Robust H∞ control of T-S fuzzy systems with input time-varying delays: A delay partitioning method

Applied Mathematics and Computation 321 (2018) 209–222

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Robust H∞ control of T-S fuzzy systems with input time-varying delays: A delay partitioning methodR Min Li a,∗, Feng Shu a, Duyu Liu a, Shouming Zhong b a b

College of Electrical and Information Engineering, Southwest Minzu University, Chengdu 610041, China School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

a r t i c l e

i n f o

Keywords: H∞ control Geometric sequence partitioning Input time-varying delays T-S fuzzy systems

a b s t r a c t Addressed in this paper is the robust H∞ control issue of T-S fuzzy systems with input time-varying delays. By means of the delay partitioning method, the delay interval is partitioned into multiple unequal subintervals whose lengths satisfy a geometric sequence. On this basis, a modified Lyapunov–Krasovskii functional is presented to analyze asymptotic stability of the open-loop system. Then a state feedback controller that ensures a prescribed H∞ performance level for the closed-loop system is proposed in linear matrix inequality format. Finally, two numerical examples are given to illustrate the effectiveness and advantages of the obtained results. © 2017 Elsevier Inc. All rights reserved.

1. Introduction In recent years, the problems of stability analysis and stabilization of nonlinear systems expressed by Takagi–Sugeno (T-S) fuzzy model have received extensive attention due to their practical applications [1–3]. T-S fuzzy model can describe complex nonlinear systems by using fuzzy sets and the membership functions, and therefore it is a very powerful and effective tool to control nonlinear systems. So far, the problems of the stability analysis and controller design of T-S fuzzy systems have been intensively investigated in [4–10]. Many studies dealing with stability analysis of nonlinear dynamic systems with delays can be roughly classified into two parts: delay-dependent ones and delay-independent ones; generally speaking, the former is less conservative than the latter [11,12]. While, time delays inevitably exist in various practical systems and may result in instability or deterioration of system performance [13–21]. For these reasons, in order to solve time-varying delays efficiently, a delay partitioning approach was first proposed in [22], and was realized by dividing the delay interval into multiple equal/unequal segments [1,23]. It was revealed that increasing the number of subintervals may produce less conservative results [1,24], and may result in requiring too many adjustable parameters. However, the computational burden rapidly increases as the subintervals get thinner. The problem of delay-dependent H∞ control for T-S fuzzy systems has been a hot topic [25–27]. Robust H∞ control design methods of T-S fuzzy delayed systems were investigated in [25]. The H∞ control problems for fuzzy systems with input R This work was partially supported by National Natural Science Foundation (61673016), SWUN Central Universities Fundamental Research Foundation Projects (2016ZYXS05), SWUN Construction Projects for Graduate Degree Programs (2017XWD-S0805), Advance Research Program of Electronic Science and Technology National Program (2017YYGZS16), Innovative Research Team of the Education Department of Sichuan Province (15TD0050), Sichuan Youth Science and Technology Innovation Research Team (2017TD0028). ∗ Corresponding author. E-mail address: [email protected] (M. Li).

https://doi.org/10.1016/j.amc.2017.10.053 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

delay were derived in [27,28]. The problems for H∞ output feedback control of T-S fuzzy systems with input/distributed delay were reported in [29,30]. However, for T-S fuzzy systems with input time-varying delays, the robust H∞ control issue by delay partitioning approach has rarely been reported, especially, based on geometric sequence partitioning method. Motivated by the previous discussions, this paper considers the robust H∞ stability problem for T-S fuzzy systems with input time-varying delays by means of delay partitioning method. Note that the delay partitioning manner in this paper is different from previous partitioning method and is a new delay partitioning approach first proposed in [31]. Compared with traditional delay partitioning method, our approach can result in less conservative conditions with lower computational complexity by dividing the delay interval into a series of geometric progression with a common ratio α . The main contribution of this paper lies in the following aspects. Based on the geometric sequence partitioning method, the delay interval is partitioned into multiple unequal subintervals with a common ratio α . By means of constructing a modified Lyapunov–Krasovskii functional, less conservative conditions are proposed, and robust H∞ fuzzy controllers are designed in the paper. Two numerical examples are provided to indicate the effectiveness and advantages of this proposed method. The rest of this paper is organized as follows: In Section 2, T-S fuzzy systems with input time-varying delays are stated, some preliminary definitions and useful lemmas are formulated. In Section 3, asymptotic stability criteria of T-S fuzzy systems with input time-varying delays is derived and robust H∞ fuzzy controllers are designed. In Section 4, two numerical examples are provided to show the effectiveness and advantages of the theoretical results. Finally, Section 5 summarizes the paper. Notations. Rn and Rn×m denote the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. I is the identity matrix with appropriate dimension; when 0 denotes a matrix, it is a zero matrix with n × n dimension, and the dimension of matrix 0p × q is the zero matrix of dimension p × q. “T”, “ − 1” stand for the transpose and inverse of A, respectively, and He(A ) = A + AT . “∗ ” implies the symmetric term in a matrix. • is the Euclidean norm in Rn . A > 0 (A ≥ 0) means that A is positive (semi-positive) definite. L2 [0, ∞) stands for the space of square-integrable vector functions over [0, ∞). 2. Preliminaries Consider the following T-S fuzzy system with input time-varying delays, for each l ∈ {1, . . . , r} (r is the number of the plant rules), the lth rule of this fuzzy model with r plant rules is described as follows: Rule l: IF θ 1 (t) is Ml1 , . . . , and θ p (t) is Ml p THEN

