Circuits Syst Signal Process DOI 10.1007/s00034-017-0606-6

Asynchronous H∞ Control of Switched Systems with Mode-Dependent Average Dwell Time Juan Gao1 · Xing Tao Wang1

Received: 13 July 2015 / Revised: 14 April 2016 / Accepted: 15 April 2016 © Springer Science+Business Media, LLC 2017

Abstract This paper is concerned with the problem of H∞ control for a class of switched systems. Time delays that appear in both the state and the output are considered. In addition, the switching of the controllers experiences a time delay with respect to that of subsystems, which is called “asynchronous switching.” By the utilization of the piecewise Lyapunov function technique, sufficient conditions that ensure the exponential stability and a weighted H∞ performance level for the closed-loop system under a mode-dependent average dwell time (MDADT) scheme is proposed. MDADT means that each subsystem has its own average dwell time (ADT), which is more general than ADT. Two types of MDADT are gained by dividing all the subsystems into two parts. Then, the asynchronous H∞ dynamical output feedback controller is designed in terms of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method. Keywords Asynchronous switching · Mode-dependent average dwell time · Time-varying delay · H∞ control · Switched system

1 Introduction Switched systems belong to an important class of hybrid systems, represented by a finite number of subsystems and a switching signal orchestrating the switching among

B

Juan Gao [email protected] Xing Tao Wang [email protected]

1

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China

Circuits Syst Signal Process

them. Due to the significance in theoretical development and practical application, the investigation of switched systems has been attracting increasing attention. A considerable number of results have been reported [2,7,8,16]. Since time-delay phenomena exist widely in many engineering systems, which may lower the system performance and even lead to system instability, switched delay systems have been extensively studied [5,13,17,35,38]. Switched systems display complicated dynamical behavior. Switched systems might be stable or unstable for different switching signal. Switched systems might be unstable even if each subsystem is stable [9]. Thus, it is necessary to co-design the switching signal and controller to obtain the performance of the system. For the switching signal design, an effective method is average dwell time (ADT) [6, 34], which has been widely used to investigate the stability and stabilization problems of switched systems [10,23,25]. Recently, the authors in [32] put forward modedependent average dwell time (MDADT) in which each subsystem has its own ADT. It has been proved that MDADT is a more general class of ADT [26,27,33]. For the controller design, the mainly used technique is linear matrix inequalities (LMIs) [1,4]. Various controllers have been designed in [10,14,15,18,22–25,30] without asynchronous switching. However, as stated in [36,37], in actual operation, it takes some time to identify the active subsystem and apply the matched controller, so there inevitably exists asynchronous switching between the subsystems and controllers. The results related to the switched systems under asynchronous switching have been reported. To mention a few, asynchronous H∞ filtering problem has been investigated in [28,29], asynchronous output feedback control problem has been addressed in [3,20], and asynchronous state feedback control problem has been studied in [11,12,21,31]. In practice, it is often not possible to obtain full information on the state variables to use them for feedback control. This makes it necessary to study the dynamical output feedback (DOF) control problem [20]. To the best of our knowledge, the asynchronous H∞ DOF control problem of switched systems with time-varying delay, especially based on the MDADT approach, has been rarely studied. The presence of time-varying delay makes the DOF control problem much more complicated. Meanwhile, its presence adds the difficulty for the design of the DOF controller. How to choose the piecewise Lyapunov function technique to establish solvable conditions for the DOF controller is a crucial issue, which has not been resolved. Thus, research in this area should be of both theoretical and practical importance, which motivates us to undertake this work. In this paper, we are interested in investigating the asynchronous H∞ DOF control for switched time-varying delay systems. By using the MDADT approach combined with the piecewise Lyapunov function technique, sufficient conditions are proposed to guarantee the exponential stability with a weighted H∞ performance for the switched closed-loop system. By dividing the subsystems into two parts, two types of MDADT are gained. Moreover, the conditions for solving the DOF controller are given in terms of LMIs. Finally, the simulation result is provided to illustrate the effectiveness of the proposed theory. The contribution of this paper is as follows: (1) The DOF controller under asynchronous switching for switched time-varying delay systems is designed; (2) the weighted H∞ performance is introduced to study the DOF control problem of switched time-varying delay systems, which has rarely been addressed before;

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(3) a more general class of switching signal, i.e., the MDADT switching signal, is considered; (4) two types of smaller MDADT are gained. The remainder of the article is organized as follows. Preliminaries and problem formulation are introduced in Sect. 2. Section 3 presents the main results. A numerical example is provided in Sect. 4. The conclusions are summarized in Sect. 5.

1.1 Notations Rn denotes the n-dimensional Euclidean space. Rm×n is the set of all real m × n matrices. P > 0 means that P is a positive definite symmetric matrix. λmin (P) (λmax (P)) is the minimum (maximum) eigenvalue of matrix P. AT denotes the transpose of matrix A. * stands for the symmetric terms in matrices. || · || refers to the Euclidean vector norm. I and 0 denote the identity matrix and the zero matrix with appropriate dimension, respectively. diag{· · · } stands for a block diagonal matrix. L 2 [0, ∞) is the space of square-integrable vector functions over [0, ∞). N represents the set of all nonnegative integers.

2 Problem Formulation and Preliminaries Consider a class of switched delay systems ⎧ x(t) ˙ = Aσ (t) x(t) + Dσ (t) x(t − d(t)) + Bσ (t) u(t) + E σ (t) ω(t), ⎪ ⎪ ⎨ y(t) = Cσ (t) x(t) + Fσ (t) x(t − d(t)) + G σ (t) ω(t), z(t) = L σ (t) x(t) + Uσ (t) x(t − d(t)) + Hσ (t) ω(t), ⎪ ⎪ ⎩ x(t) = ϕ(t), t ∈ [−h, 0],

(1)

where x(t) ∈ Rn x is the state, u(t) ∈ Rn u is the control input, y(t) ∈ R n y is the measurement output, z(t) ∈ R n z is the controller output, ω(t) ∈ R n ω is the disturbance input which belongs to L 2 [0, ∞). d(t) denotes the time-varying delay satisfying 0 ≤ ˙ ≤ h d < 1. ϕ(t) is a vector-valued initial function on [−h, 0].σ (t) : d(t) ≤ h and d(t) [t0 , ∞) → M = {1, 2, ..., M}, called the switching signal, is a piecewise right continuous function. M is the number of subsystems, and t0 is the initial time. For a switching sequence of the subsystems Σ = {(σ (to ), to ), (σ (t1 ), t1 ), ..., (σ (tk ), tk ), ...|k ∈ N}, when t ∈ [tk , tk+1 ), σ (t) = σ (tk ) = p ∈ M, we say that the pth subsystem is active. A p , D p , B p , E p , C p , F p , G p , L p , U p , and H p are known real constant matrices with appropriate dimensions. Due to the asynchronous switching between the controllers and subsystems, we consider the dynamical output feedback (DOF) controller as follows: 

x˙c (t) = Ac,σ (t−Δk ) xc (t) + Bc,σ (t−Δk ) y(t), ∀ t ∈ [tk , tk+1 ), k ∈ N u(t) = Cc,σ (t−Δk ) , xc (0) = 0,

where Δ0 = 0, and Δk < tk+1 − tk represents the delayed period.

