Extended dissipative analysis for uncertain T–S fuzzy system with time-varying delay and randomly occurring gain variations

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Journal of the Franklin Institute 356 (2019) 8542–8568 www.elsevier.com/locate/jfranklin

Extended dissipative analysis for uncertain T–S fuzzy system with time-varying delay and randomly occurring gain variations Huan Yu, Yuechao Ma∗, Junwei Liu School of Science, Yanshan University, Qinhuangdao Hebei 066004, PR China Received 27 November 2018; received in revised form 16 August 2019; accepted 18 August 2019 Available online 24 August 2019

Abstract This paper is concerned with the non-fragile dynamic output feedback control for uncertain T–S fuzzy systems with time-varying delay and randomly occurring gain variations (ROGVs). Considering the imperfect premise matching that the T–S fuzzy model and the fuzzy controller do not have the same membership function, the purpose is to enhance the robustness of the system and the flexibility of the controller design. By adjusting the free weight matrix in the concept of extended dissipative, H∞ , L2 − L∞ , passive and (Q, S, R)-dissipative performance are solved in a unified framework. Stochastic phenomenon ROGVs is considered to describe the impact of the controller gain variations in the system, which is designed into two sequences of random variables and obey the Bernoulli distribution. Based on Lyapunov–Krasovskii functional (LKF) and integral inequality technique, some less conservative sufficient conditions are obtained to guarantee the close-loop system is asymptotically stable and extended dissipative. By solving the linear matrix inequalities (LMIs), a non-fragile dynamic output feedback controller can be developed. The advantage and effectiveness of the proposed design method can be illustrated by several numerical examples. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author. E-mail address: [email protected] (Y. Ma).

https://doi.org/10.1016/j.jfranklin.2019.08.025 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction Fuzzy control [1–3], as a very important branch of intelligent control, has received widespread attention in recent decades and has become one of the most active research hotspots in the current control field. T–S fuzzy system [4] proposed by Takagi and Sugeno made the fuzzy control and modern control theory very well combined, and provides an effective framework to solve the stability and performance analysis of fuzzy control. T–S fuzzy model is based on a rigorous mathematical model, which is nonlinear in nature, and each subsystem constituting the model are linear. Such an excellent model structure makes it possible to study complex nonlinear systems using mature linear system theory. Compared with the traditional method of modeling near the equilibrium point of nonlinear system, T–S fuzzy model can approximate nonlinear system with arbitrary precision in any convex compact set, and the description of the model is simpler. Many scholars in the field of control carry out research work on T–S fuzzy systems from various aspects, such as stability analysis [5–8], observer design [9,10], filter design [11,12], event-triggered [13,14] and so on. Furthermore, time-delay exists widely in practical engineering systems, which greatly increasing the difficulty of system performance analysis and controller design. Consequently, there are numerous studies on T–S fuzzy system with time-delay [8,15–20]. In addition, in the physical system, due to the changes in physical parameters such as aging of components, changes in external environment or unknown factors that change with time, these uncertain factors may cause changes in the system model, which leads to system instability further. Up to now, uncertain T–S fuzzy system with time-varying delay was discussed in [15,16,18]. To facilitate design of fuzzy controller, a parallel distributed compensation (PDC) design technique is proposed in [21], that is, the fuzzy controller and the fuzzy model share the common premise membership function and the number of fuzzy rules. It should be noted that PDC technology reduces flexibility of fuzzy controller design and limits design freedom of controller. In this case, a controller design strategy based on the imperfect premise matching is proposed [22–25]. This method does not require fuzzy controller to have same premise membership function as fuzzy model, thus improving flexibility of controller design and a scope of application. Dissipation analysis is a significant part of fuzzy control analysis due to its unique advantages such as effectively attenuate interference, e.g. [15,17,29,52,55]. In recent years, passive analysis has also received great attention. In [18,26,53], authors prove that passive analysis can provide an effective method for system stability analysis. In addition, H∞ performance [11,12,16,31–34] and L2 − L∞ performance [27,28,30] are also extensively studied. Therefore, we introduce the concept of extended dissipative, which was first proposed by Zhang et al. [35]. By adjusting the free weight matrix of the new performance index, the extended dissipative can be transformed into some well-known performance indexes such as passive, H∞ , L2 − L∞ and (Q, S, R)-dissipative performance. A great quantity of research results for extended dissipative in a variety of dynamical systems is shown in [35–40]. In recent years, non-fragile control problem have became a topic of concern to scholars in the field of control [41–44]. Due to the inevitable uncertainty in the implementation process of the controller, small or even weak drifts and fluctuations in the parameter realization process of the designed controller may cause stability of the closed-loop system to be impaired, and the performance of the system will be reduced. In order to prevent aforementioned case from occurring, it is essential to design a non-fragile controller. In addition, owing to circumstances changes in the process of controller implementation, the controller gains may change randomly [45,46]. Under these circumstances, the controller gain variations may appear or disappear

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in the way of probability. The problem of randomly occurring gain variations has received widespread attention [47–49,51]. As far as our authors know, there are few articles on nonfragile extended dissipative control for uncertain T–S fuzzy system with time-varying delay and ROGVs. Based on the aforementioned observation and discussion, the problem of non-fragile extended dynamic output feedback control for uncertain T–S fuzzy system with time-varying delay and ROGVs is solved in this paper. The main contributions of this paper are as follows: (1) A non-fragile dynamic output feedback controller is designed for uncertain T–S fuzzy systems with time-varying delays and ROGVs under imperfect premise matching. (2) Sufficient conditions are obtained to ensure that the closed-loop system is asymptotically stable and extended dissipative. (3) Based on the results obtained, a non-fragile dynamic output feedback controller with ROGVs is designed to make the closed-loop system extended dissipative. (4) We obtain the LMI conditions that can be solved by using the MATLAB LMI toolbox. Notation: Rn denotes the n-dimensional Euclidean space, Rm × n is the set of all m × n real matrices. the superscript T denotes the matrix transpose. For a real symmetric matrix A ∈ Rn × n , A > 0(A ≥ 0) means that A is real symmetric positive defined (positive semi-defined). diag(. . . ) stands for a block-diagonal matrix. |·| denotes the Euclidean norm for vectors and · denotes the spectral norm for matrices. E{·} stands for the expectation operator. L2 [0, ∞) represents the space of square-integrable vector functions over [0, ∞). The symmetric matrix is denoted by ∗ . 2. Description of the problem and main results Consider a nonlinear system, and it can be presented as the following T–S fuzzy model: Plant rule i: IF ε1 (t) is Mi1 and . . . and εp (t) is Mip , THEN

⎧ x˙(t ) = Ai x (t ) + Adi x (t − d (t ) ) + Bi u (t ) + Bωi ω (t ), ⎪ ⎪ ⎨ z (t ) = Ci x (t ) + Cdi x (t − d (t ) ) + Di u (t ) + Dωi ω (t ), y (t ) = Cyi x (t ) + Dyi ω (t ), ⎪ ⎪ ⎩ x (t ) = ϕ (t ), t ∈ [−d2 , 0],

(1)

where i ∈  := {1, 2, . . . , r }, r is the number of IF-THEN rule, ε (t ) = [ε1 (t ), ε2 (t ), . . . , ε p (t )]T is premise variable, Mik (i = 1, 2, . . . , r, k = 1, 2, . . . , p) is the fuzzy set. x (t ) ∈ Rnx is the state vector, ω(t) ∈ Rq is the disturbance input which belongs to L2 [0, ∞). z(t) ∈ Rp is the control output, y(t) ∈ Rl is the measured output, u(t) ∈ Rm is the control input, ϕ(t) is the initial condition of the system, d(t) is a time varying continuous function that satisfies 0 ≤ d1 ≤ d(t) ≤ d2 and d˙(t ) ≤ h. Ai = Ai + Ai , Adi = Adi + Adi , Bi = Bi + Bi , Bωi = Bωi + Bωi . And Ai , Adi , Bi , Bωi , Ci , Cdi , Di , Dωi , Cyi , Dyi are known real constant matrices whit appropriate dimensions; Ai , Adi , Bi , Bωi are unknown matrices representing norm-bounded parametric uncertainties and are assumed to be of the form: [Ai Adi Bi Bωi ] = H1i F1 (t )[E1i E2i E3i E4i ],

(2)

where H1i , E1i , E2i , E3i , E4i are known real constant matrices with appropriate dimensions and F1 (t) is unknown real and possibly time-varying matrices satisfying F1T (t )F1 (t ) ≤ I . Using singleton fuzzifier, product inference, and center-average defuzzyifier, the global dynamics of the T–S fuzzy system (1) is described by the convex sum form:

