Circuits Syst Signal Process DOI 10.1007/s00034-017-0549-y

Finite-Time L 2 –L ∞ Control for Stochastic Asynchronously Switched Systems with Time-Varying Delay and Nonlinearity Hangfeng He1 · Xianwen Gao1 · Wenhai Qi2

Received: 1 June 2016 / Revised: 14 March 2017 / Accepted: 23 March 2017 © Springer Science+Business Media New York 2017

Abstract This paper deals with the problems of finite-time L 2 –L ∞ control for stochastic switched systems under asynchronous switching with time-varying delay and nonlinearity. Because of the presence of delay in the switching signal of controllers, the switching of the controllers is asynchronous with the switching of the subsystems. Firstly, based on average dwell time approach, merging switching signal technique and multiple Lyapunov function method, state feedback controllers are designed to guarantee the finite-time boundedness of stochastic switched timedelay systems under asynchronous switching by linearization techniques. Then the finite-time L 2 –L ∞ performance is analyzed. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. Keywords Stochastic switched systems · Finite-time boundedness · Finite-time L 2 –L ∞ performance · Asynchronous switching · Time-varying delay · Nonlinearity

1 Introduction Switched systems are a very important class of hybrid dynamic systems, which are composed of a set of continuous or discrete subsystems and a switching signal adjusting

B

Xianwen Gao [email protected] Hangfeng He [email protected] Wenhai Qi [email protected]

1

College of Information Science and Engineering, Northeastern University, Shenyang 110819, People’s Republic of China

2

Department of Automation, Qufu Normal University, Rizhao 276826, People’s Republic of China

Circuits Syst Signal Process

how the these subsystems run [6]. Switched systems are proposed and developed with profound theoretical and engineering background. Also, switched systems apply to many control problems of complex systems [36], such as network control systems, aircraft control systems, power systems, and chemical processes. Therefore, switched systems have received extensive attention from many researchers and have developed rapidly. With the development of computer science and electronic technology, switched systems theory has become one of the hot topics in the field of automatic control. In the actual production, the switched time-delay systems are very common, such as network control systems [19] and power systems [4]. The time lag item greatly affects the stability of the control systems because the time delay will bring instability and seriously affect the response speed of the systems; at the same time it will bring some difficulties to analyze and control the actual systems. Therefore, research on switched time-delay systems is very important, urgent, and challenging. In recent years, many important results have been achieved on switched time-delay systems [5,8,17,32,35]. The problem of exponential stability analysis for switched neutral systems with mixed time-delays and nonlinear perturbations based on descriptor system approach was addressed in [8]. Qi and Gao [17] dealt with the problem of L 1 control for positive Markovian jump systems with time-varying delays and partly known transition rates. Du et al. [5] discussed the fault diagnosis and fault tolerant control of switched linear systems. In [32], by constructing a new Lyapunov-like Krasovskii functional, sufficient conditions were derived and formulated to check the asymptotic (exponential) stability of such systems with arbitrary switching signals. Time delays may also appear in the switching signal, causing an asynchronous switching signal phenomenon between controllers and subsystems, such as communication systems [27]. The reasons for asynchronous switching may also be uncertainty or disturbance. In this paper, we consider the asynchronous switching from time delay, such as delayed switching due to detection time of subsystems switching signal, delayed switching because of channel congestion during information exchange and transfer process. In general, the phenomenon of asynchronous switching often leads to the decline of system performance or even makes the systems unstable. Zhang and Shi [34] investigated the stability and L 2 -gain problems for a class of discrete-time switched systems under asynchronous switching. In [24], stabilization of a class of switched linear neutral systems under asynchronous switching was studied. Ma and Zhao [15] investigated the network-dependent hybrid controller design for a class of networked switched linear systems. In [26], the input-to-state stability for switched nonlinear input delay systems under asynchronous switching was studied. In [37], the problem of finite-time stabilization under asynchronous switching was dealt with for a class of switched time-delay systems with nonlinear disturbances. Up to now, most of the existing literature related to stability of switched systems focus on Lyapunov asymptotic stability, which is defined over an infinite time interval [2]. However, some of the actual projects may require transient stability in a certain period of time [3]. Finite-time stability analysis and controller synthesis for switched linear systems were discussed with parameter-varying parameters [14] and stochastic time delay [1]. Finite-time boundedness and finite-time stability of switched systems with sector bounded nonlinearity and constant time delay were investigated in [12]. In

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[29], a finite-time stabilization problem was considered for a class of continuous-time Markovian jump delay systems. So far, although there are some results with respect to asynchronous switching [13,15,24,26,27,31,33,34,38], nonlinearity [9,10], and L 2 –L ∞ performance analysis [7,18,30], the problem of finite-time L 2 –L ∞ control for stochastic switched systems under asynchronous switching with time-varying delay and nonlinearity has not been investigated yet to the best of our knowledge, which motivates our investigation. For the switched systems without stochastic disturbance and time-varying delay in [38], the problem of finite-time stabilization under asynchronous switching was discussed. Wang et al. [22] studied finite-time L ∞ control for switched systems under asynchronous switching without stochastic disturbance and time-varying delay. Finite-time asynchronously switched control of switched systems with sampled-data feedback was studied in [25]. It is worth to point out that the time delays in [1,12,29] are constant delays. Moreover, though the time delays in [28] are time-varying delays, the derivatives of delays are limited to less than one, which may lead to some conservativeness. Motivated by this, we put forward a new definition of finite-time boundedness, construct a new piecewise Lyapunov function, remove some restrictive conditions on time-varying delay, and obtain results exhibiting less conservativeness. In this paper, the problem of finite-time L 2 –L ∞ control for stochastic switched systems with timevarying delay and nonlinearity under asynchronous switching is addressed. The main contributions of this paper are twofold: (i) a state feedback controller is designed to ensure that the corresponding closed-loop system is bounded in finite time; (ii) finitetime L 2 –L ∞ performance of the closed-loop system is analyzed and corresponding switched law is designed based on average dwell time. This paper is organized as follows. The problem for switched systems with timevarying delay and stochastic disturbance under asynchronous switching is formulated in Sect. 2. The main results are presented in Sect. 3. By applying the average dwell time method, a state feedback finite-time boundedness controller is designed. Then the finite-time L 2 –L ∞ performance of the closed-loop system is analyzed and corresponding switched law is designed based on average dwell time. Section 4 gives a numerical example, followed by conclusion in Sect. 5. Notation Throughout this paper, N T and N −1 denote the transpose and the inverse of any square matrix N . Symbol Rn stands for the n dimensional Euclidean space, Rn×m is the set of n × m real matrices, and S = {1, 2, . . . , N } is a set of positive numbers. The superscript I denotes the identity matrix with appropriate dimensions. Given a probability space (Ω, H, Θ), Ω is the sample space, H is the σ -algebra of subsets of the sample space and Θ is the probability measure on H . Symbol E{·} represents the mathematical expectation,  ∗  means the Euclidean norm of vectors. P > 0 (≥ 0) means P is real symmetric positive (semi-positive) definite and diag(·) denotes a block diagonal matrix. λmax (·) and λmin (·) denote, respectively, the largest and the smallest eigenvalues of the matrix inside the brackets. N denotes the set of nonnegative integer numbers and N+ = N/{0}. For simplicity, symbol ∗ is represented as an ellipsis for symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