⎧ ⎪ ⎨x˙ (t ) = (Al1 + Al1 (t ))x(t ) + (Al2 + Al2 (t ))x(t − τ (t ) ) + (Bl1 + Bl1 (t ))u(t ) +(Bl2 + Bl2 (t ))u(t − τ (t ) ) + Bωl ω (t ), t≥0 z(t ) = Cl1 x(t ) + Cl2 x(t − τ (t )) + Dl1 u(t ) + Dl2 u(t − τ (t )) ⎪ ⎩ x(t ) = ϕ (t ), t ∈ [−h2 , 0]

(2.1)

where x(t ) ∈ Rn is the state, u(t ) ∈ Ru is the control input vector, z(t ) ∈ Rm denotes the controlled output, and ω (t ) ∈ R p is the external perturbation belonging to L2 [0, ∞). θ s (t) and Mls (s = 1, . . . , p) are premise variables and the related fuzzy sets, respectively. Al1 , Al2 ∈ Rn×n , Bl1 , Bl2 ∈ Rn×u , Bωl ∈ Rn×p , Cl1 , Cl2 ∈ Rm×n , Dl1 , Dl2 ∈ Rm×u are the constant matrices, respectively. Al1 (t ), Al2 (t ) ∈ Rn×n , Bl1 (t ), Bl2 (t ) ∈ Rn×u are the uncertain matrices satisfying

[Al1 (t ), Al2 (t ), Bl1 (t ), Bl2 (t )] = Gl Fl (t )[El1 , El2 , Ebl1 , Ebl2 ] The time-varying delay τ (t) satisfies

0 < h1 ≤ τ (t ) ≤ h2 , τ˙ (t ) ≤ μ

(2.2)

with h1 , h2 , μ being constants. Then the fuzzy model can be inferred as:

⎧ r ⎪ ⎨x˙ (t ) = l=1 hl (t )((Al1 + Al1 (t ))x(t ) + (Al2 + Al2 (t ))x(t − τ (t ) ) + (Bl1 + Bl1 (t ))u(t ) (Bl2 + Bl2 (t ))u(t − τ (t ) ) + Bωl ω (t ) ), t ≥ 0 + r ⎪z(t ) = l=1 hl (t )(Cl1 x(t ) + Cl2 x(t − τ (t )) + Dl1 u(t ) + Dl2 u(t − τ (t )) ) ⎩ x(t ) = ϕ (t ), t ∈ [−h2 , 0]

where hl (t ) =

W (t ) r l , W (t ) l=1 l

Wl (t ) =

p s=1

r

Mls (θs (t ) ) with Mls (θ s (t)) being the grade of membership of θ s (t) in Mls . Since

Wl (t) ≥ 0, it holds that hl (t) ≥ 0 and l=1 hl (t ) = 1. In order to design a state feedback fuzzy controller, the lth fuzzy control rule is presented as follows: Control Rule l: IF θ 1 (t) is Ml1 , . . . , and θ p (t) is Ml p THEN

u(t ) = Kl x(t ), l = 1, . . . , r Therefore, the overall fuzzy control law is represented by

u(t ) =

r  l=1

(2.3)

hl (t )Kl x(t ), l = 1, . . . , r

M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

211

Fig. 1. Geometric sequence partitioning of the delay interval.

where Kl ∈ Ru×n (l = 1, . . . , r ) are the control gains. The close-loop system can be described as follows:

⎧

   ⎨x˙ (t ) = rl=1 ri=1 rj=1 hl (t )hi (t )h j (t ∗ ) (A¯ li + A¯ li (t ))x(t ) + (B¯ l j + B¯ l j (t ))x(t − τ (t ) ) + Bωl ω (t ) , t ≥ 0    z(t ) = rl=1 ri=1 rj=1 hl (t )hi (t )h j (t ∗ ) (Cl1 + Dl1 Ki )x(t ) + (Cl2 + Dl2 K j )x(t − τ (t )) ⎩ x(t ) = ϕ (t ), t ∈ [−h2 , 0] (2.4)

where

A¯ li = Al1 + Bl1 Ki , B¯ l j = Al2 + Bl2 K j , t ∗ = t − τ (t )

A¯ li (t ) = Al1 (t ) + Bl1 (t )Ki , B¯ l j (t ) = Al2 (t ) + Bl2 (t )K j Remark 1. For simplicity, the above notations A¯ li , B¯ l j , t ∗ , A¯ li (t ), B¯ l j (t ) are defined. Since the existence of input delays,  the term h j (t − τ (t )) are well defined, and also hold that h j (t − τ (t )) ≥ 0 and rj=1 h j (t − τ (t )) = 1 for j = 1, . . . , r. The following definition and lemmas are listed. Definition 1 ([27]). System (2.4) is said to be asymptotically stable with an H∞ norm bound γ > 0 if there exists a feedback control law u(t) such that the following two conditions hold: (1) System (2.4) with ω(t) ≡ 0 is asymptotically stable. (2) Under the zero initial condition, the controlled output z(t) satisfies constraint z(t)2 ≤ γ ω(t)2 for any nonzero ω(t) ∈ L2 [0, ∞). Lemma 1 ([32]). For n × n matrix Q > 0, scalar τ > 0, vector-valued function x˙ : [−τ , 0] −→ Rn such that the following integrations are well defined, it holds that

−τ −



τ2 2

t

t−τ





x˙ (s )Q x˙ (s ) ds ≤ x (t ) T

0



−τ

t

t+θ

T

x˙ (s )Q x˙ (s ) dsdθ ≤ T



−Q x (t − τ ) T



Q −Q

∗

τ x (t ) T

t

t−τ





−Q

x ( s ) ds T



x(t ) x(t − τ ) ∗

Q −Q





τ x(t ) t t−τ x (s ) ds

Lemma 2 ([33]). Let G, E, and F(t) being real matrices of appropriate dimensions and F(t) satisfies FT (t)F(t) ≤ I. Then, the following inequality holds for any constant ε > 0:

