# Stability Analysis and RobustH∞ Control of Switched Stochastic Systems with Time-Varying Delay

Authors: Zhengrong Xiang; Guoxin Chen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The problems of mean-square exponential stability and robustH∞ control of switched stochastic systems with time-varying delay are investigated in this paper. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criterion of such system is derived in terms of linear matrix inequalities (LMIs). Then, H∞ performance is studied and robust H∞ controller is designed. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Switched systems, a special hybrid system, are composed of a set of continuous-time or discrete-time subsystems and a rule orchestrating the switching between the subsystems. In the last two decades, there has been increasing interest in the stability analysis and control design for such switched systems since many real-world systems such as chemical systems [1], robot control systems [2], traffic systems [3], and networked control systems [4, 5] can be modeled as such systems. The past decades have witnessed an enormous interest in the stability analysis and control synthesis of switched systems [6–11].It is well known that time delay phenomenon exists in many engineering systems such as networked systems and long-distance transportation systems. Such phenomenon may cause the system unstable if it cannot be handled properly, which motivates many scientists to involve themselves in researching switched systems with time delay. Many results have been reported for stability analysis of switched systems with time delay [12, 13], where the asymptotical stability criteria are given by using common Lyapunov function approach in [12], and the exponential stability criteria under average dwell time switching signals are proposed in [13]. Moreover, the problem of delay-dependent global robust asymptotic stability of switched uncertain Hopfield neural networks with time delay in the leakage term is discussed in [14].  H∞ control of continuous-time switched systems with time delay and discrete-time switched systems with time delay are investigated in [15, 16], respectively.On the other hand, stochastic systems have attracted considerable attention during the past decades because stochastic disturbance exists in many actual operations. Many useful results on the stability analysis of stochastic systems are reported in [17–21]. The problem of robust H∞ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays is considered [22]. Based on the results of stochastic systems and switched systems, the stability analysis and stabilization of switched stochastic systems are investigated in [23, 24]. Furthermore, the problems of reliable control and reliable H∞ control for switched stochastic systems under asynchronous switching are studied in [25, 26], respectively. Recently, these results are extended to stochastic switched systems with time delay, and the exponential stability criteria are addressed [27, 28]. However, these results are very complex, which make it more difficult for us to solve many issues such as controller design under asynchronous switching and actuator failures. Therefore, there is a lot of work to do in such field. This motivates the present study.In this paper, we focus on the mean-square exponential stability analysis and robustH∞ control of switched stochastic systems with time-varying delay. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criterion is derived. Moreover, H∞ performance is studied and H∞ state feedback controller is proposed. The remainder of the paper is organized as follows. In Section 2, problem statement and some useful lemmas are given. In Section 3, based on the average dwell time method and Gronwall-Bellman inequality, the mean-square exponential stability and H∞ performance of the switched stochastic systems with time delay are investigated. Then, robust H∞ controller is designed. In Section 4, a numerical example is given to illustrate the effectiveness of the proposed approach. Finally, concluding remarks are provided in Section 5.Notation.
Throughout this paper, the superscript ‘‘T’’ denotes the transpose, and the symmetric terms in a matrices are denoted by ∗. The notation X>Y(X≥Y) means that matrix X-Y is positive definite (positive semidefinite, resp.). Rn denotes the n dimensional Euclidean space. ∥x(t)∥ denotes the Euclidean norm. L2[t0,∞) is the space of square integrable functions on [t0,∞). λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively. I is an identity matrix with appropriate dimension. diag{ai} denotes diagonal matrix with the diagonal elements ai,i=1,2,…,n.