(2)

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Let σ (tk ) = p ∈ M, σ (tk−1 ) = q ∈ M, p = q. Applying the controller (2) to system (1), we obtain the following closed-loop system ⎧ ˙¯ = A¯ σ˜ x(t) ¯ + D¯ σ˜ x(t ¯ − d(t)) + E¯ σ˜ ω(t), ⎨ x(t) ¯ ¯ (3) z(t) = L p x(t) ¯ + U p x(t ¯ − d(t)) + H¯ p ω(t), ∀t ∈ [tk , tk+1 ), k ∈ N ⎩ x(t) ¯ = ϕ(t), ¯ t ∈ [−h, 0], where 

pq, t ∈ [tk , tk + Δk ) p, t ∈ [tk + Δk , tk+1 ),        T x(t) ¯ = x T (t) xcT (t) , H¯ p = H p , L¯ p = L p 0 , U¯ p = U p 0 ,





A p B p Cc, p Dp 0 Ep ¯ ¯ ¯ , Dp = , , Ep = Ap = Bc, p C p Ac, p Bc, p G p Bc, p F p 0



A p B p Cc,q Dp 0 Ep , E¯ pq = , D¯ pq = . A¯ p = Bc,q C p Ac,q Bc,q G p Bc,q F p 0 σ˜ =

Now, we state the following definitions and lemma for latter development. Definition 1 [32] For a switching signal σ (t) and any T > t ≥ 0, let Nσ p (t, T ) be the switching numbers that the pth subsystem is activated over the interval [t, T ) and T p (t, T ) denote the total running time of the pth subsystem over the interval [t, T ), p ∈ M. We say that σ (t) has a mode-dependent average dwell time (MDADT) τ p if there exist positive numbers N0 p (we call N0 p the mode-dependent chatter bounds) and τ p such that Nσ p (t, T ) ≤ N0 p +

T p (t, T ) . τp

(4)

Definition 2 [17] The equilibrium x¯ = 0 of closed-loop system (3) with w(t) = 0 is globally uniformly exponentially stable (GUES) under certain switching signal σ (t) and initial condition x(t ¯ 0 ), if there exist constants δ > 0 and η > 0 such that the solution of the system satisfies

where ||x(t ¯ 0 )||c1

||x(t)|| ¯ ≤ δe−η(t−to ) ||x(t ¯ 0 )||c1 , ∀t ≥ t0 ,  ˙¯ 0 + θ ) . ¯ 0 + θ )|| , x(t = sup−h≤θ≤0 ||x(t

(5)

Definition 3 For the given constants α p > 0 and γ > 0, system (3) is said to be GUES with a weighted H∞ performance γ , if the following conditions are satisfied: (1) System (3) is exponentially stable with w(t) = 0; (2) Under zero initial condition, i.e., ϕ(t) ¯ = 0, t ∈ [−h, 0], it holds for any nonzero w(t) ∈ L 2 [0, ∞) that ⎧ ⎫  ∞  ∞ M ⎨   ⎬ T α p T p (t0 , t) z (t)z(t)dt ≤ γ 2 exp − w T (t)w(t)dt. (6) ⎩ ⎭ t0 t0 p=1

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Remark 1 The standard H∞ performance, which has been commonly adopted for non-switched systems, cannot be achieved in general for switched systems with an ADT switching. Thus, weighed H∞ performance with the weighted term e−αt is used in [23,34]. In this paper, since  the MDADT switching technique is used, the weighted term is replaced by exp{− M p=1 [α P T p (t0 , t)]}. It can be seen that when α p = α, ∀ p ∈ M, Definition 3 is turned into that in [23,34]. Thus, Definition 3 can be viewed as an extension of that in [23,34]. Lemma 1 [38] Let x(t) ∈ Rn be a vector-valued function with first-order continuousderivative entries. Then, the following integral inequality holds for any matrices N1 , N2 ∈ Rn×n and X = X T > 0, and a scalar function 0 ≤ d(t) ≤ h :  −

t

 x˙ (s)X x(s)ds ˙ ≤ ζ (t)

N1T + N1 −N1T + N2



ζ (t) −N2T + N2 T

  N1 X −1 N1 N2 ζ (t), + hζ T (t) T N2

T

T

t−d(t)

∗

(7)

where ζ (t) = [x T (t) x T (t − d(t))]T . Lemma 2 [1] (Schur complement) For a given symmetric matrix with the partition W =

W11 W12 , W21 W22

T = W , the following three conditions where W11 and W22 is a square matrix and W12 21 are equivalent

(1) W < 0; T W −1 W < 0; (2) W11 < 0 and W22 − W12 12 11 −1 T W12 < 0. (3) W22 < 0 and W11 − W12 W22

3 Main Results 3.1 Stability and H∞ Performance Analysis In this section, we focus on the stability and H∞ performance of the closed-loop system (3) with asynchronous behaviors. For concise notation, let T (0, t) (T (0, t)) represent the total periods that the controllers and the subsystems are matched (unmatched) during [0, t). Let T p (0, t) (T p (0, t)) denote the total running time of the pth subsystem controlled by the matched (unmatched) controller during [0, t). The following theorem presents a sufficient condition of exponential stability for the system (3) with w(t) = 0. Theorem 1 For the switched system (3) with w(t) = 0, let α p > 0, β P , μ p ≥ 1 and μˆ p ≥ 1, p ∈ M be given constants, if there exist matrices Pp > 0, Q p > 0, S p > 0, Ppq > 0, Q pq > 0 and S pq > 0, such that ∀ ( p, q) ∈ M × M, p = q,

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⎡

⎤ p p Γ11 Γ12 h A¯ Tp K T ⎢ ⎥ ⎢ ∗ Γ p h D¯ T K T ⎥ < 0, p 22 ⎣ ⎦ −1 ∗ ∗ −h S p ⎡ pq ⎤ pq Γ11 Γ12 h A¯ Tpq K T ⎢ ⎥ ⎢ ∗ Γ pq h D¯ T K T ⎥ < 0, pq 22 ⎣ ⎦ −1 ∗ ∗ −h S pq Pp ≤ μ p Ppq , Q p ≤ μ p Q pq , S p ≤ μ p S pq , Ppq ≤ μˆ p Pp , Q pq ≤ μˆ p Q p , S pq ≤ μˆ p Sq ,

(8)

(9)

(10)

where p Γ11 = A¯ Tp Pp + Pp A¯ p + α p Pp + Q p , p Γ12 = Pp D¯ p + e−α p h K T K ,