Circuits Syst Signal Process

2 Problem Statement Consider the following switched stochastic systems with time-varying delay: dx(t) = [Aσ (t) x(t) + Adσ (t) x(t − d(t)) + Bσ (t) u σ (t) (t) + Fσ (t) ν(t)]dt + f σ (t) (t, x(t), x(t − d(t)))dω(t), z(t) = Cσ (t) x(t) + Cdσ (t) x(t − d(t)), x(t) = φ(t), t ∈ [−τd , 0],

(1)

where x(t) ∈ Rn is the state vector; u σ (t) (t) ∈ Rm is the control input of the ith subsystem; ν(t) ∈ Rr is the external disturbance input; ω(t) ∈ R is a standard Wiener process; z(t) ∈ Rq is the controlled output; φ(t) represents the initial condition, and d(t) is the time-varying delay. The function σ (t) ∈ [0, +∞) → = {1, 2, . . . , N } is the piecewise constant switching signal that may depend on time t. f i ∈ Rn∗m is the stochastic disturbance which is a nonlinear function, Ai , Adi , Bi , Fi , Ci , and Cdi are constant matrices with appropriate dimensions. Corresponding to the switching signal σ (t), we have the following switching sequence {x(t0 ): (i 0 , t0 ), (i 1 , t1 ), . . . , (i k , tk ), . . . , |i k ∈ , k = 0, 1, 2, . . .} which means that the i k th subsystem is activated when t ∈ [tk , tk+1 ) and t0 is the initial time. In this paper,the control input is of the form u σ (t) (t) = K σ (t−τs (t)) x(t)

(2)

where τs (t): [0, +∞) → [0, τs ] is the switching delay, satisfying τs ≤ tk+1 − tk , k ∈ N. The following assumptions and definitions will be used in the sequel. Assumption 1 [21] We assume that the state of the system does not jump at the switching instants and that only finitely many switches can occur in any finite interval. Assumption 2 [11] d(t) satisfies either of the following conditions. (a) d(t) is differential, and the derivative is bounded ˙ ≤h 0 < d(t) ≤ τd , d(t)

(3)

(b) d(t) is continuous and bounded 0 < d(t) ≤ τd

(4)

Remark 1 As is well known, a continuous and piecewise differentiable initial condition guarantees the existence of the solutions of a non-switched linear delay system, regardless of the bound of d(t). This is still true for switched delay system (1) because the state does not jump at the switching instants.

Circuits Syst Signal Process

Assumption 3 The disturbance ν(t) is bounded in finite time and satisfies the following conditions.  0

T

ν T (θ )ν(θ )dθ ≤ dν .

(5)

Assumption 4 [11] There exist constant real matrices G 1i and G 2i of appropriate dimensions such that   trace f iT (t, x(t), x(t − d(t))) f i (t, x(t), x(t − d(t))) T T G 1i x(t) + x T (t − d(t))G 2i G 2i x(t − d(t)). ≤ x T (t)G 1i

(6)

Definition 1 [16] Given four positive constants c1 , c2 , T , d with c1 < c2 , d ≥ 0, a positive definite matrix R, and a switching signal σ (t). For ∀t ∈ [0, T ], if      ˙ 0 ) ≤ c1 ⇒ E x T (t)Rx(t) ≤ c2 E sup−τd ≤t0 ≤0 x T (t0 )Rx(t0 ), x˙ T (t0 )R x(t (7) system (1) is said to be finite-time bounded with respect to (c1 , c2 , T, d, R, σ ). Definition 2 For T > 0, γ ≥ 0, system (1) is said to be finite-time bounded with L 2 –L ∞ performance γ , if under zero initial condition φ(t) = 0, ∀t ∈ [−τd , 0], for any non-zero ν(t), it holds that    sup E z T (t)z(t) ≤ γ 2 E

∀t∈[0,T ]

T

 ν T (θ )ν(θ )dθ .

(8)

0

Definition 3 [24] For any t1 , t2 , t2 > t1 ≥ 0, let Nσ (t1 , t2 ) denotes the number of switching of σ (t) over (t1 , t2 ). If Nσ (t1 , t2 ) ≤ N0 +

t2 − t1 τa

(9)

holds for τa > 0, N0 ≥ 0, then τa is called average dwell time and N0 is called a chatter bound. Denoted by Save [τa , N0 ] the class of switching signals with average dwell time τa and chattering bound N0 . By imitating the merging signal technique in [35] to deal with the mismatched switching signal, we get a virtual switching signal σ  (t): [0, +∞) → × as follows: σ  (t) = (σ1 (t), σ2 (t)). The merging action is denoted by ⊕ such that σ  = σ1 ⊕ σ2 . The definition implies that the set of switching times of σ  is the union of the sets of switching times of σ1 and of σ2 . Lemma 1 [20] Given σ1 ∈ Save [τa , N0 ] and σ2 (t) = σ2 (t − τs (t)), then it has σ2 ∈ Save [τa , N0 + (τs /τa )], σ  ∈ Save [τa , N0 ], where τa = τa /2, N0 = 2N0 +τs /τa .

Circuits Syst Signal Process

Lemma 2 [11] For any constant matrix Z = Z T > 0, scalar τ (t) > 0 and vector function x(·) : [−τ (t), 0] → Rn such that the following Jensens integral inequality is well defined:  −τ (t)

t

t−τ (t)

 x (s)Z x(s)ds ≤ − T

t t−τ (t)

 x (s)ds Z T

t t−τ (t)

x(s)ds

(10)

3 Main Results This section will focus on the problem of finite-time boundedness and finite-time L 2 – L ∞ analysis for asynchronously switched stochastic systems with time-varying delay, nonlinearity and corresponding state feedback controllers are designed.