## 2. Problem Formulation and Preliminaries

Consider the following stochastic switched systems with time-delay:(2.1)dx(t)=[Âσ(t)x(t)+B̂σ(t)x(t-h(t))+Cσ(t)u(t)+Gσ(t)v(t)]dt+D̂σ(t)x(t)dw(t),x(t)=φ(t),t∈[t0-h,t0],z(t)=Mσ(t)x(t),
where x(t)∈Rn is the state vector, φ(t)∈Rn is the initial state function, u(t)∈Rl is the control input, v(t)∈Rp is the disturbance input which is assumed belong to L2[t0,∞], z(t)∈Rq is the signal to be estimated, w(t)∈R is a zero-mean Wiener process on a probability space (ΩFP) satisfying(2.2)E{dw(t)}=0,E{dw2(t)=dt},
where Ω is the sample space, F is σ-algebras of subsets of the sample space, P is the probability measure on F, and  E{·} is the expectation operator. h(t) is the system state delay satisfying(2.3)0≤h(t)≤h,ḣ(t)≤hd<1,
where hd is a known constant. The function σ(t):[t0,∞]→N̲={1,2,…,N} is a switching     signal which is deterministic, piecewise constant, and right continuous. The switching sequence can be described as σ:{(t0,σ(t0)),(t1,σ(t1)),…,(tk,σ(tk))}, σ(tk)∈N̲,k∈Z, where t0 is the initial time and tk denotes the kth switching instant. Moreover σ(t)=i means that the ith subsystem is activated.For each for alli∈N̲,  Ci,  Gi, and Mi are known real-value matrices with appropriate dimensions, and Âi,B̂i, and D̂i are uncertain real matrix with appropriate dimensions, which can be written as(2.4)[ÂiB̂iD̂i]=[AiBiDi]+HiFi(t)[E1iE2iE3i],
where Ai, Bi, and Di are known real-value matrices with appropriate dimensions, and  Fi(t) is unknown time-varying matrix that satisfies(2.5)FiT(t)Fi(t)≤I.Definition 2.1.
System (2.1) is said to be mean-square exponentially stable with under switching signal σ(t), if there exist scalars κ>0 and α>0, such that the solution x(t) of system (2.1) satisfies E{∥x(t)∥2}≤κe-α(t-t0)supt0-h≤θ≤t0E{∥φ(θ)∥2}, for all t>t0. Moreover, α is called the decay rate.Definition 2.2.
For anyT2>T1≥t0, let Nσ(T1,T2) denote the switching number of σ(t) on an interval (T1,T2). If
(2.6)Nσ(T1,T2)≤N0+T2-T1Tα
holds for given N0≥0, Tα>0, then the constant Tα is called the average dwell time. As commonly used in the literature, we choose N0=0.Definition 2.3.
Letγ>0 be a positive constant, for system (2.1), if there exists a controller u(t) and a switching signal σ(t), such that(1)
whenv(t)=0, system (2.1) is mean-square exponentially stable;(2)
under zero initial conditionx(t)=0, t∈[t0-h,t0], the output z(t) satisfies(2.7)E{∫t0∞e-λ(s-t0)zT(s)z(s)ds}≤γ2∫t0∞vT(s)v(s)ds,∀v(t)∈L2[t0,∞).
Then system (2.1) is said to be robustly exponentially stabilizable with a prescribed weighted H∞ performance, where λ>0.The following lemmas play an important role in the later development.Lemma 2.4 (see [29] (Gronwall-Bellman Inequality)).
Letx(t) and y(t) be real-valued nonnegative continuous functions with domain {t∣t≥t0}; a is a nonnegative scalar; if the following inequality
(2.8)x(t)≤a+∫t0tx(s)y(s)ds
holds, for t≥t0, then x(t)≤aexp(∫t0ty(s)ds).Lemma 2.5 (see [30]).
LetU, V, W, and X be constant matrices of appropriate dimensions with X satisfying X=XT, then for all VTV≤I, X+UVW+WTVTUT<0, if and only if there exists a scalar ε>0 such that X+εUUT+ε-1WTW<0.