Γ22 = −(1 − h d )e−α p h Q p − 2e−α p h K T K + he−α p h K T S −1 p K, p

pq Γ11 = A¯ Tpq Ppq + Ppq A¯ pq − β p Ppq + Q pq , pq Γ12 = Ppq D¯ pq + K T K ,

Γ22 = −(1 − h d )Q pq − 2K T K + h K T S −1 pq K . pq

Then, the closed-loop system (3) is GUES for any switching signal σ (t) with the following MDADT ln(μ p μˆ p μ˜ p ) , β p + α p ≤ 0, αp ln(μ p μˆ p μ˜ p ) + (α p + β p )Δ pM τ p ≥ τ p∗ = , β p + α p > 0, αp τ p ≥ τ p∗ =

where μ˜ p =

(11)

max {μq p }, μq p = eαq +β p , Δ pM = max T p (tk , tk+1 ), ∀k ∈ N.

q∈M,q= p

Proof According to the value of α p + β p , we divide all the subsystems into two parts: if α p + β p ≤ 0, the subsystem belongs to set Ψ1 = {1, . . . , l}; otherwise, it belongs to set Ψ2 = {l + 1, . . . , M}. For ∀t ∈ [tk , tk+1 ), k ∈ N, let t0 = 0, and define K = [I 0]. Choose a piecewise Lyapunov function of the following form:  V (t) = x¯ (t)Pσ˜ x(t) ¯ + T

 +

0 −h



t t+θ

t

t−d(t)

eκ(t−s) x¯ T (s)Q σ˜ x(s)ds ¯

˙¯ eκ(t−s) x˙¯ T (s)K T Sσ˜ K x(s)dsdθ,

(12)

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where  κ=

t ∈ [tk , tk + Δk ) βp, −α p , t ∈ [tk + Δk , tk+1 ).

Taking the derivation of the Lyapunov function, we have V˙ (t) ≤ κ V (t) − κ x¯ T (t)Pσ˜ x(t) ¯ + 2 x˙¯ T (t)Pσ˜ x(t) ¯ T T T ˙¯ + x¯ (t)Q σ˜ x(t) ¯ + h x˙¯ (t)K Sσ˜ K x(t)ds − (1 − h d )v x¯ T (t − d(t))Q σ˜ x(t ¯ − d(t))  t ˙¯ −v x˙¯ T (s)K T Sσ˜ K x(s)ds,

(13)

t−d(t)

where  v=

1, t ∈ [tk , tk + Δk ) e−α p h , t ∈ [tk + Δk , tk+1 ).

Define ξ(t) = [x¯ T (t) x¯ T (t − d(t))]T , it follows from Lemma 1 with N1 = 0, N2 = I :  −v

t

t−d(t)

˙¯ x˙¯ T (s)K T Sσ˜ K x(s)ds



 0 KTK T T T −1 ¯ − d(t)) . (14) ξ(t) + h x¯ (t − d(t))K Sσ˜ K x(t ≤ v ξ (t) ∗ −2K T K From (12)–(14), it yields that V˙ (t) ≤



β p V (t) + ξ T (t)(Γ pq + dΘ pq S pq Θ Tpq )ξ(t), t ∈ [tk , tk + Δk ) (15) −α p V (t) + ξ T (t)(Γ p + dΘ p S p Θ Tp )ξ(t), t ∈ [tk + Δk , tk+1 ),

where  Γ = p

p

p

Γ11 Γ12 ∗

p

Γ22



 ,Γ

pq

=

pq

pq

Γ11

Γ12

∗

Γ22

pq



 , Θp=

A¯ Tp K T D¯ Tp K T



 , Θ pq =

A¯ Tpq K T D¯ Tpq K T

 .

By Schur complement Lemma, (8) and (9) imply 

t ∈ [tk , tk + Δk ) β p V (t), , −α p V (t), t ∈ [tk + Δk , tk+1 )

(16)

eβ p (t−tk ) V (tk ), t ∈ [tk , tk + Δk ) e−α p (t−tk −Δk ) V (tk + Δk ), t ∈ [tk + Δk , tk+1 ).

(17)

V˙ (t) ≤ which gives that  V (t) ≤

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Using (10) and (12), we get V (tk ) ≤ μˆ p μ˜ p V (tk− ),

V (tk + Δk ) ≤ μ p V ((tk + Δk )− ).

(18)

For ∀t ∈ [tk , tk+1 ), combining (17) and (18) yields V (t) ≤ μσ (tk ) eβσ (tk ) T (tk ,t)−ασ (tk ) T (tk ,t) V (tk ) ≤ μσ (tk ) μˆ σ (tk ) μ˜ σ (tk ) eβσ (tk ) T (tk ,t)−ασ (tk ) T (tk ,t) V (tk− ) ≤ μσ (tk−1 ) μσ (tk ) μˆ σ (tk ) μ˜ σ (tk ) eβσ (tk ) T (tk ,t)−ασ (tk ) T (tk ,t) × eβσ (tk−1 ) T (tk−1 ,tk )−ασ (tk−1 ) T (tk−1 ,tk ) V (tk−1 ) ≤

k 

(μσ (ti ) μˆ σ (ti ) μ˜ σ (ti ) )eβσ (tk ) T (tk ,t)−ασ (tk ) T (tk ,t)

i=k−1 βσ (tk−1 ) T (tk−1 ,tk )−ασ (tk−1 ) T (tk−1 ,tk )

− ×e V (tk−1 ) ≤ ··· k  ≤ (μσ (ti ) μˆ σ (ti ) μ˜ σ (ti ) )eβσ (tk ) T (tk ,t)−ασ (tk ) T (tk ,t) i=1

×e



k

βσ (ti−1 ) T (ti−1 ,ti )−ασ (ti−1 ) T (ti−1 ,ti )



V (t0 ) ⎧ ⎫ M M ⎨  ⎬  = exp β p T p (0, t) − α p T p (0, t) (μ p μˆ p μ˜ p ) Nσ p(0,t) V (t0 ) ⎩ ⎭ i=1

p=1

p=1

= Ω1 Ω2 V (t0 ),

(19)

where ⎧ ⎫ l l ⎨  ⎬  β p T p (0, t) − α p T p (0, t) (μ p μˆ p μ˜ p ) Nσ p(0,t) , Ω1 = exp ⎩ ⎭ p=1 p=1 ⎧ ⎫ M M ⎨   ⎬  Ω2 = exp β p T p (0, t) − α p T p (0, t) (μ p μˆ p μ˜ p ) Nσ p(0,t) . ⎩ ⎭ p=l+1

p=l+1

For Ω1 noticing that β p ≤ −α p , together with Definition 1, we get ⎧ ⎫ l l ⎨  ⎬  Ω1 ≤ exp −α p T p (0, t) (μ p μˆ p μ˜ p ) Nσ p(0,t) ⎩ ⎭ p=1 p=1 ⎧ ⎫ ! ⎬ l ⎨ ln(μ p μˆ p μ˜ p ) ≤ exp − αp N0 p ln(μ p μˆ p μ˜ p ) + T p (0, t) . (20) ⎩ ⎭ τp p=1

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For Ω2 , noticing that T p (0, t) ≤ Δ pM Nσ p (0, t), together with Definition 1, we have ⎧ ⎫ M M ⎨  ⎬   Ω2 ≤ exp (μ p μˆ p μ˜ p ) Nσ p(0,t) −α p T p (0, t) + (α p + β p )Δ pM Nσ p (0, t) ⎩ ⎭ p=l+1 p=l+1 ⎧ ⎫ M ⎨  ⎬   ≤ exp N0 p ln(μ p μˆ p μ˜ p ) + (α p + β p )Δ pM ⎩ ⎭ p=l+1 ⎧ ⎫ !