3.1 Finite-Time Stabilization Consider the switched time-varying delay system as follows dx(t) = [ A¯ σ (t) x(t) + Adσ (t) x(t − d(t)) + Fσ (t) ν(t)]dt + f σ (t) (t, x(t), x(t − d(t)))dω(t), x(t) = φ(t), t ∈ [−τ, 0],

(11)

where A¯ σ (t) = Aσ (t) + Bσ (t) K i . In this subsection, the following theorem provides some conditions for finite-time boundedness of system (11). Theorem 1 Consider switched system (11) satisfying assumptions above mentioned. For given scalars τs > 0, τd ≥ 0, λa > 0, λb > 0, a > 0, h > 0, T > 0, κ > 1, β < 0, dν > 0. If there exist symmetric matrices P¯ii = P¯i j > 0, Q¯ ii > 0, Q¯ i j > 0, Z¯ ii > 0, Z¯ i j > 0, Mii and scalars ε¯ ii = εii−1 , ε¯ i j = εi−1 j , such that P¯ j j ≤ κ P¯i j , P¯i j ≤ κ P¯ii , Q¯ j j ≤ κ 3 Q¯ i j , Q¯ i j ≤ κ 3 Q¯ ii ,

(12)

Z¯ j j ≤ κ 3 Z¯ i j , Z¯ i j ≤ κ 3 Z¯ ii , ε¯ ii I ≤ P¯ii , ε¯ i j I ≤ P¯i j , ⎡ ⎤ T Π˜ 1ii Π˜ 2ii Fi P¯ii P¯ii AiiT + α MiiT Bii P¯ii G 1i 0 T T ⎥ ⎢ ∗ Π˜ 3ii 0 α P¯ii Adii 0 P¯ii G 2i ⎢ ⎥ T ⎢ ∗ ¯ ¯ ∗ β Pii α Pii F1i 0 0 ⎥ ⎢ ⎥ < 0, ⎢ ∗ ∗ ∗ τd Z¯ ii − 2α P¯ii 0 0 ⎥ ⎢ ⎥ ⎣ ∗ 0 ⎦ ∗ ∗ ∗ −¯εii I ∗ ∗ ∗ ∗ ∗ −¯εii I

(14)

(13) (15)

(16)

Circuits Syst Signal Process

⎡ ˜ ⎤ T Π1i j Π˜ 2i j Fi P¯i j P¯i j AiTj + α MiTj Bi j P¯i j G 1i 0 T T ⎥ ⎢ ∗ Π˜ 3i j 0 α P¯i j Adi 0 P¯i j G 2i ⎢ ⎥ j ⎢ ∗ ∗ β Pi j α P¯i j FiT 0 0 ⎥ ⎢ ⎥ < 0, ⎢ ∗ ∗ ∗ τd Z¯ i j − 2α P¯i j 0 0 ⎥ ⎢ ⎥ ⎣ ∗ 0 ⎦ ∗ ∗ ∗ −¯εi j I ∗ ∗ ∗ ∗ ∗ −¯εi j I

(17)

where Π˜ 1ii = Aii P¯ii + P¯ii AiiT + Bii Mii + MiiT BiiT + λa P¯ii + Q¯ ii − τd−1 e−λa τd Z¯ ii , Π˜ 2ii = Aii P¯ii + Bii Mii + τd−1 e−λa τd Z¯ ii , Π˜ 3ii = − (1 − h)e−λa τd Q¯ ii − τ −1 e−λa τd Z¯ ii , d

Π˜ 1i j = Ai j P¯i j + P¯i j AiTj + Bi j M j j + M Tjj BiTj − λb P¯i j + Q¯ i j − τd−1 eλb τd Z¯ i j , Π˜ 2i j = Ai j P¯i j + Bi j M j j + τd−1 eλb τd Z¯ i j , Π˜ 3i j = − (1 − h)eλb τd Q¯ i j − τd−1 eλb τd Z¯ i j . If the average dwell-time of the switching signal σ satisfies τa > τa∗ > τs and τa∗ is the numerical solution of the following equation. ξ1 e

[2 ln κ+(λa +λb )(τs +τd )] τT∗ a

+ ξ2 e

[2 ln κ+λb τs +(λa +λb )(τs +τd )] τT∗ a

= ξ3

(18)

where    1 ξ1 = c1 κe(λa +λb )τs −λa T λ1 + λ3 τd + τd2 λ5 κ 2 eλb τs +(λa +λb )τd − 1 , 2   λb τs 2 ln κ+(λa +λb )τd ξ2 = − βλ2 dν e κ +e ,     ξ3 = λ1 + λ4 τd e−λa τd κ 2 eλb τs +(λa +λb )τd − 1 c2 − βλ2 dν (1 + κeλb τs ). then switched systems (11) is finite-time bounded with respect to [c1 , c2 , T, d, R, σ ], where λ1 = min{λmin ( P˜ii ), λmin ( P˜i j )}, λ2 = max{λmax ( P˜ii ), λmax ( P˜i j )}, λ3 = max{λmax ( Q˜ ii ), λmax ( Q˜ i j )}, λ4 = min{λmin ( Q˜ ii ), λmin ( Q˜ i j )}, λ5 = λmax ( Z˜ ii ), 1 1 1 1 and P˜ii = R − 2 Pii R − 2 , P¯ii = Pii−1 , Q˜ ii = R − 2 Q ii R − 2 , Q¯ ii = P¯ii Q ii P¯ii , 1 1 Z˜ ii = R − 2 Z ii R − 2 , Z¯ ii = P¯ii Z ii P¯ii . Moreover, the state-feedback controller gains are given by K i = Mii P¯ii−1 . Proof See the “Appendix” section for the detailed proof. When the time-delay of switching signal τs = 0, which means the switching of subsystems and the switching of controllers are synchronized, we have the following corollary.

Circuits Syst Signal Process

Corollary 1 Consider a switched system (11) satisfying the assumptions above mentioned. For given scalars τd ≥ 0,λa > 0, a > 0, h > 0, T > 0, κ > 1, β < 0, dν > 0. If there exist symmetric matrices P¯ii > 0, Q¯ ii > 0, Z¯ ii > 0, Mii and scalar ε¯ ii = εii−1 , such that P¯ii ≤ κ P¯ j j , Q¯ ii ≤ κ 3 Q¯ j j , Z¯ ii ≤ κ 3 Z¯ j j , ε¯ ii I ≤ P¯ii ,

(19)

s.t. inequalities (16).