## 3. Main Results

### 3.1. Stability Analysis

In this subsection, we will focus on the exponential stability analysis of switched stochastic systems with time-varying delay.Consider the following switched stochastic system:(3.1)dx(t)=[Aσ(t)x(t)+Bσ(t)x(t-h(t))]dt+Dσ(t)x(t)dw(t),x(t)=φ(t),t∈[t0-h,t0].Theorem 3.1.
Considering system (3.1), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 satisfying
(3.2)[AiTPi+PiAi+αPi+QiPiBiDiTPi*-(1-hd)Qi0**-Pi]<0,
for all i∈N̲, then system (3.1) is mean-square exponentially stable under arbitrary switching signal with the average dwell time:
(3.3)Tα≥Tα*=ln⁡⁡μα,
where μ≥1 satisfies
(3.4)Pi≤μPj,Qi≤μQj,∀i,j∈N̲.Proof.
Consider the following Lyapunov functional for theith subsystem:
(3.5)Vi(t,x(t))=V1,i(t,x(t))+V2,i(t,x(t)),
where
(3.6)V1,i(t,x(t))=xT(t)Pix(t),V2,i(t,x(t))=∫t-h(t)txT(s)Qix(s)ds.
For the sake of simplicity, Vi(t,x(t)) is written as Vi(t) in this paper.
According to Itô formula, along the trajectory of system (3.1), we have
(3.7)dVi(t)=LVi(t)dt+2xT(t)PiDix(t)dw(t),
where
(3.8)LVi(t)=2xT(t)Pi(Aix(t)+Bix(t-h(t)))+xT(t)Qix(t)+xT(t)DiTPiDix(t)-(1-ḣ(t))xT(t-h(t))Qix(t-h(t)).
According to (2.3), we can obtain that
(3.9)LVi(t)≤2xT(t)Pi(Aix(t)+Bix(t-h(t)))+xT(t)Qix(t)+xT(t)DiTPiDix(t)-(1-hd)xT(t-h(t))Qix(t-h(t))=ξT(t)Θiξ(t),
where
(3.10)ξ(t)=[x(t)x(t-h(t))],Θi=[AiTPi+PiAi+Qi+DiTPiDiPiBi*-(1-hd)Qi].
Using Schur complement, it is not difficult to get that if inequality (3.2) is satisfied, the following inequality can be obtained:
(3.11)LVi(t)<-αV1,i(t)<0.
Combining (3.7) with (3.11) leads to
(3.12)dVi(t)=LVi(t)dt+2xT(t)PiDix(t)dw(t)<-αV1,i(t)dt+2xT(t)PiDix(t)dw(t).
Noticing (2.2) and taking the expectation to (3.12), we have
(3.13)E{dVi(t)dt}=E{LVi(t)}<-αE{V1,i(t)}<0.
According to (3.4)–(3.6), we have
(3.14)E{Vp(t)}≤μE{Vq(t)}=μE{Vq(t-)},(3.15)E{V1,p(t)}≤μE{V1,q(t)},∀p,q∈N̲.
Assume that the ith subsystem is activated during [tk,tk+1) and jth subsystem is activated during  [tk-1,tk), respectively. Using Itô formula and according to (3.13)–(3.15), we have, for any t∈[tk,tk+1),
(3.16)E{V1,i(t)}≤E{Vi(t)}=E{Vi(tk)}+E{∫tktLVi(s)ds}<μE{Vj(tk-)}-αE{∫tktV1,i(s)ds}=μE{Vj(tk-1)}+μE{∫tk-1tkLV1,j(s)ds}-αE{∫tktV1,i(s)ds}<μE{Vj(tk-1)}-αμE{∫tk-1tkV1,j(s)ds}-αE{∫tktV1,i(s)ds}≤μE{Vj(tk-1)}-αE{∫tk-1tV1,i(s)ds}≤⋯≤μNσ(t0,t)E{V(t0)}-αE{∫t0tV1,i(s)ds}.
According to Lemma 2.4 and when (3.3) holds, we have
(3.17)E{V1,i(t)}≤μNσ(t0,t)e-α(t-t0)E{V(t0)}≤κe-(α-(ln⁡⁡μ/Tα))(t-t0)E{V(t0)}.
Moreover, we can obtain
(3.18)E{‖x(t)‖2}≤κe-λ(t-t0)sup⁡-h≤θ≤0E{‖x(θ)‖2},
where κ=maxi∈N̲(λmax(Pi)+hλmax(Qi))/mini∈N̲λmin(Pi), and λ=(1/2)(α-(lnμ/Tα)) is the decay rate.
The proof is completed.Remark 3.2.
The exponential stability criterion of stochastic switched systems with time-varying delay is given in Theorem3.1. When w(t)=0, system (3.1) is degenerated to the switched system with time-varying delay, which can be described as
(3.19)ẋ(t)=Aσ(t)x(t)+Bσ(t)x(t-h(t)),x(t)=φ(t),t∈[t0-h,t0].
Using the same method, we can obtain the following exponential stability criterion of switched system (3.19).Corollary 3.3.
Considering system (3.19), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 satisfying
(3.20)[AiTPi+PiAi+αPi+QiPiBi*-(1-hd)Qi]<0,
for all i∈N̲, system (3.19) is exponentially stable under arbitrary switching signal with average dwell time satisfying (3.3).


