New criteria of input-to-state stability for nonlinear switched stochastic delayed systems with asynchronous switching

Systems & Control Letters 129 (2019) 43–50

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

New criteria of input-to-state stability for nonlinear switched stochastic delayed systems with asynchronous switching✩ Meng Zhang a , Quanxin Zhu a,b , a b

∗

School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, China

article

info

Article history: Received 6 January 2019 Received in revised form 1 April 2019 Accepted 6 May 2019 Available online 7 June 2019 Keywords: Input-to-state stability (ISS) Razumikhin-type theorem Average-dwell time Time-delays Asynchronous switching

a b s t r a c t In this paper, we discuss the problems of input-to-state stability (ISS), integral-ISS (iISS) and eλt -ISS for a class of switched stochastic delayed systems under asynchronous switching. Asynchronous switching refers to that the switching of candidate controllers does not coincident with the system modes. Different from existing works, we allow the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be time-varying and increasing during the matched time interval and unmatched time interval, respectively. Meanwhile, by using the methods of Razumikhin technique, average dwell-time (ADT) approach together with comparison equation, some desired ISStype properties are obtained. Especially, our results improve existing results regarding asynchronous switching in the literature. An example is used to demonstrate the applicability of the results. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Switched systems, as a class of hybrid systems, are composed by a family of subsystems and a switching signal which governs the switching between the system modes [1–4]. In the last few decades, investigations of switched systems have captured exceeding attention due to its practical application in a broad scope of fields, for instance, networked control systems [5], power electronics [6], communication systems [7]. On the other hand, it is inevitable to encounter disturbance in the practical systems, which means that stochastic models are more applicable to describe practical systems. Overall consideration results in our present models — switched stochastic systems. Up to now, there exist quiet a few related works concerning switched stochastic systems, see [8–11]. Stability analysis is invariably one of the primary concerns in the domain of switched stochastic systems, which have been widely examined by a host of excellent works. The notions of input-to-state stability (ISS) and integral input-to-state stability (iISS), which were firstly introduced in [12,13], have been ✩ This work was jointly supported by the National Natural Science Foundation of China (61773217, 61374080, 11531006), the Natural Science Foundation of Jiangsu Province (BK20161552), the Scientific Research Fund of Hunan Provincial Education Department (18A013) and the Construct Program of the Key Discipline in Hunan Province. ∗ Corresponding author at: MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, China. E-mail addresses: [email protected] (M. Zhang), [email protected] (Q. Zhu). https://doi.org/10.1016/j.sysconle.2019.05.004 0167-6911/© 2019 Elsevier B.V. All rights reserved.

testified extremely useful in concerning fields. ISS properties can be traced to [14] and [15,16] for deterministic inputs and stochastic inputs, respectively. Recent studies on ISS properties are fruitful, and significant contributions for the ISS properties can be referred to [17–20]. Average dwell time (ADT) [21], which stands for the number of switching in a finite interval, is an approach to investigate the switched systems. Another widely applied method to investigate switched systems is Lyapunov function (or Lyapunov–Krasovskii functionals, Lyapunov–Razumikhin functions) approach. Based on ADT and Lyapunov function (or Lyapunov–Krasovskii functionals, Lyapunov–Razumikhin functions), stability analysis and control syntheses of switched systems have been studied; see [22,23] and [24,25] (time-delay systems). In the current literature, the Lyapunov function method with an indefinite derivative was provided to examine the ISS property of the nonlinear systems in [26], which ignored time-delays. Then, a Lyapunov–Krasovskii function with indefinite derivative was exploited to verify the switched stochastic delayed systems in [27]. And, based on Razumikhin condition, [28] investigated the ISS-type property under a weaker assumption for impulsive stochastic systems. Here, we will pay our attention into switched stochastic systems by employing the method of Lyapunov–Razumikhin function with a time-varying coefficient to derive the ISS property of nonlinear system with time-delays. As far as we know, little attention has been paid to the investigation on switched stochastic delayed systems with the above method, which motivates our research. In the aforementioned results [3,11,23], there needed a general hypothesis that the switches were required to occur at the