(20)

where Π˜ 1ii = Aii P¯ii + P¯ii AiiT + Bii Mii + MiiT BiiT + λa P¯ii + Q¯ ii − τd−1 e−λa τd Z¯ ii , Π˜ 2ii = Aii P¯ii + Bii Mii + τd−1 e−λa τd Z¯ ii , Π˜ 3ii = − (1 − h)e−λa τd Q¯ ii − τ −1 e−λa τd Z¯ ii . d

If the average dwell-time of the switching signal σ satisfies τa > τa∗ =

T ln κ ln ξ4

(21)

where ξ4 =

c2 (1 − κ)(λ6 + λ8 τd e−λa τd ) + βdν eλa T (1 − κ)[( 21 τd2 λ5 + λ6 + λ7 τd )c1 ] + βκdν

.

then switched systems with τs = 0 is finite-time bounded with respect to [c1 , c2 , T, d, R, σ ], where λ6 = λmin ( P˜ii ), λ7 = λmax ( Q˜ ii ), λ8 = λmin ( Q˜ ii ), λ5 = 1 1 1 1 λmax ( Z˜ ii ), and P˜ii = R − 2 P¯ii R − 2 , P¯ii = P −1 , Q˜ ii = R − 2 Q¯ ii R − 2 , Q¯ ii = ii

1 1 P¯ii Q ii P¯ii , Z˜ ii = R − 2 Z¯ ii R − 2 , Z¯ ii = P¯ii Z ii P¯ii . Moreover, the state-feedback controller gains are given by K i = Mii P¯ii−1 .

3.2 Finite-Time L 2 –L ∞ Analysis Theorem 2 Consider switched system (11) satisfying assumptions above mentioned. For given scalars τs > 0, τd ≥ 0, λs > 0, λu > 0, a > 0, h > 0, T > 0, κ > 1, β < 0, dν > 0. If there exist symmetric matrices P¯ii = P¯i j > 0, Q¯ ii > 0, Q¯ i j > 0, Z¯ ii > 0, Z¯ i j > 0, Mii and scalar ε¯ ii = εii−1 , ε¯ i j = εi−1 j , such that min λ2 , s.t. inequalities (16), (17), P¯ j j ≤ κ P¯i j , P¯i j ≤ κ P¯ii , Q¯ j j ≤ κ 3 Q¯ i j , Q¯ i j ≤ κ 3 Q¯ ii ,

(22) (23) (24)

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Z¯ j j ≤ κ 3 Z¯ i j , Z¯ i j ≤ κ 3 Z¯ ii , ε¯ ii I ≤ P¯ii , ε¯ i j I ≤ P¯i j , ⎡ ⎤ − P¯ii 0 P¯ii CiiT ⎣ ∗ − P¯ii P¯ii C T ⎦ < 0, dii ∗ ∗ −I ⎡ ¯ ⎤ − Pi j 0 P¯i j CiTj ⎣ ∗ − P¯i j P¯i j C T ⎦ < 0,  

∗

∗

−λ2 I I ∗ − P¯ii −λ2 I I ∗ − P¯i j

(25) (26) (27)

(28)

di j



−I

< 0,

(29)

< 0,

(30)



where Π˜ 1ii = Aii P¯ii + P¯ii AiiT + Bii Mii + MiiT BiiT + λa P¯ii + Q¯ ii − τd−1 e−λa τd Z¯ ii , Π˜ 2ii = Aii P¯ii + Bii Mii + τd−1 e−λa τd Z¯ ii , Π˜ 3ii = −(1 − h)e−λa τd Q¯ ii − τ −1 e−λa τd Z¯ ii , d

Π˜ 1i j = Ai j P¯i j + P¯i j AiTj + Bi j M j j + M Tjj BiTj − λb P¯i j + Q¯ i j − τd−1 eλb τd Z¯ i j , Π˜ 2i j = Ai j P¯i j + Bi j M j j + τd−1 eλb τd Z¯ i j , Π˜ 3i j = −(1 − h)eλb τd Q¯ i j − τd−1 eλb τd Z¯ i j . system (11) is finite-time bounded respect to [c1 , c2 , T, d, R, σ ] with L 2 –L ∞ performance γ under the following asynchronous switching τa > τa∗ = max{τa1 , τa2 },

(31)

where τa1 is the solution of ξ1 e

[2 ln κ+(λa +λb )(τs +τd )] τT

a1

+ ξ2 e

[2 ln κ+λb τs +(λa +λb )(τs +τd )] τT

a1

= ξ3 ,

and ξ5 , − ln ξ6 ξ5 = [2 ln κ + λb τs + (λa + λb )τd ]T,

τa2 =

ln γ 2

ξ6 = − 2βλ2 κeλb τs . Moreover, the state-feedback controller gains are given by K i = Mii P¯ii−1 .

Circuits Syst Signal Process

Proof According to Theorem 1 and zero initial condition, from the “Appendix” section in the proof of Theorem 1, we can get that when t ∈ [tk + τs , tk+1 ), E{x T (t)Pii x(t)} ≤ E{V (t)} < −βλ2 e

[2lnκ+λb τs +(λa +λb )τd ] τTa

 E

T

 ν (θ )ν(θ )dθ . T

(32)

0

and when t ∈ [tk , tk + τs ), E{x T (t)Pi j x(t)} ≤ E{V (t)} T

< −βλ2 κeλb τs e[2lnκ+λb τs +(λa +λb )τd ]( τa −1)  T  E ν T (θ )ν(θ )dθ .

(33)

0

It is also true that   T E x T (t − d(t))Pii x(t − d(t)) < − βλ2 e[2lnκ+λb τs +(λa +λb )τd ] τa  T  E ν T (θ )ν(θ )dθ .