GF (t )E + E T F T (t )GT ≤ ε GGT + ε −1 E T E Lemma 3 ([34, Finsler’s lemma]). Let ζ ∈ Rn ,  = T ∈ Rn×n , and B ∈ Rm×n with rank(B) < n. The following statements are equivalent: (i) ζ T ζ < 0, ∀Bζ = 0, ζ = 0; (ii) B⊥ T B⊥ < 0; (iii) ∃L ∈ Rn×m :  + He(LB ) < 0; where B⊥ ∈ Rn×(n−rank(B )) is the right orthogonal complement of B. 3. Main results This section first introduces a new delay partitioning approach, then gives stability condition of T-S fuzzy delayed systems and designs the robust H∞ fuzzy controller. For any integral N ≥ 1, the delay interval [h1 , h2 ] is partitioned into N unequal geometric subintervals. The positive comδ

mon ratio α may be more than, or equal to, less than 1, and satisfying δ k = α , ∀k ∈ {2, . . . , N}, δ k is the length of the kth k−1 subinterval, as illustrated in Fig. 1.



δk = τk − τk−1 = α k τk = τk−1 + α k , k = 1, . . . , N

(3.1)

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M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

where τ0 = h1 , τN = h2 . It is obvious that [h1 , h2 ] =

N

I ,I k=1 k 1

= [τ0 , τ1 ), ..., IN−1 = [τN−2 , τN−1 ), IN = [τN−1 , τN ].

Remark 2. This is a new delay partitioning method that divides the delay interval [h1 , h2 ] into N unequal subintervals based on a positive common ratio α , Fig. 1 shows α > 1. The following augment vector will be used in the sequel:



T ξ (t ) = η0T (t ), η1T (t ), η2T (t ), η3T (t ) ∈ R(3N+4)n where



T

T η0 (t ) = xT (t ), x˙ T (t ), xT (t − τ0 ), xT (t − τ (t )) , η1 (t ) = xT (t − τ1 ), . . . , xT (t − τN )  t T  t−τ T  t  t−τN−1 0 T T T T η2 (t ) = x (s )ds, . . . , x (s )ds , η3 (t ) = x (s )ds, . . . , x ( s )d s t−τ0

t−τN−1

t−τ1

t−τN

Theorem 1. Assume that condition (2.2) holds. Fix scalar α > 0, γ > 0, positive integer N, and given matrices Ki ∈ Ru×n (i = 1, . . . , r ). The system (2.4) is robustly asymptotically stable with an H∞ norm bound γ if there exist symmetric positive definite , P ∈ Rn×n , matrices Mq ∈ Rn×n (q = 1, . . . , 4 ), and scalar ε > 0 such that the matrices Zk , Rk , Q1k , Q2k ∈ Rn×n (k = 1, . . . , N ), R lij following linear matrix inequalities (LMIs) hold, for 1 ≤ l, i, j ≤ r:

⎡ T 4 4 + 0 +  + εli j E¯liT E¯li ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣

ϕ1 −1 ∗ ∗ ∗ ∗

∗ ∗

where

⎡ 011 ⎢∗ ⎢ 0 = ⎢∗ ⎣∗ 022 =

N 

P

022

∗ ∗

δk Zk +

k=1

0 0 R1 − Zδ1 1 ∗

N 

τk2−1

k=1



2

Q1k +

ϕ2

ϕ3

0 −2 ∗ ∗ ∗

0 0

−3 ∗ ∗

MBωl 0 0 0 −γ 2 I ∗

⎤

MGl 0 ⎥ 0 ⎥ ⎥<0 0 ⎥ ⎦ 0 −εli j I

(3.2)

⎤

0 0

⎥ N N   2δk ⎥ 2 Q1k − Q ⎥, 011 = R˜ − τ + τk−1 2k k k=1 k=1 044 ⎦ Z1

δ1

τk2 − τk2−1

N 

2

k=1

Q2k , 044 = −(1 − μ )R˜ − 2

N  Zk k=1

δk

 Z Z Z Z Z Z Z 1 = diag −R2 + R1 + 2 + 1 , −R3 + R2 + 3 + 2 , . . . , −RN + RN−1 + N + N−1 , RN + N δ2 δ1 δ3 δ2 δN δN−1 δN    2Q  2Q11 2Q12 2 Q1N 2Q22 2 Q2N 21 2 = diag , , . . . , ,  = diag , , . . . , 3 2 δ1 (τ1 + τ0 ) δ2 (τ2 + τ1 ) δN (τN + τN−1 ) τ02 τ12 τN−1



4 = Cl1 + Dl1 Ki 0 0 Cl2 + Dl2 K j , E¯li = El1 + Ebl1 Ki 0 0 El2 + Ebl2 K j ,

T M = M1

⎡

M2

M1T A¯ li

⎢ =⎣ ⎡

ϕ1 = ⎣ Zδ22 +

+ ∗ ∗ ∗

Z1

δ1

M3

A¯ Tli M1

Z3

M4

A¯ Tli M2 − M1T −M2T − M2 ∗ ∗ Z2

δ3 + δ2

03n×Nn ...