⎬ M ⎨  ln(μ p μˆ p μ˜ p ) + (α p + β p )Δ pM − α p T p (0, t) (21) × exp . ⎩ ⎭ τp p=l+1

Define π1 =

min

p,q∈M, p=q

{λmin (Pp ), λmin (Ppq )},

π2 = max {λmax (Pp )} + h max {λmax (Q p )} + p∈M

p∈M

h2 max {λmax (S p )}. 2 p∈M

Set " δ=

⎧ l ⎨1 

π2 exp ⎩2 π1

N0 p ln(μ p μˆ p μ˜ p )



p=1

⎫ ⎬ # $ 1 + N0 p ln(μ p μˆ p μ˜ p ) + (α p + β p )Δ pM ) , ⎭ 2 p=l+1    ln(μ p μˆ p μ˜ p ) 1 − αp , η = − max max p∈Ψ1 2 τp   ln(μ p μˆ p μ˜ p ) + (α p + β p )Δ pM max − αp . p∈Ψ2 τp M 



Then, from (11) and (19)–(21), we can obtain ||x(t)|| ¯ ≤ δe−η(t−t0 ) ||x(t ¯ 0 )||c1 .

(22)

By Definition 2, we can conclude that the closed-loop system (3) with w(t) = 0 is GUES for any switching signal with MDADT (11). This completes the proof.

 Remark 2 To facilitate the latter design of the DOF controller, in Theorem 1, a matrix K = [I 0] is added into the third term of the piecewise Lyapunov function (12). Remark 3 A unique feature of the approaches in this paper is the utilization of MDADT. Different from the ADT approach adopted in [12,18–20], where the parameters are mode-independent, and the ADT for all the subsystems are required to be

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larger than a common constant τa , the parameters selected in this paper are modedependent, and we only require the ADT among the intervals associated with the pth subsystem to be larger than τ p , where the intervals are not adjacent. Remark 4 Different form most existing results on asynchronous control problem [3, 11,12,20,28,29,31], in which α and β are positive, in this paper, β p , p ∈ M can be negative. Based on the value of α p + β p , p ∈ M, we get two types of MDADT (11). It can be seen that for the same parameters α and μ if only β p < 0, p ∈ M exist, the MDADT (11) is smaller than that in [3,11,12,20,28,29,31]. Now, we are in a position to give the weighted H∞ performance analysis for the system (3). Theorem 2 For the switched system (3), let γ > 0, α p > 0, β p , μ p ≥ 1 and μˆ p ≥ 1, p ∈ M be given constants, if there exist matrices Pp > 0, Q p > 0, S p > 0, Ppq > 0, Q pq > 0 and S pq > 0, ∀( p, q) ∈ M × M, p = q, such that (10) and the following inequalities hold ⎡

⎤ p p Γ11 Γ12 Pp E¯ p L¯ Tp h A¯ Tp K T ⎢ ⎥ ⎢ ∗ Γ22p 0 U¯ pT h D¯ Tp K T ⎥ ⎢ ⎥ ⎢ ⎥ ∗ −γ 2 I H¯ pT h E¯ Tp K T ⎥ < 0, ⎢ ∗ ⎢ ⎥ ⎢ ∗ ⎥ ∗ ∗ −I 0 ⎣ ⎦ ∗ ∗ ∗ ∗ −h S −1 p ⎡ pq ⎤ pq Γ11 Γ12 Ppq E¯ pq L¯ Tp h A¯ Tpq K T ⎢ ⎥ ⎢ ∗ Γ22pq 0 U¯ pT h D¯ Tpq K T ⎥ ⎢ ⎥ ⎢ ⎥ ∗ −γ 2 I H¯ pT h E¯ Tpq K T ⎥ < 0, ⎢ ∗ ⎢ ⎥ ⎢ ∗ ⎥ ∗ ∗ −I 0 ⎣ ⎦ ∗ ∗ ∗ ∗ −h S −1 pq

(23)

(24)

then the closed-loop system (3) is GUES with a weighted H∞ performance level γ˜ √ for any switching signal σ (t) with MDADT satisfying (11), where γ˜ = γ ρ and ρ = M l exp{ p=1 [N0 p ln(μ p μˆ p μ˜ p )] + p=l+1 [((α p + β p )Δ pM + ln(μ p μˆ p μ˜ p ))N0 p ]}. Proof (8) and (9) can be concluded from (23) and (24). By Theorem 1, the exponential stability of the system (3) with w(t) = 0 is guaranteed. Next, we will show the weighted H∞ performance of the system. Constructing the Lyapunov function (12) and using the same method in Theorem 1, it gives V˙ (t) ≤



t ∈ [tk , tk + Δk ) β p V (t) − Υ (t), −α p V (t) − Υ (t), t ∈ [tk + Δk , tk+1 )

where Υ (t) = z T (t)z(t) − γ 2 w T (t)w(t).

(25)

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Integrating both sides of (25), it holds that ⎧ % β p (t−tk ) V (t ) − t eβ p (t−s) Υ (s)ds, t ∈ [t , t + Δ ) ⎪ k k k k ⎨e tk V (t) ≤ e−α p (t−tk%−Δk ) V (tk + Δk ) ⎪ t ⎩ − tk+Δ e−α p (t−s) Υ (s)ds, t ∈ [tk + Δk , tk+1 ). k

For ∀t ∈ [tk , tk+1 ), it follows from (18) and (26) that V (t) ≤ μσ (tk ) eβσ (tk ) T (tk ,t)−ασ (tk ) T (tk ,t) V (tk )  t − eβσ (tk )T (s,t)−ασ (tk ) T (s,t) Υ (s)ds tk

≤ μσ (tk ) μˆ σ (tk ) μ˜ σ (tk ) eβσ (tk ) T (tk ,t)−ασ (tk ) T (tk ,t) V (tk− )  t β T (s,t)−ασ (tk ) T (s,t) − e σ (tk )  Υ (s)ds tk βσ (tk ) T (tk ,t)−ασ (t ) T (tk ,t)

≤ μσ (tk−1 ) μσ (tk ) μˆ σ (tk ) μ˜ σ (tk ) e ×e

βσ (tk−1 ) T (tk−1 ,tk )−ασ (t



−μσ (tk ) μˆ σ (tk ) μ˜ σ (tk )

k−1

) T (tk−1 ,tk )

t

e

k

V (tk−1 )