(34)

0

From inequalities (32) and (34), we have  T     T  x(t) Pii 0 x(t) E ν T (θ )ν(θ )dθ , (35) < ξ7 E x(t − d(t)) 0 Pii x(t − d(t)) 0 where T

ξ7 = −2βλ2 e[2lnκ+λb τs +(λa +λb )τd ] τa . Multiplying both sides by diag{Pii , Pii }, LMI (27) is equivalent to    T  P 0 Cii  Cii Cdii < 0. − ii + T 0 Pii Cdii

(36)

(37)

Multiplying the left side by diag{x T (t), x T (t − d(t))} and right side by diag {x(t), x(t −d(t))} and combining (35) and (37), we derive that when t ∈ [tk +τs , tk+1 ),  E{z (t)z(t)} < ξ7 E T

T

 ν (t)ν(t)dt . T

(38)

0

Similarly, we can get that when t ∈ [tk , tk + τs ),  E{z T (t)z(t)} < ξ8 E 0

T

 ν T (t)ν(t)dt ,

(39)

Circuits Syst Signal Process

where T

ξ8 = −2βλ2 κeλb τs e[2lnκ+λb τs +(λa +λb )τd ]( τa −1) . Let ξ9 > ξ8 , ξ9 > ξ7 , we can obtain that for ∀t ∈ [0, T ], 

T

E{z T (t)z(t)} < ξ9 E

 ν T (t)ν(t)dt ,

(40)

0 T

ξ9 = −2βλ2 κeλb τs e[2lnκ+λb τs +(λa +λb )τd ] τa .

(41)

Therefore, system (1) is finite-time bounded with respect to [c1 , c2 , T, d, R, σ ] √ with L 2 –L ∞ performance γ = ξ9 . The proof is completed. Remark 2 It is noted from (41) that if the parameters α, λa , λb , κ, β are given, performance γ changes following λ2 . We can get the allowable minimum τa , through solving the minimum λ2 . By increasing κ and α, decreasing β and τd , adjusting λa and λb , we can get the feasible solution as far as possible. Remark 3 Theorem 2 gives finite-time L 2 –L ∞ analysis of the switched system (11). For a given target L 2 –L ∞ performance γ ∗ , the method to make system (11) meet the performance is in the following. If there is no restriction on the ADT τa , we can increase τa to make the system meet the performance as Theorem 2. Otherwise, we can put τ¯a (τ¯a is the maximum ADT) and other parameters into formula (41) to obtain the maximum λ2 , which transforms the constraint on the performance index γ ∗ into a constraint on λ2 (Table 1). Corollary 2 Consider a switched system (11) with τs = 0 satisfying the assumptions mentioned above. For given scalars τd ≥ 0, λa > 0, a > 0, h > 0, T > 0, κ > 1, β < 0, dν > 0. If there exist symmetric positive definite matrices P¯ii > 0, Q¯ ii > 0, Z¯ ii > 0, Mii and scalar ε¯ ii = εii−1 , such that min λ2 ,

(42)

P¯ii ≤ κ P¯ j j , Q¯ ii ≤ κ 3 Q¯ j j , Z¯ ii ≤ κ 3 Z¯ j j , ε¯ ii I ≤ P¯ii , s.t. inequalities (16), (27), (29),

(43) (44) (45)

where Π˜ 1ii = Aii P¯ii + P¯ii AiiT + Bii Mii + MiiT BiiT + λa P¯ii + Q¯ ii − τd−1 e−λa τd Z¯ ii , Π˜ 2ii = Aii P¯ii + Bii Mii + τd−1 e−λa τd Z¯ ii , Π˜ 3ii = −(1 − h)e−λa τd Q¯ ii − τ −1 e−λa τd Z¯ ii . d

Circuits Syst Signal Process Table 1 Algorithm 1

system (11) is finite-time bounded with L 2 –L ∞ performance γ under asynchronous switching. τa > τa∗ = max



 T ln κ (2 ln κ + λa τd )T . , ln ξ4 ln γ 2 − ln(−2βλ2 κ)

(46)

4 Numerical Example In this section, we present a numerical example to show the effectiveness of the proposed theorems in the previous section. Consider system (1) as follows: dx(t) = [Aσ (t) x(t) + Adσ (t) x(t − d(t)) + Bσ (t) u σ (t) (t) + Fσ (t) ν(t)]dt + f σ (t) (t, x(t), x(t − d(t)))dω(t), z(t) = Cσ (t) x(t) + Cdσ (t) x(t − d(t)), x(t) = φ(t), t ∈ [−τd , 0], which consists of two subsystems with R = I , τs = 0.33, d(t) = 0.05 sin(30t), τd = 0.05, h = 1.5. Subsystem 1 is described as follows: f 1 (t, x(t), x(t − d(t))) = 0.71 sin(t)(G 11 x(t) + G 21 x(t − d(t))),

Circuits Syst Signal Process

⎡

⎤ ⎡ ⎤ −1.7 1.7 0.1 1.5 −1.7 0.1 A11 = A12 = ⎣ 1.3 −1 0.7⎦ , Ad11 = Ad12 = ⎣−1.3 1 −0.3⎦ , −0.7 1 0.6 −0.7 1 0.6 ⎡ ⎤ ⎡ ⎤ 0.1 0.2 0.3 0.3 0.2 0.1 C11 = C12 = ⎣0.2 0.3 0.1⎦ , Cd11 = Cd12 = ⎣0.1 0.3 0.2⎦ , 0.3 0.2 0.1 0.2 0.1 0.3 ⎡ ⎤ ⎡ ⎤ 0.1 0.2 0.1 0.1 0.2 0.3 B11 = B12 = ⎣0.2 0.3 0.4⎦ , F11 = ⎣0.2 0.1 0.05⎦ , 0.3 0.2 0.1 0.1 0.3 0.2 ⎡ ⎤ 0.1 0.2 0.3 F21 = ⎣0.4 0.3 0.2⎦ , G 11 = −0.5I, G 21 = −0.2I, 0.1 0.2 0.4 Subsystem 2 is described as follows: f 2 (t, x(t), x(t − d(t))) = 0.71 cos(t)(G 12 x(t) + G 22 x(t − d(t))), ⎡ ⎤ ⎡ ⎤ 1 −1 −1.1 −1 0 0.1 A22 = A21 = ⎣−0.7 0.1 −0.6⎦ , Ad22 = Ad21 = ⎣1.3 −0.1 0.6⎦ , 1.7 −1.1 −1.7 1.5 0.1 1.8 ⎡ ⎤ ⎡ ⎤ 0.1 0.3 0.2 0.2 0.1 0.3 C22 = C21 = ⎣0.3 0.2 0.1⎦ , Cd22 = Cd21 = ⎣0.1 0.3 0.2⎦ , 0.2 0.1 0.3 0.3 0.2 0.1 ⎡ ⎤ ⎡ ⎤ 0.4 0.2 0.4 0.4 0.1 0.2 B22 = B21 = ⎣0.3 0.2 0.1⎦ , F22 = ⎣0.3 0.2 0.1⎦ , 0.1 0.2 0.3 0.2 0.2 0.3 ⎡ ⎤ 0.2 0.2 0.1 F12 = ⎣0.3 0.2 0.1⎦ , G 22 = −0.3I, G 12 = −0.1I, 0.1 0.3 0.4 Choose the initial state x(t) = [0.2; 0.15; 0.15], the disturbance ν(t) = [0.1e−0.01t sin(π t); 0.1e−0.01t sin(π t); 0.1e−0.01t sin(π t)], c1 = 0.1, c2 = 50. Given scalar T = 5, γ = 2.8, β = −0.5, α = 0.5, κ = 1.001, λa = 0.1, λb = 0.3, by using LMI Toolbox, we can get feasible solution following Theorem 2 with the parameters (Table 2). τa∗ = 0.5508, ⎡ ⎤ ⎡ ⎤ 16.4410 −22.7095 −6.3796 −11.5269 9.6490 4.3749 K 1 = ⎣−10.7258 14.2212 −6.1099⎦ , K 2 = ⎣ 10.3594 −26.7701 −1.7559⎦ . 2.3548 −20.7739 6.9481 2.9597 3.0598 −8.2738 Figure 1 depicts state trajectories of system. When time t tends to infinity, states tend to zero, which means the system is stable. But the transient states exceed prescribed bounds, which is not allowed in some of the practical engineering. Therefore, the