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8545

⎧ r    ⎪ ⎪ x˙(t ) = hi (ε (t ) ) Ai x (t ) + Adi x (t − d (t ) ) + Bi u (t ) + Bωi ω (t ) , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ r ⎪ ⎨z (t ) =  hi (ε (t ) )[Ci x (t ) + Cdi x (t − d (t ) ) + Di u (t ) + Dωi ω (t )], (3) i=1 ⎪ r   ⎪  ⎪ ⎪ y (t ) = hi (ε (t ) ) Cyi x (t ) + Dyi ω (t ) , ⎪ ⎪ ⎪ i=1 ⎪ ⎩ x (t ) = ϕ (t ), t ∈ [−d2 , 0], where hi (ε (t ) ) = rβi (βε (i (t ε))(t )) , βi (ε (t ) ) = pj=1 Mi j (ε (t ) ), and Mij (ε(t)) is the grade of memi=1 r bership r of εj (t) in Mij . Then, β i (ε(t)) ≥ 0 and i=1 βi (ε (t ) ) ≥ 0. Hence, we have hi (ε(t)) ≥ 0 and i=1 hi (ε (t ) ) = 1. The fuzzy controller with r rules is considered here: Plant rule j: IF ε1 (t) is Nj1 and. . .and εq (t) is Njq , THEN ⎧ ⎨x˙u (t ) = Au j xu (t ) + Bu j y (t ), (4) u (t ) = K j xu (t ) + Ky j y (t ), ⎩ xu (t ) = ϕ (t ), t ∈ [−d2 , 0], where j ∈  := {1, 2, . . . , r }, r is the number of IF-THEN rule, N jk ( j = 1, 2, . . . , r, k = 1, 2, . . . , q) is the fuzzy set, xu (t ) ∈ Rnu is the controller state. Au j = Au j + α1 (t )Au j , Bu j = Bu j + α2 (t )Bu j , K j = K j + K j and Ky j = Ky j + Ky j . Auj , Buj and Kj , i ∈ R are the nominal controller gain matrices to be determined. Auj , Buj , Kj and Kyj are perturbed matrices. Two perturbation types are considered in this section. Type 1: the additive form Au j = H2 j F2 (t )E5 j , Bu j = H2 j F2 (t )E6 j , K j = H3 j F2 (t )E7 j , Ky j = H3 j F2 (t )E8 j . Type 2: the multiplicative form Au j = H2 j F2 (t )E5 j Au j , Bu j = H2 j F2 (t )E6 j Bu j , K j = H3 j F2 (t )E7 j K j , Ky j = H3 j F2 (t )E8 j Ky j , where H2j , H3j , E5j , E6j , E7j and E8j are known real constant matrices, F2 (t) is an unknown matrix satisfying F2T (t )F2 (t ) ≤ I . In this paper, we consider type 1, then type 2 can be obtained similarly. The full-fragile dynamic output feedback fuzzy controller is represented by: ⎧ r    ⎪ ⎪ x ˙ t = g j (ε (t ) ) Au j xu (t ) + Bu j y (t ) , ( ) ⎪ u ⎪ ⎨ i=1 r    (5) u t = g j (ε (t ) ) K j xu (t ) + Ky j y (t ) , ( ) ⎪ ⎪ ⎪ i=1 ⎪ ⎩ xu (t ) = ϕ (t ), t ∈ [−d2 , 0], β (ε (t ) ) where g j (ε (t ) ) = r j β j (ε (t )) , β j (ε (t ) ) = qk=1 N jk (ε (t ) ), and Njk (ε(t)) is the grade of memj=1  bership of εk (t) in Mjk . Then, β j (ε(t)) ≥ 0 and rj=1 β j (ε (t ) ) ≥ 0. Hence, we have gj (ε(t)) ≥ 0  and rj=1 g j (ε (t ) ) = 1. The stochastic variable αi (t ) = , i = 1, 2 characterizes the phenomena of randomly occurring uncertainties, which is assumed to obey the Bernoulli distribution [45,46] as Pr = {ai (t ) = 1} = 1 − ρi , Pr = {ai (t ) = 0} = ρi where ρ i ∈ [0, 1] is a known constant. Remark 1. Randomly occurring controller gain fluctuations are common and inevitable due to the existence of disturbances. Random variables α 1 (t) and α 2 (t) have been introduced to model

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the probability distribution of ROGVs. Specifically, when ρ1 = 0 and ρ2 = 0, the controller (5) reduces to general non-fragile dynamic out feedback controller. It is worth mentioning that randomly occurring controller gain fluctuations are rarely used in dynamic out feedback controllers. From Eqs. (3) and (5), the closed-loop system can be obtained: ⎧ r  r

  ⎪ x ˙ t = hi (ε (t ) )g j (ε (t ) )[ Ai + Bi Ky j Cyi x (t ) + Adi x (t − d (t ) ) + Bi K j xu (t ) ( ) ⎪ ⎪ ⎪ i=1 ⎪

j=1  ⎪ ⎪ ⎪ + Bωi + Bi Ky j Dyi ω (t )], ⎪ ⎪ ⎨ r  r

  z (t ) = hi (ε (t ) )g j (ε (t ) )[ Ci + Di Ky j Cyi x (t ) + Cdi x (t − d (t ) ) + Di K j xu (t ) ⎪ i=1 ⎪

j=1  ⎪ ⎪ ⎪ Dωi + Di Ky j Dyi ω (t )], + ⎪ ⎪ r  r ⎪    ⎪ ⎪ hi (ε (t ) )g j (ε (t ) ) Bu j Cyi x (t ) + Au j xu (t ) + Bu j Dyi ω (t ) . ⎩x˙u (t ) =

(6)

i=1 j=1

 Denote xˆ(t ) = x T (t )

T xu T (t ) , and



Ai Bi K j , hi (t )g j (t ) Bu j Cyi Au j i=1 j=1 i=1 j=1

r  r r  r   A 0 Aˆ d (t ) = hi (t )g j (t )Aˆ di (t ) = hi (t )g j (t ) di , 0 0 i=1 j=1 i=1 j=1

r  r r  r   B + Bi Ky j Dyi , Bˆ ω (t ) = hi (t )g j (t )Bˆ ωi j (t ) = hi (t )g j (t ) ωi Bu j Dyi i=1 j=1 i=1 j=1 r  r r  r     Cˆ (t ) = hi (t )g j (t )Cˆi j (t ) = hi (t )g j (t ) Ci + Di Ky j Cyi Di K j , Aˆ (t ) =

r  r 

hi (t )g j (t )Aˆ i j (t ) =

i=1 j=1 r  r 

Cˆd (t ) =

Dˆ ω (t ) =

r  r 

i=1 j=1 r  r 

hi (t )g j (t )Cˆdi (t ) =

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

 hi (t )g j (t ) Cdi

 0,

i=1 j=1

hi (t )g j (t )Dˆ ωi j =

r  r 

  hi (t )g j (t ) Dωi + Di Ky j Dyi .

i=1 j=1

Then the closed-loop fuzzy system can be written as follows:  x˙ˆ(t ) = Aˆ (t )x (t ) + Aˆ d (t )x (t − d (t ) ) + Bˆ ω (t )ω (t ), z (t ) = Cˆ (t )x (t ) + Cˆd (t )x (t − d (t ) ) + Dˆ ω (t )ω (t ).

(7)

The following section is about assumption, definitions and lemmas, which will be used to develop the main results in sequel. Assumption 1 [35]. Matrices , 1 , 2 and 3 satisfy the following conditions: (1) (2) (3) (4) (5)

= T , 1 = 1 T and 3 = 3 T ; ≥ 0 and 1 ≤ 0; Dωi  ·   = 0, ∀i ∈ S; ( 1  +  2  )  = 0; T T Dωi

1 Dωi + Dωi

2 + 2T Dωi + 3 > 0, ∀i ∈ S.

Definition 1 [35]. For given matrices , 1 , 2 and 3 satisfying Assumption 1, system (5) (or system (6)) is said to be extended dissipative if there exists a scalar ρ such that the

H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

following inequality holds for any tf ≥ 0 and all ω(t) ∈ L2 [0, ∞):  tf   J (s )dt − sup zT (t ) z (t ) ≥ ρ,

8547

(8)

0≤t≤t f

0

where J (t ) = zT (t ) 1 z (t ) + 2zT (t ) 2 ω (t ) + ωT (t ) 3 ω (t ). Lemma 1 [16]. Let matrices ϒ 1 , ϒ 2 and are real matrices of appropriate dimensions with satisfying = T > 0, for all T (t)(t) ≤ I, + ϒ1 (t )ϒ2 + ϒ1 T (t )ϒ2 < 0, if and only if there exists a scalar ς > 0, such that + ς −1 ϒ1 ϒ1 T + ς ϒ2 ϒ2 T < 0. Lemma 2 [16]. Let symmetric matrice and matrics ψ 1 and ψ 2 have appropriate dimensions. Then + ψ1 ψ2 T + ψ2 ψ1 T < 0, if there exists a scalar λ > 0 satisfying

λ−1 ψ1 + λψ2 < 0. ∗ −2I Lemma 3 [15]. For any constant matrices of appropriate dimensions Ni , Mi , Lj , Hj , i = 1, 2, j = 1, 2, 3, positive-definite symmetric matrices Q1 and Q2 , then:  t   − x˙T (s )Q1 x˙(s )ds ≤ ξ T (t ) 1 + d1Y1 T Q1 −1Y1 ξ (t ); t−d1