βσ (tk−1 ) T (s,tk )−ασ (t

T (s,tk ) k−1 ) 

tk−1 )T (tk ,t) −ασ (tk ) T (tk ,t)

β

× e σ (tk Υ (s)ds  t β T (s,t)−ασ (t ) T (s,t) k − e σ (tk )  Υ (s)ds tk

≤

k 

(μσ (ti ) μˆ σ (ti ) μ˜ σ (ti ) )e

i=k−1 βσ (tk−1 ) T (tk−1 ,tk )−ασ (t

×e



−μσ (tk ) μˆ σ (tk ) μ˜ σ (tk ) βσ (tk )T (t

βσ (tk ) T (tk ,t)−ασ (t ) T (tk ,t) k

T (t ,t ) k−1 )  k−1 k

t

e

− V (tk−1 )

βσ (tk−1 ) T (s,tk )−ασ (t

T (s,tk ) k−1 ) 

tk−1

−ασ (tk ) T (tk ,t) k ,t)

×e Υ (s)ds  t β T (s,t)−ασ (t ) T (s,t) k − e σ (tk )  Υ (s)ds ≤ ···

tk

⎫ ⎧ M M ⎨  ⎬  β p T p (0, t) − α p T p (0, t) (μ p μˆ p μ˜ p ) Nσ p(0,t) V (t0 ) ≤ exp ⎭ ⎩ p=1 p=1 ⎧ ⎫  t M ⎨  ⎬ − β p T p (s, t) − α p T p (s, t) exp ⎩ ⎭ t0 p=1

×

M  p=1

(μ p μˆ p μ˜ p ) Nσ p(s,t) Υ (s)ds.

(26)

Circuits Syst Signal Process

Therefore, under the zero initial condition, we have 

t t0

⎧ ⎫ M M ⎨  ⎬  β p T  p (s, t) − α p T p (s, t) exp (μ p μˆ p μ˜ p ) Nσ p (s,t) Υ (s)ds ≤ 0. ⎩ ⎭ p=1

p=1

(27) That is ⎧ l ⎨   exp −α p T p (s, t) + f (s, t) ⎩ t0 p=1 ⎫ M   ⎬ + −α p T p (s, t) + f (s, t) Υ (s)ds ≤ 0, ⎭



t

(28)

p=l+1

where f (s, t) = (α & p + β p )T p (s, t) + ' Nσ p (s, t) ln(μ p μˆ p μ˜ p ). M Multiplying exp − p=1 f (t0 , t) on both sides of (28) yields 

t

 exp{Φ1 + Φ2 }z T (s)z(s)ds ≤ γ 2

t0

t

exp{Φ1 + Φ2 }w T (s)w(s)ds,

(29)

t0

  where Φ1 = lp=1 [−α p T p (s, t)− f (t0 , s)], Φ2 = M p=l+1 [−α p T p (s, t)− f (t0 , s)]. For Φ1 , from Definition 1 and (11), and noticing that −(α p + β p ) ≥ 0, we get  l  Φ1 ≥ Σ −α p T p (s, t) − Nσ p (t0 , s) ln(μ p μˆ p μ˜ p ) p=1

l ln(μ p μˆ p μ˜ p )T p (t0 , s) ≥ Σ −α p T p (s, t) − N0 p ln(μ p μˆ p μ˜ p ) − τp p=1  l  ≥ Σ −α p T p (t0 , t) − N0 p ln(μ p μˆ p μ˜ p ) . (30) p=1

For Φ2 , from Definition 1 and (11), and noticing that T p (t0 , s) ≤ Δ pM Nσ p (t0 , s), we have   −α p T p (s, t) − ((α p + β p )Δ pM + ln(μ p μˆ p μ˜ p ))Nσ p (t0 , s) p=l+1 !

M T p (t0 , s) −α p T p (s, t) − ((α p + β p )Δ pM +ln(μ p μˆ p μ˜ p )) N0 p + ≥ Σ τp p=l+1  M  (31) ≥ Σ −α p T p (t0 , t) − ((α p + β p )Δ pM + ln(μ p μˆ p μ˜ p ))N0 p .

Φ2 ≥

M

Σ

p=l+1

Circuits Syst Signal Process

Combining (30) and (31), and noticing that − f (to , s) ≤ 0, we obtain 

M

l

Σ [−α p T p (t0 , t)] − Σ [N0 p ln(μ p μˆ p μ˜ p )] p=1  M − Σ [((α p + β p )Δ pM + ln(μ p μˆ p μ˜ p ))N0 p ] p=l+1   M ≤ exp {Φ1 + Φ2 } ≤ exp Σ [−α p T p (s, t)] .

exp

p=1

(32)

p=1

From (29) and (32), it follows that ⎫ ⎬ T −α p T p (t0 , t) z (s)z(s)ds exp ⎩ ⎭ to p=1 ⎧ ⎫  t M ⎨ ⎬   ≤ γ 2ρ −α p T p (s, t) w T (s)w(s)ds. exp ⎩ ⎭ to



⎧ M ⎨ 

t

(33)

p=1

M [((α p + β p )Δ pM + ln where ρ = exp{Σ lp=1 [N0 p ln(μ p μˆ p μ˜ p )] + Σ p=l+1 (μ p μˆ p μ˜ p ))N0 p ]}. Integrating both sides of (33) from t = t0 to ∞ yields



∞ t0

⎧ ⎫  ∞ M ⎨   ⎬ T α p T p (t0 , s) z (s)z(s)ds ≤ γ˜ 2 exp − w T (s)w(s)ds, ⎩ ⎭ t0

(34)

p=1

√ where γ˜ = γ ρ. This means that system (3) achieves a weighted H∞ performance level γ˜ . The proof is completed.



3.2 Controller Design In this section, based on the proposed weighted H∞ performance condition, we will give the design method of the DOF controller for the system (1). Theorem 3 For the switched system (1), let γ > 0, α p > 0, β p , ε p > 0, μ p ≥ 1 and μˆ p ≥ 1, p ∈ M be given constants, if there exist matrices Ac, p , Bc, p , Cc, p , P1 p > 0, X1 p > 0, L p > 0, Q p > 0, I p > 0, Ppq > 0, Q pq > 0 and S pq > 0, such that ∀( p, q) ∈ M × M, p = q,

Circuits Syst Signal Process



X1 p I ⎡ p Ξ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗ ⎡ pq Γ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

I P1 p

> 0,

(35)