A¯ Tli M3 −M3 0 ∗ ZN

ZN−1

δN + δN−1

⎤ A¯ Tli M4 + M1T B¯ l j T ¯ M2 Bl j − M4 ⎥ ⎦, M3T B¯ l j T ¯ T ¯ M4 Bl j + Bl j M4 ⎤

⎦ δN , ϕ3 =

ZN



 ϕ2 =

2Q21

τ1 +τ0

2Q11

τ0

2Q22

τ2 +τ1

2Q12

τ1

... 03n×Nn

2Q1N

... 03n×Nn

2Q2N



τN−1



τN +τN−1

Proof. For any t ≥ 0, the new LKF is as follows:

V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ) + V4 (xt ) where

V1 (xt ) = xT (t )P x(t )  t N   x(s )ds + V2 (xt ) = xT ( s )R t −τ (t )

k=1

t

t−δk

xT (s − τk−1 )Rk x(s − τk−1 )ds

(3.3)

M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

V3 (xt ) =

N  

−τk

k=1

V4 (xt ) =



−τk−1

N  

x˙ T (s )Zk x˙ (s ) ds dλ

t+λ

 t

t

t−τk−1

k=1

t

θ

213

t

λ

x˙ T (s )Q1k x˙ (s ) ds dλ dθ +

N   k=1

 t

t−τk−1

t−τk

θ

t

x˙ T (s )Q2k x˙ (s ) ds dλ dθ

λ

The derivative of V˙ 1 (xt ) is given as

V˙ 1 (xt ) = 2xT (t )P x˙ (t )

(3.4)

The derivative of V˙ 2 (xt ) is derived as

V˙ 2 (xt ) ≤ xT (t )R˜x(t ) − (1 − μ )xT (t − τ (t ))R˜x(t − τ (t )) +

N 

N 

xT (t − τk−1 )Rk x(t − τk−1 ) −

k=1

xT (t − τk )Rk x(t − τk )

(3.5)

k=1

The derivative of V3 (xt ) is expressed as



V˙ 3 (xt ) = x˙ T (t )



N 

δk Zk x˙ (t ) −

k=1

N   k=1

t−τk−1

x˙ T (s )Zk x˙ (s )ds

t−τk

(3.6)

By Lemma 1,

−

N   k=1

t−τk−1

t−τk

x˙ T (s )Zk x˙ (s )ds = −

N 

τk − τk−1 δk k=1

≤−



t−τk−1

t−τk

 N  τ (t ) − τk−1

δk

k=1

x˙ T (s )Zk x˙ (s )ds

t−τk−1

t −τ (t )

x˙ T (s )Zk x˙ (s )ds



N 

τk − τ (t ) t −τ (t ) T x˙ (s )Zk x˙ (s )ds δk t−τk k=1  T    N  1 x(t − τk−1 ) −Zk Zk x(t − τk−1 ) ≤ −Zk x(t − τ (t )) δk x(t − τ (t )) k=1      T N  1 x(t − τ (t )) −Zk Zk x(t − τ (t )) + −Zk x(t − τk ) δk x(t − τk ) −

(3.7)

k=1

Then, it follows from (3.6)–(3.7) that



N 

V˙ 3 (xt ) ≤ x˙ (t ) T



δk Zk x˙ (t ) −

k=1

+2

N  1 k=1

+2

δk

N  1 k=1

δk

N  1

δk k=1

xT (t − τk−1 )Zk x(t − τk−1 )

xT (t − τk−1 )Zk x(t − τ (t )) − 2

N  1 k=1

xT (t − τ (t ))Zk x(t − τk ) −

N  1 k=1

δk



N 

V˙ 4 (xt ) = x˙ (t )

k=1

−

τk2−1 2

N  t−τk−1  t−τk

k=1

Q1k +

N  k=1



t

θ

−

N   k=1

t

t−τk−1



t

θ

τk2 − τk2−1 2

x˙ T (s )Q2k x˙ (s ) ds dθ

By using Lemma 1,

x˙ T (s )Q1k x˙ (s ) ds dθ

xT (t − τ (t ))Zk x(t − τ (t ))

xT (t − τk )Zk x(t − τk )

The derivative of V4 (xt ) is presented as T

δk

 Q2k x˙ (t ) −

N   k=1

t

t−τk−1

(3.8)

 θ

t

x˙ T (s )Q1k x˙ (s ) ds dθ

(3.9)

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M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

k=1

−



N  2

≤

τk2−1

N   k=1 N 

≤

k=1

T 

τ

x(t ) t−τk−1 x (s ) ds

 kt−1

t−τk−1



t−τk

t

θ

2 τk2 − τk2−1

−Q1k ∗



Q1k −Q1k



τ

x(t ) t−τk−1 x (s ) ds

 kt−1

x˙ T (s )Q2k x˙ (s ) ds dθ



T 

δ k x(t )

t−τk−1 t−τk

x ( s ) ds

−Q2k ∗

Q2k −Q2k





δ x(t )  kt−τk−1

(3.10)

x ( s ) ds

t−τk

Hence, from (3.4)–(3.10), one can get that



N 

V˙ (xt ) ≤ 2x (t )P x˙ (t ) + x˙ (t ) T

T

+ xT (t ) R˜ −

N 

2 Q1k −

k=1

× −(1 − μ )R˜ − 2 N 

N 

! xT (t )

 N  2 k=1

−

N  k=1

τ

2 k−1

"

τk−1

k=1

−

2 Q1k

t

2 τk2 − τk2−1



δk

t

t−τk−1

t−τk

τk2 − τk2−1 2

k=1





N 

xT (t − τk−1 ) Rk −

k=1

Zk

t−τk−1

Q1k +

N 

 Q2k x˙ (t )

2δk Q x(t ) + xT (t − τ (t )) τk + τk−1 2k

x(t − τk ) + 2

N 

xT (t − τk−1 )

k=1

x ( s )d s + 2

xT (s )dsQ1k

t−τk−1

2

k=1

x(t − τ (t )) +

δk

k=1





N  Zk

xT (t − τk ) −Rk −

k=1

+2

N 

k=1



+

δk Zk +

k=1



τk2−1

N 



N 

xT (t − τ (t ))

t−τk−1

xT (s )dsQ2k

x ( s )d s + 2 

N  k=1

t−τk−1

t−τk

x(t − τk−1 )

