Robust H∞ control of Takagi–Sugeno fuzzy systems with state and input time delays

Fuzzy Sets and Systems 160 (2009) 403 – 422 www.elsevier.com/locate/fss

Robust H ∞ control of Takagi–Sugeno fuzzy systems with state and input time delays Bing Chena,∗ , Xiaoping Liub , Chong Lina , Kefu Liub a Institute of Complexity Science, Qingdao University, Qingdao 266000, PR China b Faculty of Electrical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1

Received 13 April 2007; received in revised form 25 March 2008; accepted 28 March 2008 Available online 8 April 2008

Abstract This paper addresses the robust H∞ fuzzy control problem for nonlinear uncertain systems with state and input time delays through Takagi–Sugeno (T–S) fuzzy model approach. The delays are assumed to be interval time-varying delays, and no restriction is imposed on the derivative of time delay. Based on Lyapunov–Krasoviskii functional method, delay-dependent sufficient conditions for the existence of an H∞ controller are proposed in linear matrix inequality (LMI) format. Illustrative examples are given to show the effectiveness and merits of the proposed fuzzy controller design methodology. Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved. Keywords: H∞ control; T–S fuzzy system; Time delay; Delay-dependent stabilization; Uncertainty

1. Introduction During the last two decades, the problem of stability analysis and stabilization of nonlinear systems in Takagi and Sugeno [21] fuzzy model has been extensively studied, and a lot of significant results on stabilization and H∞ control via linear matrix inequality (LMI) approach have been reported, see [1–3,22,23,7,8,5,18,25,35,34,27], and the reference therein. So far, Takagi–Sugeno (T–S) fuzzy model has become a popular and effective approach to control complex and ill-defined systems for which the application of conventional techniques is infeasible. In view of time delays frequently occurring in practical dynamic systems, T–S fuzzy model was first used to deal with the stability analysis and control synthesis of nonlinear time delay systems in [4]. Afterwards, many people have devoted a great deal of effort to both theoretical research and implementation techniques for T–S fuzzy systems with time delays. In these works, stability analysis and fuzzy controller design methods are provided in the sense of delay-independent stability [14,33], and in the sense of delay-dependent stability [9,15,16,27]. Generally speaking, the delay-independent approach provides the stability conditions irrespective of the size of the delay. As a result, it may be lead to comparatively conservative stability analysis results, particularly when the delay is small. More recently, further delay-dependent stabilization results are presented for T–S fuzzy systems with interval time delay in [13,24,12]. The robust stabilization problem has been addressed in [24], and LMI-based stability criteria and stabilization conditions have been proposed based on Lyapunov–Krasoviskii functional method, while the H∞ control ∗ Corresponding author. Tel.: +86 532 85953607.

E-mail address: [email protected] (B. Chen). 0165-0114/$ - see front matter Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2008.03.024

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B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

problem has been studied in [13,12], and some sufficient conditions for the existence of an H∞ controller have been presented by means of LMI format. However, all the aforementioned results have not considered the effect of control input delay on the resulting systems. Hence, these approaches mentioned above may be invalid when they are used to control the systems with input delays. As well-known, input delays are often encountered in many industrial processes. The presence of input delay, if not taken into account in the controller deign, may cause instability or serious deterioration in the performance of the resulting control systems. In addition, in modern industrial systems, sensors, controllers and plants are often connected over a network medium. The controller signals are transmitted through a network, therefore, the network-induced time delay is usually inevitable. In view of this, more recently, many controller design schemes have been presented for linear systems with input delay [19,32,31]. Then, there are a few results on stabilization of T–S fuzzy systems with state and input delays. To the best of author’s knowledge, the literature by Lin et al. [17] is the only one, which deals with the stability and stabilization of T–S fuzzy systems with state and input delays. Based on Razumikhin Theorem, the delay-dependent stability and stabilization conditions have been developed in terms of LMIs. However, uncertainties and external perturbations have not been considered in [17]. Motivated by the above concerns, we will consider the problem of robust H∞ control for T–S fuzzy systems with both state and input delays. It is assumed that state and input delays are time-varying delays with both upper and lower bounds. there is no restriction imposed on the derivative of time delay. Delay-dependent sufficient conditions for stability and stabilization will be derived in terms of LMI format, and systematic design procedure for an H∞ fuzzy controller will be proposed for uncertain T–S fuzzy systems with state and input delays. At last, three examples are given to illustrate the effectiveness and feasibility of our results. Throughout this paper, identity matrix, of appropriate dimensions, will be denoted by I . The notation X > 0 (respectively, X 0), for X ∈ Rn×n means that the matrix X is real symmetric positive definite (respectively, positive semi-definite). If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. The symbol “ *” in a matrix A ∈ Rn×n stands for the transposed elements in the symmetric positions. The superscripts “ T” and “ −1” denote the matrix transpose and inverse, respectively. 2. Problem statement Consider a nonlinear time delay system represented by T–S fuzzy model as follows: Plant Rule i: IF 1 (t) is Ni1 · · ·  p (t) is Ni p THEN x(t) ˙ = (Ai + A1i (t))x(t) + (A1i + A1i (t))x(t − (t)) + Bwi w(t) +(Bi + Bi (t))u(t) + (B1i + B1i (t))u(t − (t)), z(t) = Ci x + C1i x(t − (t)) + Di u(t) + D1i u(t − (t)), x(t) = (t), t ∈ [−d, 0], where Ni j is the fuzzy set, x(t) ∈ Rn stands for the state vector, u(t) ∈ Rm denotes the control input vector, z(t) ∈ Rq is the controlled output, and w(t) ∈ R p , which is assumed to belong to L 2 [0, ∞), denotes the external perturbation. Ai , A1i ∈ Rn×n , Bi , B1i ∈ Rn×m , Bwi ∈ Rn×d , Ci and C1i ∈ Rq×s , Di and D1i ∈ Rq×m are known constant matrices. Constant d is the upper bound of (t) and (t). Scalar k is the number of IF–THEN rules. 1 (t), 2 (t), . . . ,  p (t) are the premise variables. It is assumed that the premise variables do not depend on the input u(t). Ai (t), A1i (t), Bi (t) and B1i (t) denote the time-varying uncertain matrices satisfying [Ai , A1i , Bi , B1i ] = G i Fi (t)[E i , E 1i , E bi , E b1i ], where G i , E i , E 1i , E bi and E b1i are known constant matrices of appropriate dimensions. F(t) is an unknown matrix function satisfying the inequality F T (t)F(t) I . (t) and (t) denote state and input delays, respectively. In this paper, delays (t) and (t) are assumed to be interval type delay, i.e., there exist known positive constants m ,  M , m and  M such that m  (t) M , m  (t) M ,

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

405

and let d = max{ M ,  M }. Given a pair of (x(t), u(t)), the final output of the fuzzy system is inferred as follows: x(t) ˙ =

k 

h i ((t))[(Ai + Ai )x(t) + (A1i + A1i )x(t − (t))

i=1

+(Bi + Bi )u(t) + (B1i + B1i )u(t − (t)) + Bwi w(t)], z(t) =

k 

h i ((t))[Ci x + C1i x(t − (t)) + Di u(t) + D1i u(t − (t))],

i=1

x(t) = (t), t ∈ [−d, 0], (1) k p where h i ((t)) = i ((t))/ i=1 i ((t)), i ((t)) = j=1 Ni j ( j (t)) and Ni j ( j (t)) is the degree of the membership k of  j (t) in fuzzy set Ni j . In this paper, it is assumed that i ((t))0 for i = 1, 2, . . . , k and i=1 i ((t)) > 0 for all k t. Therefore, h i ((t))0 (for i = 1, 2, . . . , k) and i=1 h i ((t)) = 1. For simplicity, the following notations will be used: A¯ i j = Ai j + Ai j ,