44

M. Zhang and Q. Zhu / Systems & Control Letters 129 (2019) 43–50

same time for controllers and system modes, i.e., synchronous switching. However, asynchronous switching, which is opposed to the synchronous switching, is caused by the detection delay of switching signal which results in the mismatched period of designed controller in each subsystem. In recent years, asynchronous phenomena can be found in many fields, such as hybrid systems [29–32], Markov systems [33,34], stochastic systems [35] and so on. Different from above asynchronous switching works, [36] allowed the Lyapunov function to increase not only in the period of mode-identifying process but also in normal-working period with matched controller. However, it only focused on deterministic systems. Hence, the method used in the above papers cannot be directly applied for our stochastic systems. Then, [37] allowed the Lyapunov function to be indefinite derivative during the matched time, which generalized the results in [35], but they ignored time-delays in the state. Based on [37], the time delays were added in [27] for switched system under asynchronous switching by using Lyapunov–Krasovskii function. Unlike [27], here we consider the Lyapunov–Razumikhin function to deal with stochastic delayed systems for asynchronous switching, which allow the Lyapunov function be time-varying in the matched interval. As is well-known, little attention has been paid to the investigation on asynchronous switching with above method, which is another motivation for our research. The objective of this work is to investigate the ISS-type property for switched stochastic delayed systems with asynchronous switching, which allows the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be time-varying during the matched time interval. The contributions of this paper lie in three aspects: (i) Compared with [28], we extend the result to switched stochastic systems together with asynchronous switching, and adopt the Razumikhin-type theorem and comparison principle to obtain the ISS-type property. (ii) Though [27] considered the Lyapunov–Krasovskii function to be time-varying and increasing, respectively, during the time when the active subsystem and the controller match each other. The results obtained in this paper are less conservative than those results given in [27], which required the ADT condition to have a parameter ρ . (iii) Time delays appear not only in the state of continuous dynamics, but also in the switching signal of the controller, which generalize the results in [35,37]. The remainder of the paper is organized as follows. Section 2 provides a few notions and introduces the definitions of ISStype, K, K∞ functions. Section 3 provides sufficient conditions to prove the ISS-type property of the nonlinear systems with asynchronous switching. Section 4 presents one example. Finally, Section 5 elaborates the conclusions of the study. 2. Model description and notations Throughout this work, unless otherwise specified, we use the following notions. Set N = {1, 2, . . .}, R = (−∞, +∞), R+ = [0, +∞). Rn and Rn×m denote, respectively, n-dimensional Euclidean space and n × m-dimensional Euclidean matrix space. For given real numbers x and y, then x ∨ y stands for the maximum of x and y, x ∧ y stands for the minimum of x and y. For x ∈ Rn , |x| defines the Euclidean vector norm. PC([a, b]; Rn ) denotes the class of piecewise continuous functions from [a, b] to Rn . For φ ∈ PC([a, b]; Rn ), its norm denoted by ∥φ∥[a,b] = supa≤s≤b |φ (s)|. For τ > 0, let C 1,2 denote the family of all nonnegative functions V (t , x) on [t0 − τ , ∞) × Rn that are continuously once differentiable in t and twice in x. Let (Ω , F , {Ft }t ≥0 , P) be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all P-null sets). For given τ > 0, let PCbF0 ([−τ , 0]; Rn ) (PCbFt ([−τ , 0]; Rn )) denote the family of all bounded F0 (Ft )-measurable, PC-valued random

variable φ , satisfying ∥φ∥ = sup−τ ≤θ ≤0 E|φ (θ )| < ∞, where E denotes the expectation of stochastic process. For function ψ : R → R, denote ψ (t − ) = lims→0− ψ (t + s), and the Dini derivative of ψ (t) is defined as D+ ψ (t) = lim sups→0+ (ψ (t + s) − ψ (t))/s. In this paper, we consider the following switched stochastic system: dx(t) = fσ (t) (t , xt , u(t))dt

+ gσ (t) (t , xt , u(t))dω(t),

t ≥ t0 ,

(1)

where x(t) is the system state, u(t) ∈ PC([t0 , ∞); R ) denotes the disturbance input; xt is denoted by xt = x(t − τ (t)), τ (t) : [t0 , +∞) → [0, τ ] is a continuous function. σ (t) : R+ → H = {1, 2, . . . , N} stands for the switching function, which is assumed to be a piecewise constant function continuous from the right, and tl , l ∈ R+ represents the switching instant. ω(t) = (ω1 (t), ω2 (t), . . . , ωm (t))T stands for an m-dimensional Ft -adapted Brownian motion. For any i ∈ H, fi : R+ × Rn × Rm → Rn , gi : R+ × Rn × Rm → Rn×m are continuous with respect to x and u and uniformly continuous in u. The initial data is given by {x(θ ) : −τ ≤ θ ≤ 0} = φ ∈ PC([−τ , t0 ]; Rn ). For stability analysis, we suppose that fi (t , 0, u) ≡ 0, gi (t , 0, u) ≡ 0 for all (t , u) ∈ R+ × Rm and i ∈ H, which admits x(t) = 0 is the trivial solution. In addition, denote by x(t) = x(t , t0 , φ ) the solution of (1) such that x(t0 ) = φ . We further assume that fi and gi are locally Lipschitz with respect to x and t, which means the solution process x(t , x(t0 ), φ ) is only defined on the certain finite interval [t0 , tmax ) with tmax > t0 . However, all the subsequent results are still valid for this case. In practical systems, if the switching of candidate controller is not coincided with the system modes, it will lead to asynchronous switching. The candidate controller is considered as: u(t) = hσ (t −d) (t , xt , w (t)), where d is the switching delay and w (t) is the reference input which can stabilize the corresponding system (1). The delay d is smaller than the corresponding switching internal, i.e., if the ith subsystem is active in [tl , tl+1 ), l ∈ N+ , then d⟨κ = inf{tl+1 − tl }⟩0. Due to the existence of switching delay d, we have the switching sequence {x(t0 ) : (i0 , t0 + d), . . . , (il , tl + d), . . . |il ∈ σ (t), l ∈ N} which means that the il -th controller is active during t ∈ [tl + d, tl+1 + d), l ∈ N. For any i ∈ H, let Vi ∈ C 1,2 , from [38] we denote an operator LV : [t0 , ∞) × Rn → Rn for system (1) as follows: m

LVi (t , xt ) = Vt (t , x, i) + Vx (t , x, i)fi (t , xt , u(t))