δk

x(t − τ (t ))

k

δk

Z

x(t − τk )

k

δk

k=1 t

Z

Zk

2 xT (t )Q2k τk + τk−1



t−τk−1

t−τk

x ( s )d s

x T ( s )d s

= η0T (t )0 η0 (t ) − η1T (t )1 η1 (t ) − η2T (t )2 η2 (t ) − η3T (t )3 η3 (t ) ! " t Z N N   2 Q1k k T T +2 x (t − τk−1 ) x(t − τ (t )) + 2 x (t ) x ( s )d s

δk

k=2

+2

N 

xT (t − τ (t ))

Z

k

δk

k=1

τk−1

k=1

x(t − τk ) + 2

N  k=1

t−τk−1

2 xT (t )Q2k τk + τk−1



t−τk−1

t−τk

x ( s )d s

where 0 , 1 , 2 , 3 are defined in Theorem 1. In order to establish the delay dependent stability condition, define MT = M1

 0 = 2η0T (t )M

r  r  r 

(3.11)

M2

M3

M4 , and note that







hl (t )hi (t )h j (t ∗ ) (A¯ li + A¯ li (t ))x(t ) + (B¯ l j + B¯ l j (t ))x(t − τ (t ) ) + Bωl ω (t ) − x˙ (t )

l=1 i=1 j=1

=

r  r  r 



hl (t )hi (t )h j (t ∗ )2η0T (t )M A¯ li

−I

0

B¯ l j

η0 (t ) +

l=1 i=1 j=1



r  r  r 

hl (t )hi (t )h j (t ∗ )2η0T (t )MGl Fl (t )

l=1 i=1 j=1

× El1 + Ebl1 Ki

0

0

El2 + Ebl2 K j

η0 (t ) +

r  r  r 

hl (t )hi (t )h j (t ∗ )2η0T (t )MBωl ω (t )

l=1 i=1 j=1

=

r  r  r  l=1 i=1 j=1

hl (t )hi (t )h j (t ∗ )η0T (t )( + )η0 (t ) +

r  r  r 

hl (t )hi (t )h j (t ∗ )2η0T (t )MBωl ω (t )

(3.12)

l=1 i=1 j=1

where  is defined in Theorem 1, and  = MGl Fl (t )E¯li + E¯liT FlT (t )GTl MT , by applying Lemma 2,  ≤ εli−1j MGl GTl MT +

εli j E¯liT E¯li , E¯li = El1 + Ebl1 Ki 0 0 El2 + Ebl2 K j .

M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

215

Considering (3.11) and (3.12), it holds that

V˙ (xt ) ≤

r  r  r 

hl (t )hi (t )h j (t ∗ )η0T (t )(0 +  + )η0 (t ) − η1T (t )1 η1 (t ) − η2T (t )2 η2 (t ) − η3T (t )3 η3 (t )

l=1 i=1 j=1

+2

N 

x (t − τk−1 ) T

k=2

× x(t − τk ) + 2

N  k=1

≤

r  r  r 

Z

k

δk

x(t − τ (t )) + 2

2 xT (t )Q2k τk + τk−1



N 

! x (t ) T

2 Q1k

k=1 t−τk−1

t−τk

"

τk−1

x ( s )d s +

t

t−τk−1

r  r  r 

x ( s )d s + 2

N 

xT (t − τ (t ))

Z

k=1

k

δk

hl (t )hi (t )h j (t ∗ )2η0T (t )MBωl ω (t )

l=1 i=1 j=1

hl (t )hi (t )h j (t ∗ )η0T (t )(0 +  + )η0 (t ) − η1T (t )1 η1 (t ) − η2T (t )2 η2 (t ) − η3T (t )3 η3 (t )

l=1 i=1 j=1

+ 2η0T (t )ϕ1 η1 (t ) + 2η0T (t )ϕ2 η2 (t ) + 2η0T (t )ϕ3 η3 (t ) +

r  r  r  l=1 i=1 j=1

hl (t )hi (t )h j (t ∗ )η0T (t )MBωl BTωl MT η0 (t )γ −2 + γ 2 ωT (t )ω (t )

⎡ 0 +  + εli−1j MGl GTl MT + εli j E¯liT E¯li ⎢ r  r  r ∗  ⎢ ≤ hl (t )hi (t )h j (t ∗ )ξ T (t )⎢ ∗ ⎣ l=1 i=1 j=1 ∗ +

r  r  r  l=1 i=1 j=1

ϕ1 −1

ϕ2

ϕ3

0 −2 ∗

∗ ∗

⎤

⎥ ⎥ ⎥ξ (t ) −3 ⎦ 0 0

hl (t )hi (t )h j (t ∗ )η0T (t )MBωl BTωl MT η0 (t )γ −2 + γ 2 ωT (t )ω (t )

(3.13)

where ϕ 1 , ϕ 2 , ϕ 3 are defined in Theorem 1. According to the definition of z(t), it yields that

zT (t )z(t ) − γ 2 ωT (t )ω (t ) =

r  r  r 



hl (t )hi (t )h j (t ∗ ) (Cl1 + Dl1 Ki )x(t ) + (Cl2 + Dl2 K j )x(t − τ (t ))

T

l=1 i=1 j=1

×

r  r  r 



hl (t )hi (t )h j (t ∗ ) (Cl1 + Dl1 Ki )x(t ) + (Cl2 + Dl2 K j )x(t − τ (t ))



l=1 i=1 j=1

− γ 2 ωT (t )ω (t ) ≤

r  r  r 

hl (t )hi (t )h j (t ∗ )η0T (t )T4 4 η0 (t ) − γ 2 ωT (t )ω (t ) + V˙ − V˙

l=1 i=1 j=1

≤

r  r  r 



hl (t )hi (t )h j (t ∗ )η0T (t ) T4 4 + 0 +  + 

η0 (t ) − η1T (t )1 η1 (t )

l=1 i=1 j=1

− η2T (t )2 η2 (t ) − η3T (t )3 η3 (t ) + 2η0T (t )ϕ1 η1 (t ) + 2η0T (t )ϕ2 η2 (t ) + 2η0T (t )ϕ3 η3 (t ) +

r  r  r  l=1 i=1 j=1

≤

r  r  r 

hl (t )hi (t )h j (t ∗ )η0T (t )MBωl BTωl MT η0 (t )γ −2 − V˙ hl (t )hi (t )h j (t ∗ )ξ T (t )ξ (t ) − V˙

l=1 i=1 j=1

where

⎡ T 4 4 + 0 +  + εli−1j MGl GTl MT + εli j E¯liT E¯li + γ −2 MBωl BTωl MT ⎢ ∗ =⎣ ∗ ∗