### 3.2.H∞ Performance Analysis

In this subsection, we will investigate theH∞ performance of switched stochastic systems with time-varying delay.Consider the following switched stochastic system:(3.21)dx(t)=[Aσ(t)x(t)+Bσ(t)x(t-h(t))+Gσ(t)v(t)]dt+Dσ(t)x(t)dw(t),x(t)=φ(t),t∈[t0-h,t0],z(t)=Mσ(t)x(t).Theorem 3.4.
Considering system (3.21), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 such that
(3.22)[AiTPi+PiAi+αPi+QiPiBiPiGiDiTPiMiT*-(1-hd)Qi000**-γ2I00***-Pi0****-I]<0
hold for all i∈N̲, system (3.21) is said to have weighted H∞ performance γ under arbitrary switching signal with the average dwell time:
(3.23)Tα≥Tα*=ln⁡⁡μα,
where μ≥1 satisfies
(3.24)Pi≤μPj,Qi≤μQj,∀i,j∈N̲.Proof.
By Theorem3.1, we can readily obtain that system (3.21) is mean-square exponential stable when v(t)=0.
Assume that theith subsystem is activated during [tk,tk+1). Choose the following Lyapunov functional candidate for the ith subsystem:
(3.25)Vi(t)=V1,i(t)+V2,i(t),
where
(3.26)V1,i(t)=xT(t)Pix(t),V2,i(t)=∫t-h(t)txT(s)Qix(s)ds.
Using Itô formula, along the trajectory of system (3.21); we have
(3.27)dVi(t)=LVi(t)dt+2xT(t)PiDix(t)dw(t),
where
(3.28)LVi(t)=2xT(t)Pi(Aix(t)+Bix(t-h(t))+Giv(t))+xT(t)Qix(t)+xT(t)DiTPiDix(t)-(1-ḣ(t))xT(t-h(t))Qix(t-h(t))
Let Γ(t)=zT(t)z(t)-γ2vT(t)v(t); we have
(3.29)LVi(t)+Γ(t)=ςT(t)Φiς(t),
where ς(t)=[xT(t)xT(t-h(t))vT(t)]T, and
(3.30)Φi=[AiTPi+PiAi+Qi+MiTMi+DiTPiDiPiBiPiGi*-(1-hd)Qi0**-γ2I].
Combining (3.22) with (3.29)–(3.30), and using Schur complement, we have
(3.31)LVi(t)+Γ(t)<-αV1,i(t)<0.
Noticing (2.2) and taking the expectation to (3.27), we have
(3.32)E{dVi(t)dt}=E{LVi(t)}.
According to (3.25)–(3.27), we have
(3.33)E{Vp(t)}≤μE{Vq(t-)},∀p,q∈N̲.
Using Itô formula, we have, for any t∈[tk,tk+1),
(3.34)E{Vi(t)}=E{Vi(tk)}+∫tktE{LVi(s)+Γ(s)}ds-E{∫tktΓ(s)ds}<μE{Vi-1(tk-)}-E{∫tktΓ(s)ds}=μE{Vi-1(tk-1)}+μE{∫tk-1tk(LVi-1(s)+Γ(s))ds}-μE{∫tk-1tkΓ(s)ds}-E{∫tktΓ(s)ds}<μE{Vi-1(tk-1)}-μE{∫tk-1tkΓ(s)ds}-E{∫tktΓ(s)ds}<⋯<μNσ(t0,t)E{V(t0)}-μNσ(t0,t)E{∫t0t1Γ(s)ds}-⋯-E{∫tktΓ(s)ds}=μNσ(t0,t)E{V(t0)}-E{∫t0teNσ(s,t)ln⁡μΓ(s)ds}.
Under zero initial condition, we can obtain
(3.35)E{∫t0teNσ(s,t)ln⁡μΓ(s)ds}<0.
Moreover, we have
(3.36)E{∫t0teNσ(s,t)ln⁡μzT(s)z(s)ds}<γ2∫t0teNσ(s,t)ln⁡μvT(s)v(s)ds.
Multiplying both sides of (3.36) by e-Nσ(t0,t)lnμ leads to
(3.37)E{∫t0te-Nσ(t0,s)ln⁡μzT(s)z(s)ds}<γ2∫t0te-Nσ(t0,s)ln⁡μvT(s)v(s)ds.
Noticing Nσ(t0,s)≤((s-t0)/Tα) and Tα≥Tα*=lnμ/α, we have
(3.38)E{∫t0te-α(s-t0)zT(s)z(s)ds}<γ2∫t0tvT(s)v(s)ds.
When t→∞, it leads to
(3.39)E{∫t0∞e-α(s-t0)zT(s)z(s)ds}<γ2∫t0∞vT(s)v(s)ds.
The proof is completed.Remark 3.5.
Whendw(t)=0, system (3.21) is reduced to a switched delay system, which can be described as
(3.40)ẋ(t)=Aσ(t)x(t)+Bσ(t)x(t-h(t))+Gσ(t)v(t),x(t)=φ(t),t∈[t0-h,t0].
Using the method proposed in Theorem 3.4, we can obtain the following conclusion.Corollary 3.6.
Considering system (3.40), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 such that
(3.41)[AiTPi+PiAi+αPi+QiPiBiPiGiMiT*-(1-hd)Qi00**-γ2I0***-I]<0
hold for all i∈N̲, system (3.40) is said to have weighted H∞ performance γ under arbitrary switching signal with the average dwell time scheme (3.23).