 −

t−d1

  x˙T (s )Q2 x˙(s )ds ≤ ξ T (t ) 2 + (d2 − d1 )Y2 T Q2 −1Y2 ξ (t ),

t−d2

where

    N2 L1 L2 L3 , Y2 = H1 M1 H2 M2 H3 , Y1 = N1   ξ T (t ) = xˆT (t ) xˆT (t − d1 ) xˆT (t − d (t ) ) xˆT (t − d2 ) ωT (t ) , ⎡ ⎤ ⎡ 0 H1 N1 + N1T −N1 + N2T L1T L2T L3T ⎢ ⎢∗ M1 + M1T ∗ −N2 − N2T −L1T −L2T −L3T ⎥ ⎢ ⎥ ⎢ ∗ 1 = ⎢ ∗ ∗ 0 0 0 ⎥, 2 = ⎢∗ ⎣ ⎣∗ ∗ ∗ ∗ 0 0 ⎦ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗

0 H2T 0 ∗ ∗

−H1 −M1 + M2T −H2 −M2 − M2T ∗

⎤ 0 T H3 ⎥ ⎥ 0 ⎥ −H3T ⎦ 0

Theorem 1. Given positive scalars d1 , d2 , h, ε(0 < ε < 1), δi (i = 1, 2 ), and matrices ,

1 , 2 and 3 , for system (1) and non-fragile output feedback controller (5), the closedloop system (7) with uncertainties and time-delay is asymptotically stable and extended dissipative, if the membership functions of the fuzzy model and fuzzy controller satisfy h j (ε (t ) ) − ϑ j g j (ε (t ) ) ≥ 0, where 0 < ϑj < 1, and there exist matrices P > 0, Qm > 0, Zl > 0,  p = Tp > 0, Nk , Mk , Lq , Hq , m = 1, 2, l = 1, 2, k = 1, 2, q = 1, 2, 3, p = 1, 2, . . . , r with appropriate dimensions such that the follow matrix inequalities hold: i j − i < 0,

(9)

ϑi ii − ϑi i + i < 0,

(10)

ϑ j i j + ϑi ji − ϑ j i − ϑi  j + i +  j < 0, i < j,

(11)

i =

εP − CˆiT Cˆi ∗

−CˆiT Cˆdi > 0, T Cˆdi (1 − ε )P − Cˆdi

(12)

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where

⎡

⎢ ⎢ ⎢ ⎢ i j (t ) = ⎢ ⎢ ⎢ ⎣

i j +

⎡

 k=1,2

√ d 1 1 i j T

k

∗ ∗ ∗ ∗ ∗

( 1, 1 ) ⎢ ∗ ⎢ i j = ⎢ ⎢ ∗ ⎣ ∗ ∗

0 (2, 2 ) ∗ ∗ ∗

22 ∗ ∗ ∗ ∗ ( 1, 3 ) 0 ( 3, 3 ) ∗ ∗

0 0 0 (4, 4 ) ∗

√

d12 1i j T 0 33 ∗ ∗ ∗

√

d1Y1T

0 0 44 ∗ ∗

√

d12Y2T 0 0 0 55 ∗

2i j T 0 0 0 0 −I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

⎤ ( 1, 5 ) 0 ⎥ ⎥ (3, 5 )⎥ ⎥, 0 ⎦ ( 5, 5 )

where

    ¯ 1Cˆi j 0

¯ 1Cˆdi 0 0 0

¯ 1 Dˆ ωi , 1i j = PAˆ E 0 PAˆ di 0 0 0 PBˆ ωi , 2i j =

22 = −2δ1 P + δ12 Q1 , 33 = −2δ2 P + δ22 Q2 , 44 = −Q1 , 55 = −Q2 , (1, 1 ) = PAˆ E + Aˆ TE P +

3 

Zi , (1, 3 ) = PAˆ di ,

i=1 T

2 , (4, 4 ) = −Z3 , (1, 5 ) = PBˆ ωi − Cˆ T 2 , (2, 2 ) = −Z1 , (3, 3 ) = −(1 − h )Z21 , (3, 5 ) = −Cˆdi  T T (5, 5 ) = −Dˆ ωi 2 − 2 Dˆ ωi − 3 , 1 = − 1 , d12 = d2 − d1 .

The scalar in Definition 1 can be designed as follows ρ = −V (0 ) − P sup

−d2 ≤σ ≤0

|ϕ (σ )|2 .

(13)

Proof. Choose an augment LKF as following: V (t ) =

3 

Vi (t ),

(14)

i=1

where V1 (t ) =  xT (t )P x (t ), t t t V2 (t ) = t−d1  x T (s )Z1 x (s )d s + t −d (t )  x T (s )Z2 x (s )d s + t−d2  x T (s )Z3 x (s )d s, 0 t T  −d1  t T ˆ ˆ ˆ ˆ V3 (t ) = −d1 t+θ x˙ (s )Q1 x˙(s )d sd θ + −d2 t+θ x˙ (s )Q2 x˙(s )d sd θ , The time derivative of V1 (t) is    T  E V˙1 (t ) = E 2 x (t )P x˙(t )    = E 2 x T (t )P Aˆ (t )x (t ) + Aˆ d (t )x (t − d (t ) ) + Bˆ ω (t )ω (t ) = 2 x T (t )PAˆ E (t )x (t ) + 2 x T (t )PAˆ d (t )x (t − d (t ) ) + 2 x T (t )PBˆ ω (t )ω (t ),   where Aˆ E (t ) = E Aˆ (t ) .

(15)

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8549

The time derivative of V2 (t) is 3    

E V˙2 (t ) = xˆT (t )Zi xˆ(t ) − xˆT (t − d1 )Z1 xˆ(t − d1 ) − 1 − d˙(t ) xˆT (t − d (t ) )Z2 xˆ(t − d (t ) ) i=1

− xˆT (t − d2 )Z3 xˆ(t − d2 ) 3  ≤ xˆT (t )Zi xˆ(t ) − xˆT (t − d1 )Z1 xˆ(t − d1 ) − (1 − h )xˆT (t − d (t ) )Z2 xˆ(t − d (t ) ) i=1

− xˆT (t − d2 )Z3 xˆ(t − d2 ).

(16)

The time derivative of V3 (t) is  T     T E V˙3 (t ) = E d1 xˆ˙ (t )Q1 xˆ˙(t ) + d12 xˆ˙ (t )Q2 xˆ˙(t ) −  −

t

T xˆ˙ (s )Q1 xˆ˙(s )ds

t−d1 t−d1

T x˙ˆ (s )Q2 xˆ˙(s )ds.

(17)

t−d2

Applying Lemma 3, we have  t   − x˙T (s )Q1 x˙(s )ds ≤ ξ T (t ) 1 + d1Y1 T Q1 −1Y1 ξ (t ),

(18)

t−d1

 −

t−d1

  x˙T (s )Q2 x˙(s )ds ≤ ξ T (t ) 2 + d12Y2 T Q2 −1Y2 ξ (t ).

(19)

t−d2

Among them, we can compute easily that V˙ (t ) = 2xˆT (t )PAˆ E (t )xˆ(t ) + 2xˆT (t )PAˆ d (t )xˆ(t − d (t ) ) + 2xˆT (t )PBˆ ω (t )ω (t ) 3  + xˆT (t )Zi xˆ(t ) − xˆT (t − d1 )Z1 xˆ(t − d1 ) − (1 − h )xˆT (t − d (t ) )Z2 xˆ(t − d (t ) ) i=1

 T  T − xˆT (t − d2 )Z3 xˆ(t − d2 ) + E d1 xˆ˙ (t )Q1 xˆ˙(t ) + d12 xˆ˙ (t )Q2 xˆ˙(t ) ⎧ ⎫ ⎨ ⎬ + ξ T (t ) k + d1Y1 T Q1 −1Y1 + d12Y2 T Q2 −1Y2 ξ (t ), ⎩ ⎭

(20)

k=1,2

and   V˙ (t ) − J (t ) = V˙ (t ) − zT (t ) 1 z (t ) + 2zT (t ) 2 ω (t ) + ωT (t ) 3 ω (t ) r  r  hi g j ξ T (t )[i j + 1 + 2 + d1 1Tij (−22 )−1 1ij + d12 1Tij (−33 )−1 1ij ≤ i=1 j=1

+ d1Y1T (−44 )−1Y1 + d12Y2T (−55 )−1Y2 + zT (t ) 1 z (t )]ξ (t ).

(21)

It can be seen from Eq. (21), if r  r  i=1 j=1

hi (ε (t ) )g j (ε (t ) ) i j < 0,

(22)

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then V˙ (t ) − J (t ) ≤ 0. Similar to [23], a slack matrix is introduced r  r 

 hi h j − g j i = 0,

(23)

i=1 j=1

where  p = Tp ∈ 22n×22n > 0, p = 1, 2, . . . , r. From Eqs. (22) and (23), we can find that V˙ (t ) =

r  r 

hi (ε (t ) )g j (ε (t ) ) i j

i=1 j=1

=

r  r r  r  

 hi h j − g j + ϑ j h j − ϑ j h j i + hi g j i j i=1 j=1

≤

r 

i=1 j=1

hi2 (ϑi ii − ϑi i + i ) +

i=1

+

r  r 



 hi g j − ϑ j h j i j − i

i=1 j=1

r  

 hi h j ϑ j i j + ϑi ji − ϑ j i − ϑi  j + i +  j ,

(24)

i=1 i< j

with h j (ε (t ) ) − ϑ j g j (ε (t ) ) ≥ 0 for all j and ε(t). By Schur complement, Eq. (22) is equivalent to inequality (9)–(11), which implies that V˙ (t ) − J (t ) ≤ 0.