Ai j = Ai + Bi K j , Ai j = Ai + Bi K j ,

A¯ 1i = A1i + A1i ,

B¯ 1i = B1i + B1i , h i = h i ((t)),

h i () = h i ((t − )),  = (t),  = (t). To design a state feedback fuzzy controller, the ith fuzzy control rule is presented as follows: Control Rule i: IF 1 (t) is Ni1 · · ·  p (t) is Ni p THEN u(t) = K i x(t), i = 1, 2, . . . , k. Hence, the overall fuzzy control law is represented by u(t) =

k 

h i ((t))K i x(t),

(2)

i=1

where K i (i = 1, 2, . . . , k) are the local control gains. k h i ((t −))K i x(t −). Therefore, a natural and essential Remark 1. The existence of input delay leads to the term i=1 k h i ((t)) = 1 with assumption is that all h i ((t))’s are well defined for t ∈ [− M , 0], and also satisfy the equality i=1 h i ((t))0 for t ∈ [− M , 0] for i = 1, 2, . . . , k. Associated with the control law (2), the resulting closed-loop system can be expressed as follows: ⎧⎛ ⎞ k k ⎨   x(t) ˙ = h i ((t)) ⎝ Ai + Ai + h j ((t))(Bi + Bi )K j ⎠ x(t) ⎩ i=1 j=1 ⎫ k ⎬  +(A1i + A1i )x(t − ) + h s ((t − ))(B1i + B1i )K s x(t − ) + Bwi w(t) ⎭ s=1

=

k  k k  

h i h j h s ()( A¯ i j x(t) + A¯ 1i x() + B¯ 1i K s x(t − ) + Bwi w(t)),

(3)

h i h j h s ()((Ci + Di K j )x + C1i x(t − ) + D1i K s x(t − )).

(4)

i=1 j=1 s=1

z(t) =

k  k  k  i=1 j=1 s=1

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B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

Definition. Systems (3)–(4) is said to be robustly asymptotically stable with an H∞ norm bound  > 0 if there exists a feedback control law u(t) such that the following holds: (1) System (3) with w(t) ≡ 0 is asymptotically stable. (2) Under the assumption of zero initial condition, the controlled output z(t) satisfies z(t)2  w(t)2 for any nonzero w(t) ∈ L 2 [0, ∞). k K i x(t) such that systems (3)–(4) The objective of this paper is to determine the feedback control law u(t) = i=1 to be robust asymptotically stable with an H∞ norm bound  > 0. To state our main results, the following lemmas are useful in the proofs of our results. Lemma 1 (Park and Kwon [20]). For any constant matrix M > 0, any scalars a and b with a < b, and any vector function x(t) : [a, b] → Rn such that the integrals concerned are well defined, then  b T  b   b x(s) ds M x(s) ds (b − a) x T (s)M x(s) ds. a

a

a

Lemma 2 (Wang et al. [26]). Let D, E, and F(t) be real matrices of appropriate dimensions, and F(t) satisfying F T (t)F(t) I. Then, the following inequality holds for any constant  > 0: D F(t)E + E T F T (t)D T  D D T + −1 E T E. 3. H∞ performance analysis In this section, we will derive the delay-dependent LMI conditions for H∞ performance analysis for the systems (3)–(4). To this end, define 0 = 21 ( M + m ), = 21 ( M − m ), 0 = 21 ( M + m ), = 21 ( M − m ).

(5)

Then, by the above notations, we have that (t) ∈ [0 − , 0 + ], 0  , (t) ∈ [0 − , 0 + ] and 0  . Apparently, when = 0, i.e., m =  M , (t) becomes a constant delay. If = 0 , then the routine case for the time delay, i.e., 0 (t) M , is covered. Obviously, the similar results also hold for (t) in the cases of = 0 and = 0 . In the following, it is assumed that the feedback gain matrices K i (i = 1, 2, . . . , k) are known. Then, the following conclusion can be obtained. Theorem 1. For given scalars 0 > 0, > 0, 0 > 0, > 0 and  > 0, as well as the given matrices K i (i = 1, 2, . . . , k), systems (3)–(4) are robustly asymptotically stable with an H∞ norm bound  if there exist matrices Tl > 0, Hl > 0, P11 , P12 , P13 , P22 , P23 , P33 , Q q , Nq , L q , Mq and Wq (l = 1, 2, 3, 4, q = 1, 2, . . . , 6) so that the following LMIs hold, for 1i, j, s  k: ⎡ ⎤ (rl ) +  + T Q Bwi ⎢ ⎥ ∗ − 0 ⎦ < 0, (6) ⎣ ⎡

∗ P11 P12 P13

⎢ ⎣ ∗ ∗

⎤

∗

−2 I

⎥ P22 P23 ⎦ > 0, ∗ P33

where  = diag



 1 1 1 1 1 1 T2 , T3 , T4 , H2 , H3 , H4 , 0 0 0 0

(7)

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

407

= [Z 1 − N − L Z 2 − M − W ], = [Ci + Di K j C1i 0 0 D1i K s 0], T T T Z 1T = [P22 + P23 0 − P22 P12 0 − P23 ],

Q T = [Q T1 Q T2 Q T3 Q T4 Q T5 Q T6 ],

T T T T T Z 2T = [P23 + P33 0 − P23 P13 0 − P33 ],

N T = [N1T N2T N3T N4T N5T N6T ],

L = T

[L T1

L T2 L T3 L T4 L T5 [W1T W2T W3T W4T

=

L T6 ], M T W5T W6T ],

[M1T

M2T M3T M4T M5T M6T ],

WT =  = QG i Fi (t)E i j + (QG i Fi (t)E i j )T , E i j = [E i + E bi K j E 1i 0 0 E b1i K s 0], and (rl ) is a symmetric block-matrix with rl being its element at the position (r, l), and defined as follows for 1r , l 6: T T + P13 + P13 + T1 + H1 + 0 T2 + 0 H2 + N1 + N1T + M1 + M1T + Q 1 Ai j + AiTj Q T1 , 11 = P12 + P12

12 = N2T + M2T − L 1 + Q 1 A1i + AiTj Q T2 , 13 = −P12 − N1 + N3T + M3T + L 1 + AiTj Q T3 , 14 = P11 + N4T + M4T − Q 1 + AiTj Q T4 , 15 = N5T + M5T − W1 + Q 1 B1i K s + AiTj Q T5 , 16 = −P13 + N6T − M1 + M6T + W1 + AiTj Q T6 , 22 = −L 2 − L T2 + Q 2 A1i + AT1i Q T2 , 23 = −N2 + L 2 − L T3 + AT1i Q T3 , 24 = −L T4 − Q 2 + AT1i Q T4 , 25 = −L T5 − W2 + Q 2 B1i K s + AT1i Q T5 , 26 = −M2 − L T6 + W2 + AT1i Q T6 , 33 = −N3 − N3T + L 3 + L T3 − T1 , 34 = −N4T + L T4 − Q 3 , 35 = −N5T + L T5 − W3 + Q 3 B1i K s , 36 = −N6T − M3 + L T6 + W3 , 44 = 0 T3 + 0 H3 + 2 T4 + 2 H4 − Q 4 − Q T4 , 45 = −W4 + Q 4 B1i K s − Q T5 , 46 = −M4 + W4 − Q T6 , T T Q5 , 55 = −W5 − W5T + Q 5 B1i K s + K sT B1i T T Q6 , 56 = −M5 + W5 − W6T + K sT B1i