Circuits Syst Signal Process

Fig. 1 State trajectories of stable closed-loop system

Fig. 2 System mode and controller mode

Fig. 3 State trajectories of closed-loop system

Circuits Syst Signal Process

Fig. 4 x T (t)Rx(t) trajectories of closed-loop system Table 2 τa∗ with different h and γ

h

0.1

0.5

0.5

1.5

1.5

2

γ

2.8

2.8

2

2

2.8

2.8

τa∗

0.5236

0.5170

1.2167

1.4219

0.5508

0.5850

finite-time bounded is very meaningful. Figures 2, 3 and 4 depict the system mode and asynchronous controller mode σ (t) and σ (t − τs ), State x(t) and x T (t)Rx(t) trajectories of the closed-loop system with time-varying delay and nonlinearity. From Table 1, systems have solutions either for h < 1 and h > 1. From Fig. 4, systems satisfy x T (t)Rx(t) < c2 . And if choose the zero initial state x(t) = [0; 0; 0], systems satisfy L 2 –L ∞ performance in the finite time T , the proposed method in this example is effective.

5 Conclusions This paper deals with the problem of finite-time bounded and finite-time L 2 –L ∞ control for stochastic switched systems under asynchronous switching with time-varying delay and nonlinearity. By merging switching signal technique, multiple Lyapunov function method and free-weighting matrix, controllers which ensure that the closedloop system is finite-time bounded and finite-time L 2 –L ∞ bounded are designed in linear matrix inequalities. In future work, the results in this paper can be extended to finite-time L 2 –L ∞ control for stochastic time-delay switched systems with general nonlinearity and actuator saturation. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grants (61573088), (61573087) and (61433004).

Circuits Syst Signal Process

Appendix Because of the switching delay, after the jth subsystem has been switched to the ith subsystem, the controller K j is still active. Thus, we rewrite the closed-loop system as   dx(t) = A¯ σ  (t) x(t) + A¯ dσ  (t) x(t − d(t)) + Fσ  (t) ν(t) dt + f σ  (t) (t, x(t), x(t − d(t)))dω(t), x(t) = φ(t), t ∈ [−τd , 0],

(47)

where A¯ ii = Aii + Bii K ii = Ai + Bi K i , A¯ i j = Ai j + Bi j K i j = Ai + Bi K j , ¯ Adii = Adi j = Adi , Fii = Fi j = Fi , f ii (t, x(t), x(t − d(t))) = f i j (t, x(t), x(t − d(t))) = f i (t, x(t), x(t − d(t))). For system (47), the piecewise Lyapunov–Krasovskii function candidate is chosen as follows: V (t) = Vσ  (t) (t) = x T (t)Pσ  (t) x(t)  t + x T (s)eλσ  (t) (s−t) Q σ  (t) x(s)ds t−d(t) 0  t

 +

−τd

t+θ

y T (s)eλσ  (t) (s−t) Z σ  (t) y(s)dsdθ,

(48)

where Pσ  (t) , Q σ  (t) , Z σ  (t) are positive definite matrices to be determined and λii = λa , λi j = −λb , y(t)dt = dx(t). Introducing the free-weighting matrix Tii and Ti j yields Λii (t) = 2y T (t)TiiT {[ A¯ ii x(t) + A¯ dii x(t − d(t)) + Fi ν(t) − y(t)]dt + f i (t, x(t), x(t − d(t)))dω(t)} = 0, Λi j (t) = 2y T (t)TiTj {[ A¯ i j x(t) + A¯ di j x(t − d(t)) + Fi ν(t) − y(t)]dt + f i (t, x(t), x(t − d(t)))dω(t)} = 0.

(49) (50)

By Ito’s ˆ formula [23], we have dVii (t) = LVii (t)dt + 2x T (t)Pii f i (t, x(t), x(t − d(t)))x(t)dω(t),

(51)

Circuits Syst Signal Process

dVi j (t) = LVi j (t)dt + 2x T (t)Pi j f i (t, x(t), x(t − d(t)))x(t)dω(t).

(52)

where  ∂2V 1  ii LVii (t) = tr x T (t) f iT (t, x(t), x(t − d(t)) f i (t, x(t), x(t − d(t)))x(t)) 2 ∂x2  T ∂ Vii ∂ Vii (t) + (t) + × [(Ai + Bi K i )x(t) + Adi x(t − d(t)] ∂t ∂x ≤ 2x T (t)Pii [ A¯ ii x(t) + A¯ dii x(t − d(t)) + Fi ν(t)] −λa τ T ˙ x (t − d(t))Q ii x(t − d(t)) + x T (t)Q ii x(t) − (1 − d(t))e

+ trace[ f iT (t, x(t), x(t − d(t)))Pii f i (t, x(t), x(t − d(t)))]  0  t − λa y T (s)eλa (s−t) Z ii y(s)dsdθ −τd t

 − λa −e

t+θ

x T (θ )eλa (s−t) Q ii x(θ )dθ

t−d(t)  t −λa τd

y T (θ )Z ii y(θ )dθ

t−d(t)