(25)

Integrating from 0 to t for Eq. (25), we can derive  t J (s )ds ≥ V (t ) − V (0 ).

(26)

0

Recalling Eq. (14), we have V (t ) ≥ xˆT (t )Pxˆ(t ) ≥ 0.

(27)

Also, notice from Eq. (13) that −V (0 ) ≥ ρ. Thus, it follows from Eq. (26) that  t J (s )ds ≥ ρ + xˆT (t )Pxˆ(t ), ∀t ≥ 0.

(28)

0

In order to prove that the system (6) satisfies Definition 1, we will discuss the two cases of   = 0 and   = 0, respectively. First of all, we consider the case of   = 0. It follows from Eq. (28) that, for ∀tf ≥ 0,  tf J (s )ds ≥ ρ + xˆT (t )Pxˆ(t ) ≥ ρ. (29) 0

Owing to zT (t) z(t) ≡ 0, Eq. (29) is equivalent to inequality (8). In the next place, we consider the case of   = 0. From Assumption 1, we obtain that  1  +  2  = 0 and D2i  = 0, ∀i ∈ S, which implies that 1 = 0, 2 = 0 and 3 > 0. Thus, we have J (s ) = ωT (t ) 3 ω (t ) ≥ 0.

(30)

H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

By Eqs. (30) and (27), we obtain that, for ∀t ≥ 0 and ∀tf ≥ 0 with tf ≥ t,  t  tf J (s )ds ≥ J (s )ds ≥ ρ + xˆT (t )Pxˆ(t ). 0

8551

(31)

0

For ∀t ≥ 0 and ∀tf ≥ 0 with tf ≥ t, when t ≥ d(t), it is obvious that 0 ≤ t − d (t ) ≤ t f . Thus, when t ≥ d(t), it follows from Eq. (31) that  tf J (s )ds ≥ ρ + xˆT (t − d (t ) )Pxˆ(t − d (t ) ). (32) 0

For ∀t ≥ 0 and ∀tf ≥ 0 with tf ≥ t, when t ≤ d(t), we have that −d2 ≤ −d (t ) ≤ t − d (t ) ≤ 0. Under these circumstances, it can be verified that # #2 ρ + xˆT (t − d (t ) )Pxˆ(t − d (t ) ) ≤ ρ + P#xˆ(t − d (t ) )# ≤ ρ + P sup

−d2 ≤σ ≤0

|ϕ (σ )|2

= − V (0 )  tf ≤ J (s )ds.

(33)

0

Therefore, from Eqs. (32) and (33), for any 0 < ε < 1, the following condition holds:  tf J (s )ds ≥ ρ + ε xˆT (t )Pxˆ(t ) + (1 − ε )xˆT (t − d (t ) )Pxˆ(t − d (t ) ).

(34)

0

Recalling Eq. (7) with D2i  = 0, ∀i ∈ S, for ∀t satisfying 0 ≤ t ≤ tf , we have we have

T

xˆ(t ) xˆ(t ) zT (t ) z (t ) = − i xˆ(t − d (t ) ) xˆ(t − d (t ) ) + ε xˆT (t )Pxˆ(t ) + (1 − ε )xˆT (t − d (t ) )Pxˆ(t − d (t ) ),

(35)

where i is defined in Eq. (12), which shows that i > 0. Thus, for any t ≥ 0, zT (t ) z (t ) ≤ ε xˆT (t )Pxˆ(t ) + (1 − ε )xˆT (t − d (t ) )Pxˆ(t − d (t ) ).

(36)

Thus, the inequality (8) can be derived from Eqs. (35) and (36). In summary, according to Definition 1, we can conclude that the system (7) is extended dissipative.  T ˆ˙ we introduce δ 1 > 0 such that the following Remark 2. To deal with the nonlinear term xˆ˙ Q1 x, inequality holds,

(P − δ1 Q1 )T Q1−1 (P − δ1 Q1 ) = PQ1−1 P − 2δ1 P + δ12 Q1 ≥ 0. Conservativeness can be reduced by adjusting positive scalars δ 1 . The intention of δ 2 is similar to δ 1 . ) In addition, the numbers of decision variables of Theorem 1 is (6 + r ) n(n+1 + 8n2 + 2n × 2 nq n = nx + nu , where r is the number of IF-THEN rule. Theorem 2. Given positive scalars d1 , d2 , h, ε(0 < ε < 1), δi (i = 1, 2 ), α 1 and α 2 (α 1 and α 2 are of opposite sign), and matrices , 1 , 2 and 3 , for system (1) and non-fragile output feedback controller (5), the closed-loop system (7) with uncertainties and time-delay is asymptotically stable and extended dissipative, if the membership functions of the fuzzy model and fuzzy controller satisfy h j (ε (t ) ) − ϑ j g j (ε (t ) ) ≥ 0, where 0 < ϑj < 1, and there exist

8552

H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568 T

matrices R > 0, Qm > 0, Z l > 0,  p =  p > 0, N k , M k , L q , H q , m = 1, 2, l = 1, 2, k = 1, 2, q = 1, 2, 3, p = 1, 2, . . . , r with appropriate dimensions such that the follow matrix inequalities hold: i j − i < 0,

(37)

ϑi ii − ϑi i + i < 0,

(38)

ϑ j i j + ϑi ji − ϑ j i − ϑi  j + i +  j < 0, i < j,

(39)

⎡

11 ⎣ ∗ ∗ where

0 22 ∗

⎤ 13 23 ⎦ > 0, 33

(40)

√ √ √ √  ⎡¯ T ⎤ ¯k d1 ¯ 1Ti j d12 ¯ 1Ti j d1Y¯1T d12Y¯2T ¯ 2i i j +  j k=1,2 ⎢ ⎥ ⎢ ¯ 22 ∗  0 0 0 0 ⎥ ⎢ ⎥ ⎢ ¯ 33 ∗ ∗  0 0 0 ⎥ i j (t ) = ⎢ ⎥, ⎢ ¯ 44 ∗ ∗ ∗  0 0 ⎥ ⎢ ⎥ ⎣ ¯ 55 ∗ ∗ ∗ ∗  0 ⎦ ∗ ∗ ∗ ∗ ∗ −I ⎡ ⎤ 0 0 ( 1, 1 ) ( 1, 3 ) ( 1, 5 ) ⎢ ∗ 0 0 0 ⎥ (2, 2 ) ⎢ ⎥ ⎢ i j = ⎢ ∗ ∗ 0 ( 3, 3 ) (3, 5 )⎥ ⎥, ⎣ ∗ ∗ ∗ 0 ⎦ (4, 4 ) ∗ ∗ ∗ ∗ ( 5, 5 ) $ % Ai R + α1 Bi K j + Bi Ky j Cyi R αAi R + αBi Ky j Cyi R K= ,   K3 αAi R + α 2 α2 RE Bu j Cyi R + αBi Ky j Cyi R where

    K 3 = Ai R + αα2 RE Bu j Cyi R + α1 Bi K j + αα1 α2 RE Au j + Bi Ky j Cyi R,



αR αR R R + δ12 Q1 , 33 = −2δ2 + δ22 Q2 , 44 = −Q1 , 55 = −Q2 , 22 = −2δ1 αR αR αR αR

3    A R αAdi R , (2, 2 ) = −Z 1 , (3, 3 ) = −(1 − h )Z 2 , (1, 1 ) = sym K + Z i , (1, 3 ) = di A αAdi R di R i=1 $ %



T

T Bωi + Bi Ky j Dyi − RCiT 2 − α1 Di K j 2 − R Di Ky j Cyi 2 RCdiT 2

T , , ( 3, 5 ) = − ( 1, 5 ) = αRCdiT 2 Bωi + Bi Ky j D yi + αα2 R Bu j Dyi − αRCiT 2 − αR Di Ky j Cyi 2  T αR αR R R , 22 = (1 − ε ) , 13 = Ci R + α1 Di K j αCi R , (4, 4 ) = −Z 3 , 11 = ε αR αR αR αR T

  T

(5, 5 ) = − Dωi + Di Ky j Dyi 2 − 2T Dωi + Di Ky j Dyi − 3 , 33 = −1 , 23 = Cdi R αCdi R ,



Adi R αAdi R Bωi + Bi Ky j Dyi  1i j = K¯ 0 0 0 0 , Adi R αAdi R Bωi + Bi Ky j Dyi + αα2 RBu j Dyi ⎡$ ⎤

T

T %T

T T T ¯ ¯ ¯

 T RCi 1 + α1 Di K j 1 + R Di KyiCyi 1 RCdi 1 ¯ ⎦.  2i j = ⎣

 0 ¯ 1 0 0 0 1 Dωi + Di Kyi Dyi ¯ 1 + αRCiT 2 + αR Di Ky j Cyi T

¯1 αRCdiT

αRCiT

H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

8553

In this case, the scalar ρ is the same as in Theorem 1. Proof. First, P and its inverse can be designed as