66 = −H1 − M6 − M6T + W6 + W6T . Proof. Consider the following Lyapunov function candidate: V = V1 + V2 + V3 , where

(8) 



t

t



V1 = x P11 x + 2x P12 x(s) ds + 2x P13 x(s) ds + t−0 t−0  t  t  t  t T T x (s) ds P23 x(s) ds + x (s) ds P33 +2 T

 V2 =

T

t−0

t

t−0  t

+

T

t−0  0

x T (s)T1 x(s) ds +

t−0

−0



x T (s)H1 x(s) ds +



t−0

t

x T ()T2 x() d ds

t+s 0  t −0

t+s

x T ()H2 x() d ds,



t

T

t−0

t−0

x (s) ds P22

x(s) ds,

t

t−0

x(s) ds

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B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

 V3 =



0

−0



+



t

x˙ ()T3 x() ˙ d ds + T

t+s 0

−0



t

−0 +  t

x˙ T ()T4 x() ˙ d ds

−0 − t+s  −0 +  t

x˙ T ()H3 x() ˙ d ds +

−0 −

t+s

x˙ T ()H4 x() ˙ d ds.

t+s

Differentiating V gives T T V˙ = x T (t)(P12 + P12 + P13 + P13 + T1 + H1 + 0 T2 + 0 H2 )x(t) T T ˙ − 2x (t)P12 x(t − 0 ) − 2x T (t)P13 x(t − 0 ) +2x (t)P11 x(t) −x T (t − 0 )T1 x(t − 0 ) − x T (t − 0 )H1 x(t − 0 ) ˙ +x˙ T (t)(0 T3 + 2 T4 + 0 H3 + 2 H4 )x(t)  t  t T ) x(s) ds + 2x T (t)(P23 + P33 ) x(s) ds +2x T (t)(P22 + P23 t−0 t−0  t  t x(s) ds − 2x T (t − 0 )P23 x(s) ds −2x T (t − 0 )P22 t−0 t−0  t  t T x(s) ds − 2x T (t − 0 )P33 x(s) ds −2x T (t − 0 )P23 t−0 t−0  t  t +2 x˙ T (t)P12 x(s) ˙ ds + 2 x˙ T (t)P13 x(s) ˙ ds t−0 t−0  t  t x T (s)T2 x(s) ds − x T (s)H2 x(s) ds −

 −  −

t−0 t t−0 t

t−0

 x˙ T (s)T3 x(t) ˙ ds −

t−0 t−0 +

x˙ T (s)T4 x(t) ˙ ds

t−0 − t−0 +

 x˙ T (s)H3 x(t) ˙ ds −

t−0 −

x˙ T (s)H4 x(t) ˙ ds.

(9)

Define eT (t) = [x T (t) x T (t − ) x T (t − 0 ) x˙ T (t) x T (t − ) x T (t − 0 )], T T T Z 1T = [P22 + P23 0 − P22 P12 0 − P23 ], T T T T T Z 2T = [P23 + P33 0 − P23 P13 0 − P33 ].

Then, (9) can be rewritten as follows:   0  1 e(t) V˙ = eT (t) ∗ 2  t  x(s) ds + 2e(t)T Z 2 +2e(t)T Z 1  −  −

t−0

t t−0 t t−0



x T (s)H2 x(s) ds −  x˙ T (s)H3 x(t) ˙ ds −

t

t t−0

 x(s) ds − 

x˙ T (s)T3 x(t) ˙ ds −

t−0 t−0 +

t−0 −

t−0 −

x˙ T (s)H4 x(t) ˙ ds

T T 0 = P12 + P12 + P13 + P13 + T1 + H1 + 0 T2 + 0 H2 ,

2 = diag(0, −T1 , , 0, −H1 ),  = 0 T3 + 2 T4 + 0 H3 + 2 H4 .

x T (s)T2 x(s) ds

t−0 t−0 +

with 1 = [0 − P12 P11 0 − P13 ],

t

x˙ T (s)T4 x(t) ˙ ds (10)

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

409

To establish the delay-dependent stability conditions, the free-weighting matrix approach [10,11] will be used in the following. By Newton–Leibniz formula, we have the equality below:  t 0 = x(t) − x(t − 0 ) − x(s) ˙ ds. t−0

Furthermore, define a matrix N T = [N1T N2T N3T N4T N5T N6T ] with appropriate dimensions. Then, the following equality can be verified easily:    t T 0 = 2e (t)N x(t) − x(t − 0 ) − x(s) ˙ ds t−0  t = 2eT (t)N [I 0 − I 0 0 0]e(t) − 2eT (t)N x(s) ˙ ds t−0  t = eT (t)( N + TN )e(t) − 2eT (t)N x(s) ˙ ds (11) t−0

with  N = [N 0 − N 0 0 0]. Similarly, the following equalities (12)–(15) can be obtained. For matrix L T = [L T1 L T2 L T3 L T4 L T5 L T6 ] with L i having compatible dimensions,    t−0 T x(s) ˙ ds 0 = 2e (t)L x(t − 0 ) − x(t − ) −  t−t−0 x(s) ˙ ds, = eT (t)( L + TL )e(t) − 2eT (t)L

(12)

where  L = [0 − L L 0 0 0]. For matrix M T = [M1T M2T M3T M4T M5T M6T ] with Mi being of compatible dimensions,    t 0 = 2eT (t)M x(t) − x(t − 0 ) − x(s) ˙ ds t−0 t T T T x(s) ˙ ds, = e (t)( M +  M )e(t) − 2e (t)M

(13)

where  M = [M 0 0 0 0 − M]. For matrix W T = [W1T W2T W3T W4T W5T W6T ] with Wi having compatible dimensions,    t−0 x(s) ˙ ds 0 = 2eT (t)W x(t − 0 ) − x(t − ) − t−  t− 0 = eT (t)(W + TW )e(t) − 2eT (t)W x(s) ˙ ds,

(14)

t−

t−0

t−

where W = [0 0 0 0 − W W ]. Finally, for matrix Q T = [Q T1 Q T2 Q T3 Q T4 Q T5 Q T6 ] with Q i having compatible dimensions, ⎛ ⎞ k  k k   0 = 2eT (t)Q ⎝ h i h j h s ()( A¯ i j x(t) + A¯ 1i x(t − ) + B¯ 1i K s x(t − ) + Bwi w(t)) − x(t) ˙ ⎠ i=1 j=1 s=1

=

k  k k  

h i h j h s ()2eT (t)Q[Ai j A1i 0 − I B1i K s 0]e(t)

i=1 j=1 s=1 k  k k  

+

h i h j h s ()2eT (t)Q Mi Fi (t)[E i + E bi K j E 1i 0 0 E b1i K s 0]e(t)

i=1 j=1 s=1

−

k  k k  

h i h j h s ()2QeT (t)Bwi w(t)

i=1 j=1 s=1

=

k  k  k  i=1 j=1 s=1

h i h j h s ()eT (t)( + )e(t) −

k  k  k  i=1 j=1 s=1

h i h j h s ()2QeT (t)Bwi w(t),

(15)

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B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

where  = QG i Fi (t)E i j + (QG i Fi (t)E i j )T with E i j = [E i + E bi K j E 1i 0 0 E b1i K s 0], and  = (rl ) is a symmetric block-matrix and its block-elements rl are defined as follows for 1 r , l 6: 11 = Q 1 Ai j + AiTj Q T1 , 12 = Q 1 A1i + AiTj Q T2 , 13 = AiTj Q T3 , 14 = −Q 1 + AiTj Q T4 , 15 = Q 1 B1i K s + AiTj Q T5 , 16 = AiTj Q T6 , 22 = Q 2 A1i + AT1i Q T2 , 23 = AT1i Q T3 , 24 = −Q 2 + AT1i Q T4 , 25 = Q 2 B1i K s + AT1i Q T5 , 26 = AT1i Q T6 , 33 = 0, 34 = −Q 3 , 35 = Q 3 B1i K s , 36 = 0, 44 = −Q 4 − Q T4 , 45 = Q 4 B1i K s − Q T5 , T T T T 46 = −Q T6 , 55 = −Q 5 B1i K s − K sT B1i Q 5 , 56 = K sT B1i Q 6 , 66 = 0.