1

+ trace[giT (t , xt , u(t))Vxx (t , x, i)gi (t , xt , u(t))]. 2

where

( ∂ V (t , x) ) i , ∂t ( ∂ V (t , x) ∂ Vi (t , x) ) i Vx (t , x, i) = ,..., , ∂ x1 ∂ xn ( ∂ 2 V (t , x) ) i Vxx = . ∂ xl ∂ xm n×n

Vt (t , x, i) =

Definition 1. A continuous function α : R+ → R+ is said to belong to class K if α is strictly increasing and α (0) = 0. If α is also unbounded, then we say that it belongs to class K∞ . Moreover, the class υ K(υ K∞ ) function and CK(CK∞ ) function are the subsets of class K(K∞ ), which are convex and concave. A continuous function β : R+ × R+ → R+ is said to belong to class KL if β (·, t) ∈ K for each fixed t ≥ t0 and β (r , t) is decreasing to zero as t → ∞ for each fixed r ≥ 0. Definition 2 ([21]). For a switching signal σ and any t > s ≥ t0 , let Nσ (t , s) be the switching numbers of σ over the interval [s, t).

M. Zhang and Q. Zhu / Systems & Control Letters 129 (2019) 43–50

For given N0 > 0 and τa > 0, N0 and τa are called the chatter bound and the average dwell time (ADT), respectively, if they satisfy Nσ (t , s) ≤ N0 +

t −s

τa

45

Since ELWi (t) is continuous in interval [tl−1 , tl ), l ∈ N+ , we obtain D+ EWi (t) = ELWi (t) = ELVi (t , xt ). Define c1 = mini∈H {c1i }, c2 = maxi∈H {c2i }, ρ (s) = maxi∈H {ρi (s)}, and

.

{

ϕ (t), t ∈ [tN(t ,t0 )−1 + d, tN(t ,t0 ) ), η, t ∈ [tN(t ,t0 )−1 , tN(t ,t0 )−1 + d).

Definition 3 ([28]). System (1) is said to be (1) input-to-state stable (ISS), if there exist functions ρ ∈ KL and α, γ ∈ K∞ such that

χ (t) =

α (E(|x(t)|)) ≤ ϖ (E∥φ∥, t − t0 ) + γ (∥u∥[t0 ,t ] ),

c1 (|x|) ≤ Vi (t , x) ≤ c2 (|x|),

(2)

LVi (t , x) ≤ χ (t)Vi (t , x) + ρ (|u(t)|).

(3)

t ≥ t0 ;

(2) integral input-to-state stable (iISS), if there exist functions ϖ ∈ KL and α, γ ∈ K∞ such that

α (E(|x(t)|)) ≤ ϖ (E∥φ∥, t − t0 ) ∫ t + γ (|u(s)|)ds,

Thus, conditions (D1 )–(D2 ) can be rewritten as

Similarly, χq (t) can be denoted as follows:

{

χq (t) =

t ≥ t0 ;

t0

(3) eλt -weighted input-to-state stable (eλt -ISS), if there exist constant λ > 0, functions ϖ ∈ KL and α1 , α2 , γ ∈ K∞ such that eλ(t −t0 ) α1 (E|x(t)|) ≤ α2 (E∥φ∥)

+ sup {eλ(s−t0 ) γ (|u(s)|)}, t0 ≤s≤t

t ≥ t0 .

ϕq (t), t ∈ [tN(t ,t0 )−1 + d, tN(t ,t0 ) ), η, t ∈ [tN(t ,t0 )−1 , tN(t ,t0 )−1 + d),

where χq (t) = χ (t) ∨ (− τ ). Noting that c1 ∈ VK∞ , c2 ∈ CK∞ , it follows from (2) that ln q

c1 (E|x|) ≤ Ec1 (|x|) ≤ EWi (t)

(4)

and Ec2 (∥φ∥) ≤ c2 (E∥φ∥).

(5)

3. Main results

For all θ ∈ [−τ , 0], one can see

In this section, by employing the method of Lyapunov– Razumikhin and ADT condition, we will obtain some Razumikhin-type criteria on ISS, iISS and eλt -ISS for system (1).

EWσ (t0 ) (t0 + θ ) ≤ E[c2 (|φ (θ )|)]

Theorem 1. If there exist constants q > 1, µ ≥ 1, η > 0, 0 ≤ d < κ and functions Vi ∈ C 1,2 , c1i ∈ VK∞ , c2i ∈ CK∞ , ρi ∈ K∞ , ϕ (t) ∈ PC([t0 − τ , ∞]; R), such that for all i ∈ H

We first prove

≤ c2 (E(|φ (θ )|)) ≤ c2 (E(∥φ∥)).