+ y T (t)(τd Z ii − TiiT − Tii )y(t), ∂ 2 Vi j 1 LVi j (t) = tr (x T (t) f iT (t, x(t), x(t − d(t))) f i (t, x(t), x(t − d(t)))x(t)) 2 ∂x2  T ∂ Vi j ∂ Vi j (t) + (t) + × [(Ai + Bi K j )x(t) + Adi x(t − d(t)] ∂t ∂x ≤ 2x T (t)Pi j [ A¯ i j x(t) + A¯ di j x(t − d(t)) + Fi ν(t)] λb τ T ˙ x (t − d(t))Q i j x(t − d(t)) + x T (t)Q i j x(t) − (1 − d(t))e

+ trace[ f iT (t, x(t), x(t − d(t)))Pi j f i (t, x(t), x(t − d(t)))]  0  t + λb y T (s)e−λb (s−t) Z i j y(s)dsdθ −τd t

 + λb

t+θ

x T (θ )e−λb (s−t) Q i j x(θ )dθ

t−d(t)  t

− eλb τd

y T (θ )Z i j y(θ )dθ

t−d(t)

+ y T (t)(τd Z i j − TiTj − Ti j )y(t). Combining Assumption 4 and (15), there is εii > 0 for f iT (t, x(t), x(t − d(t)))Pii f i (t, x(t), x(t − d(t))) T T G 1i x(t) + x T (t − d(t))G 2i G 2i x(t − d(t))]. ≤ εii [x T (t)G 1i

(53)

Circuits Syst Signal Process

By Lemma 2, we can get  −

t

y T (θ )Z ii y(θ )dθ

t−d(t)

≤ =

−τd−1 τd−1





t

 y (θ )dθ Z ii T

t−d(t)

T 

x(t) x(t − d(t))

t

y(θ )dθ

t−d(t)

−Z ii Z ii Z ii −Z ii



 x(t) . x(t − d(t))

(54)

Construct function Γii , we have Γii (t) = LVii (t) + λa Vii (t) + βν T (t)Pii ν(t) + Λii (t) − 2y T (t)TiiT f i (t, x(t), x(t − d(t)))dω(t) ≤ 2[x T (t)Pii + y T (t)TiiT ][ A¯ ii x(t) + A¯ dii x(t − d(t)) + Fi v(t)] − (1 − h)e−λa τd x T (t − d(t))Q ii x(t − d(t)) + x T (t)(Q ii + λa Pii )x(t) T G x(t) + x T (t − d(t))G T G x(t − d(t))] + εii [x T (t)G 1i 1i 2i 2i  T    x(t) −Z ii Z ii x(t) + τd−1 e−λa τd x(t − d(t)) Z ii −Z ii x(t − d(t))

+ y T (t)(τd Z ii − TiiT − Tii )y(t) + βν T (t)Pii ν(t) = η T (t)Φii η(t),

(55)

  where η T (t) = x T (t) x T (t − d(t)) ν T (t) y T (t) and ⎤ A¯ iiT Tii Π1ii Π2ii Pii Fi T T ⎥ ⎢ ∗ Π3ii 0 A¯ dii ii ⎥, Φii = ⎢ ⎦ ⎣ ∗ ∗ β Pii FiT Tii ∗ ∗ ∗ τd Z ii − TiiT − Tii T Π1ii = Pii A¯ ii + A¯ iiT Pii + εii G 1i G 1i + Q ii + λa Pii − τd−1 Z ii e−λa τd , Π2ii = Pii A¯ ii + τ −1 Z ii e−λa τd , ⎡

d

T Π3ii = εii G 2i G 2i − (1 − h)e−λa τd Q ii − τd−1 Z ii e−λa τd .

Multiplying both sides of Φii by Δ = diag{ P¯ii , P¯ii , P¯ii , P¯ii } and letting Mii = K ii P¯ii , Tii = α Pii , we derive ⎤ Π¯ 1ii Π¯ 2ii Fi P¯ii P¯ii AiiT + α MiiT Bii T ⎥ ⎢ ∗ Π¯ 3ii 0 α P¯ii Adii ⎥, ΔT Φii Δ = ⎢ ⎦ ⎣ ∗ ∗ β P¯ii α P¯ii FiT ¯ ¯ ¯ ∗ ∗ ∗ τd Pii Z ii Pii − 2α Pii ⎡

(56)

Circuits Syst Signal Process

where T G 1i P¯ii + P¯ii Q ii P¯ii Π¯ 1ii = Aii P¯ii + P¯ii AiiT + Bii Mii + MiiT BiiT + εii P¯ii G 1i + λa P¯ii − τ −1 P¯ii Z ii P¯ii e−λa τd , d

Π¯ 2ii = Aii P¯ii + Bii Mii + τd−1 P¯ii Z ii P¯ii e−λa τd , T G 2i P¯ii − (1 − h)e−λa τd P¯ii Q ii P¯ii − τd−1 P¯ii Z ii P¯ii e−λa τd . Π¯ 3ii = εii P¯ii G 2i By using the Schur complement Lemma, letting Q¯ ii = P¯ii Q ii P¯ii , Z¯ ii = P¯ii Z ii P¯ii , ε¯ ii = εii−1 , (56) is equivalent to ⎡

Π˜ 1ii ⎢ ∗ ⎢ ⎢ ∗ ΔT Φii Δ = ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

⎤ T Π˜ 2ii Fi P¯ii P¯ii AiiT + α MiiT Bii P¯ii G 1i 0 T T ⎥ Π˜ 3ii 0 α P¯ii Adii 0 P¯ii G 2i ⎥ T ¯ ¯ ∗ β Pii α Pii Fi 0 0 ⎥ ⎥. ∗ ∗ τd Z¯ ii − 2α P¯ii 0 0 ⎥ ⎥ 0 ⎦ ∗ ∗ ∗ −¯εii I ∗ ∗ ∗ ∗ −¯εii I

(57)

where Π˜ 1ii = Aii P¯ii + P¯ii AiiT + Bii Mii + MiiT BiiT + λa P¯ii + Q¯ ii − τd−1 e−λa τd Z¯ ii , Π˜ 2ii = Aii P¯ii + Bii Mii + τd−1 e−λa τd Z¯ ii , Π˜ 3ii = − (1 − h)e−λa τd Q¯ ii − τ −1 e−λa τd Z¯ ii . d

From (51), (57) and Theorem 1, we can obtain LVii (t) + λa Vii (t) + βν T (t)Pii ν(t) − 2y T (t)TiiT f i (t, x(t), x(t − d(t)))dω(t) < 0 (58) Similarly, we can get LVi j (t) − λb Vi j (t) + βν T (t)Pi j ν(t) − 2y T (t)TiTj f i (t, x(t), x(t − d(t)))dω(t) < 0 (59) Considering formula (23) and formula (32), we can get 

t

tk +τs

d[eλa θ Vii (θ )] =



t

λa eλa θ Vii (θ )dθ +

tk +τs



<−  −

t tk +τs t

tk +τs



t tk +τs

eλa θ dVii (θ )

2eλa θ [x T (θ )Pii − y T (t)TiiT ] f i (θ, x(θ ), x(θ − d(θ )))dω(θ ) βeλa θ ν T (θ )Pii ν(θ)dθ.