& M R S N −1 , . P= P = MT T NT Y

(41)

And we can set new variables 1 and 2 as



& I R I S , 2 = . 1 = MT 0 0 NT

(42)

Then, it is clear to see that P 1 = 2 . It follows that

R I T T > 0. 1 P 1 = 1 2 = I & S

(43)

Note that the identity that PP−1 = I gives to M N T = I − R& S holds. Through Eq. (43) and the Schur complement, we obtain R − & S −1 > 0 which ensures that I − R& S is invertible. Hence, the invertible matrices N and M, which satisfy M N T = I − R& S , always exist. Pre- and postmultiplying Eqs. (9)–(11) by   diag T1 , T1 , T1 , T1 , I , T1 , T1 , T1 , T1 , I and its transpose, respectively. We, then, obtain condition (44)–(46), ˆ j < 0, ˆ ij − 

(44)

ˆ ii − ϑi  ˆi+ ˆ i < 0, ϑi

(45)

ˆ i j + ϑi ˆ ji − ϑ j  ˆ i − ϑi  ˆ j + ˆi+ ˆ j < 0, i < j, ϑj

(46)

where

⎡

⎢ ⎢ ⎢ ⎢ ˆ i j (t ) = ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

ˆ ij + 

1, 1 ) ( ⎢ ∗ ⎢ ˆ ij = ⎢  ⎢ ∗ ⎢ ⎣ ∗ ∗ $

 k=1,2

ˆk 

∗ ∗ ∗ ∗ ∗

0  (2, 2 ) ∗ ∗ ∗

d1 ˆ 1Ti j

d12 ˆ 1Ti j

d21Yˆ1T

d22Yˆ2T

ˆ 22i j  ∗ ∗ ∗ ∗

0

ˆ 33i j  ∗ ∗ ∗

0 0

0 0 0

1, 3 ) ( 0 3, 3 ) ( ∗ ∗

0 0 0  (4, 4 ) ∗

ˆ 44i j  ∗ ∗ ⎤ 1, 5 ) ( 0 ⎥ ⎥ ⎥ 3, 5 )⎥, ( ⎥ 0 ⎦ 5, 5 ) (

ˆ 55i j  ∗

Ai R + Bi K j M T K=     & S Ai R + N E Bu j Cyi R + & S Bi K j M T + N E Au j M T

T ˆ 2i j

⎤

⎥ ⎥ 0 ⎥ ⎥ 0 ⎥, ⎥ 0 ⎥ ⎥ 0 ⎦ −I

% Ai ,   & S Ai + N E Bu j Cyi

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H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

Adi , ( 2, 2 ) & S Adi i=1 T RCdi 2 , = −Zˆ1 , ( 3, 3 ) = −(1 − h )Zˆ2 , ( 3, 5 ) = T Cdi

2 T



4, 4 ) = −Zˆ3 − Qˆ 2 , ( 5, 5 ) = − Dωi + Di Ky j Dyi 2 − 2T Dωi + Di Ky j Dyi − 3 , (

R I ˆ ˆ ˆ ˆ ˆ + δ12 Qˆ 1 , ( 44 = −Q2 , 55 = −Q2 , 22 = −2δ1 1, 5 ) I & S $

T

T % Bωi + Bi Ky j Dyi − RCiT 2 − M Di K j 2 − R Di Ky j Cyi 2 = , T

& S Bωi + & S Bi Ky j Dyi + & S α2 Bu j Dyi − CiT 2 − Di Ky j Cyi 2

ˆ 33 = −2δ2 R I + δ22 Qˆ 2 , ˆ 1i j  I & S



Adi R & S Adi Bωi + Bi Ky j Dyi = K 0 0 0 0 , & S Adi R Adi Bωi + & S Bi Ky j Dyi + N Bu j Dyi ⎡$

T

T %T T ¯ ¯1 D D RC

+ M K + R K C



1 i j i ˆ 2i j = ⎣ 0

1 T i yi yi T ¯ ¯ Ci 1 + Di Ky j Cyi 1 % T T ¯1

T RCdi

¯ 0 0 0 1 Dωi + Di Kyi Dyi , T ¯ Cdi

1 1, 1 ) = sym{K } + (

3 

A R Zˆi , ( 1, 3 ) = & di S Adi R

T1 Qi 1 = Qˆ i , T1 Z j 1 = Zˆ j , T1 l 1 = l , T1 Ni 1 = Nˆ i , T1 Mi 1 = Mˆ i , T1 Li 1 = Lˆ i , T1 Hi 1 = Hˆ i , ⎡ ⎤   ˆ k , i = 1, 2, 3, j = 1, 2, l = 1, 2, . . . , r. T1 ⎣ k ⎦ 1 =  k=1,2

k=1,2



I 0  , Let S= S= ST ,  S T Qˆ i  S = Qi ,  S T Zˆ j  S = Zj,  S T Nˆ i  S = Ni ,  S T Mˆ i  S = Mi ,  S T Hˆ i  S= 0 & S −1    ˆ T  Hi ,  S T Lˆ i  S = Li , T1 Pre- and post-multiplying k=1,2 k 1 = k=1,2 k , S l S = l .   (44)–(46) by diag  S,  S,  S,  S, I ,  S,  S,  S,  S , I and its transpose, respectively. We, then, obtain condition (37)–(39). In order to obtain the LMI condition, we suppose M = α1 I and N = α2 I . Here, α 1 and α 2 are given constants. According to the expressions of P and P−1 in Eq. (41), it is easy to obtain that & S −1 = (1 − α1 α2 )−1 R. Let α = (1 − α1 α2 )−1 , we have seen the positive definite matrices R and & S satisfy R& S − I > 0, which implies  0 < α < 1. Furthermore, pre- and post-multiplying (12) by diag  S T1 ,  S T1 and its transpose, respectively, yields that ⎡ ' ( ' ( ⎤   S ε T2 1 − T1 CˆiT Cˆi 1  S S − T1 CˆiT Cˆdi 1  S ⎣ ' ( ⎦>0 (47) T  ∗ S (1 − ε ) T2 1 − T1 Cˆdi Cˆdi 1  S is equivalent to Eq. (12). By the Schur Complement Formula, Eq. (47) is equivalent to Eq. (40). Hence, Eqs. (37)–(39) and (40) can guarantee that closed-loop system (6) is asymptotically stable and extended dissipative. 

H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

8555

Remark 3. In Theorem 2, we assume that the expression of matrices M and N are M = α1 I and N = α2 I , respectively. Its purpose is to ensure that the conditions are LMIs. Owing to the establishment of M N T = I − R& S , it is necessary to obtain that 0 < α = (1 − α1 α2 )−1 < 1. Furthermore we can get that α 1 α 2 < 0 and 0 < α < 1 are equivalent. For the sake of simplicity, we can choose some proper constant a1 and a2 be of opposite sign in Theorem 3 and the following Theorems. There are uncertainties in the above theorem, and the uncertainties will be dealt with in Theorem 3. Theorem 3. Given positive scalars d1 , d2 , h, ε(0 < ε < 1), δi (i = 1, 2 ), λ pi j (p = 1, 2, 3), α 1 and α 2 (α 1 and α 2 are of opposite sign), and matrices , 1 , 2 and 3 , for system (1) and non-fragile output feedback controller (5), the closed-loop system (7) with uncertainties and time-delay is asymptotically stable and extended dissipative, if the membership functions of the fuzzy model and fuzzy controller satisfy h j (ε (t ) ) − ϑ j g j (ε (t ) ) ≥ 0, where 0 < ϑj < 1, and there T exist matrices R > 0, Qm > 0, Z l > 0,  p =  p > 0, N k , M k , L q , H q , m = 1, 2, l = 1, 2, k = 1, 2, q = 1, 2, 3, p = 1, 2, . . . , r with appropriate dimensions such that the follow matrix inequalities hold: i j − i < 0,

(48)

ϑi ii − ϑi i + i < 0,

(49)

ϑ j i j + ϑi ji − ϑ j i − ϑi  j + i +  j < 0, i < j,

(50)

⎡

11 ⎢ ∗ ⎢ ⎣ ∗ ∗ where ⎡

0 22 ∗ ∗

ˆ 13  23 33 ∗

⎤ 14 0 ⎥ ⎥ > 0, 34 ⎦ 44

⎤ ⎡ 2 ⎤ ⎡ 1 3 i j Y1 Y3T i j U1 U3T Oˆ 1 3 2 ⎢ ⎥ ⎢ ⎥ ⎢ ij i j = ⎣ ∗ , = , = −λ3i j 0 ⎦ ij ⎣ ∗ −λ2i j 0 ⎦ ij ⎣ ∗ −λ1i j −1 ∗ ∗ −λ−1 ∗ ∗ −λ ∗ ∗ 3i j 2i j ⎡ T T T⎤ T T √ √ √ √  ¯k i j + d1  1i j d12  1i j d1Y 1 d12Y 2  2i j  ⎢ ⎥ k=1,2 ⎢ ⎥ ⎢ ⎥ ∗  0 0 0 0 22 1 ⎢ ⎥ ⎥, i j = ⎢ ∗ ∗  0 0 0 33 ⎢ ⎥ ⎢ ⎥ ∗ ∗ ∗  0 0 44 ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ 55 0 ⎦ ∗ ∗ ∗ ∗ ∗ −I ⎡ ⎤ 0 0 ( 1, 1 ) ( 1, 3 ) ( 1, 5 ) ⎢ ⎥ ⎢ ∗ 0 0 0 ⎥ (2, 2 ) ⎢ ⎥ i j = ⎢ ∗ ∗ 0 ( 3, 3 ) (3, 5 )⎥, ⎢ ⎥ ⎣ ∗ ∗ ∗ 0 ⎦ (4, 4 ) ∗ ∗ ∗ ∗ ( 5, 5 ) 4