∫ ∫

provided xt ∈ PCFt ([−τ , 0]; Rn ) satisfying that Vi (t + θ, xt (θ )) ≤ ln q qVi (t , x(t)) for all θ ∈ [−τ , 0], and function ϕq (t) = ϕ (t) ∨ (− τ ) is continuous. (D3 )Vσ (tl ) (tl , x(tl )) ≤ µVσ (tl−1 ) (tl , x(tl )), l ∈ N+ ;

∫∞ (D4 ) t ϕq+ (s)ds < ∞, and there exists a positive constant δ > 0 0∫ t such that t ϕq− (s)ds ≥ δ (t − t0 ), for sufficiently large t, where 0 ϕq+ (t) = ϕq (t) ∨ 0, ϕq− (t) = (−ϕq (t)) ∨ 0. And the average dwell time satisfies τa > (ln µ + (δ + η)d)/δ . Then, system (1) is ISS, iISS and eλt -ISS, where λ ∈ (0, δ − ln µ+(δ+η)d ). τ a

Proof. For any φ ∈ PCbF0 ([−τ , 0]; Rn ), we define the solution x(t , t0 , φ ) of (1) by x(t). Letting Wσ (t) (t) = Vσ (t) (t , x(t)), then it follows from Itoˆ differential formula [39] that dWi (t) = LWi (t)dt + Vx (t , x(t), i)gi (t , xt , u(t))dω(t), t ∈ [tl−1 , tl ), l ∈ N+ . We can calculate that

ρ (|u(s)|) exp(

t

∫

χq (v )dv )ds,

Let ∆t be small enough such that t + ∆t ∈ (tl−1 , tl ). Then, by using Lemma 3.2 in [40] and Fubini’s Theorem, we get

∫ t

t +∆t

ELWi (s)ds.

t ∈ [t0 , t1 ).

(6)

s

t0

For any positive constant ς , consider the following comparison equation

{

y˙ i (t) = χq (t)yi (t) + ρ (|u(t)|) + ς, yσ (t0 ) (t0 ) = δ0 ,

t0 ≤ t < t1 ,

(7)

where δ0 = c2 (E(∥φ∥)) + ς . Solving (7) yields that yσ (t0 ) (t) = δ0 exp(

∫

t

χq (s)ds) t0

∫

t

+

∫ t [ρ (|u(s)|) + ς] exp( χq (v )dv )ds, t ∈ [t0 , t1 ).

t0

s

We claim that the following holds EWσ (t0 ) (t) < yσ (t0 ) (t), t ∈ [t0 , t1 ).

(8)

Suppose (8) is not true, i.e., there is some t > t0 satisfying EWσ (t0 ) (t ∗ ) ≥ yσ (t0 ) (t ∗ ). Let t ∗ = inf{t > t0 : EWσ (t0 ) (t) ≥ yσ (t0 ) (t)}. By the continuity of EWσ (t0 ) (t) and EW (t0 ) < yσ (t0 ) (t0 ), we obtain EWσ (t0 ) (t) < yσ (t0 ) (t) for all t0 ≤ t < t ∗ , EWσ (t0 ) (t ∗ ) = yσ (t0 ) (t ∗ ) and EWσ (t0 ) (t) ≥ yσ (t0 ) (t) for all t ∈ (t ∗ , t ∗ + ∆t), where ∆t > 0 is sufficiently small. Hence, it follows that t ∈ (t ∗ , t ∗ +∆t). ∗

EWσ (t0 ) (t) − EWσ (t0 ) (t ∗ )

LWi (t) = LVi (t , xt ).

EWi (t + ∆t) − EWi (t) =

t

+

(D2 ) LVi (t , xt ) ≤

ϕ (t)Vi (t , x) + ρi (|u(t)|) t ∈ [tl + d, tl+1 ), ηVi (t , x) + ρi (|u(t)|), t ∈ [tl , tl + d);

χq (s)ds) t0

(D1 ) c1i (|x|) ≤ Vi (t , x) ≤ c2i (|x|);

{

t

EWσ (t0 ) (t) ≤ c2 (E(∥φ∥)) exp(

t − t∗

≥

yσ (t0 ) (t) − yσ (t0 ) (t ∗ ) t − t∗

.

Thus D+ EWσ (t0 ) (t ∗ ) ≥ χq (t ∗ )yσ (t0 ) (t ∗ ) + ς

> χq (t ∗ )yσ (t0 ) (t ∗ ).

(9)

46

M. Zhang and Q. Zhu / Systems & Control Letters 129 (2019) 43–50

Let χ − (t) = (−χ (t)) ∨ 0 and χq− (t) = (−χq (t)) ∨ 0. Then, we have χq− (t) = χq− (t) ∧ lnτ q . Noting that lnτ q ≥ −χq (t), χq (t) ≥ −χq− (t), q=

∫t

ln q

τ

t −τ

ds for any θ ∈ [−τ , 0], if t ∗ + θ ≥ t0 , we obtain

qEWσ (t0 ) (t ∗ ) ≥ exp(

t

∫

ln q

t +θ

τ

ds)yσ (t0 ) (t ∗ )

t∗

∫ ≥ exp(−

t ∗ +θ

χq (s)ds){δ0 exp(

[ρ (|u(s)|) + ς ] exp(

+

qEWσ (tr ) (t∗ )

χq (s)ds)

∫ t∗ ∫ t∗ χq (s)ds)yσ (tr ) (tr ) ≥ exp( χq− (s)ds){exp( t +θ tr ∫ t∗ ∫ t∗ ∗ [ρ (|u(s)|) + ς ] exp( χq (v )dv )ds} +

t0

t ∗ +θ

∫

t∗

∫

t∗

∫

χq (v )dv )ds} s

t0

= δ0 exp(

t ∗ +θ

∫

[ρ (|u(s)|) + ς ] exp(

χq (v )dv )ds s

= yσ (t0 ) (t ∗ + θ ) ≥ EWσ (t0 ) (t ∗ + θ ).