Then, combining (10)–(15) yields the following equality: V˙ =

k  k  k 

h i h j h s ()eT (t)((rl ) + )e(t)

i=1 j=1 s=1

 t  t  t +2e(t)T Z 1 x(s) ds + 2e(t)T Z 2 x(s) ds − 2eT (t)N x(s) ˙ ds t−0 t−0 t−0  t−0  t  t−0 −2eT (t)L x(s) ˙ ds − 2eT (t)M x(s) ˙ ds − 2eT (t)W x(s) ˙ ds t− t− t−0  t  t  t x T (s)T2 x(s) d − x T (s)H2 x(s) ds − x˙ T (s)T3 x(s) ˙ ds −  − −

t−0 t

t−0  t−0 +

x˙ (s)H3 x(s) ˙ ds − T

t−0 k  k  k 

t−0 −

t−0



x˙ (s)T4 x(s) ˙ ds −

t−0 +

T

t−0 −

x˙ T (s)H4 x(s) ˙ ds

h i h j h s ()2eT (t)Q Bwi w(t),

(16)

i=1 j=1 s=1

where matrix (rl ) and  are the same ones defined in (6). Note that T4 > 0. Thus when  − 0 0, we have  t−0   t−0 + x˙ T (s)T4 x(s) ˙ ds = − x˙ T (s)T4 x(s) ˙ ds − − t−0 −

−  −

t−0 −  t−0 +

t−  t− t−0

t− t−0

x˙ T (s)T4 x(s) ˙ ds

x˙ T (s)T4 x(s) ˙ ds

x˙ T (s)T4 x(s) ˙ ds.

 t− ˙ ds yields Since  ∈ [0 − , 0 + ], applying Lemma 1 to − t−0 x˙ T (s)T4 x(s)  t−   t−   t− 1 − x˙ T (s)T4 x(s) ˙ ds  − x˙ T (s) ds T4 x(s) ˙ ds t−0 t−0 t−  t−0   t−0 0  1 = − x˙ T (s) ds T4 x(s) ˙ ds . t− t− Consequently, (17) and (18) imply that for  − 0  0,   t−0   t−0  t−0 + 1 − x˙ T (s)T4 x(s) ˙ ds  − x˙ T (s) ds T4 x(s) ˙ ds . t− t−0 − t− On the other hand, it can be clearly seen that (19) is also true for the case of  − 0 > 0. Similar to (19), the following inequality holds for any  ∈ [0 − , 0 + ]:    t−0  t−0  t−0 +

1 − x˙ T (s)H4 x(s) ˙ ds  − x˙ T (s) ds H4 x(s) ˙ ds .

t− t−0 −

t−

(17)

(18)

(19)

(20)

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

411

In addition, by applying Lemma 2 to the corresponding integral terms in (16), the following inequalities are obtained:   t   t  t 1 T T − x (s)T2 x(s) ds  − x (s) ds T2 x(s) ds , (21) 0 t−0 t−0 t−0      t t t 1 x T (s)H2 x(s) ds  − x T (s) ds H2 x(s) ds , (22) − 0 t−0 t−0 t−0      t t t 1 x˙ T (s)T3 x(s) ˙ ds  − x˙ T (s) ds T3 x(s) ˙ ds , (23) − 0 t−0 t−0 t−0      t t t 1 − x˙ T (s)H3 x(s) ˙ ds  − x˙ T (s) ds H3 x(s) ˙ ds . (24) 0 t−0 t−0 t−0 Then, substituting (19)–(24) into (16) results in that  T    k  k k   (rl ) +  e(t) e(t) V˙  h i h j h s () (t) ∗ − (t) i=1 j=1 s=1

+

k  k k  

T h i h j h s ()eT (t)Q Bwi Bwi Qe(t)−2 + 2 w T (t)w(t),

(25)

i=1 j=1 s=1 T Qe(t)−2 + 2 w T (t)w(t) is used, (t) is defined by where the inequality 2eT (t)Q Bwi w(t)eT (t)Q Bwi Bwi  t   t  t−0  t  t  t−0 T T T T T T T x (s) ds x˙ (s) ds x˙ (s) ds x (s) ds x˙ (s) ds x˙ (s) ds ,  (t) = t−0

t−0

t−

t−0

t−0

t−

and the matrices and  are the same ones defined in (6). Let = [Ci + Di K j C1i 0 0 D1i K s 0]. Then, according to the definition of z(t), we have ⎞ ⎛ ⎛ ⎞ k  k  k k  k k    z T (t)z(t) = ⎝ h i h j h s ()eT (t) T ⎠ × ⎝ h l h p h q () e(t)⎠ i=1 j=1 s=1



k  k k  

l=1 p=1 q=1

h i h j h s ()eT (t) T e(t).

i=1 j=1 s=1

Furthermore, z T (t)z(t) − 2 w T (t)w(t) k  k k    h i h j h s ()eT (t) T e(t) − 2 w T (t)w(t) + V˙ − V˙ i=1 j=1 s=1



k  k k  

h i h j h s ()eT (t) T e(t) − 2 w T (t)w(t)

i=1 j=1 s=1

+

k  k  k 

 h i h j h s ()

i=1 j=1 s=1

+

k  k  k  i=1 j=1 s=1

=

k  k  k  i=1 j=1 s=1

e(t) (t)

T 

(rl ) +  ∗ 



e(t) (t)



T h i h j h s ()−2 eT (t)Q Bwi Bwi Qe(t) + 2 w T (t)w(t) − V˙



e(t) h i h j h s () (t)

T 

 ∗ 



T QT. with  = (rl ) +  + T + −2 Q Bwi Bwi

 e(t) − V˙ (t)

(26)

412

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

At present stage, by applying Schur complement to (6), we have that    <0 ∗ 

(27)

which means that z T (t)z(t) − 2 w T (t)w(t) − V˙ . Consequently, integrating both sides of (28) from 0 to T gives  T  T z T (t)z(t) dt − 2 w T (t)w(t) dt  − V (T ) + V (0) 0

(28)

(29)

0

which, together with zero initial condition, implies that  ∞  ∞ z T (t)z(t) dt  2 w T (t)w(t) dt. 0

0

As a result, z(t)2 w(t) holds. When w(t) ≡ 0, it follows from (25) that V˙ < 0, which ensures the asymptotical stability of the closed-loop system. The proof is thus completed.  Because of the existence of the parameter uncertainty  in (6), Theorem 1 cannot be directly used to determine the performance of the closed-loop system. However, the following theorem provides a sufficient condition for the system to be robustly asymptotically stable with an H∞ norm bound . Theorem 2. For given scalars 0 > 0, > 0, 0 > 0, > 0 and  > 0, as well as the given matrices K i (i = 1, 2, . . . , k), systems (3)–(4) are robustly asymptotically stable with an H∞ norm bound  > 0 if there exist matrices Tl > 0, Hl > 0, P11 , P12 , P13 , P22 , P23 , P33 , Q q , Nq , L q , Mq and Wq (l = 1, 2, 3, 4, q = 1, 2, . . . , 6), and scalars i js > 0 so that the following LMIs hold, for i = 1i, j, s  k: ⎡ ⎤ (rl ) + T + i js E iTj E i j Q Bwi QG i ⎢ ∗ − 0 0 ⎥ ⎢ ⎥ < 0, (30) 2 ⎣ ∗ ∗ − I 0 ⎦ ∗ ∗ ∗ −i js I ⎤ ⎡ P11 P12 P13 ⎣ ∗ P22 P23 ⎦ > 0, (31) ∗ ∗ P33 where (i j ), E i j , and  are defined as in Theorem 1. Proof. In light of the structure of , we have (rl ) +  = (rl ) + QG i F(t)E i j + E iTj F T (t)G iT Q.