(51)

⎤ Oˆ 3T ⎥ 0 ⎦, −λ−1 1i j

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H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

$ Ai R + α1 BiY3 j + BiY4 j Cyi R K= K3

% αAi R + αBiY4 j Cyi R , αAi R + α 2 α2Y2 j Cyi R + αBiY4 j Cyi R

where 3    K 3 = Ai R + αα2Y2 j Cyi R + α1 BiY3 j + αα1 α2Y1 j + BiY4 j Cyi R, (1, 1 ) = sym K + Zi,



A R αAdi R , (1, 3 ) = di Adi R αAdi R $

T

T % Bωi + BiY4 j Dyi − RCiT 2 − α1 DiY3 j 2 − R DiY4 j Cyi 2

T , ( 1, 5 ) = Bωi + BiY4 j Dyi + αα2Y2 j Dyi − αRCiT 2 − αR DiY4 j Cyi 2



Adi R αAdi R Bωi + BiY4 j Dyi 0 0 0 ,  1i j = K 0 Adi R αAdi R Bωi + BiY4 j Dyi + αα2Y2 j Dyi ⎡$

T

T %T T ¯ ¯1 D D RC

+ α Y + R Y C



1 1 i 3 j 1 i 4 j yi i

T  2i j = ⎣ 0 T ¯ ¯1 αRCi 1 + αR DiY4 j Cyi

%

T T ¯

T

1 RCdi ¯ 1 Dωi + DiY4 j Dyi 0 0 0

, T ¯ αRCdi

1

ς4 0 . . . 0 E8 j Dyi 0 . . . 0 U3 = ) *+ , ) *+ , , $ 0 U1 = ) Oˆ 1 = Oˆ 3 = Y1 = Y3 =

ς1 $ ς2 ς5 ς6

3

... *+

0 ,

T

2 Di H3 j

7

0 )

... *+

4

0 )

 ¯ 1 Di H3 j T

0 ,

4

... *+

0 ,

d1 ς1

d12 ς1

0

E4i + E3iY2 j Dyi

0

0

0

4

0

ς3

... *+

0 )

5

0 )

... *+

0 )

... *+

0 ,

d1 ς5

0 ,

E6 j Dyi

d12 ς5

0 )

4

3

... *+

0

i=1

T %T E3i H3 j

,

T , % 0 , ,

T 0 , ,

7

0 )

... *+ 9

 0 , , ς1 = H1i

 H1i ,

    ς2 = E1i R + α1 E3iY3 j + E3iY4 j Cyi R α1 E1i R + αE3iY4 j Cyi R , ς3 = E2i R αE2i R ,    

T ς4 = α1 E7i + E8 j R αE8 j R , ς5 = 0 , αα2 RH2 j   ς6 = (1 − ρ2 )E6iCyi R + (1 − ρ1 )α1 E5 j (1 − ρ2 )αE6iCyi R , O2 = F1 (t ), U2 = Y2 = F2 (t ),

  ˆ 13 = Ci R + α1 Di K j αCi R T , 14 = 0 E7 j ,  0 0

  0 −λ4i j , 34 = α1 Di H3 j 0 , 44 = ∗ −λ−1 4i j

H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

¯i ¯¯ =   i ∗

8557

05×11 . 011×11

In this case, controller gains are given by Au j = R−1Y1 j , Bu j = R−1Y2 j , K j = Y3 j and Ky j = Y4 j , the scalar ρ is the same as in Theorem 1. &1 (t ) +  &1 (t ). Then, by Proof. From Theorem 2, it can be obtained easily that = Lemma 1, there exist scalars −λ1i j > 0, such that &1 = O1 O2 O3 + (O1 O2 O3 )T ≤ λ1i j O3T O3 + λ−1 O1 O1T .  1i j By the Schur complement, we have ⎡ ⎤ &1 O1 O3T &2 = ⎣ ∗ −λ1i j 0 ⎦ < 0, ∗ ∗ −λ−1 1i j

(52)

(53)

where



αAi R + αBi K¯y j Cyi R Ai R + α1 Bi K j + Bi K¯y j Cyi R &  , K1 = &3 K αAi R + α 2 α2 RE Bu j Cyi R + αBi K¯y j Cyi R     &3 = Ai R + αα2 RE Bu j Cyi R + α1 Bi K j + αα1 α2 RE Au j + Bi Ky j Cyi R, ( K 1, 1 )1 3    &1 + = sym K Z i, i=1

$

T

T %

Bωi − RCiT 2 − M Di K j 2 − R Di Ky j Cyi 2 Adi R αAdi R   , (1, 5 )1 = , (1, 3 )1 =

T Adi R αAdi R Bωi − αRCiT 2 − αR Di Ky j Cyi 2



Adi R αAdi R Bωi + Bi Ky j Dyi & & 0 0 0 , 1i j = K1 0 Adi R αAdi R Bωi + Bi Ky j Dyi + αα2 Bu j Dyi $ % ς2 0 ς3 0 E4i + E3i Bu j Dyi 0 . . . 0 O3 = ) *+ , , O1 = Oˆ 1 ,

 ς&2 = E1i R + α1 E3i K j + E3i Ky j Cyi R

5

 α1 E1i R + αE3i Ky j Cyi R . 2

&1 are the same as in Theorem 2. Furthermore, &2 = i j + The remaining items of 2

 i j , where 2

 i j = U1U2U3 + (U1U2U3 )T + Y1Y2Y3 + (Y1Y2Y3 )T ,

(54)

by Lemma 1 and the Schur complement, Eqs. (48)–(50) holds. By Lemma 1 and Schur complement, Eq. (51) is equivalent to inequality (40).  Remark 4. Theorem 3 gives the sufficient conditions for the extended dissipation of the closed-loop system. However, there are nonlinear terms αα 2 Y2j Cyi R, α 2 α 2 Y2j Cyi R, Bi Y4j Cyi R and E3i Y4j Cyi R in the condition (48)–(50). This means that Eqs. (48)–(50) is a non-linear matrix inequality, which can not be solved by MATLAB. By using Lemma 2, the condition (48)–(50) will be transformed into LMI in Theorem 4.

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H. Yu, Y. Ma and J. Liu / Journal of the Franklin Institute 356 (2019) 8542–8568

Theorem 4. Given positive scalars d1 , d2 , h , ε(0 < ε < 1), δi (i = 1, 2 ), λ pi j ( p = 1, 2, 3, 4, 5 ), α 1 and α 2 (α 1 and α 2 are of opposite sign), and matrices , 1 , 2 and 3 , for system (1) and non-fragile output feedback controller (5), the closed-loop system (7) with uncertainties and time-delay is asymptotically stable and extended dissipative, if there exist maT trices R > 0, Qm > 0, Z l > 0,  p =  p > 0, N k , M k , L q , H q , m = 1, 2, l = 1, 2, k = 1, 2, q = 1, 2, 3, p = 1, 2, . . . , r with appropriate dimensions such that the follow matrix inequalities hold: i j −  i < 0,

(55)

 ϑi ii − ϑi  i + i < 0,

(56)

   ϑ j i j + ϑi  ji − ϑ j  i − ϑi  j + i +  j < 0, i < j,

(57)

where

⎡

⎤ i4j λ−1 λ−1 4i j ψ1 j + λ4i j ψ2i 5i j ψ3 j + λ5i j ψ4i ⎦, i j = ⎣ ∗ −2I 0 ∗ ∗ −2I ⎡ 3 ⎤ ⎡ 2 ⎤ ⎡ 1 i j Y1 Y3T i j U1 U3T i j Oˆ 1 4 3 2 ⎣ ⎦ ⎣ ⎦ ⎣ −λ3i j 0 −λ2i j 0 i j = ∗ , i j = ∗ , i j = ∗ −λ1i j −1 ∗ ∗ −λ−1 ∗ ∗ −λ ∗ ∗ 3i j 2i j ⎡  ⎤ T   T T ¯ i j + d1   d12   d21Y 1 T d22Y 2 T  2i j  1 1 i j i j k ⎢ ⎥ k=1,2 ⎢ ⎥ ⎢ ⎥ ∗  0 0 0 0 22i j ⎢ ⎥ ⎥, i1j = ⎢ ∗ ∗  0 0 0 33i j ⎢ ⎥ ⎢ ∗ ∗ ∗ 44i j 0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ 55i j 0 ⎦ ∗ ∗ ∗ ∗ ∗ −I ⎡ ⎤ 0 0 (1, 1 ) ( 1, 3 ) ( 1, 5 ) ⎢ ⎥ ⎢ ∗ 0 0 0 ⎥ (2, 2 )  ⎢ ⎥ i j = ⎢ ∗ ∗ 0 ( 3, 3 ) (3, 5 )⎥, ⎢ ⎥ ⎣ ∗ ∗ ∗ 0 ⎦ (4, 4 ) ∗ ∗ ∗ ∗ ( 5, 5 )