= δ0 exp(

χq− (s)ds) exp(

t∗

∫

≥ δ0 exp(

t ∗ −τ

t ∗ +θ t0

χq− (s)ds) exp(−

tr

χq (s)ds) t0

t ∗ +θ

(11)

From (10), (11) and (3), it follows that D+ EWσ (t0 ) (t) ≤ χ (t ∗ )E Wσ (t0 ) (t ∗ ) + ρ (|u(t ∗ )|) ≤ χq (t ∗ )EWσ (t0 ) (t ∗ ) + ρ (|u(t ∗ )|), which is a contradiction. Therefore, (8) holds. Let ς → 0, we obtain (6). Now, we assume that for l ≤ r , r ∈ N+ , i.e. for l = 1, 2, . . . , r, we have EWσ (tl−1 ) (t) < yσ (t) (t) t

∫ = exp(

χq (s)ds)yσ (tl−1 ) (tl−1 )

∫

t ∈ [tl−1 , tl ).

(12)

Obviously, for l = 1, from (8), (12) is true. From (D3 ) and (12), we have EWσ (tr ) (tr ) ≤ µEWσ (tr −1 ) (tr )

≤ µ exp(

∫

tr

If t∗ + θ < tr , then either t∗ + θ ∈ [t0 − τ , t0 ) or t∗ + θ ∈ [tk−1 , tk ) for some k ∈ {1, 2, . . . , r }. For t∗ + θ ∈ [t0 − τ , t0 ), we can see qEWσ (tr ) (t∗ )

+µ

tr

χq (s)ds)yσ (tr −1 ) (tr −1 ) tr

∫

tr − 1

χq (s)ds)yσ (tr ) (tr )] χq (s)ds)[exp( t t +θ ∫ ∗t∗ ∫ rt∗ −χq− (s)ds)yσ (tr ) (tr ) ≥ exp( χq− (s)ds) exp( tr t +θ ∫ ∗tr − χq (s)ds)yσ (tr ) (tr ) = exp(

qEWσ (tr ) (t∗ )

∫ t∗ ∫ t∗ ≥ exp(− χq (s)ds){exp( χq (s)ds)yσ (tr ) (tr ) t +θ t ∫ t∗ ∗ ∫ tr∗ [ρ (|u(s)|) + ς ] exp( χq (v )dv )ds} + s

s

(13)

[ρ (|u(s)|) + ς ] exp(

+

[ρ (|u(s)|) + ς ] exp(

tr

t∗ +θ

∫

χq (s)ds) ∗

≥ exp( tr tr

χq (s)ds)yσ (tr −1 ) (tr −1 )

tr − 1

+µ

tr

∫

[ρ (|u(s)|) + ς ] exp(

tr − 1

t

χq (v )dv )ds. s

[ρ (|u(s)|) + ς ] exp( tr

(14)

= µ exp(

tr

∫

χq (v )dv )ds + ς}

s

t∗

∫ +

We will prove that, for all t ∈ [tr , tr +1 ) EWσ (tr ) (t) < yσ (tr ) (t).

χq (v )dv )ds s

∫

∫

t∗ +θ

∫

tr

{µ exp(

χq (s)ds)yσ (tr ) (tr )

tr t

∫

t∗

∫ +

t

yσ (tr ) (t) = exp(

χq (s)ds)yσ (tr ) (tr ) tr

For all tr ≤ t < tr +1 , (13) can be solved as

∫

t∗ +θ

= exp(

χq (v )dv )ds.

y˙ i (t) = χq (t)yi (t) + ρ (|u(t)|) + ς, yi (t) = µyj (t) + ς .

(16)

tr

Analogous to the proof of (8), we consider the following comparison equation

{

t∗

∫

−

≥ exp(

∫

[ρ (|u(s)|) + ς ] exp(

t∗

∫

tr − 1

∫

(15)

If t∗ +θ ∈ [tk−1 , tk ) for some k ∈ {1, 2, . . . , r }, then tk−1 ≤ t∗ +θ < tk ≤ tr . Hence, we have

s

tl−1

χq (v )dv )ds

s

≥ δ0 ≥ EWσ (tr ) (t∗ + θ ).

∫ t [ρ (|u(s)|) + ς] exp( χq (v )dv )ds,

t

t∗ +θ

t∗ +θ

tl − 1

+

∫

= yσ (tr ) (t∗ + θ ) ≥ EWσ (tr ) (t∗ + θ ).

χq− (s)ds)

χq− (s)ds) ≥ δ0 ≥ EWσ (t0 ) (t ∗ + θ ).

[ρ (|u(s)|) + ς ]) exp( tr

t0

∫

t∗ +θ

∫ + exp(

t∗

∫

χq (s)ds)yσ (tr ) (tr )