### 3.3. Design of RobustH∞ Controller

In this subsection, the following robustH∞ controller(3.42)u(t)=Kσ(t)x(t)
will be designed for system (2.1). Then the corresponding closed-loop system can be described as(3.43)dx(t)=[(Âσ(t)+Cσ(t)Kσ(t))x(t)+B̂σ(t)x(t-h(t))+Gσ(t)v(t)]dt+D̂σ(t)x(t)dw(t),x(t)=φ(t),t∈[t0-h,t0],z(t)=Mσ(t)x(t).Theorem 3.7.
Considering system (2.1), for given scalars α, εi>0, hd<1, if there exist matrix Zi, symmetric positive definite matrices Xi, Yi> 0 such that
(3.44)[Σ11iBiXiGiXiDiTXiMiTXiE1iTXiE3iT*Σ22i000XiE2iT0**-γ2I0000***Σ44i000****-I00*****-εiI0******-εiI]<0
holds for all i∈N̲, with the average dwell time:
(3.45)Tα≥Tα*=ln⁡⁡μα,
where μ≥1 satisfies
(3.46)Xj≤μXi,Xi-1YiXi-1≤μXj-1YjXj-1,∀i,j∈N̲,
then, there exists a robust H∞ controller:
(3.47)u(t)=Kσ(t)x(t),Ki=ZiXi-1,
which can render the corresponding closed-loop system (3.43) mean-square exponentially stable with weighted H∞ performance γ, where
(3.48)∑11i=XiAiT+ZiTCiT+AiXi+CiZi+αXi+Yi+εiHiHiT,∑22i=-(1-hd)Yi,∑44i=εiHiHiT-Xi.Proof.
By Theorem3.4, system (3.43) is mean-square exponentially stable with weighted H∞ performance γ if the following inequalities are satisfied:
(3.49)[Λ11iPiB̂iPiGiD̂iTPiMiT*-(1-hd)Qi000**-γ2I00***-Pi0****-I]<0,
where Λ11i=(Âi+CiKi)TPi+Pi(Âi+CiKi)+αPi+Qi.
Then usingΛPi=diag{Pi-1,Pi-1,I,Pi-1,I} to pre- and postmultiply Λi, we have
(3.50)T̂i=ΛPiΛiΛPi<0.
Furthermore,(3.51)T̂i=[T̂11iB̂iXiGiXiD̂iTXiMiT*T̂22i000**-γ2I00***-Xi0****-I]<0,
where T̂11i=Xi(Âi+CiKi)T+(Âi+CiKi)Xi+αXi+Yi, T̂22i=-(1-hd)Yi, Xi=Pi-1, and Yi=Pi-1QiPi-1.
Combining (3.51) with (2.4), we have
(3.52)T̂i=Ti+ΔTi,
where
(3.53)T̂i=[T11iBiXiGiXiDiTXiMiT*T22i000**-γ2I00***-Xi0****-I]<0,ΔTi=[XiE1iTFiTHiT+HiFiE1iXiHiFiE2iXi0XiE3iTFiTHiT0XiE2iTFiTHiT000000000HiFiE3iXi000000000]T11i=Xi(Ai+CiKi)T+(Ai+CiKi)Xi+αXi+Yi,andT22i=-(1-hd)Yi.
Moreover,(3.54)ΔTi=[Hi000000Hi00][Fi00Fi][XiE1iTXiE3iTXiE2iT0000000]T+[XiE1iTXiE3iTXiE2iT0000000][Fi00Fi]T[Hi000000Hi00]T.
According to Lemma 2.5, we have
(3.55)ΔTi≤εi[Hi000000Hi00][Hi000000Hi00]T+εi-1[XiE1iTXiE3iTXiE2iT0000000][XiE1iTXiE3iTXiE2iT0000000]T.
Substituting (3.55) to (3.52), and using Schur complement, we can obtain that (3.52) is equivalent to (3.44). Denoting  Xi=Pi-1, and Yi=Pi-1QiPi-1, it is easy to get that (3.46) is equivalent to (3.24).
The proof is completed.Remark 3.8.
Theorem3.4 presents the sufficient conditions which could guarantee that the switched stochastic delay system is stable with H∞ performance; when the robust H∞ control problem is considered, we can solve the problem by substituting the closed-loop uncertain parameters Âi+CiKi, B̂i, D̂i to Theorem 3.4.


