0 I M T1 Cð1; 1Þ ¼ G þ ½ M1 Gþ þ I A1i I 0 0 T1 þ T2 ; þ s1 W 1 þ s2 W 2 þ s1 Q1 þ s2 Q2 0 T
0 A1i
"
M T2 0 þ 0
# þ ½ M2
0
Cð1; 2Þ ¼ GT ½ 0 AT2i T M 1 ; Cð1; 3Þ ¼ GT ½ 0 ½Bi K j T T M 2 : 0 0 , M 2 ¼ d2 GT , where d1 ; d2 are scalars. Multiplying (17) by diagfðG1 ÞT ; X ; Let M 1 ¼ d1 GT Bi K j A2i X g and diagfG1 ; X ; X g on the left and on the right, respectively, by Schur complement the result inequality is obtained as following: 2 6 6 6 6 Kij ¼ 6 6 6 6 4
Kð1; 1Þ
Kð1; 2Þ
0
0
Kð2; 2Þ Kð2; 3Þ ð1 l1 ÞXT 1 X
Kð2; 4Þ 0 ð1 l2 ÞXT 2 X
3 s2 Z T s2 Y T 7 7 7 7 0 7; 7 0 7 7 5 0
s1 Z T s1 Y T 0 0 s1 Q1 1
ð18Þ
s2 Q1 2
where Kð1; 1Þ ¼ Z þ Z T þ X ðT 1 þ T 2 ÞX þ s1 W 11 þ s2 W 21 ; Kð1; 2Þ ¼ Y þ X ðAT1i þ DAT1i Þ Z T þ d1 XAT2i þ d2 XK Tj BTi þ s1 W 12 þ s2 W 22 ; Kð2; 2Þ ¼ Y Y T þ s1 W 13 þ s2 W 23 ; Kð2; 3Þ ¼ ð1 d1 ÞA2i X þ DA2i X ; Kð2; 4Þ ¼ ð1 d2 ÞBi K j X þ DBi K j X ; 1 1 : X ¼ P 1 ; Y ¼ P 1 2 ; Z ¼ P 2 P 1 P
Then (18) can be expressed as follows: 2 Pð1; 1Þ Pð1; 2Þ 0 6 Pð2; 2Þ Pð2; 3Þ 6 6 6 l1 XT 1 X Kij ¼ 6 6 6 6 4
0 Pð2; 4Þ 0
s1 Z T s1 Y T
l2 XT 2 X
0 s1 Q1 1
0
3 s2 Z T s2 Y T 7 7 7 7 0 7 þ MDN ij þ N T DT M T ; ij 7 0 7 7 5 0 s2 Q1 2
ð19Þ
2336
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
where Pð1; 1Þ ¼ Z þ Z T þ X ðT 1 þ T 2 ÞX þ s1 W 11 þ s2 W 21 ; Pð1; 2Þ ¼ Y þ XAT1i Z T þ d1 XAT2i þ d2 XK Tj BTi þ s1 W 12 þ s2 W 22 ; Pð2; 2Þ ¼ Y Y T þ s1 W 13 þ s2 W 23 ; Pð2; 3Þ ¼ ð1 d1 ÞA2i X ; Pð2; 4Þ ¼ ð1 d2 ÞBi K j X ; M T ¼ ½ 0 DT 0 0 0 0 ; N ij ¼ ½ E1ai X 0 E2ai X
0 0 : Applying Lemma 2, the resulting inequality (19) is equivalent to (12), that is V_ ðtÞ < 0. It means that the closed-loop system (10) is asymptotically stable. In the following, what we need to do is to consider the constraint of (15) and (16). Multiplying (15) and (16) by diagfðG1 ÞT ; X g and diagfG1 ; X g on the left and on the right, respectively, we have 3 3 2 2 W 11 W 12 0 W 21 W 22 0 7 7 6 6 ð20Þ W 13 d1 A2i X 5 P 0; 4 W 23 d2 Bi K j X 5 P 0: 4
X Q1 1 X
Ebi K j X
X Q1 2 X
The resulting inequality (20) is (13). The proof of the Theorem 1 is completed.