≥ exp(

t∗

∫

t∗ +θ

∫

If t ∗ + θ < t0 , then t ∗ − τ ≤ t ∗ + θ < t0 < t ∗ . We also have qEWσ (t0 ) (t ∗ ) ≥ δ0 exp(

s

tr

(10)

t∗

t∗

∫

χq (s)ds)yσ (tr ) (tr ) ≥ exp(− χq (s)ds){exp( tr t∗ +θ ∫ t∗ ∫ t∗ [ρ (|u(s)|) + ς ] exp( χq (v )dv )ds} +

t ∗ +θ

∫

t0

∫

t∗

∫

t ∗ +θ

+

s

tr

χq (s)ds) t0

∫

Suppose (14) does not hold. Define t∗ = {t ∈ [tr , tr +1 ) : EWσ (tr ) ≥ yσ (t) (t)} t∗ ∈ [tr , tr +1 ) such that EWσ (tr ) (t) < yσ (tr ) (t) for all tr ≤ t < t∗ , EWσ (tr ) (t∗ ) = yσ (tr ) (t∗ ) and D+ Wσ (tr ) (t∗ ) > χq (t∗ )EWσ (tr ) (t∗ ) + ρ (|u(t∗ )|). For all θ ∈ [−τ , 0], if t∗ + θ ≥ tr , then tr ≤ t∗ + θ ≤ t∗ , we have

t∗ +θ

∫

χq (v )dv )ds s

∫

t∗ +θ

χq (s)ds)yσ (tr −1 ) (tr −1 ) tr − 1

M. Zhang and Q. Zhu / Systems & Control Letters 129 (2019) 43–50 tr

∫

+µ

[ρ (|u(s)|) + ς ] exp(

tr − 1

+ ς exp(

t∗ +θ

∫

tN(t ,t0 ) > t − d, for any positive number M ≤ m ≤ N(t , t0 ), we have

χq (v )dv )ds s

t ∗ +θ

∫

∫ χq (s)ds)

tr t∗ +θ

∫

[ρ (|u(s)|) + ς ] exp(

+

χq (v )dv )ds

≥ µ exp(

χq (s)ds)yσ (tr −1 ) (tr −1 )

tr − 1 t∗ +θ

∫

[ρ (|u(s)|) + ς ] exp(

= yσ (tr ) (t∗ + θ ) ≥ EWσ (tr ) (t∗ + θ ).

(17)

From (15)–(17), we obtain that qEWσ (t) (t∗ ) ≥ EWσ (t) (t∗ + θ ), ∀θ ∈ [−τ , 0]. By (3), we also have D+ EWσ (t) (t∗ ) ≤ χq (t∗ )EWσ (t) (t∗ ), which yields a contradiction. And thus (14) holds. Hence, by the mathematical induction and comparison equation (13), then (12) implies the following. EWσ (tl−1 ) (t) < exp(

t

∫

t

[ρ (|u(s)|) + ς ] exp(

+

χq (v )dv )ds

≤ δ0 µl−1 exp(

t

∫

χq (s)ds) t0

+

l−1 ∑

µ

l−i

ti

∫

∫

[ρ (|u(s)|) + ς ] exp(

χq (v )dv )ds s

[ρ (|u(s)|) + ς ] exp(

+

χq (v )dv )ds.

EWσ (tl ) (t) ≤ c2 (E(∥φ∥))µ exp(

+

∑

µl−i

χq (s)ds)

ρ (|u(s)|) exp(

t0

and K1 =

χq (v )dv )ds

∫t

µN(t ,s) e

s

t

∫

tl−1

) 0)

χq (v )dv )ds, t ≥ t0 .

(18)

∫

t

∫t

t0

χq (s)ds

N(t ,s)

µ

+

e

s

χq (v )dv

ρ (|u(s)|)ds

t0

= Λ1 + Λ2 ,

t ≥ t0 .

(19)

c2 (E(∥φ∥)).

∫t

(21)

.

(22)

+

ϕ (s)ds+((η+δ )d/τa −δ )(t −tM )

∫ tM

.

t0

χq (s)ds

Λ1 = c2 (E(∥φ∥))µN(t ,t0 ) e t0 , ∫ t ∫t Λ2 = µN(t ,s) e s χq (v)dv ρ (|u(s)|)ds.

χ (v )dv

∫t

≤ µN(t ,t0 ) e ∫ tM

∫

Case I: When t ∈ [tN(t ,t0 ) , tN(t ,t0 ) + d). In this case, the candidate controller does not coincident with the system modes. In view of

s

χq (v )dv

s

χq (s)ds −ϵ (tM −t0 )

e

+

∫ tM t0

χ (v )dv

µN(t ,tM ) e

χ (v )dv

is a positive constant,

And ∫t

µN(t ,tM ) e

tM

∫t

tM

χq (v )dv χq (v )dv

χq+ (v )dv (η+δ )h(1+N0 )

e

∫∞ + t ϕ (v )dv

e

M

and C2 =

t

e−ϵ (t −s) ρ (|u(s)|)ds.

(24)

t0

Substituting (23) and (24) into (19) yields EWσ (t) (t) ≤ C1 e−ϵ (t −t0 ) c2 (E(∥φ∥))

t0

In the light of ∫ tcondition (iv), there exists some positive number M such that t ϕq− (s)ds ≥ δ (t − t0 ) when t ≥ tM . 0 Due to the existence of asynchronous switching, we discuss the proof in two cases.