(32)

Applying Lemma 2 to (32) produces T T T (rl ) +  (rl ) + i−1 js QG i G i Q + i js E i j E i j .

Therefore, a sufficient condition for (6) holding is ⎡ ⎤ T T (rl ) + T + i−1 js QG i G i Q + i js E i j E i j Q Bwi ⎣ ∗  0 ⎦ < 0. ∗ ∗ −2 I

(33)

Obviously, by Schur complement, (30) is equivalent to (33). This means that under conditions (30) and (31), (6) and (7) hold. Thus, the proof is completed by Theorem 1. 

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

413

4. Robust H∞ fuzzy controller design Based on Theorem 2, we will present a systematic procedure for designing the robust H∞ fuzzy controller. The suggested fuzzy controller (2) can guarantee systems (3)–(4) to be robustly asymptotically stable with an H∞ norm bound  > 0. Theorem 3. For given scalars 0 > 0, > 0, 0 > 0, > 0,  > 0 and a p ( p = 2, . . . , 6, a4  0), systems (3)–(4) is robust asymptotically stable with an H∞ norm bound  > 0, and the feedback gain matrices are given by K i = Fi X −1 , i = 1, 2, . . . , k, if there exist matrices T¯l > 0, H¯ l > 0, P¯11 , P¯12 , P¯13 , P¯22 , P¯23 , P¯33 , N¯ q , L¯ q , M¯ q , W¯ q (l = 1, 2, 3, 4, q = 1, 2, . . . , 6), and X, as well as matrices Fi and scalars i js > 0 so that the following LMIs hold, for 1i, j, s k: ⎡ ⎤ ¯ ¯1 ¯ ¯    ⎢ ∗ − ¯ 0 0 ⎥ ⎢ ⎥ < 0, (34) ⎣ ∗ ∗ −2 I 0 ⎦ ¯1 ∗ ∗ ∗ − ⎡ ⎤ P¯11 P¯12 P¯13 ⎣ ∗ P¯22 P¯23 ⎦ > 0, (35) ∗ ∗ P¯33 where ¯ T = [B T a2 B T a3 B T a4 B T a5 B T a6 B T ],  wi wi wi wi wi   wi T E i X + E bi F j E 1i X 0 0 E b1i Fs 0 ¯ 1 = , Ci X + Di F j C1i X 0 0 D1i Fs 0   1 ¯ 1¯ 1 ¯ 1 ¯ 1 ¯ 1 ¯ ¯ T2 , T3 , T4 , H2 , H3 , H4 ,  = diag 0 0 0 0

¯  = diag(i js I, I ), ¯ = [ Z¯ 1 − N¯ − L¯ Z¯ 2 − M¯ − W¯ ],

T T ¯T P12 0 − P¯23 ], + P¯23 0 − P¯22 Z¯ 1T = [ P¯22 T T T T T T 0 − P¯33 ], Z¯ 2 = [ P¯23 + P¯33 0 − P¯23 P¯13 T T ¯T ¯T ¯T ¯T ¯T ¯ ¯ N = [ N1 N2 N3 N4 N5 N6 ], L¯ T = [ L¯ T1 L¯ T2 L¯ T3 L¯ T4 L¯ T L¯ T6 ], 5

M¯ T = [ M¯ 1T M¯ 2T M¯ 3T M¯ 4T M¯ 5T M¯ 6T ], W¯ T = [W¯ 1T W¯ 2T W¯ 3T W¯ 4T W¯ 5T W¯ 6T ]

¯ being its element at the position (r, l), and defined as follows for ¯ ) is a symmetric block-matrix with  ¯ = ( ¯  and  rl rl 1r, l 6: ¯ = P¯12 + P¯ T + P¯13 + P¯ T + T¯1 + H¯ 1 + 0 T¯2 + 0 H¯ 2 + N¯ 1 + N¯ T  11 12 13 1 + M¯ 1 + M¯ 1T + Ai X + Bi F j + X AiT + F jT BiT + i js G i G iT , ¯ = N¯ T + M¯ T − L¯ 1 + A1i X + a2 X AT + a2 F T B T + a2 i js G i G T ,  12 2 2 i j i i ¯ = − P¯12 − N¯ 1 + N¯ T + M¯ T + L¯ 1 + a3 X AT + a3 F T B T + a3 i js G i G T ,  13 3 3 i j i i ¯ = P¯11 + N¯ T + M¯ T − X + a4 X AT + a4 F T B T + a4 i js G i G T ,  14

4

4

i

j

i

i

¯ = N¯ T + M¯ T − W¯ 1 + B1i Fs + a5 X AT + a5 F T B T + a5 i js G i G T ,  15 i j i i 5 5 ¯ = − P¯13 + N¯ T − M¯ 1 + M¯ T + W¯ 1 + a6 X AT + a6 F T B T + a6 i js G i G T , 16 6 6 i j i i ¯ = − L¯ 2 − L¯ T + a2 A1i X + X AT a2 + a2 a2 i js G i G T ,  22 2 1i i

414

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

¯ 23 = − N¯ 2 + L¯ 2 − L¯ T3 + X AT1i a3 + a2 a3 i js G i G iT , ¯ = − L¯ T − a2 X + X AT a4 + a2 a4 i js G i G T ,  24 4 1i i ¯ = − L¯ T − W¯ 2 + a2 B1i Fs + X AT a5 + a2 a5 i js G i G T , 25

1i

5

¯ 26 = − M¯ 2 − L¯ T6 + W¯ 2 + X AT1i a6 + a2 a6 i js G i G iT , ¯ = − N¯ 3 − N¯ T + L¯ 3 + L¯ T − T¯1 + a3 a3 i js G i G T , 33

3

3

i

i

¯ = − N¯ T + L¯ T − a3 X + a3 a4 i js G i G T ,  34 4 4 i ¯ 35 = − N¯ 5T + L¯ T5 − W¯ 3 + a3 B1i Fs + a3 a5 i js G i G iT , ¯ = − N¯ T − M¯ 3 + L¯ T + W¯ 3 + a3 a6 i js G i G T , 36

6

6

i

¯ = 0 T¯3 + 0 H¯ 3 + 2 T¯4 + 2 H¯ 4 − a4 X − a4 X + a4 a4 i js G i G T ,  44 i ¯ = −W¯ 4 + a4 B1i Fs − a5 X + a4 a5 i js G i G T , 45

i

¯ 46 = − M¯ 4 + W¯ 4 − a6 X + a4 a6 i js G i G iT , T a5 + a5 a5 i js G i G iT , ¯ 55 = −W¯ 5 − W¯ 5T + a5 B1i Fs + FsT B1i ¯ = − M¯ 5 + W¯ 5 − W¯ T + F T B T a6 + a5 a6 i js G i G T , 56

6

s

1i

i

¯ 66 = − H¯ 1 − M¯ 6 − M¯ 6T + W¯ 6 + W¯ 6T + a6 a6 i js G i G iT .