h
If D of (2) is F(t), that is J ¼ 0, the uncertainties in system are norm-bounded parameter uncertainties. For D ¼ F ðtÞ, we get the following theorem: Theorem 2. Consider the closed-loop system (10) and scalars s1 , s2 ; d1 ; d2 . If there exist X > 0, S 1 > 0, S 2 > 0, Q1 > 0, Q2 > 0, e > 0 and real matrices Y ; Z, W 11 ; W 12 ; W 13 ; W 21 ; W 22 ; W 23 ; K i satisfying the following equations: 3 2 s2 Z T s1 Z T U12 0 0 XET1ai U11 6 U þ eDDT s1 Y T s2 Y T U23 U24 0 7 7 6 22 7 6 T 7 6 l S 0 0 0 XE 1 1 2ai 7 6 6 l2 S 2 0 0 K Tj ETbi 7 ð21Þ 7 < 0; i; j ¼ 1; 2; . . . ; r; 6 7 6 7 6 s1 Q1 0 0 7 6 7 6 4 s2 Q2 0 5 2
W 11 6 4
W 12 W 13
3
0 7 d1 A2i X 5 P 0; X Q1 1 X
2 W 21 6 4
W 22 W 23
3
eI
0 7 d2 Bi K j 5 P 0; X Q1 2 X
i; j ¼ 1; 2; . . . ; r:
ð22Þ
where e ¼ e1 , U11 ; U12 ; U22 ; U23 ; U24 are same as Theorem 1. And ‘‘*’’ denotes the transposed elements in the symmetric positions. Then, system (10) is asymptotically stable for any 0 6 si ðtÞ 6 si ði ¼ 1; 2Þ; moreover, the control gain matrix K i is given by K i ¼ K i X 1 ;
i ¼ 1; 2; . . . ; r:
It is noted that the resulting conditions for robust stabilization in Theorems 1 and 2 are no longer LMIs conditions 1 because of the nonlinear term X Q1 1 X ; X Q2 X in (13) and (22). Because the way to solve Theorem 1 is similar to the way to solve Theorem 2, we mainly analysis the way to solve Theorem 1. An easy way to solve (12) and (13) is simply to set Q1 ¼ a1 X , Q2 ¼ a2 X , where a1 ; a2 > 0 is a tuning parameter. Then by applying a simple one-dimensional search over the positive variable a1 ; a2 , we can find the feasible solution of the resulting LMI conditions. However, this method may increase the conservativeness. Similar to [21], another less conservative approach can be developed using an iterative algorithm next. For the sake of simplicity, we only consider how to apply the algorithm to solve matrix inequalities (12) and (13). First, we need to introduce a new variable L1 ; L2 > 0 such that 1 X Q1 1 X P L1 ; X Q2 X P L2 ;
and replace the condition (13) with 3 2 W 11 W 12 0 7 6 W 13 d1 A2i X 5 P 0; 4
L1
ð23Þ (23) and 2 W 21 W 22 6 W 23 4
3 0 7 d2 Bi K j 5 P 0: L2
ð24Þ
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
2337
1 1 1 1 1 Since (23) is equivalent to X 1 Qi X 1 6 L1 , and i ði ¼ 1; 2Þ, by setting L1 ¼ L1 ; L2 ¼ L2 ; Q1 ¼ Q1 ; Q2 ¼ Q2 ; P ¼ X using Schur complement, condition (13) can be replaced with (24) and " # " # L1 P L2 P P 0; P 0; ð25Þ P Q1 P Q2
L1 L1 ¼ I;
L2 L2 ¼ I;
XP ¼ I;
Q1 Q1 ¼ I;
Q2 Q2 ¼ I:
ð26Þ
Note that condition (26) still includes nonlinear conditions. However, using the idea in a cone complementary linearization algorithm of [22], the problem of finding a feasible solution of the non-convex matrix inequalities (12), (24), (25), (26) can be considered as a cone complementary problem involving LMIs conditions Minimize TraceðL1 L1 þ L2 L2 þ XP þ Q1 Q1 þ Q2 Q2 Þ; Subject to ð12Þ; ð24Þ; ð25Þ and L1 I L2 I X I P 0; P 0; P 0; I L1 I L2 I P " # " # Q1 I Q2 I P 0; P 0: I Q1 I Q2