## 3.1. Stability Analysis

In this subsection, we will focus on the exponential stability analysis of switched stochastic systems with time-varying delay.Consider the following switched stochastic system:(3.1)dx(t)=[Aσ(t)x(t)+Bσ(t)x(t-h(t))]dt+Dσ(t)x(t)dw(t),x(t)=φ(t),t∈[t0-h,t0].Theorem 3.1.
Considering system (3.1), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 satisfying
(3.2)[AiTPi+PiAi+αPi+QiPiBiDiTPi*-(1-hd)Qi0**-Pi]<0,
for all i∈N̲, then system (3.1) is mean-square exponentially stable under arbitrary switching signal with the average dwell time:
(3.3)Tα≥Tα*=ln⁡⁡μα,
where μ≥1 satisfies
(3.4)Pi≤μPj,Qi≤μQj,∀i,j∈N̲.Proof.
Consider the following Lyapunov functional for theith subsystem:
(3.5)Vi(t,x(t))=V1,i(t,x(t))+V2,i(t,x(t)),
where
(3.6)V1,i(t,x(t))=xT(t)Pix(t),V2,i(t,x(t))=∫t-h(t)txT(s)Qix(s)ds.
For the sake of simplicity, Vi(t,x(t)) is written as Vi(t) in this paper.
According to Itô formula, along the trajectory of system (3.1), we have
(3.7)dVi(t)=LVi(t)dt+2xT(t)PiDix(t)dw(t),
where
(3.8)LVi(t)=2xT(t)Pi(Aix(t)+Bix(t-h(t)))+xT(t)Qix(t)+xT(t)DiTPiDix(t)-(1-ḣ(t))xT(t-h(t))Qix(t-h(t)).
According to (2.3), we can obtain that
(3.9)LVi(t)≤2xT(t)Pi(Aix(t)+Bix(t-h(t)))+xT(t)Qix(t)+xT(t)DiTPiDix(t)-(1-hd)xT(t-h(t))Qix(t-h(t))=ξT(t)Θiξ(t),
where
(3.10)ξ(t)=[x(t)x(t-h(t))],Θi=[AiTPi+PiAi+Qi+DiTPiDiPiBi*-(1-hd)Qi].
Using Schur complement, it is not difficult to get that if inequality (3.2) is satisfied, the following inequality can be obtained:
(3.11)LVi(t)<-αV1,i(t)<0.
Combining (3.7) with (3.11) leads to
(3.12)dVi(t)=LVi(t)dt+2xT(t)PiDix(t)dw(t)<-αV1,i(t)dt+2xT(t)PiDix(t)dw(t).
Noticing (2.2) and taking the expectation to (3.12), we have
(3.13)E{dVi(t)dt}=E{LVi(t)}<-αE{V1,i(t)}<0.
According to (3.4)–(3.6), we have
(3.14)E{Vp(t)}≤μE{Vq(t)}=μE{Vq(t-)},(3.15)E{V1,p(t)}≤μE{V1,q(t)},∀p,q∈N̲.
Assume that the ith subsystem is activated during [tk,tk+1) and jth subsystem is activated during  [tk-1,tk), respectively. Using Itô formula and according to (3.13)–(3.15), we have, for any t∈[tk,tk+1),
(3.16)E{V1,i(t)}≤E{Vi(t)}=E{Vi(tk)}+E{∫tktLVi(s)ds}<μE{Vj(tk-)}-αE{∫tktV1,i(s)ds}=μE{Vj(tk-1)}+μE{∫tk-1tkLV1,j(s)ds}-αE{∫tktV1,i(s)ds}<μE{Vj(tk-1)}-αμE{∫tk-1tkV1,j(s)ds}-αE{∫tktV1,i(s)ds}≤μE{Vj(tk-1)}-αE{∫tk-1tV1,i(s)ds}≤⋯≤μNσ(t0,t)E{V(t0)}-αE{∫t0tV1,i(s)ds}.
According to Lemma 2.4 and when (3.3) holds, we have
(3.17)E{V1,i(t)}≤μNσ(t0,t)e-α(t-t0)E{V(t0)}≤κe-(α-(ln⁡⁡μ/Tα))(t-t0)E{V(t0)}.
Moreover, we can obtain
(3.18)E{‖x(t)‖2}≤κe-λ(t-t0)sup⁡-h≤θ≤0E{‖x(θ)‖2},
where κ=maxi∈N̲(λmax(Pi)+hλmax(Qi))/mini∈N̲λmin(Pi), and λ=(1/2)(α-(lnμ/Tα)) is the decay rate.
The proof is completed.Remark 3.2.
The exponential stability criterion of stochastic switched systems with time-varying delay is given in Theorem3.1. When w(t)=0, system (3.1) is degenerated to the switched system with time-varying delay, which can be described as
(3.19)ẋ(t)=Aσ(t)x(t)+Bσ(t)x(t-h(t)),x(t)=φ(t),t∈[t0-h,t0].
Using the same method, we can obtain the following exponential stability criterion of switched system (3.19).Corollary 3.3.
Considering system (3.19), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 satisfying
(3.20)[AiTPi+PiAi+αPi+QiPiBi*-(1-hd)Qi]<0,
for all i∈N̲, system (3.19) is exponentially stable under arbitrary switching signal with average dwell time satisfying (3.3).