Proof. Notice that (34) implies that X is a nonsingular matrix. So, let Q 1 = X −1 . Thus, to prove Theorem 3, we can show that (34) and (35) imply (30) and (31). To this end, let Q p = a p Q 1 ( p = 2, . . . , 6), i js = i−1 js , and define the following variables: T1 = Q 1 T¯1 Q T1 , T2 = Q 1 T¯2 Q T1 , T3 = Q 1 T¯3 Q T1 , T4 = Q 1 T¯4 Q T1 , H1 = Q 1 H¯ 1 Q T1 , H2 = Q 1 H¯ 2 Q¯ 1 , H3 = Q 1 H¯ 3 Q T1 , H4 = Q 1 H¯ 4 Q T1 , P11 = Q 1 P¯11 Q T1 , P12 = Q 1 P¯12 Q T1 , P13 = Q 1 P¯13 Q T1 , P22 = Q 1 P¯22 Q T1 , P23 = Q 1 P¯23 Q T1 , P33 = Q 1 P¯33 Q T1 , K i = Fi Q 1 , i = 1, 2, . . . , k, Nl = Q 1 N¯ l Q T1 , L l = Q 1 L¯ l Q¯ T1 , Ml = Q 1 M¯ l Q T1 , Wl = Q 1 W¯ l Q T1 , l = 1, 2, . . . , 6. ⎞ ⎛ Then, pre- and post-multiplying both sides of (34) by  = diag ⎝ Q 1 , · · · Q 1 , I, I, I ⎠ and its transpose, respec   12

tively, shows that (34)T < 0. Then, it can be verified by Schur complement that (34)T < 0 is equivalent to (33). Consequently, from the proof of Theorem 2, (33) ensures that (30) is true. In addition, (31) is immediately obtained by pre- and post-multiplying both sides of (35) with diag(Q 1 , Q 1 , Q 1 ) and its transpose. Therefore, by Theorem 2, the proof is completed.  Remark 2. Unlike in Theorem 2, in order to determine the feedback gain matrices, we have to set Q i = ai Q 1 (i = 2, 3, 4, 5, 6) to obtain the LMI conditions. The parameters ai (i = 2, 3, 4, 5, 6) should be given prior to solve LMI (34). How to choose these design parameters to optimize is still an open problem. If the common form for Q i (i = 1, 2, 3, 4, 5, 6) are retained (i.e., ai s are not introduced), one has to employ nonlinear programming to solve nonrestrict LMI conditions which brings much computational burden and may lead to conservative results. Remark 3. To reduce the conservatism of the stability criteria, some slack variables are introduced. Then, it should be pointed out that the works in [28–30] show that when the slack variables are introduced, some ones among them maybe not useful for improving the results. And the redundant slack variables will lead to the burdensome computation. Therefore, it is important to known which slack variables are redundant to reduce the conservatism of the stability criteria. However, to expressly known the effect of each slack variable to improvement of the stability criteria is very difficult, and is still an opened problem which will be studied further. In what follows, we consider the robust stabilization problem of system (3) with w(t) ≡ 0. And the following corollary can be derived from Theorem 3 directly.

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

415

Corollary 1. For given scalars 0 > 0, > 0, 0 > 0, > 0, and scalars a ( = 2, 3, 4, 5, 6, a4  0), if there exist matrices T¯l > 0, H¯ l > 0, P¯11 , P¯12 , P¯13 , P¯22 , P¯23 , P¯33 , N¯ q , L¯ q , M¯ q , W¯ q (l = 1, 2, 3, 4, q = 2, 3, 4, 5, 6) and X, as well as matrices Fi and scalars i js > 0 such that for 1i, j, s k, ⎡

⎤ ˆ ¯ ¯   ⎣ ∗ − ¯ 0 ⎦ < 0, ∗ ∗ −i js I

(36)

⎡

⎤ P¯11 P¯12 P¯13 ⎣ ∗ P¯22 P¯23 ⎦ > 0, ∗ ∗ P¯33

(37)

T

ˆ = [E i X + E bi F j E 1i X 0 0 E b1i Fs 0], and , ¯ are defined as in Theorem 3, then the feedback gain ¯ ¯ and  where  matrices K i are given by K i = Fi X −1 , i = 1, 2, . . . , k and the resulting closed-loop system (3) is asymptotically stable. In addition, another especial case of (3) is u(t − ) ≡ 0. For this case, the following corollary can be obtained from the proof of Theorems 1–3 by setting P13 = 0, P23 = 0, P33 = 0, Hl = 0 (for l = 1, 2, 3, 4), M = 0, W = 0, as well as Ni = 0, L i = 0 and Q i = 0 for i = 5, 6. Corollary 2. For given scalars 0 > 0, > 0,  > 0 and a p ( p = 2, 3, 4), systems (3)–(4) are robust asymptotically stable with an H∞ norm bound  > 0 and the feedback gain matrices are given by K i = Fi X −1 , i = 1, 2, . . . , k, if there exist matrices T¯l > 0, P¯11 , P¯12 , P¯22 , N¯ q , L¯ q (l = 1, 2, 3, 4, q = 1, 2, 3, 4), and X, as well as matrices Fi and scalars i j > 0 so that the following LMIs hold, for 1i < j  k: ⎤ ⎡ ¯ ¯ ¯ I¯ Bwi (i, j)  ⎢ ∗ − ¯ 0 0 ⎥ ⎥ < 0, 1i  k, ⎢ (38) ⎣ ∗ ∗ −2 I 0 ⎦ ¯ ∗ ∗ ∗ − ⎡ ⎤ ¯ ¯ j, i) ¯ ¯ j, i) Bwi + Bwj (i, j) + ( (i, j) + ( ¯ ¯

I ⎢ ⎥ 2 2 2 ⎢ ⎥ ¯ ⎢ ⎥ < 0, 1i < j  k, ∗ − 0 0 (39) ⎢ ⎥ 2 ⎣ ⎦ ∗ ∗ − I 0 ¯ ∗ ∗ ∗ −   P¯11 P¯12 > 0, (40) ∗ P¯22 where

 E i X + E bi F j E 1i X 0 0 , Ci X + Di F j C1i X 0 0 T T ¯T 0 − P¯22 Z¯ T = [ P¯22 P12 ], N¯ T = [ N¯ 1T N¯ 2T N¯ 3T N¯ 4T ],   ¯ = diag 1 T2 1 T3 1 T4 , L¯ T = [ L¯ T1 L¯ T2 L¯ T3 L¯ T4 ],  0 0 ¯ = diag(i j I, I ) ¯I T = [I a2 I a3 I a4 I ], 

¯T = ¯ = [ Z¯ − N¯ − L], ¯





416

and

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

⎡¯ ¯ ¯ 11  12 13 ¯ ¯ ⎢ ∗  22 23 ¯ (i, j) = ⎢ ⎣ ∗ ¯ ∗ 33 ∗ ∗ ∗

¯ ⎤  13 ¯ ⎥  24 ⎥ ¯ ⎦  34 ¯ 44

with ¯ = P¯12 + P¯ T + T¯1 + 0 T¯2 + N¯ 1 + N¯ T + Ai X + Bi F j + X AT + F T B T + i j G i G T ,  11 12 1 i j i i ¯ = N¯ T − L¯ 1 + a2 A1i X T + a2 X AT + a2 F T B T + a2 i j G i G T , 12 2 i j i i ¯ = − P¯12 − N¯ 1 + N¯ T + L¯ 1 + a3 X AT + a3 F T B T + a3 i j G i G T , 13 3 i j i i ¯ = P¯11 + N¯ T − X + a4 X AT + a4 F T B T + a4 i j G i G T ,  14 4 i j i i ¯ = − L¯ 2 − L¯ T + a2 A1i X T + X AT a2 + a2 a2 i j G i G T ,  22 2 1i i ¯ = − N¯ 2 + L¯ 2 − L¯ T + X AT a3 + a2 a3 i j G i G T ,  23