Ai R + α1 BiY3 j Ai R + α1 BiY3 j + αα1 α2Y1 j 3    Z i ,  1i j (1, 1 ) = sym K  + K =

i=1

ψ1 j

Adi R = K 0 Adi R $  0 ... 0 = ) *+ , 4

⎤ Oˆ 3T 0 ⎦, −λ−1 1i j

αAi R , αAi R



αAdi R Bωi + BiY4 j Dyi 0 0 0 , αAdi R Bωi + BiY4 j Dyi + αα2Y2 j Dyi % √ √ d1  d12  0 . . . 0 ) *+ , , 9

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T

T  R T  = BiY4 j , ψ4i = αα2Y2 j + BiY4 j $ %  Cyi R αCyi R 0 ... 0 ψ2i = ) *+ , , Oˆ 3

=

$ ς2

0

ς3

E4i + E3iY2 j Dyi

0

0 )

T

0 − DiY4 j Cyi 2 0 , )

4

T E3iY4 j Cyi

⎤ 0 )

 ς2 = E1i R + α1 E3iY3 j

... *+

... *+

0 )

... *+

0 , ,

% ¯i 0  = , ,  i ∗

... *+ 5

... *+



17

15

⎡ ⎢ ψ 3 j = ⎣0 )

αR

8559

0 ,



DiY4 j Cyi

T

013×5 , 013×13

¯1 0

4

⎥ 0 ⎦, ,

6

 α1 E1i R .

In this case, controller gains are given by Au j = R−1Y1 j , Bu j = R−1Y2 j , K j = Y3 j and Ky j = Y4 j , the scalar ρ is the same as in Theorem 1. Proof. By the Schur complement and Lemma 2, if there exists a positive scalar λij such that Eqs. (55)–(57) hold, then one has i4j + ψ1 j ψ2iT + ψ2i ψ1Tj + ψ3 j ψ4iT + ψ4i ψ3Tj < 0,

(58)

On the other hand, Eqs. (48)–(50) and (58) are equivalent. Therefore, if Eqs. (55)–(57) holds, then Eqs. (48)–(50) holds. The proof is thus completed. 

3. Numerical examples

Example 1. The mass-spring-damper system is modeled as a nonlinear model as follows [50] M x¨ + g(x, x˙ ) + f (x ) = φ(x˙ )u, where f (x ) = c3 x + c4 x 3 , g(x, x˙ ) = b(c1 x + c2 x˙ ) and φ(x˙ ) = 1 + c5 x˙3 . The model parameters are chosen, such as M = 1, b = 1, c1 = 0.5, c2 = 1.726, c3 = 0.5, c4 = 0.67 and c5 = 0. Suppose that x ∈ [−1, 1], x˙ ∈ [−1, 1].

 Select membership functions are h1 (θ (t ) ) = 1 − x12 (t ) ; h2 (θ (t ) ) = x12 (t ). Then the T–S fuzzy system is obtained as follows ⎧ 2  ⎨˙ x¯(t ) = βi (θ ){Ai x¯(t ) + Adi x¯(t − d (t ) ) + Bi u (t ) + Bωi ω (t )}, i=1 ⎩ y (t ) = Cyi x¯(t ) + Dyi ω (t ),

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 T  T where x˙¯ = x1 x2 = x x˙ is the state vector, u(t) is the input and ω(t) is the external disturbance.





1 1 0 0 0 0.1 , A2 = , Ad1 = Ad2 = , A1 = −1 −1.726 −1.67 −1.726 0 0.1  T    T  T B1 = B2 = 0 1 , Cy1 = Cy2 = 1 0 , Bω1 = 1 1.1 . Bω2 = 1 1 Dy1 = 0.01, Dy2 = 0. The control output z(t) is taken with the following parameter values:       C1 = C2 = −0.1 0 , Cd1 = 0.1 0 , Cd2 = 0.2 0 , D1 = D2 = 1, Dω1 = Dω2 = 0. Select membership functions of the dynamic output feedback controller are g1 (θ (t ) ) = 5x 2 5x 2 1 − e− 0.85 ; g2 (θ (t ) ) = e− 0.85 . In addition, we choose the other parameters involved in the simulation are taken as h = 0.2, d1 = 0.15, d2 = 0.3, a1 = −17, a2 = 0.01, λ111 = λ112 = λ121 = λ122 = 0.6, λ211 = λ212 = λ221 = λ222 = 0.1, λ311 = λ312 = λ321 = λ322 = 0.1, λ411 = λ412 = λ421 = λ422 = 0.01, λ511 = λ512 = λ521 = λ522 = 0.1, δ1 = 1.2, δ2 = 1.2, ρ1 = ρ2 = 0.5, ϑ1 = ϑ1 = 0.1. And use non-fragile dynamic output feedback controller (5) with perturbed matrices:





0 0 0 0.1 0.1 0.1 , E21 = , E22 = , E11 = E12 = 0 0.1 0 0.1 0 0.2 0.1 , E31 = E32 = E41 = E42 = 0   E51 = E52 = E71 = E72 = 0.1 0 , E61 = 0.1, E62 = 0.01,

0 0.1 , E81 = E82 = 0, H11 = 0 0.1

    0 0.1 0.1 , H21 = H22 = , H31 = 0.1 0 , H32 = 0.01 0 . H12 = 0 0.01 0 H∞ performance : Free weight matrices are designed in the concept of extended dissipation as follows: = 0, 1 = −1, 2 = 0, 3 = γ 2 , γ = 1, ρ = 0.

 Suppose ω (t ) = sin t ∗ e−t ∈ L2 [0, ∞ ), and the initial condition ϕ T (t ) = 0.5 [−1, 0]. Then, we can obtain the controller gain matrices:





0.1504 0.5137 −1.2467 0.3508 −0.0131 , Au2 = , Bu1 = , Au 1 = 0.8042 −1.5619 −1.2918 0.4736 −0.0045

    0.0040 , K1 = −0.0042 0.0205 , K2 = 0.0108 0.0235 , Bu2 = 0.0012

 −1 , t ∈

Ky1 = 5.3467e−4 , Ky2 = 2.5294e−4 . Fig. 1 plots the state trajectories in H∞ performance. Fig. 2 plots the states of controller and control input in H∞ performance. Passivity performance: Free weight matrices are designed in the concept of extended dissipation as follows: = 0, 1 = 0, 2 = 1, 3 = γ , γ = 1, ρ = 0.

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Fig. 1. H∞ : the state trajectories (Example 1).

Fig. 2. H∞ : the states of controller and control input (Example 1).

 Suppose ω (t ) = sin t ∗ e−t ∈ L2 [0, ∞ ), and the initial condition ϕ T (t ) = 0.5 [−1, 0]. Then, we can obtain the controller gain matrices:

 −1 , t ∈







0.0940 −0.6128 −1.1082 0.8793 −0.0631 , Au2 = , Bu1 = , Au 1 = 0.4866 −1.7021 −1.6820 1.2643 −0.0618

    −0.0161 , K1 = −0.0902 −0.0048 , K2 = −0.0596 0.0189 , Bu2 = −2.1814e−4 Ky1 = 0.0058, Ky2 = 0.0028. Fig. 3 plots the state trajectories in Passivity performance. Fig. 4 plots the states of controller and control input in Passivity performance. (Q, S, R)- dissipative performance: Free weight matrices are designed in the concept of extended dissipation as follows: = 0, 1 = −0.01, 2 = 0.1, 3 = 6 − γ , γ = 1, ρ = 0.

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Fig. 3. Passivity: the state trajectories (Example 1).

Fig. 4. Passivity: the states of controller and control input (Example 1).

  Suppose ω (t ) = sin t ∗ e−t ∈ L2 [0, ∞ ), and the initial condition ϕ T (t ) = 0.5 −1 , t ∈ [−1, 0]. Then, we can obtain the controller gain matrices:





0.0564 −1.0338 −0.7797 −0.6329 −0.0058 , Au2 = , Bu1 = , Au 1 = −0.0507 −2.1427 −0.4990 1.4572 −4.0113e−4

    −0.0023 0.0989 , K2 = −0.0603 0.2879 , Bu2 = −4 , K1 = 0.2560 −4.3794e Ky1 = −3.7031e−4 , Ky2 = 0.0021. Fig. 5 plots the state trajectories in (Q, S, R)- dissipative performance. Fig. 6 plots the states of controller and control input in (Q, S, R)- dissipative performance. Example 2. Consider T–S fuzzy system (1) with four fuzzy rules with the following parameters:







0 0 0 1 1 2.5 1.2 3 , A2 = , A3 = , A4 = , A1 = −1 −1 −2.3 −1 −2.3 −1 −3 −1.2

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Fig. 5. (Q, S, R)- dissipative: the state trajectories (Example 1).