ϕ1 −1 ∗ ∗

ϕ2

ϕ3

0 −2 ∗

0 0

−3

⎤ ⎥ ⎦

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M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

And then, by using Schur complement to (3.2), it obtains that

⎡

T4 4 + 0 +  + εli−1j MGl GTl MT + εli j E¯liT E¯li + γ −2 MBωl BTωl MT

⎢ ⎣

ϕ1 −1

∗ ∗ ∗

∗ ∗

ϕ2

ϕ3

0 −2 ∗

0 0

⎤ ⎥ ⎦<0

(3.14)

−3

which means that

zT (t )z(t ) − γ 2 ωT (t )ω (t ) ≤ −V˙

(3.15)

Therefore, integrating both sides of (3.15) from 0 to T gives



T

0

zT (t )z(t )dt −



T

γ 2 ωT (t )ω (t )dt ≤ −V (T ) + V (0 )

0

Consider zero initial condition, and it implies that



∞

0

zT (t )z(t )dt ≤



∞

γ 2 ωT (t )ω (t )dt

0

In a word, z(t)2 ≤ γ ω(t)2 holds. When ω(t) ≡ 0, it follows from (3.2), it is easy to obtain that V˙ is negative definite. Therefore, the system (2.4) is asymptotically stable. This completes the proof.  t t xT (s )ds]T , η3 (t ) = Remark 3. Augmented vectors η1 (t ) = [xT (t − τ1 ), . . . , xT (t − τN )]T , η2 (t ) = [ t−τ xT (s )ds, . . . , t−τ 0 N−1  t−τ0 T  t−τN−1 T T [ t−τ x (s )ds, . . . , t−τ x (s )ds] show that more information about τ (t) may be utilized in the proposed work. 1

N

Next, based on the H∞ performance analysis for system (2.4) in Theorem 1, state feedback fuzzy controllers are designed as follows: Theorem 2. Assume that condition (2.2) holds. Fix scalar α > 0, γ > 0, positive integer N, and scalars β p ( p = 2, 3, 4 ). The closedloop system (2.4) is robustly asymptotically stable with an H∞ norm bound γ if there exist symmetric positive definite matrices ¯ , P¯ ∈ Rn×n , matrices X ∈ Rn×n , F ∈ Ru×n , and scalar σ > 0 such that the following LMIs  Z¯ k , R¯ k , Q¯ 1k , Q¯ 2k ∈ Rn×n (k = 1, . . . , N ), R i lij hold, for 1 ≤ l, i, j ≤ r:

⎡ ¯  ⎢ ∗ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣

ϕ¯ 1 ¯1 − ∗ ∗ ∗ ∗ ∗

∗ ∗

ϕ¯ 2

ϕ¯ 3

¯

0 ¯2 − ∗ ∗ ∗ ∗

0 0 ¯3 − ∗ ∗ ∗

0 0 0

−γ 2 I ∗ ∗

E¯li∗T 0 0 0 0 −σli j I ∗

¯T  4 0 0 0 0 0 −I

⎤ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎦

(3.16)

Moreover, the feedback gain matrices are given by Ki = Fi X −1 (i = 1, . . . , r ) where

¯ 11 = R˜¯ − 

N 

2Q¯ 1k −

N 

k=1

2δk Q¯ 2k + Al1 X + Bl1 Fi + X T ATl1 + FiT BTl1 + σli j Gl GTl τ + τ k k−1 k=1

¯ 12 = P¯ + β2 AT + β2 F T BT − X + β2 σli j Gl GT ,  ¯ 13 = β3 X T AT + β3 F T BT + β3 σli j Gl GT  i l1 i l1 l1 l l1 l ¯ 14 = β4 F T BT + β4 X T AT + Al2 X + Bl2 Fj + β4 σli j Gl GT  i l1 l1 l ¯ 22 = 

N 

δk Z¯ k +

k=1

N 

τk2−1

k=1

2

Q¯ 1k +

N 

τk2 − τk2−1

k=1

2

Q¯ 2k − β2 X − β2 X T + β22 σli j Gl GTl

¯ 23 = −β3 X T + β2 β3 σli j Gl GT ,  ¯ 24 = β2 Al2 X + β2 Bl2 Fj − β4 X T + β2 β4 σli j Gl GT  l l ¯ ¯ ¯ 33 = R¯ 1 − Z1 + β 2 σli j Gl GT ,  ¯ 34 = Z1 + β3 Al2 X + β3 Bl2 Fj + β3 β4 σli j Gl GT  3 l l δ1 δ1 ¯ 44 = −(1 − μ )R˜¯ − 2 

N  Z¯ k k=1

⎡ ϕ¯ 1 = ⎣ δ2 + Z¯ 2

Z¯ 1

δ1

Z¯ 3

Z¯ 2

δ3 + δ2

δk

+ β4 Al2 X + β4 Bl2 Fj + β4 FjT BTl2 + β4 X T ATl2 + β42 σli j Gl GTl 03n×Nn ...