3

1i

i

¯ = − L¯ T − a2 X + X AT a4 + a2 a4 i j G i G T ,  24 4 1i i ¯ = − N¯ 3 − N¯ T + L¯ 3 + L¯ T − T¯1 + a3 a3 i j G i G T , 33 3 3 i ¯ = − N¯ T + L¯ T − Q¯ 3 + a3 a4 i j G i G T ,  34

4

4

i

¯ = 0 T¯3 + 2 T¯4 − a4 X − a4 X + a4 a4 i j G i G T .  44 i

5. Computer simulation In this section, some examples are used to illustrate the effectiveness of the proposed results, and to compare with the existing results. Example 1. Consider the following uncertain T–S fuzzy system with state and input delays: Rule 1: IF (t) = x2 (t) + a(v t¯/2L)x1 (t) + (1 − a)(v t¯/2L)x1 (t − ) is about 0, THEN x(t) ˙ = (A1 + A1 )x(t) + (A11 + A11 )x(t − (t)) + (B1 + B1 )u(t) + (B11 + B11 )u(t − (t)). Rule 2: IF (t) = x2 (t) + a(v t¯/2L)x1 (t) + (1 − a)(v t¯/2L)x1 (t − ) is about  or −, THEN x(t) ˙ = (A2 + A2 )x(t) + (A21 + A21 )x(t − (t)) + (B2 + B2 )u(t) + (B21 + B21 )u(t − (t)), where

⎡

⎡ ⎤ ⎤ v t¯ v t¯ −(1 − a) 0 0 0 0 ⎢ ⎢ ⎥ ⎥ Lt0 Lt0 ⎢ ⎢ ⎥ ⎥ v t¯ v t¯ ⎢ ⎢ ⎥ ⎥ A1 = ⎢ a 0 0 ⎥ , A11 = ⎢ (1 − a) 0 0⎥, ⎢ ⎢ ⎥ ⎥ Lt0 Lt0 ⎣ ⎣ ⎦ ⎦ v 2 t¯2 v t¯ v 2 t¯2 −a (1 − a) 0 0 0 2Lt0 t0 2Lt0 ⎡ ⎡ ⎤ ⎤ v t¯ v t¯ −a −(1 − a) 0 0 0 0 ⎢ ⎢ ⎥ ⎥ Lt0 Lt0 ⎢ ⎢ ⎥ ⎥ v t¯ v t¯ ⎢ ⎢ ⎥ ⎥ A2 = ⎢ a 0 0 ⎥ , A21 = ⎢ (1 − a) 0 0⎥, ⎢ ⎢ ⎥ ⎥ Lt0 Lt0 ⎣ ⎣ ⎦ ⎦ v 2 t¯2 dv t¯ dv 2 t¯2 0 0 0 −ad (1 − a) 2Lt0 ⎡ ⎤2Lt0 t0 ⎡ ⎤ v t¯ v t¯ ⎢ lt ⎥ ⎢ lt ⎥ 0 ⎥ ⎢ 0⎥ B1 = ⎢ ⎣ 0 ⎦ , B2 = ⎣ 0 ⎦ , B11 = 0.1B1 , B21 = 0.1B2 , 0 0 −a

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

417

80 x1 x2 x3

70 60 50 40 30 20 10 0 −10

0

10

20

30 Time (Sec)

40

50

60

Fig. 1. Response of state for Example 1.

G i = [0.255 0.255 0.255]T , E i = E i1 = [0.1 0 0], E bi = 0.15, E bi1 = 0.05, i = 1, 2 and l = 2.8, L = 5.5, v = −1.0, t¯ = 2.0, t0 = 0.5, a = 0.7, d = 10t0 /. When all the uncertainties are removed from the above system, this system is just Example 5.2 in [17]. According to [17], take   1 h1 = , h2 = 1 − h1. 1 + exp(−3((t) + 0.5)) For the case where Ai = 0, A1i = 0, Bi = 0, and B1i = 0, the following fuzzy control law is proposed in [17] for  =  = 0.1: u(t) = h 1 [3.6857 − 9.1238 0.8632]x(t) + h 2 [3.7138 − 9.7537 0.9086]x(t),

(41)

which guarantees the asymptotic stability of the resulting closed-loop system. Because the uncertainties have not been taken into account, the control law (41) cannot, theoretically, be used to stabilize this uncertain system. Then, Corollary 1 can be used to solve the robust stabilization problem for the above system. By setting the parameter as a p = 0.0125 ( p = 2, 3, 5, 6) and a4 = 1.5, and 0 = 1.2135 and 0 = 1, solving LMIs (36)–(37) gives max = 1.2135, max = 1 and the feedback gains as follows: K 1 = [3.6776 − 0.8520 0.0154],

K 2 = [3.6793 − 0.8500 0.0155],

which implies that the regarding fuzzy controller guarantees the asymptotic stability of the closed-loop system for (t) ∈ [0, 2.4270] and (t) ∈ [0, 2]. The simulation is run under the initial conditions as (t) = [4, −1, 2] for t ∈ [−2.4270, 0], and u(t) = 0 for t ∈ [−2, 0] with (t) = 1 + sin(t) and (t) = 1.2 + 1.2 sin(t). The simulation results are shown by Figs. 1, 2. Fig. 1 shows the state response of the system. Fig. 2 displays the control input signal. In addition, for given m and m by using Corollary 1 in this paper, we can get  M ,  M and the feedback gains as shown in Table 1. Example 2. Consider the following T–S fuzzy system: Plant Rule i: IF x2 (t) is h i , Then x(t) ˙ = (Ai + Ai )x + (A1i + A1i )x(t − (t)) + (Bi + Bi )u(t) +(B1i + B1i )u(t − (t)) + Bwi w(t), z(t) = Ci x(t) + C1i x(t − (t)) + Di u(t) + D1i u(t − (t)), i = 1, 2,

(42)

418

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

1

u

0 −1 −2 −3 −4 −5 −6

0

10

20

30 Time (Sec)

40

50

60

Fig. 2. Control input curve for Example 1. Table 1

m

M

m

M

Feedback gain matrices

0 0.5 1

2 2.9205 4.7610

0 0.5 1

2.4270 3.8210 4.8209

[3.6776 − 0.8520 0.0154], [3.6793 − 0.8500 0.0155] [2.6259 − 0.2866 0.0022], [2.6239 − 0.2859 0.0022] [2.2239 − 0.0830 0.0002], [2.2238 − 0.0830 0.0002]