(23)

∞ (η+δ )d(1+N0 )+ t ϕq+ (s)ds M e .

where K2 = µM +N0 e K2 e−ϵ (t0 −tM ) . Thus,

Λ 2 = C2

where ∫t

χq (s)ds

tM

≤ µM e t0 q ≤ K2 e−ϵ (t −tM ) = C2 e−ϵ (t −s) ,

c2 (E(∥φ∥))

∫t

∫t

χ (s)ds

∫ tM

s

(t) ≤ µN(t ,t0 ) e

c2 (E(∥φ∥))

µN(t ,tM ) e

= µM e

From Definition 2 and inequality (18), we have EWσ (tN(t ,t

χq (s)ds

χq (s)ds

where C1 = µM +N0 K1∫e

s

ρ (|u(s)|) exp(

+

ti

∫

t0

Λ1 ≤ C1 e−ϵ (t −t0 ) c2 (E(∥φ∥)),

t

ti−1

i=1 t

∫

ti

∫ tM

∫t

where K1 (t) = e tM q Substituting (22) into (21) yields

t0

∫

According to (19), we get

∫t

Letting ς → 0, we have

∫

(20)

(β+δ )d(1+N0 )+ t ϕq+ (s)ds −ϵ (t −tM ) M e

s

l

− δ )(t − tm ).

τa

≤ µN0 e

t

∫

tl − 1

l−1

+(

tm +d

(η + δ )d

∫t

ti−1

i=1 t

χq (s)ds tN(t ,t ) 0

µN(t ,tM ) e tM q = µ(N0 +(t −tM )/τa ) e(η+δ)d(1+N0 ) K1 (t)

t

∫

t

On the other hand, in view of τa > ((ln µ + (η + δ )h)/δ ), there exists ϵ ∈ (0, δ − (ln µ/τa ) − ((η + δ )d/τa )) such that (ln µ/τa ) < δ − ((η + δ )d/τa ) − ϵ . Thus, it follows from inequality (20) that

s

tl−1

χq (s)ds +

∫

≤ (η + δ )dN(t , tm ) + η(t − tN(t ,t0 ) ) ∫ tN(t ,t ) 0 − δ (tN(t ,t0 ) − tm ) + ϕq+ (s) tm +d ∫ tN(t ,t ) 0 ϕq+ (s)ds ≤ (η + δ )d(1 + N0 ) +

= µM e

t

∫

χq (s)ds

tm tN(t ,t ) 0

Λ1 = µN(t ,t0 ) e

χq (s)ds)yσ (tl−1 ) (tl−1 ) tl−1

∫

tm +d

∫

tN(t ,t )−1 +d 0

χq (v )dv )ds s

tr − 1

+ς

t∗ +θ

∫

χq (s)ds = tm ∫ + ··· +

≤ ηd(N(t , t0 ) − m) + η(t − tN(t ,t0 ) ) ∫ tN(t ,t ) 0 + ··· + ϕq (s)ds

t∗ +θ

∫

t

tN(t ,t )−1 +d 0

s

tr

+µ

t∗

∫

47

t

∫

e−ϵ (t −s) ρ (|u(s)|)ds.

+ C2

(25)

t0

From (4), we obtain c1 (E|x|) ≤ C1 e−ϵ (t −t0 ) c2 (E(∥φ∥))

∫

t

e−ϵ (t −s) ρ (|u(s)|)ds.

+ C2 t0

(26)

48

M. Zhang and Q. Zhu / Systems & Control Letters 129 (2019) 43–50

Let α (r) = c1 (r), ϖ (r , s) = C1 c2 (r)e−ϵ s , γ (r) = C2 ρ (r). Thus, system (1) is i-ISS. From (26), it can be easily checked that c1 (E|x|) ≤ C1 e−ϵ (t −t0 ) c2 (E(∥φ∥)) 1 + C2 ρ (∥u(t)∥[t0 ,t ] )ds.

(27)

ϵ

Let α (r) = c1 (r), ϖ (r , s) = C1 c2 (r)e−ϵ s , γ (r) = 1ϵ C2 ρ (r). Hence, system (1) is ISS. Let ϵ = ϵ1 + λ, where ϵ1 and λ are positive constants, From (26), we obtain eλ(t −t0 ) c1 (E|x|) ≤ C1 e−ϵ1 (t −t0 ) c2 (E(∥φ∥)) t

∫

e−ϵ1 (t −s) eλ(t −t0 ) ρ (|u(s)|)ds

+ C2 t0

≤ C1 c2 (E(∥φ∥)) + C2

1

sup {eρ (s−t0 ) γ (|u(s)|)},

ϵ1

(28)

t0 ≤s≤t

where α1 (r) = c1 (r), α2 (r) = C1 c2 (r), γ (r) = λt

C ρ (r). ϵ1 2 1

Then,

system (1) is e -ISS. Case II: When t ∈ [tN(t ,t0 ) + d, tN(t ,t0 )+1 ). In this case, the candidate controller matches the system modes. Similarly, we can show that for any positive number M ≤ m ≤ N(t , t0 ),

∫

t

tm +d

∫

χq (s)ds =

χq (s)ds

tm tN(t ,t ) +d 0

tm

∫ + ··· +

χq (s)ds +

∫

tN(t ,t ) 0

χq (s)ds tN(t ,t ) +d 0

≤ (η + δ )d(1 + N0 ) + +(

t

∫

t

ϕq+ (s)ds

− δ )(t − tm ),

τa ∫ tM

∫t

χ (s)ds

∫ tM

where C = µ e 1

Λ2 =

∫

t

M

µN(t ,s) e

t0

∫t s

χ (s)ds

χq (v )dv

χq (s)ds

(29) e

≤ C2

t

∫

∫

µN(t ,tM ) e

.