(60)

Circuits Syst Signal Process

Taking expectations, we can get  E

t tk +τs

d[e

λa θ



Vii (θ )] = eλa t E{Vii (t)} − eλa (tk +τs ) E{Vii (tk + τs )} 
t tk +τs

 −βeλa θ ν T (θ )Pii ν(θ )dθ .

(61)

Namely E{Vii (t)} < e

λa (tk +τs −t)

 E{Vii (tk + τs )} − β E



t

ν (θ )Pii ν(θ )dθ . T

tk +τs

(62)

Similarly, we can get E{Vi j (t)} < e−λb (tk −t) E{Vi j (tk )} − βeλb τs E



t

 ν T (θ )Pi j ν(θ )dθ .

(63)

tk

Considering (48) with Theorem 1, we have E{Vii (tk + τs )} ≤κ E{Vi j (tk + τs )}, E{Vi j (tk )} ≤κe(λa +λb )τd E{V j j (tk )}.

(64)

Combining (62), (63), (64) and Assumption 3, by some mathematical manipulation, we can get that when t ∈ [tk + τs , tk+1 ), E{V (t)} < e

−λa (t−tk −τs )

< κe

 E{Vii (tk + τs )} − β E

−λa (t−tk −τs )

t t +τs t

 ν (θ )Pii ν(θ )dθ T

k E{Vi j (tk + τs )} − β E

tk +τs

 ν (θ )Pii ν(θ )dθ T

< κe−λa (t−tk −τs )+λb τs E{Vi j (tk )}  tk +τs   ν T (θ )Pi j ν(θ )dθ − β E − βκeλb τs E tk

< κ 2 e−λa (t−tk −τs )+λb τs +(λa +λb )τd E{V j j (tk )}  tk +τs   λb τs T E ν (θ )Pi j ν(θ )dθ − β E − βκe tk

t tk +τs t tk +τs

 ν T (θ )Pii ν(θ )dθ  ν (θ )Pii ν(θ )dθ T

< ··· < κ (2k+1) e−λa [t−t0 +(k+1)τs ]+kλb τs +(k+1)(λa +λb )τd E{V (t0 )} − βλ2 [1 + κ 2 eλb τs +(λa +λb )τd + · · · + κ 2k ekλb τs +k(λa +λb )τd ]dν − βλ2 κeλb τs [1 + κ 2 eλb τs +(λa +λb )τd + · · · + κ 2(k−1) e(k−1)λb τs +(k−1)(λa +λb )τd ]dν

Circuits Syst Signal Process 

N +1+

1 − [κ 2 eλb τs +(λa +λb )τd ] 0 < −βλ2 dν 1 − κ 2 eλb τs +(λa +λb )τd − βλ2 κeλb τs dν

t−t0 τa



  t−t N0 + τa0 2 λ τ +(λ +λ )τ s a b b d 1 − [κ e ]

1 − κ 2 eλb τs +(λa +λb )τd

+ κe(λa +λb )τs e[2 ln κ+(λa +λb )(τs +τd )]N0 

e



2 ln κ 1 τa +[(λa +λb )(τs +τd ) τa

−λa ](t−t0 )

E{V (t0 )}.

(65)

Similarly, we can get that when t ∈ [tk , tk + τs ), E{V (t)} < − βλ2 dν

  t−t N0 + τa0 2 λ τ +(λ +λ )τ s a b b d 1 − [κ e ]

1 − κ 2 eλb τs +(λa +λb )τd

− βλ2 κeλb τs dν +e 

e

  t−t N0 + τa0 2 λ τ +(λ +λ )τ s a b b d 1 − [κ e ]

1 − κ 2 eλb τs +(λa +λb )τd

(λa +λb )τs [2 ln κ+(λa +λb )(τs +τd )]N0

e

2 ln κ 1 τa +[(λa +λb )(τs +τd )] τa

 −λa (t−t0 )

E{V (t0 )}.

(66)

Considering (48), we can get E{V (t)} ≥(λ1 + λ4 τd e−λa τd )E{x T (t)Rx(t)},   1 E{V (t0 )} ≤ λ1 + λ3 τd + τd2 λ5 c1 . 2

(67) (68)

Combining (66), (67) and (68), we can get E{x T (t)Rx(t)} <[κe(λa +λb )τs e[2 ln κ+(λa +λb )(τs +τd )]N0 

e

2 ln κ 1 τa +[(λa +λb )(τs +τd )] τa

 −λa (t−t0 )

E{V (t0 )}



N +1+

1 − [κ 2 eλb τs +(λa +λb )τd ] 0 − βλ2 dν 1 − κ 2 eλb τs +(λa +λb )τd − βλ2 κeλb τs dν

t−t0 τa



  t−t N0 + τa0 2 λ τ +(λ +λ )τ s a b b d 1 − [κ e ]

(λ1 + λ4 τd e−λa τd )−1
1 − κ 2 eλb τs +(λa +λb )τd

]

(69)

Simplifying formula (69), we get τa > τa∗ and τa∗ is the numerical solution of the following equation. ξ1 e

[2 ln κ+(λa +λb )(τs +τd )] τT∗ a

+ ξ2 e

[2 ln κ+λb τs +(λa +λb )(τs +τd )] τT∗ a

= ξ3

(70)

Circuits Syst Signal Process

where ξ1 = c1 κe

(λa +λb )τs −λa T

  1 2 λ1 + λ3 τd + τd λ5 (κ 2 eλb τs +(λa +λb )τd − 1), 2

ξ2 = − βλ2 dν eλb τs (κ + e2 ln κ+(λa +λb )τd ), ξ3 = (λ1 + λ4 τd e−λa τd )(κ 2 eλb τs +(λa +λb )τd − 1)c2 − βλ2 dν (1 + κeλb τs ). Therefore, if the ADT satisfies (70), by Definition 1, system (11) is finite-time bounded with respect to [c1 , c2 , T, d, R, σ ]. The proof is completed.  

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