⎤ Z¯ N

Z¯ N−1

δN + δN−1

Z¯ N

δN

⎡

⎦, ϕ¯ 2 = ⎣

2Q¯ 11

τ0

2Q¯ 12

τ1

... 03n×Nn

2Q¯ 1N

τN−1

⎤ ⎦

M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

⎡ ϕ¯ 3 = ⎣

2Q¯ 21

2Q¯ 22

τ1 +τ0

2Q¯ 2N

...

τ2 +τ1

τN +τN−1

03n×Nn

⎤

 T ⎦, ¯ T = Bωl

β2 BTωl

β3 BTωl

β4 BTωl

217



  ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 = diag −R¯ 2 + R¯ 1 + Z2 + Z1 , −R¯ 3 + R¯ 2 + Z3 + Z2 , . . . , −R¯ N + R¯ N−1 + ZN + ZN−1 , R¯ N + ZN  δ2 δ1 δ3 δ2 δN δN−1 δN     ¯ ¯ ¯ ¯ ¯ ¯ 2Q21 2Q22 2 Q2N ¯ 2 = diag 2Q11 , 2Q12 , . . . , 2Q1N ,  ¯ 3 = diag  , , . . . , 2 δ1 (τ1 + τ0 ) δ2 (τ2 + τ1 ) δN (τN + τN−1 ) τ02 τ12 τN−1     Cl1 X + Dl1 Fi 0 0 Cl2 X + Dl2 Fj E X + Ebl1 Fi 0 0 El2 X + Ebl2 Fj ¯4 =  , E¯li∗ = l1 Proof. It is clear to see that (3.16) implies that X is a nonsingular matrix. Thus, define M1 = X −1 , M2 = β2 M1 , M3 = β3 M1 , and M4 = β4 M1 . Let σli j = εli−1j , define the following variables:

R˜¯ = X T R˜X, P¯ = X T P X, Z¯k = X T Zk X, R¯ k = X T Rk X, Q¯ 1k = X T Q1k X, Q¯ 2k = X T Q2k X, k = 1, . . . , N

⎧ ⎨

Then pre- and post-multiplying both sides of (3.16) with diag M1T , . . . , M1T , IT , IT , IT

⎩#



%$&

⎫ ⎬ ⎭

and its transpose, respectively, one can get

3N+4

(3.14) by Schur complement, which is equivalent to (3.2). According to Theorem 1, the proof is completed.



Remark 4. Consider the open-loop fuzzy model without uncertainties.



 x˙ (t ) = rl=1 hl (t )(Al1 x(t ) + Al2 x(t − τ (t ) ) ), x(t ) = ϕ (t ), t ∈ [−h2 , 0]

t≥0

(3.17)

The asymptotic stability condition for the open-loop system (3.17) is stated below. Corollary 1. Assume that condition (2.2) holds. Fix scalar α > 0 and positive integer N. The system (3.17) is asymptotically stable , P ∈ Rn×n , and matrix Y ∈ R(3N+4 )n×n if there exist symmetric positive definite matrices Zk , Rk , Q1k , Q2k ∈ Rn×n (k = 1, . . . , N ), R such that the following LMIs hold

 + He(Y l ) < 0, l = 1, . . . , r where

⎡ 0 ⎢∗ =⎣ ∗

ϕ1 −1 ∗ ∗

∗

l = Al1

−I

0

⎤

ϕ2

ϕ3

0 −2 ∗

0 0

−3

Al2

0n×3Nn

⎥ ⎦

Proof. The same LKF (3.3) for system (3.17) is adopted for stability analysis. From (3.11), and define  , where 0 , 1 , 2 ,  3 , ϕ 1 , ϕ 2 , ϕ 3 are defined in Theorem 1. Therefore, it follows from V˙ (xt ) ≤ rl=1 hl (t )ξ T (t ) ξ (t ). According to fuzzy system r T (3.17), it means that 0 = l=1 hl (t )ξ (t )l ξ (t ), where l is defined in Corollary 1. Hence, the asymptotic stability condition for system (3.17) is expressed as

V˙ (xt ) ≤

r 

hl (t )ξ T (t ) ξ (t ) < 0

(3.18)

l=1

subject to : 0 =

r 

hl (t )ξ T (t )l ξ (t )

l=1

Consequently, by using Finsler’s lemma (Lemma 3), there exists a matrix Y with appropriate dimensions such that (3.18) is equivalent to

V˙ (xt ) ≤

r 



hl (t )ξ T (t )  + He(Y l )

ξ (t ) < 0

(3.19)

l=1

According to Lyapunov–Krasovskii stability theorem, it is obvious that system (3.17) is asymptotically stable. This completes the proof. 

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M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

Fig. 2. The state response of system (4.1) with h2 = 1.6, μ = 0.1.

Fig. 3. Control input curve for (4.1).

4. Numerical example Example 1. Consider the following T-S fuzzy system with input time-varying delays:

M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

219

Fig. 4. The state response of system (4.2) with h2 = 6.2104, N = 10, μ = 0.

x˙ (t ) =

2 

hl (t )( (Al1 + Al1 (t ))x(t ) + (Al2 + Al2 (t ))x(t − τ (t ) ) + (Bl1 + Bl1 (t ))u(t )

l=1

+(Bl2 + Bl2 (t ))u(t − τ (t ) ) + Bωl ω (t ) ) z(t ) =

2 

hl (t )(Cl1 x(t ) + Cl2 x(t − τ (t )) + Dl1 u(t ) + Dl2 u(t − τ (t )) )