F12

ρ (|u(s)|)ds

∫ tM s

M

ρ (|u(s)|)ds

e−ϵ (t −s) ρ (|u(s)|)ds,

(30) ∫∞ + t c (v )dv

χ + (v )dv

where K3 = µM +N0 e t0 q e(η+δ )d(1+N0 ) and C 2 = K3 e −ϵ (t0 −tM ) e . From (19), (29) and (30), we have

M

EWσ (t) (t) ≤ C 1 e−ϵ (t −t0 ) c2 (E(∥φ∥))

+C

∫

− 1)x,

G11 = (

G12 =

1 2

|cos t |x2t 1/2 ) , 1 + t2

xt .

−ϵ (t −s)

ρ (|u(s)|)ds.

3

F22

x+

1

u,

G21 = (

2(1 + t 2 ) 2 1 1 = x + u, G22 = xt . 4 2

x2t 1 + t2

)1/2 ,

Choosing Lyapunov functions V1 = x2 , V2 = 2x2 , let fσ (t) (t , ψ, u(t)) = Fσ (t) (t , ψ (0), u(t)), gσ (t) (t , ψ ) = Gσ (t) (t , ψ (−0.25|sin t |)), q = 2. For xt ∈ LFt ([−τ , 0]; R2 ), which satisfies E|x(t + θ )|2 ≤ 2E|x(t)|2 for all θ ∈ [−τ , 0], we can calculate LVi (t , xt ) ≤ χ (t)Vi (t , x) + ρ (|u|),

t

e

t2

1+ 1 = x + u, 2

F21 =

t0

2

2

and ∫ χq (v )dv+ tt χq (v )dv

t0 t

∫ tM

where τ (t) = 0.25|sin t |, τ = 0.25, σ (t) = {1, 2}. Let F11 = (

t0

≤ µM

Consider the following switched stochastic delayed

dx(t) = Fσ (t) (t , x(t), u(t))dt + Gσ (t) (t , x(t − τ (t)))dω(t) c2 (E(∥φ∥))

N(t ,tM ) −(t0 −tM )

µ

Remark 3. In [37], they put forward the assumption that the Lyapunov function is time-varying during the matched time, which generalized the results in [35]. Based on [37], the time delays were added in [27] for switched system under asynchronous switching by using the Lyapunov–Krasovskii function. In this paper, we consider Lyapunov–Razumikhin function, and the results in this paper are less conservative than the main results in [27], which required the ADT condition to have a parameter ρ .

Example. system:

which means that

Λ1 = µM e t0 q µN(t ,tM ) e tM ≤ C 1 e−ϵ (t −t0 ) c2 (E(∥φ∥)),

Remark 2. (i) The rate coefficient of the Lyapunov function in matched time is negative in previous asynchronous switching [29,30,32,33,35], we generalize it to time-varying, see condition (D2 ). However, [29,30,32] only focused on deterministic systems. Then, the stochastic systems with asynchronous switching were considered in [33,35]. (ii) Compared with [33,35], iISS, eλt -ISS were not considered, and the time delays in the state were ignored in [35], and so the method in [35] cannot be applied in our results. The asynchronous switching was studied for stochastic delayed systems in [33] with Razumikhin-type condition, but the Lyapunov function in matched time was negative in [33].

4. Numerical example

tm

(η + δ )d(1 + N0 )

with asynchronous switching. The time delay between the candidate controllers and the system modes is d, which is smaller than κ . From condition (D2 ), it implies that during the switch interval [tl , tl+1 ), the system modes are mismatched with the candidate controllers in the interval [tl , tl + d), the subsystems and the candidate controllers are matched in the interval [tl + d, tl+1 ). If d = 0, then Theorem 1 in this paper is analogous to Theorem 1 in [28].

(31)

t0

Similar to the proof of Case I, we can obtain the ISS-type property of system (1).

where ϕ (t) = 8/(1 + t 2 ) − 2, η = 1.5, ρ (r) = r 2 . Thus, we have ϕq (t) = ϕ (t). And ∞

∫

∫

∞

8

Therefore, according to the analysis of the above two cases, system (1) satisfies ISS-type property.

ϕ (s)ds ≤ ds = 4π < ∞, 1 + s2 ∫0 t ∫ 0t ϕ − (s)ds ≥ 2ds = 2t .

Remark 1. The method in this paper is based on [28], but we extend the results in [28] to switched stochastic systems together

Given µ = 2, d = 0.1, from the condition (D4 ), we can obtain ln 2+(1.5+2)∗0.1 the average dwell time τa > = 0.52. 2

+

0

0

M. Zhang and Q. Zhu / Systems & Control Letters 129 (2019) 43–50

49

References

Fig. 1. State trajectory with w (t) = sin t.

Fig. 2. State trajectory with w (t) = 0.

Remark 4. In Fig. 1, we let the reference input w (t) = sin t, it can be seen that the state x(t) will not converge to zero, but will remain bounded in probability. In Fig. 2 if we take w (t) = 0, the state x(t) is more easy to converge to zero in a fast speed. 5. Conclusion In this work, the ISS-type property has been studied for a class of switched stochastic delayed systems under asynchronous switching. Employing the method of Razumikhin technique, average dwell-time (ADT) approach together with comparison equation, then some desired ISS-type properties are obtained. Especially, our results improve some existing results regarding asynchronous switching in the literature.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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