ð27Þ
The above cone complementary problem (27) can be solved using the following algorithm. Algorithm 1. (i) Find a feasible solution fq0 ; Y 0 ; Z 0 ; K i0 ; W 110 ; W 120 ; W 130 ; W 210 ; W 220 ; W 230 ; S 10 ; S 20 ; X 0 ; P 0 ; L10 ; L10 ; L20 ; L20 ; Q10 ; Q10 ; Q20 ; Q20 g of the LMIs in (27). If there are none, then exit. Set k ¼ 0. (ii) Solve the following LMI optimization problem for the variables fq; Y ; Z; K i ; W 11 ; W 12 ; W 13 ; W 21 ; W 22 ; W 23 ; S 1 ; S 2 ; X ; P ; L1 ; L1 ; L2 ; L2 ; Q1 ; Q1 ; Q2 ; Q2 g: Minimize
Trace Nk ;
subject to LMIs in ð27Þ; where Nk ¼ ðL1k L1 þ L1k L1 þ L2k L2 þ L2k L2 þ X k P þ P k X þ Q1k Q1 þ Q1k Q1 þ Q2k Q2 þ Q2k Q2 Þ. Then set L1kþ1 ¼ L1 ; L1kþ1 ¼ L1 ; L2kþ1 ¼ L2 ; L2kþ1 ¼ L2 ; X kþ1 ¼ X ; P kþ1 ¼ P ; Q1kþ1 ¼ Q1 ; Q1kþ1 ¼ Q1 ; Q2kþ1 ¼ Q2 ; Q2kþ1 ¼ Q2 . (iii) If the above LMI optimization problem leads to positive definite matrices X ; Q such that the resulting LMIs (12) and (13) are feasible, then exit. Otherwise, set k ¼ k þ 1, and go to step (ii). Remark 2. Theorems 1 and 2 provide some new sufficient conditions of the delay-dependent fuzzy systems with state and input delays. The methods are based on the descriptor model transformation [17,18] and a recent result on bounding of cross products of vectors. We would like to emphasize that, to the best of our knowledge, so far there has been no result appeared in the literature for the fuzzy controller design by using the descriptor model transformation. Our results given here are less conservative for the case. Remark 3. By Algorithm 1, we can get optimal delay upper bounds with the cone complementary linearization algorithm [21,22] for solving the stabilization problems of time-varying fuzzy systems.
4. Numerical examples In this section, we present several numerical examples to illustrate the usefulness of the proposed theoretical results. Example 1. Consider the nonlinear mass-spring-damper mechanical systems in [13]. The state-space is represented as following: x_ 1 ðtÞ 0:75x31 ðtÞ 0:5x1 ðtÞ þ 0:1ð1 þ 0:11ðtÞÞx2 ðtÞ þ uðt sðtÞÞ ; ð28Þ ¼ x_ 2 ðtÞ x1 ðtÞ where j1ðtÞj2 6 1. The system (28) has one nonlinear term 0:75x31 ðtÞ. Now by adopting fuzzy sets, the T–S fuzzy system of (28) can be constructed as follows:
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L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
Table 1 A comparison for allowable input delay Methods
In [13]
In [14]
In this paper
s
0.4223
1.75
3.06