## 3.2.H∞ Performance Analysis

In this subsection, we will investigate theH∞ performance of switched stochastic systems with time-varying delay.Consider the following switched stochastic system:(3.21)dx(t)=[Aσ(t)x(t)+Bσ(t)x(t-h(t))+Gσ(t)v(t)]dt+Dσ(t)x(t)dw(t),x(t)=φ(t),t∈[t0-h,t0],z(t)=Mσ(t)x(t).Theorem 3.4.
Considering system (3.21), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 such that
(3.22)[AiTPi+PiAi+αPi+QiPiBiPiGiDiTPiMiT*-(1-hd)Qi000**-γ2I00***-Pi0****-I]<0
hold for all i∈N̲, system (3.21) is said to have weighted H∞ performance γ under arbitrary switching signal with the average dwell time:
(3.23)Tα≥Tα*=ln⁡⁡μα,
where μ≥1 satisfies
(3.24)Pi≤μPj,Qi≤μQj,∀i,j∈N̲.Proof.
By Theorem3.1, we can readily obtain that system (3.21) is mean-square exponential stable when v(t)=0.
Assume that theith subsystem is activated during [tk,tk+1). Choose the following Lyapunov functional candidate for the ith subsystem:
(3.25)Vi(t)=V1,i(t)+V2,i(t),
where
(3.26)V1,i(t)=xT(t)Pix(t),V2,i(t)=∫t-h(t)txT(s)Qix(s)ds.
Using Itô formula, along the trajectory of system (3.21); we have
(3.27)dVi(t)=LVi(t)dt+2xT(t)PiDix(t)dw(t),
where
(3.28)LVi(t)=2xT(t)Pi(Aix(t)+Bix(t-h(t))+Giv(t))+xT(t)Qix(t)+xT(t)DiTPiDix(t)-(1-ḣ(t))xT(t-h(t))Qix(t-h(t))
Let Γ(t)=zT(t)z(t)-γ2vT(t)v(t); we have
(3.29)LVi(t)+Γ(t)=ςT(t)Φiς(t),
where ς(t)=[xT(t)xT(t-h(t))vT(t)]T, and
(3.30)Φi=[AiTPi+PiAi+Qi+MiTMi+DiTPiDiPiBiPiGi*-(1-hd)Qi0**-γ2I].
Combining (3.22) with (3.29)–(3.30), and using Schur complement, we have
(3.31)LVi(t)+Γ(t)<-αV1,i(t)<0.
Noticing (2.2) and taking the expectation to (3.27), we have
(3.32)E{dVi(t)dt}=E{LVi(t)}.
According to (3.25)–(3.27), we have
(3.33)E{Vp(t)}≤μE{Vq(t-)},∀p,q∈N̲.
Using Itô formula, we have, for any t∈[tk,tk+1),
(3.34)E{Vi(t)}=E{Vi(tk)}+∫tktE{LVi(s)+Γ(s)}ds-E{∫tktΓ(s)ds}<μE{Vi-1(tk-)}-E{∫tktΓ(s)ds}=μE{Vi-1(tk-1)}+μE{∫tk-1tk(LVi-1(s)+Γ(s))ds}-μE{∫tk-1tkΓ(s)ds}-E{∫tktΓ(s)ds}<μE{Vi-1(tk-1)}-μE{∫tk-1tkΓ(s)ds}-E{∫tktΓ(s)ds}<⋯<μNσ(t0,t)E{V(t0)}-μNσ(t0,t)E{∫t0t1Γ(s)ds}-⋯-E{∫tktΓ(s)ds}=μNσ(t0,t)E{V(t0)}-E{∫t0teNσ(s,t)ln⁡μΓ(s)ds}.
Under zero initial condition, we can obtain
(3.35)E{∫t0teNσ(s,t)ln⁡μΓ(s)ds}<0.
Moreover, we have
(3.36)E{∫t0teNσ(s,t)ln⁡μzT(s)z(s)ds}<γ2∫t0teNσ(s,t)ln⁡μvT(s)v(s)ds.
Multiplying both sides of (3.36) by e-Nσ(t0,t)lnμ leads to
(3.37)E{∫t0te-Nσ(t0,s)ln⁡μzT(s)z(s)ds}<γ2∫t0te-Nσ(t0,s)ln⁡μvT(s)v(s)ds.
Noticing Nσ(t0,s)≤((s-t0)/Tα) and Tα≥Tα*=lnμ/α, we have
(3.38)E{∫t0te-α(s-t0)zT(s)z(s)ds}<γ2∫t0tvT(s)v(s)ds.
When t→∞, it leads to
(3.39)E{∫t0∞e-α(s-t0)zT(s)z(s)ds}<γ2∫t0∞vT(s)v(s)ds.
The proof is completed.Remark 3.5.
Whendw(t)=0, system (3.21) is reduced to a switched delay system, which can be described as
(3.40)ẋ(t)=Aσ(t)x(t)+Bσ(t)x(t-h(t))+Gσ(t)v(t),x(t)=φ(t),t∈[t0-h,t0].
Using the method proposed in Theorem 3.4, we can obtain the following conclusion.Corollary 3.6.
Considering system (3.40), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,Qi>0 such that
(3.41)[AiTPi+PiAi+αPi+QiPiBiPiGiMiT*-(1-hd)Qi00**-γ2I0***-I]<0
hold for all i∈N̲, system (3.40) is said to have weighted H∞ performance γ under arbitrary switching signal with the average dwell time scheme (3.23).