where h 1 = (1 − 1/(1 + exp(−6x2 + 1.5)))(1/(1 + exp(−6x2 − 1.5))) and h 2 = 1 − h 1 , and the system matrices are         0 1 0.1 0 0 0 , A11 = , B1 = , B11 = , A1 = 0.1 −2 0.1 −0.2 1 0.25     0 1 0.1 0.1 A2 = , A12 = , B2 = B1 , B12 = B11 , 0.1 −0.75 0 −0.15   0 Bw1 = Bw2 = , C1 = [0.5 0.15], C2 = [0.35 0.25], D1 = D2 = 0.1, 1 C11 = [0.05 0.015], C12 = [0.035 0.025], D11 = D12 = 0.01,     −0.3 0 −0.15 0.2 , E1 = E2 = , G1 = G2 = 0 0.3 0 0.04   −0.05 −0.35 E 11 = E 12 = , E b1 = E b2 = [0.05 0.15], 0.08 −0.45 E b11 = E b12 = [0.025 0.075]. Note that both the state delay and the input delay appear in the system dynamics and the control output equations. Given  = 1, then for m = 0 and m = 0, we apply Theorem 3 with a p = 0.0125 ( p = 2, 3, 5, 6) and a4 = 1.5 to solve (34) and (35), and it was found that the maximal upper bounds of (t) and (t) are  M = 1.0996 and  M = 1.0324, and the feedback gain matrices are K 1 = [−0.9735 − 0.9788], K 2 = [−0.9679 − 0.9748], for m = 1 and m = 1, we have that  M =1.6841 and  M =1.3782 and the feedback matrices are K 1 = [−1.0421 − 1.2718], K 2 = [−1.0384 − 1.2698]. Thus for the case of 1  (t)1.6841 and 1  (t)1.3782, take the disturbance input as w(t) = 4e−0.1t sin(t), (t) = 1 + |0.37 cos(2t)| and (t) = 1 + |0.68 cos(2t)|, the simulation is carried out under the initial conditions (t) = [4, −3] and u(t) = 0 for t 0. Fig. 3 shows the state response of the closed-loop systems (42) and Fig. 4 displays the control input curve. It is seen from Fig. 3 that the closed-loop system is asymptotically stable.

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

419

4 x1 x2

3 2 1 0 −1 −2 −3

0

5

10

15

20 25 30 Time (Sec)

35

40

45

50

Fig. 3. Response of state for Example 2.

3

u

2 1 0 −1 −2 −3 −4

0

5

10

15

20 25 30 Time (Sec)

35

40

45

50

Fig. 4. Control input curve for Example 2.

Example 3. For the special case of u(t −(t)) ≡ 0, the robust H∞ fuzzy control problem is studied in [13], in which the H∞ fuzzy controller design procedure is proposed. The following example is taken from [13]. Consider the uncertain nonlinear time-delay system described as follows: x˙1 = −x1 (2 + sin2 x2 ) + x2 + 0.1x1 (t − (t)) + 0.2x2 (t − (t)) +c(t)x1 cos2 x2 + u 1 + (1 + sin2 (x2 ))w(t), x˙2 = x12 − x2 (1 − cos2 x2 ) + 0.2x1 (t − (t)) sin2 x2 − 0.5x2 (t − (t)) + 0.5u 2 + 0.1c(t)x2 , where c(t) is an uncertain parameter satisfying c(t) ∈ [−0.2, 0.2]. According to [13], by selecting the membership functions as M1 (x2 ) = sin2 x2 ,

M2 (x2 ) = cos2 x2 ,

then the above nonlinear system can be represented by the following T–S fuzzy model.

420

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

1.5 1

x1 x2

0.5 0 −0.5 −1 −1.5 −2 −2.5

0

5

10

15

20 25 30 Time (Sec)

35

40

45

50

Fig. 5. Response of state for Example 3.

8

u1 u2

7 6 5 4 3 2 1 0 −1

0

5

10

15

20 25 30 Time (Sec)

35

40

45

50

Fig. 6. Control input curve for Example 3.

Rule 1. IF x2 is M1 , THEN x˙ = (A1 + A1 )x + (A11 + A11 )x(t − (t)) + (B1 + B1 )u + Bw1 w(t), z = C1 x + D1 u(t), Rule 2. IF x2 is M2 , THEN x˙ = (A2 + A2 )x + (A21 + A21 )x(t − (t)) + (B2 + B2 )u + Bw2 w(t), z = C2 x + D2 u(t), where

       −3 1 0.1 0 1 0 2 , A11 = , B1 = , Bw1 = , A1 = 1 −1 0.2 −0.5 0 0.5 0         −2 1 0.1 0.2 1 0 1 A2 = , A21 = , B2 = , Bw2 = , 1 0 0 −0.5 0 0.5 0 

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403 – 422

421

Table 2

m

0

2

4

6

8

10

M

1.7435

3.7005

5.6999

7.7005

9.6995

11.6983



   0.1 0 1 0 C1 = C2 = , D1 = D2 = I, G 1 = G 2 = , 0 0.1 0 0     0 0.2 0.2 0 E1 = , E2 = , E 1i = 0, E bi = 0, i = 1, 2. 0 0 0 0 For this system, setting  = 1 and 0 = 1, the method in [13] suggests the maximum allowed bound = 0.2358. This implies that the allowable interval for (t) is [0.7642, 1.2358]. Then, applying Corollary 2 with ai = 0.015 (i = 2, 3) and a4 = 0.25, it was found that the maximal allowable value of is 0.8536. When setting = 0.2358, the maximum allowed bound of 0 is 9.8799 and the feedback gain matrices are     −1.0176 −1.1074 −0.7408 −1.5285 K1 = , K2 = , −0.6989 −2.1621 −0.6989 −2.0718 which means that the regarding fuzzy controller guarantees the asymptotic stability of the closed-loop system for (t) ∈ [9.6441 10.1157]. Take the disturbance input as w(t) = 2e−0.1t sin(t), and time-varying delay as (t) = 9.65 + |0.36 sin(2t)| which satisfies 9.6441(t)10.1157. Then, the simulation is carried out under the initial conditions (t) = [1.5, −2.5] for t < 0. Fig. 5 shows the state response of the resulting closed-loop systems and Fig. 6 displays the control input curve. In addition, for given different m , by using Corollary 2 we have the corresponding  M , as shown in Table 2. 6. Conclusion In this paper, we have studied the H∞ control problem for T–S fuzzy systems with state and input delays. Both the state time delay and input time delay are time-varying delays and no restriction is imposed on the derivative of time delays. Based on Lyapunov–Krasoviskii functional method, a new design scheme of H∞ fuzzy controller is derived. The main contribution lies in that delay-dependent conditions are presented in terms of LMI format. The effectiveness of the proposed results has been illustrated by some examples. Acknowledgment This work is partially supported by the Natural Sciences and Engineering Research Council of Canada, the National Natural Science Foundation of China (Nos. 60674055, 60774047), the Taishan Scholar programme and the Natural Science Foundation of Shandong Province (No. Y2006G04). References [1] W. Assawinchaichote, S.K. Nguang, P. Shi, H∞ output feedback control design for uncertain fuzzy singularly perturbed systems: an lmi approach, Automatica 40 (12) (2004) 2147–2152. [2] S.G. Cao, N.W. Rees, G. Feng, Analysis and design for a class of fuzzy control systems using dynamic fuzzy-state-space models, IEEE Trans. Fuzzy Systems 7 (2) (1999) 192–200. [3] S.G. Cao, N.W. Rees, G. Feng, H∞ control of uncertain fuzzy continuous-time systems, IEEE Trans. Fuzzy Systems 8 (2) (2000) 171–190. [4] Y.Y. Cao, P.M. Frank, Analysis and synthesis of nonlinear time-delay system via fuzzy control approach, IEEE Trans. Fuzzy Systems 8 (2) (2000) 200–211. [5] C.L. Chen, G. Feng, D. Sun, Y. Zhu, H∞ output feedback control of discrete-time fuzzy systems with application to chaos control, IEEE Trans. Fuzzy Systems 13 (4) (2005) 531–543. [7] G. Feng, H∞ controller design of fuzzy dynamic systems based on piecewise Lyapunov functions, IEEE Trans. Systems Man Cybernet. Part B 34 (1) (2004) 283–292. [8] G. Feng, H∞ controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and based on bilinear matrix inequalities, IEEE Trans. Fuzzy Systems 13 (1) (2005) 94–103.

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