Rule 1: If x1 ðtÞ is about C1 ; then x_ ðtÞ ¼ ðA11 þ DA11 ÞxðtÞ þ ðB1 þ DB1 Þuðt sðtÞÞ, Rule 2: If x1 ðtÞ is about C2 , then x_ ðtÞ ¼ ðA12 þ DA12 ÞxðtÞ þ ðB2 þ DB2 Þuðt sðtÞÞ, where the associated matrices are given by 0:5 0:1 1 0:1 0 0:11ðtÞ 1 ; A12 ¼ ; DA11 ¼ DA12 ¼ ; B1 ¼ B2 ¼ ; A11 ¼ 1 0 1 0 0 0 0 " pffiffiffiffiffiffiffi # pffiffiffiffiffiffiffi 0:1 ; E1a1 ¼ E1a2 ¼ ½ 0 D1 ¼ D2 ¼ 0:1 ; Eb1 ¼ Eb2 ¼ ½ 0 0 ; J ¼ 0: 0 Applying our algorithm to this paper, system (28) achieves stability for all delay 0 < s < 3:06 when d2 ¼ 0:95. Table 1 lists the maximum upper bound of input delay for the system (28) in [13,14] and this paper. When s < 3:06, the feedback gain is K 1 ¼ ½ 0:4669 0:2856 ; K 2 ¼ ½ 0:0501 0:0679 . Example 2. In the following, we will consider the truck-trailer system [15], which can be described by the following T–S fuzzy system: vt x1 ðtÞ is about 0, THEN Rule 1: IF hðtÞ ¼ x2 ðtÞ þ 2L x_ ðtÞ ¼ ðA1 þ DA1 ÞxðtÞ þ ðB1 þ DB1 Þuðt sÞ: vt Rule 2: IF hðtÞ ¼ x2 ðtÞ þ 2L x1 ðtÞ is about p or p, THEN x_ ðtÞ ¼ ðA2 þ DA2 ÞxðtÞ þ ðB2 þ DB2 Þuðt sÞ,
where 2 6 A1 ¼ 6 4
Ltvt0 vt Lt0 2 2
v t 2Lt 0
3
2
7 0 07 5; vt 0 t0
6 A2 ¼ 6 4
0 0
3
Ltvt0
0
0
vt Lt0
0
7 07 5;
2 2
v t d 2Lt 0
dvt t0
2
vt lt0
3
6 7 B1 ¼ B2 ¼ 4 0 5;
0
0
with a ¼ 0:7; v ¼ 1:0; t ¼ 2:0; t0 ¼ 0:5; L ¼ 5:5; l ¼ 2:8; d ¼ 10 tp0 . D1 ¼ D2 ¼ ½ 0:255 0:255
0:255 T ;
E1a1 ¼ E1a2 ¼ ½ 0:1 0 0 ;
Eb1 ¼ Eb2 ¼ ½0:15;
The following fuzzy rules are employed 1 1 h1 ðxÞ ¼ 1 and 1 expð3ðhðtÞ 0:5pÞÞ 1 expð3ðhðtÞ þ 0:5pÞÞ
J ¼ 0:
h2 ðxÞ ¼ 1 h1 ðxÞ:
Applying our algorithm to this paper, the maximal allowable input delay is 5.92 for the proposed fuzzy systems with input delay when d2 ¼ 1:9. But in [15], the maximal allowable input delay is s ¼ 0:75 for the same fuzzy systems. In order to show our result less conservative, we compare the feedback gain matrices and the maximal allowable input delay in Table 2.
Table 2 Comparison of the feedback gain for different input delay Methods
s
K1
K2
Chen et al. [15] Our result
0.75 0.75 5.92
½ 3:4227 0:3535 0:0045 ½ 0:2007 0:0859 1:1886 ½ 0:2282 0:0674 19:8924
½ 3:5215 0:3617 0:0056 ½ 0:2040 0:0990 1:4765 ½ 0:2262 0:1140 20:0603
L. Li, X. Liu / Chaos, Solitons and Fractals 40 (2009) 2329–2339
2339
5. Conclusion By using a type of new Lyapunov function and the descriptor system method, we establish some new delay-dependent criteria for robust stabilization of uncertain fuzzy systems with state and input delays. The numerical algorithm underlying our controller design methodology involves solving a series of parameterized LMI problems. Numerical examples are given to demonstrate the superiority of the proposed method. Furthermore, our controller design methodology leads to a state feedback controller which can stabilize a larger class of uncertain systems than the existing methods.
Acknowledgements This work is supported in part by the National Natural Science Foundation of China under Grant 60575039 and in part by the National Key Basic Research and Development Program of China under Grant 2002CB312201-06.
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