Fig. 6. (Q, S, R)- dissipative: the states of controller and control output (Example 1).

1 1 1.2 2 , B2 = , B3 = , B4 = , 0 0 1 0







1 0 1 0 1 0.8 1 0 , Bω2 = , Bω3 = , Bω4 = , Bω1 = 0 1 0 1 0 1 2 1

    0 0.5 , Cy1 = Cy2 = 10 2 , Cy3 = Cy4 = 9 2 , C1 = C2 = C3 = C4 = 0 0.5







0 0 0 0.1 0.1 1 0.6 0.4 , Dω2 = , Dω3 = , Dω4 = , Dω1 = 0 0.01 0 0.2 0.5 0.8 0.3 1 0.1 0.2 0 0.2 , D2 = , D3 = , D4 = , D1 = 0.1 0.3 1 1         Dy1 = 0 0.1 , Dy2 = 0.1 0 , Dy3 = 0.1 0.2 , Dy4 = 0.5 0.8 . B1 =

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Table 1 γ for different d2 for Example 3. d2 γ

0 1.2017

0.03 2.3465

0.06 3.7150

In this example, we consider cases without uncertainty and non-fragility. Select membership functions are as follows: 0.5 0.5 , h2 (ε (t ) ) = 0.5 − , 1 + e(−x1 +1 ) 1 + e(−x1 −1 ) 0.5 0.5 h3 (ε (t ) ) = , h4 (ε (t ) ) = . −x −1 ( ) 1 1+e 1 + e(−x1 +1 ) h1 (ε (t ) ) = 0.5 −

And select membership functions of the dynamic output feedback controller are as follows: 0.5 0.5 , g2 (ε (t ) ) = 0.5 − , −x −1 ( ) 1+e 1 1 + e(−x1 −1 ) 0.5 0.5 g3 (ε (t ) ) = , g4 (ε (t ) ) = . 1 + e(−x1 −1 ) 1 + e(−x1 −1 ) g1 (ε (t ) ) = 0.5 −

In addition, we choose the other parameters involved in the simulation are taken as h = 0.1, a1 = −10, a2 = 0.45, λ1i j = λ2i j = λ3i j = 0.2, λ41 j = λ42 j = 0.35, λ43 j = λ43 j = 0.3, λ51 j = λ52 j = 2.7, λ53 j = 2.4, λ53 j = 2.3, δ1 = 1, δ2 = 2, ρ1 = ρ2 = 1, ϑ1 = ϑ1 = 0.1, i = j = 1, . . . , 4. In this example, we consider H∞ performance, the other cases can be similarly obtained. Choose the free weight matrix in the concept of extended dissipation as follow: = 0, 1 = −1, 2 = 0, 3 = γ 2 , γ = 1, ρ = 0. Table 1 shown the feasible and minimum H∞ performance level γ for different upper bounds of time-varying delay d2 . In the literature [54], the γ value is 4 without considering the timevarying delay, and in Theorem 4, the γ value is 1.2017 in the case of d2 = 0. Obviously, when the upper bound of time-delay is the same (d2 = 0), the upper bound of smaller than [54] can be obtained by the method proposed in this paper. The smaller the γ value, the stronger the robustness of the system and the stronger the ability to resist external interference. Accordingly, the method is superior to [54] in performance analysis.   We choose the system and controller states under initial conditions ϕ T (t ) = −3 3 . For ω (t ) = 0, γ = 4, d2 = 0.0512 and d1 = 0.0256. By solving the LMIs obtained in Theorem 4 and using Matlab LMI tool box, we can obtain feasible solutions and the controller gain matrices:



−4.4138 −1.1100 −0.3007 1.3409 , Au2 = , Au1 = −0.3319 0.7642 −1.7867 0.9086



5.8637 −0.7510 −6.1957 5.2837 , Au4 = , Au3 = −2.2698 −0.6661 −1.5645 0.7245







8.6860 0.7474 1.9457 8.8133 , Bu2 = , Bu3 = , Bu4 = , Bu1 = −0.3563 −0.9380 −2.0327 0.3304       K1 = 0.1307 −0.0035 , K2 = 0.0960 −0.0404 , K3 = 0.0650 −0.0466 ,   K4 = 0.1133 −0.0171 ,

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Fig. 7. The state trajectories (Example 2).

Fig. 8. The states of controller (Example 2).

Ky1 = −0.2501, Ky2 = −0.1701, Ky3 = −0.0501, Ky4 = 0.1500. Fig. 7 plots the state trajectories in Example 2. Fig. 8 plots the states of controller in Example 2. Remark 5. In Example 1, we designed non-fragile dynamic output feedback controllers and obtained the results of H∞ , passive and dissipative performance analysis for the mass-springdamper system. It can be seen from Example 1 that the proposed method can solve H∞ , passive, and dissipative performance under a unified framework. In Example 2, the minimum lower bound of the H∞ index γ is significantly smaller than the literature [54] without considering the time-varying delay and other parameters being the same. Therefore, the proposed method has stronger resistance to external interference than the literature [54] in system performance analysis. In addition, in the above example, due to the application of the imperfect premise matching method, the membership functions of fuzzy controller are simpler than the model using the PDC controller [15,16,41]. Therefore, the proposed model improves the flexibility of the fuzzy controller design and reduces the controller implementation cost.

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4. Conclusion This paper is concerned with the non-fragile dynamic output feedback control for uncertain T–S fuzzy systems with time-varying delay and ROGVs under imperfect premise matching. Based on LKF and integral inequality technique, some sufficient conditions with less conservative are obtained to guarantee asymptotically stable and extended dissipative of the close-loop system. By solving LMI, a non-fragile dynamic output feedback controller can be developed. The advantage and effectiveness of the proposed design method can be illustrated by several numerical examples. In this paper, the non-fragile dynamic output feedback control for uncertain T–S fuzzy systems with time-varying delay and ROGVs solves the problems of H∞ , L2 − L∞ , passive and dissipative performance under the unified framework. In addition, choosing some new integral inequalities to reducing the conservativeness of the model can be regarded as a future work. In future work, we intend to design an adaptive event-triggered dynamic output feedback controller to save computational resource. Choosing some new integral inequalities to reduce the conservativeness of the model can also be regarded as a future work. Acknowledgment This work is partially supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2018203099. References [1] D. Zhai, L. An, J. Dong, Q. Zhang, Switched adaptive fuzzy tracking control for a class of switched nonlinear systems under arbitrary switching, IEEE Trans. Fuzzy Syst. 26 (2018) 585–597. [2] Y. Yang, J. Ruan, J. Li, B. Liu, X. Kong, Route choice behavior model with guidance information based on fuzzy analytic hierarchy process, Int. J. Innov. Comput. Inf. Control 14 (1) (2018) 363–370. [3] V. Vorachart, H. Takagi, Evolving fuzzy logic rule-based game player model for game development. international journal of innovative computing, Inf. Control 13 (6) (2017) 1941–1951. [4] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern. 15 (1985) 116–132. [5] C. Ting, Stability analysis and design of Takagi–Sugeno fuzzy systems, Inf. Sci. 176 (19) (2006) 2817–2845. [6] H. Dong, Y. Joo, M. Tak, Local stability analysis of continuous-time Takagi–Sugeno fuzzy systems: a fuzzy Lyapunov function approach, Inf. Sci. 257 (2014) 163–175. [7] Y. Zhao, H. Gao, J. Lam, B. Du, Stability and stabilization of delayed T–S fuzzy systems: a delay partitioning approach, IEEE Trans. Fuzzy Syst. 17 (4) (2009) 750–762. [8] X. Zhao, C. Lin, B. Chen, Q. Wang, A novel Lyapunov–Krasovskii functional approach to stability and stabilization for T–S fuzzy systems with time delay, Neurocomputing 313 (2018) 288–294. [9] H. Hassani, J. Zarei, M. Chadli, J. Qiu, Unknown input observer design for interval type-2 T–S fuzzy systems with immeasurable premise variables, IEEE Trans. Cybern. 47 (2017) 2639–2650. [10] X. Tang, L. Deng, N. Liu, S. Yang, J. Yu, Observer-based output feedback MPC for T–S fuzzy system with data loss and bounded disturbance, IEEE Trans. Cybern. (PP(99)) (2018) 1–14. [11] B. Jiang, Z. Mao, P. Shi, H∞ -filter design for a class of networked control systems via T–S fuzzy-model approach, IEEE Trans. Fuzzy Syst. 18 (1) (2010) 201–208. [12] D. Hellani, A. Hajjaji, R. Ceschi, Finite frequency H∞ filter design for T–S fuzzy systems: new approach, Signal Process. 143 (2018) 191–199. [13] X. Su, F. Xia, J. Liu, L. Wu, Event-triggered fuzzy control of nonlinear systems with its application to inverted pendulum systems, Automatica 94 (2018) 236–248. [14] X. Su, F. Xia, L. Wu, C. Chen, Event-triggered fault detector and controller coordinated design of fuzzy systems, IEEE Trans. Fuzzy Syst. 26 (4) (2018) 2004–2016.

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