(4.1)

l=1

where

 A11 =

 B11 =

 Bω 2 =

 C22 =

 G1 =

 E21 =

 Eb21 =

−2 0.2







1 0.1 , A12 = −2 0.01











0 −2 , A21 = −0.2 0.12











1 0.1 , A22 = −0.2 0



0 0.03 0 0.01 0.5 , B12 = , B21 = , B22 = , Bω 1 = 0.1 −0.2 0.1 −0.1 0 0.5 0





0.04 −0.01 −0.3 0 −0.15 0







0.15 0.05 , C12 = 0.25 −0.01

0 1 , C11 = 0.1 0.25



*



−0.01 0.2 , D11 = , D12 = 0.08 0.05





0 −0.3 , G2 = 0.3 0











0 0.25





−0.015 1 , C21 = 0.05 0.25

0.15 0.25

   0.05 0.2 0.05 0.015 , D21 = , D22 = 0.08 0.012



0 −0.15 , E11 = 0.3 0

0.2 −0.05 , E22 = 0.04 0.08





+





0.01 −0.1







0.2 −0.05 , E12 = 0.04 0.08







 



−0.035 −0.045

−0.035 0.05 −3 , Eb11 = , Eb12 = −0.045 −0.015 0.1

0.05 −3 , Eb22 = −0.015 0.1

In two rules, the membership functions are h1 (θ (t )) = sin2 (θ (t )), h2 (θ (t )) = 1 − h1 (θ (t )). Given γ = 0.8, we consider Theorem 2 with β2 = β3 = β4 = 0.1, σli j = 0.3, μ = 0.1, N = 3. It can be find the maximal upper bounds of τ (t) is h2 =





1.6 with h1 = 0.5, and the feedback gain matrices are K1 = −3.6964 −1.0057 , K2 = −3.4230 −1.9700 . N = 4, it can be also find the maximal upper bounds of τ (t) is h2 = 3.0 with h1 = 1.0, and the feedback gain matrices are K1 =

220

M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

Fig. 5. The state response of system (4.2) with h2 = 3.471, μ = 0.1. Table 1 Upper bounds of h2 for different values of μ.



−3.6133

T



−0.5173 , K2 = −3.3321

Methods

μ=0

μ = 0.1

[35] [36] [37] (Theorem 2) [38] (N = 3) [1] (N = 3) [39] [40] [41] (Theorem 1) Corollary 1 (N = 3) Corollary 1 (N = 4)

1.5974 1.5974 1.6609 1.956 2.002 2.0689 2.2943 2.5932 2.8036 3.6756

1.4847 – 1.5332 1.733 1.8090 1.8447 – 2.3268 2.4965 3.1844

−3.3867 . Consider τ (t) satisfying 0.5 ≤ τ (t) ≤ 1.6, under the initial condition ϕ (t ) =



T

1 −1 with N = 3, the disturbance ω (t ) = 1 1 e−0.08t sin(5π /6 ) and τ (t ) = 1 + 0.1| sin(t )|, and the state responses of closed-loop systems (4.1) are showed in Fig. 2. Fig. 3 displays the control input curve. It is clearly seen that the closedloop system is asymptotically stable from Fig. 2. Example 2. Consider the following system:

Rule 1 : If θ1 (t ) is ± π /2, then x˙ (t ) = A11 x(t ) + A12 x(t − τ (t ) ) Rule 2 : If θ2 (t ) is ± 0, then x˙ (t ) = A21 x(t ) + A22 x(t − τ (t ) ) where

 A11 =

−2 0





0 −1 , A12 = −0.9 −1





0 −1 , A21 = −1 0





0.5 −1 , A22 = −1 0.1

(4.2)



0 −1

In the Rules 1 and 2, the membership functions are h1 (θ (t )) = sin2 (θ (t )), h2 (θ (t )) = 1 − h1 (θ (t )). For different delay derivative rate μ, comparing the upper bounds of h2 with some previous results in Table 1. It is observed from Table 1 that Corollary 1 in this paper is less conservative than other results. More simulation trials have been conducted in Table 2, it shows that the conservatism can be reduced for a higher number of partitioning subintervals. In general, the increasing rates of maximum delay bounds decrease when the delay partitioning number increases. Choose h2 = 6.2104 with N = 10 and μ = 0, let τ (t ) = 6.2104 exp(−0.5t ), the initial condition

ϕ (t ) = 1

T

−1 , and the state responses of T-S fuzzy system (4.2) are presented in Fig. 4.

M. Li et al. / Applied Mathematics and Computation 321 (2018) 209–222

221

Table 2 Upper bounds of h2 for different values of μ. Corollary 1

N=2

N=3

N=4

N=5

N=6

N=7

N=8

N=9

N = 10

μ=0 μ = 0.1

1.9011 1.7411

2.8036 2.4965

3.6756 3.1844

4.4860 3.8108

5.3189 4.1579

5.8821 4.4322

5.9999 4.4975

6.0997 4.5773

6.2104 4.6856

Table 3 Upper bounds of h2 for μ = 0.1 and different values of h1 . Methods

h1 = 0.4

h1 = 0.8

h 1 = 1.2

Corollary 1 (N = 3) Corollary 1 (N = 4)

3.315 4.260

3.076 3.646

2.997 3.471

Furthermore, let delay derivative rate μ = 0.1, comparing the maximum bounds of h2 in Table 3. Choose h2 = 3.471 with

μ = 0.1, the initial condition ϕ (t ) = 1

T

−1 , and the state responses of T-S fuzzy system (4.2) are conducted in Fig. 5. From Table 3, it is obvious that increasing the number of delay partitioning subintervals, less conservative results will be obtained, which shows the effectiveness of this paper. 5. Conclusions This paper addresses the problem of robust H∞ control for T-S fuzzy systems with input time-varying delays. Based on a new delay partitioning method, by constructing a modified Lyapunov–Krasovskii functional, less conservative conditions are obtained, and robust fuzzy controllers are designed via satisfying H∞ performance level. Two numerical examples are provided to demonstrate the effectiveness and advantages of this proposed approach. Due to the existence of input timevarying delays, exponential synchronization and non-fragile control problem of T-S fuzzy system is full of challenges and will be investigated in the future work. References [1] H.-B. Zeng, J.H. Park, J.-W. Xia, S.-P. 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