⎤ p p p Ξ12 Ξ13 0 E p X1 p L Tp Ξ17 ε p X1 p p p p Ξ22 Ξ23 0 Ξ25 L Tp h ATp 0 ⎥ ⎥ p T T ∗ Ξ33 0 0 Up h Dp 0 ⎥ ⎥ p ∗ ∗ Ξ44 0 0 0 0 ⎥ ⎥ < 0, H pT h E Tp 0 ⎥ ∗ ∗ ∗ −γ 2 I ⎥ ∗ ∗ ∗ ∗ −I 0 0 ⎥ ⎥ ∗ ∗ ∗ ∗ ∗ −hI p 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ −ε p I ⎤ pq Γ12 Ppq E¯ pq L¯ Tp h A¯ Tpq K T pq Γ22 0 U¯ pT h D¯ Tpq K T ⎥ ⎥ 2 ∗ −γ I H¯ pT h E¯ Tpq K T ⎥ ⎥ < 0, ⎦ ∗ ∗ −I 0 −1 ∗ ∗ ∗ −h S pq

(36)

(37)

Y p J p−1 ≤ μ p Ppq , diag{ε p I, Q p } ≤ μ p Q pq , I p−1 ≤ μ p S pq , Ppq ≤ μˆ p Yq Jq−1 , Q pq ≤ μˆ p diag{εq I, Qq }, S pq ≤ μˆ p Iq−1 ,

(38)

where p

Ξ11 = A p X1 p + X1 p ATp + B p Cc, p + Cc,T p B Tp + α p X1 p + L p , Ξ12 = A p + Ac,Tp + α p I + ε p X1 p , Ξ13 = D p + e−α p h X1 p , p

p

p

p

T T Ξ17 = hX1 p ATp + hCcp B p , Ξ25 = P1 p E p + Bcp G p , p

T Ξ22 = P1 p A p + ATp P1 p + Bc, p C p + C Tp Bc, p + α p P1 p ε p I,

Ξ23 = P1 p D p + Bc, p F p + e−α p h I, Ξ44 = −(1 − h d )e−α p h Q p , p

p

Ξ33 = −(1 − h d )e−α p h ε p I − 2e−α p h I + he−α p h I p . p

Then, the closed-loop system (3) is GUES with a weighted H∞ performance level √ γ˜ for any switching signal σ (t) with MDADT satisfying (11), where γ˜ = γ ρ and l M ρ = exp{Σ p=1 [N0 p In(μ p μˆ p μ˜ p )]+Σ p=l+1 [((α p +β p )Δ pM + In(μ p μˆ p μ˜ p ))N0 p ]}. Moreover, the controller gains are given by −T Ac, p = P2−1 p [Ac, p − P1 p A p X1 p − Bc, p C p X1 p − P1 p B p Cc, p ]X2 p ,

Bc, p = P2−1 p Bc, p , Cc, p = Cc, p X2−T p . (39)

Circuits Syst Signal Process

Proof Partition Pp and its inverse as Pp =



P1 p P2 p X 1 p X2 p −1 , P , = p P2Tp P3 p X2Tp X3 p

(40)

where P3 p > 0, X3 p > 0, and P2 p , X2 p are invertible matrices. Define the following matrices Jp =





X1 p I I P1 p εp I 0 , Y , Q . = = p p 0 P2Tp , X2Tp 0 0 Qp

(41)

By computation, we can get P1 p X1 p + P2 p X2Tp = I, Pp J p = Y p .

(42)

Multiplying diag {J pT , I, I, I, I } by pre- and post-(23), we can obtain ⎤ ⎡ ˜p ˜p Γ11 Γ12 J pT Pp E¯ p J pT L¯ Tp hJ pT A¯ Tp K T ⎢ ∗ Γp 0 U pT h D¯ Tp K T ⎥ 22 ⎥ ⎢ 2 T ⎢ ∗ ∗ −γ I H¯ p h E¯ Tp K T ⎥ ⎥ < 0, ⎢ ⎦ ⎣ ∗ ∗ ∗ −I 0 −1 ∗ ∗ ∗ ∗ −h S p

(43)

where p Γ˜11 = J pT ( A¯ Tp Pp + Pp A¯ p + α p Pp + Q p )J p , p Γ˜12 = J pT (Pp D¯ p + e−α p h K T K ).

Define the following matrices: Ac, p = P1 p A p X1 p + P2 p Bc, p C p X1 p + P1 p B p Cc, p X2Tp + P2 p Ac, p X2Tp , Bc, p = P2 p Bc, p , Cc, p = Cc, p X2Tp , L p = X2 p Q p X2Tp , L p = S −1 p . From (40), we get

Ap A p X1 p + B p Cc, p , Ac, p P1 p A p + Bc, p C p

ε p X1 p X1 p + L p ε p X1 p J pT Q p J p = , ε p X1 p εp I



I 0 X1 p Dp , , J pT Pp D¯ p = J pT Pp J p = I P1 p P1 p D p + Bc, p F p 0

J pT Pp A¯ p J p =



(44)

Circuits Syst Signal Process



X1 p 0 Ep , , J pT Pp E¯ p = P1 p E p + Bc, p G p I 0



X1 p L Tp X1 p ATp + Cc,T p B Tp T ¯T T J pT L¯ Tp = , J . K = A p p L Tp ATp

J pT K T K =

(45)

Substituting (45) into (43) and applying Schur complement Lemma, we can obtain (36). Thus, (36) is equivalent to (23). Notice that (37) is equivalent to (24), and (38) is equivalent to (10). The proof is completed. 

Remark 5 The asynchronous DOF control problem was also studied in [20] for a class of switched delay systems based on the ADT approach. However, the state delay is time invariant, and the switching delay only involves in partial controller gain matrices. The advantages of the result in this paper are that the state delay considered is time varying, and the switching delay appears in all the controller gain matrices. On the other hand, not only the stability but also the H∞ performance for the switched system is studied, especially based on the MDADT approach, which brings more flexibility to find the feasible controller. Notice that the inequality conditions in Theorem 3 are mutually dependent, and we present the following computational algorithm to obtain the DOF controller and the MDADT. Algorithm 1 Step 1 ∀ p ∈ M, given constants α p and ε p , solve (35) and (36) to obtain Ac, p , Bc, p , Cc, p , P1 p , X1 p , L p , Q p and I p . Step 2 Compute the invertible matrices X2 p satisfying L p = X2 p Q p X2Tp by the function fsolve (· · · ) in MATLAB. Then P2 p can be obtained from P1 p X1 p + P2 p X2Tp = I . Step 3 Compute the matrices Y p and J p by (41). Step 4 According to (39), the controller matrices Ac, p , Bc, p and Cc, p can be obtained. Step 5 Upon substituting the matrices obtained from Step 1–Step 4 to (37) and (38), they can be transformed into LMIs with respect to Ppq , Q pq and S pq . Step 6 Solve (37) and (38) for the given constants β p , μ p and μˆ p . Step 7 Use μ˜ p = max {μq p } with μq p = eαq +β p to obtain μ˜ p . q∈M,q= p

Step 8 Substitute α p , β p , μˆ p , and μ˜ p to (11) to obtain τ p∗ . Remark 6 It can be seen that a smaller α p will be favorable to the feasibility of (36) and a larger β p will be favorable to the feasibility of (37). In view of this, for the choice of α p in Algorithm 1, for the first time, we can choose a larger α p , if (36) is unfeasible, we can decrease α p appropriately. Repeat this until (36) is feasible. For the choice of β p in Algorithm 1, for the first time, we can choose a β p < −α p , if (37) is unfeasible, we can increase β p appropriately. Repeat this until (37) is feasible.

Circuits Syst Signal Process

4 Example In this section, we present a numerical example to demonstrate the effectiveness of the proposed method. Consider system (1) consisting of three subsystems, Subsystem 1: ⎡

A1

E1 C1 L1

⎤ ⎡ ⎤ ⎡ ⎤ −1.9 0 1.1 0.2 0 0 1.1 −0.9 0.3 ⎦ , D1 = ⎣ 0.1 0 0.1 ⎦ , B1 = ⎣ 0.4 ⎦ , =⎣ 0 −0.2 0.1 0.3 0 0.1 0.2 2 ⎡ ⎤ 0.1 = ⎣ 0.2 ⎦ , 0.5       = 2.2 3 3 , F1 = 0 0.1 0 , G 1 = 0.8 ,       = 0 0.5 0.5 , U1 = 0.2 0 0.4 , H1 = 0.3 .

Subsystem 2: ⎡

A2

E2 C2 L2

⎤ ⎡ ⎤ ⎡ ⎤ −1.8 0 0.1 0.2 0 0.1 0.2 = ⎣ 0.1 −2.4 0 ⎦ , D2 = ⎣ 0.1 0.1 0.1 ⎦ , B2 = ⎣ 0.5 ⎦ , 0 0.1 0.1 0 0.1 0.2 2 ⎡ ⎤ 0.4 = ⎣ 0 ⎦, 0.6       = 0.6 1 2 , F2 = 0.1 0.1 1 , G 2 = 0.3 ,       = 0.2 1.3 0 , U2 = 0.1 0.4 0 , H2 = 0.2 .

Subsystem 3: ⎡

A3

E3 C3 L3

⎤ ⎡ −2.2 0 0.3 0.2 −1.9 0.3 ⎦ , D3 = ⎣ 0.1 =⎣ 0 0.2 0 0.4 0 ⎡ ⎤ 0 = ⎣ 0.1 ⎦ , 0.6     = 1.2 4 2 , F3 = 0.1 0.1 0 ,     = 0.4 0 0 , U3 = 0.3 0.1 0 ,

⎤ ⎡ ⎤ 0 0 0.4 0 0.1 ⎦ , B3 = ⎣ −0.4 ⎦ , 0.1 0.2 3

  G 3 = 0.6 ,   H3 = 0.1 ,

Considering d(t) = 0.9 + 0.1 sin(t), we can get that h = 1, h d = 0.1. Taking α1 = 1.3, α2 = 1.2, α3 = 1.4, ε1 = 0.5, ε2 = 1, ε3 = 0.1, and γ = 1. Following Step 1–Step 4 of Algorithm 1, we can obtain the DOF controller gains

Circuits Syst Signal Process 4 System Mode Controller Mode

3.5 3 2.5 2 1.5 1 0.5 0

0

2

4

6

8

10

12

Time (s) Fig. 1 Switching signals

⎡ Ac,1 = Cc,1 = Ac,2 = Cc,2 = Ac,3 = Cc,3 =

⎤ ⎡ ⎤ −4.9484 −7.2571 −16.2287 1.4895 ⎣ −9.1137 −32.8127 −70.5270 ⎦ , Bc,1 = ⎣ 7.2964 ⎦ , −17.2043 −59.1784 −138.7735 14.2323   0.3397 −1.5057 −4.0909 , ⎡ ⎤ ⎡ ⎤ −1.0910 −0.8166 4.8126 −0.5375 ⎣ 1.5152 −8.2070 8.7754 ⎦ , Bc,2 = ⎣ −1.1477 ⎦ , −4.7626 10.8616 −48.9274 4.6752   0.1740 1.2542 −3.8081 , ⎡ ⎤ ⎡ ⎤ −8.5193 −4.9855 −3.7357 0.3924 ⎣ −5.0012 −108.5742 −8.8421 ⎦ , Bc,3 = ⎣ 6.9366 ⎦ , −11.8495 −240.5768 −24.4987 15.5646   0.1387 −1.9609 −2.4129 .

Then, choosing β1 = −1.33, μ1 = 2, μˆ 1 = 6.5, β2 = −0.7, μ2 = 8.7, μˆ 2 = 8.5, β3 = −1.44, μ3 = 7.5 and μˆ 3 = 7, and following Step 5 and Step 6 of Algorithm 1, we can seek the feasible solutions Ppq , Q pq and S pq of (37) and (38). Following Step 7 of Algorithm 1, we can get μ˜ 1 = 1.0725, μ˜ 2 = 2.0138 and μ˜ 3 = 0.8694. Assume that Δ1M = 0.8, Δ2M = 0.3 and Δ3M = 0.2, following Step 8 of Algorithm 1, we can obtain τ1∗ = 2.0268, τ2∗ = 4.2945 and τ3∗ = 2.7292. Remark 7 Although the matrix inequalities (35)–(38) are coupled. According to Algorithm 1, we can firstly solve (35) and (36) to gain Ac, p Bc, p , Cc, p , P1 p , X1 p , L p , Q p and I p , and compute the matrices X2 p , P2 p , Y p and Y p by Step 2 and Step 3. Then, we solve (37) and (38) by substituting the matrices obtained into (37) and

Circuits Syst Signal Process 10 x

1

9

x2 8

x

3

7 6 5 4 3 2 1 0

0

2

4

6

8

10

12

Time (s) Fig. 2 States of the open-loop system

(38). By adjusting the parameters β p , μ p and μˆ p appropriately, we seek the feasible solutions Ppq , Q pq and S pq such that (37) and (38) hold. In the simulation, we choose the initial condition being ϕ(t) = [0.2 0.2 0.1]T , τ1 = 2.1, τ2 = 4.3 and τ3 = 2.8. Figure 1 describes the switching signals of system and controller. The states of the open-loop system are shown in Fig. 2. Figure 3 detects the states of the closed-loop system. The states of the DOF controller are given in Fig. 4. It can be seen that the open-loop system is unstable, and the closed-loop system is exponentially stable, which indicates that the designed controller in (39) under the admissible switching signals is effective despite asynchronous switching. Let N0 p = 0, p = 1, 2, 3, according to Theorem 3, the resulting closedloop system is exponentially stable with a weighted H∞ performance γ˜ = 1.0779. Remark 8 The parameters α p , β p , ε p , μ p and μˆ p in this paper are mode dependent. When we solve (35)–(38), we can adjust any of them to ensure the feasibility. Thus, it will be more feasible in practice to design a MDADT switching than a ADT switching [12,18–20].

5 Conclusion In this paper, the asynchronous H∞ control problem for switched time-varying delay systems with MDADT has been studied. By adopting MDADT approach and the piecewise Lyapunov function technique, the exponential stability and weighted H∞ performance results for switched systems are proposed. Based on the value of α p +β p ,

Circuits Syst Signal Process 0.2 x1 x2

0.15

x3

0.1

0.05

0

−0.05

−0.1

−0.15

0

2

4

6

8

10

12

Time (s) Fig. 3 States of the closed-loop system 0.14 x

c1

0.12

x

c2

x

c3

0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04

0

2

4

6

8

10

12

Time (s) Fig. 4 States of the controller

p ∈ M, two types of MDADT are obtained. Moreover, the corresponding solvability conditions for desired DOF controller are established, and the computational algorithm for the design of the DOF controller and MDADT is presented. Finally, an example is given to illustrate the effectiveness of the proposed design method. In fact, the main approaches utilized in this work can be used to deal with the problem of asynchronous finite-time DOF control of switched systems, which could be our future work.

Circuits Syst Signal Process

References 1. S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994) 2. M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998) 3. D. Ding, G. Yang, H∞ static output feedback control for discrete-time switched linear systems with average dwell time. IET Control Theory Appl. 4(3), 381–390 (2010) 4. L. Gurvits, Stability of discrete linear inclusion. Linear Algebra Appl. 231(4), 47–85 (1995) 5. L. Hetel, J. Daafouz, C. Iung, Stabilization of arbitrary switched linear systems with unknown timevarying delays. IEEE Trans. Autom. Control 51(10), 1668–1674 (2006) 6. J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time. in Proceedings of the 38th IEEE Conference on Decision and Control, pp. 2655—2660 (1999) 7. D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems. IEEE Contr. Syst. Mag. 19(5), 59–70 (1999) 8. H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009) 9. D. Liberzon, Switching in Systems and Control (Birkhauser, Boston, 2003) 10. J. Lian, M. Wang, Sliding-mode control of switched delay systems with nonlinear perturbations: average dwell time approach. Nonlinear Dyn. 62(4), 791–798 (2010) 11. J. Lin, S. Fei, Z. Gao, Stabilization of discrete-time switched singular time-delay systems under asynchronous switching. J. Frankl. Inst. 349(5), 1808–1827 (2012) 12. H. Liu, Y. Shen, Asynchronous finite-time stabilisation of switched systems with average dwell time. IET Control Theory Appl. 6(9), 1213–1219 (2012) 13. S. Liu, Z. Xiang, Exponential H∞ output tracking control for switched neutral system with timevarying delay and nonlinear perturbations. Circuits Syst. Signal Process. 32(1), 103–121 (2013) 14. X. Liu, K. Zhang, S. Li, H. Wei, Optimal control for switched delay systems. Int. J. Innov. Comput. Inf. Control 10(4), 1555–1566 (2014) 15. V.F. Montagner, V.J.S. Leite, R.C.L.F. Oliveira, P.L.D. Peres, State feedback control of switched linear systems: an LMI approach. J. Comput. Appl. Math. 194(2), 192–206 (2006) 16. K. Narendra, J. Balakrishnan, A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Trans. Autom. Control 39(12), 2469–2471 (1994) 17. X. Sun, J. Zhao, D.J. Hill, Stability and L 2 -gain analysis for switched delay systems: a delay-dependent method. Automatica 42(10), 1769–1774 (2006) 18. X. Su, P. Shi, L. Wu, Y. Song, Fault detection filtering for nonlinear switched stochastic systems. IEEE Trans. Autom. Control (2015). doi:10.1109/TAC.2015.2465091 19. X. Su, L. Wu, P. Shi, C. Chen, Model approximation for fuzzy switched systems with stochastic perturbation. IEEE Trans. Fuzzy Syst. 23(5), 1458–1473 (2015) 20. R. Wang, Z. Wu, P. Shi, Dynamic output feedback control for a class of switched delay systems under asynchronous switching. Inform. Sci. 225(10), 72–80 (2013) 21. B. Wang, H. Zhang, D. Xie, G. Wang, C. Dang, Asynchronous stabilisation of impulsive switched systems. IET Control Theory Appl. 7(16), 2021–2027 (2013) 22. C. Wang, Redundant input guaranteed cost switched tracking control for omnidirectional rehabilitative training walker. Int. J. Innov. Comput. Inf. Control 10(3), 883–895 (2014) 23. L. Wu, J. Lam, Weighted H∞ filtering of switched systems with time-varying delay: average dwell time approach. Circuits Syst. Signal Process. 28(6), 1017–1036 (2009) 24. L. Wu, R. Yang, P. Shi, X. Su, Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings. Automatica 59, 206–215 (2015) 25. L. Wu, T. Qi, Z. Feng, Average dwell time approach to L 2 -L ∞ control of switched delay systems via dynamic output feedback. IET Control Theory Appl. 3(10), 1425–1436 (2009) 26. D. Xie, H. Zhang, H. Zhang, B. Wang, Exponential stability of switched systems with unstable subsystems: a mode-dependent average dwell time approach. Circuits Syst. Signal Process. 32(6), 3093–3105 (2013) 27. M. Xiang, Z. Xiang, H.R. Karimi, Asynchronous L 1 control of delayed switched positive systems with mode-dependent average dwell time. Inform. Sci. 278(10), 703–714 (2014) 28. W. Xiang, J. Xiao, H∞ filtering for switched nonlinear systems under asynchronous switching. Int. J. Syst. Sci. 42(5), 751–765 (2011)

Circuits Syst Signal Process 29. Z. Xiang, C. Qiao, M.S. Mahmoud, Robust H∞ filtering for switched stochastic systems under asynchronous switching. J. Frankl. Inst. 349(3), 1213–1230 (2012) 30. X. Yu, C. Wu, F. Liu, L. Wu, Sliding mode control of discrete-time switched systems with time-delay. J. Frankl. Inst. 350(1), 19–33 (2013) 31. X. Zhao, P. Shi, L. Zhang, Asynchronously switched control of a class of slowly switched linear systems. Syst. Control Lett. 61(12), 1151–1156 (2012) 32. X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with modedependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012) 33. X. Zhao, H. Liu, Z. Wang, Weighted H∞ performance analysis of switched linear systems with modedependent average dwell time. Int. J. Syst. Sci. 44(11), 2130–2139 (2013) 34. G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Disturbance attenuation properties of time-controlled switched systems. J. Frankl. Inst. 338(7), 765–779 (2001) 35. G. Zhang, C. Han, Y. Guan, L. Wu, Exponential stability analysis and stabilization of discrete-time nonlinear switched systems with time delays. Int. J. Innov. Comput. Inf. Control 8(3(A)), 1973–1986 (2012) 36. L. Zhang, H. Gao, Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010) 37. L. Zhang, P. Shi, Stability, L 2 -gain and asynchronous H∞ control of discrete-time switched systems with average dwell time. IEEE Trans. Autom. Control 54(9), 2192–2199 (2009) 38. X. Zhang, M. Wu, J. She, Y. He, Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41(8), 1405–1412 (2005)