## 3.3. Design of RobustH∞ Controller

In this subsection, the following robustH∞ controller(3.42)u(t)=Kσ(t)x(t)
will be designed for system (2.1). Then the corresponding closed-loop system can be described as(3.43)dx(t)=[(Âσ(t)+Cσ(t)Kσ(t))x(t)+B̂σ(t)x(t-h(t))+Gσ(t)v(t)]dt+D̂σ(t)x(t)dw(t),x(t)=φ(t),t∈[t0-h,t0],z(t)=Mσ(t)x(t).Theorem 3.7.
Considering system (2.1), for given scalars α, εi>0, hd<1, if there exist matrix Zi, symmetric positive definite matrices Xi, Yi> 0 such that
(3.44)[Σ11iBiXiGiXiDiTXiMiTXiE1iTXiE3iT*Σ22i000XiE2iT0**-γ2I0000***Σ44i000****-I00*****-εiI0******-εiI]<0
holds for all i∈N̲, with the average dwell time:
(3.45)Tα≥Tα*=ln⁡⁡μα,
where μ≥1 satisfies
(3.46)Xj≤μXi,Xi-1YiXi-1≤μXj-1YjXj-1,∀i,j∈N̲,
then, there exists a robust H∞ controller:
(3.47)u(t)=Kσ(t)x(t),Ki=ZiXi-1,
which can render the corresponding closed-loop system (3.43) mean-square exponentially stable with weighted H∞ performance γ, where
(3.48)∑11i=XiAiT+ZiTCiT+AiXi+CiZi+αXi+Yi+εiHiHiT,∑22i=-(1-hd)Yi,∑44i=εiHiHiT-Xi.Proof.
By Theorem3.4, system (3.43) is mean-square exponentially stable with weighted H∞ performance γ if the following inequalities are satisfied:
(3.49)[Λ11iPiB̂iPiGiD̂iTPiMiT*-(1-hd)Qi000**-γ2I00***-Pi0****-I]<0,
where Λ11i=(Âi+CiKi)TPi+Pi(Âi+CiKi)+αPi+Qi.
Then usingΛPi=diag{Pi-1,Pi-1,I,Pi-1,I} to pre- and postmultiply Λi, we have
(3.50)T̂i=ΛPiΛiΛPi<0.
Furthermore,(3.51)T̂i=[T̂11iB̂iXiGiXiD̂iTXiMiT*T̂22i000**-γ2I00***-Xi0****-I]<0,
where T̂11i=Xi(Âi+CiKi)T+(Âi+CiKi)Xi+αXi+Yi, T̂22i=-(1-hd)Yi, Xi=Pi-1, and Yi=Pi-1QiPi-1.
Combining (3.51) with (2.4), we have
(3.52)T̂i=Ti+ΔTi,
where
(3.53)T̂i=[T11iBiXiGiXiDiTXiMiT*T22i000**-γ2I00***-Xi0****-I]<0,ΔTi=[XiE1iTFiTHiT+HiFiE1iXiHiFiE2iXi0XiE3iTFiTHiT0XiE2iTFiTHiT000000000HiFiE3iXi000000000]T11i=Xi(Ai+CiKi)T+(Ai+CiKi)Xi+αXi+Yi,andT22i=-(1-hd)Yi.
Moreover,(3.54)ΔTi=[Hi000000Hi00][Fi00Fi][XiE1iTXiE3iTXiE2iT0000000]T+[XiE1iTXiE3iTXiE2iT0000000][Fi00Fi]T[Hi000000Hi00]T.
According to Lemma 2.5, we have
(3.55)ΔTi≤εi[Hi000000Hi00][Hi000000Hi00]T+εi-1[XiE1iTXiE3iTXiE2iT0000000][XiE1iTXiE3iTXiE2iT0000000]T.
Substituting (3.55) to (3.52), and using Schur complement, we can obtain that (3.52) is equivalent to (3.44). Denoting  Xi=Pi-1, and Yi=Pi-1QiPi-1, it is easy to get that (3.46) is equivalent to (3.24).
The proof is completed.Remark 3.8.
Theorem3.4 presents the sufficient conditions which could guarantee that the switched stochastic delay system is stable with H∞ performance; when the robust H∞ control problem is considered, we can solve the problem by substituting the closed-loop uncertain parameters Âi+CiKi, B̂i, D̂i to Theorem 3.4.


## 4. Numerical Example

In this section, a numerical example is given to illustrate the effectiveness of the proposed approach. Consider system (2.1) with the following parameters:(4.1)A1=[20.91.4-2],B1=[-0.300.1-0.5],C1=[2132],D1=[-0.300.10.2],G1=[-0.800-0.1],M1=[21-13],E11=[0.700.90],E21=[0.2000.8],E31=[0.6000],H1=[0.4000.8],A2=[-210-3],B2=[-0.40.10-0.8],C2=[1213],D2=[-0.20.050-0.4],G2=[-0.100-0.9],M2=[1-231],E12=[0.2000],E22=[0.300.10],E32=[00.400.5],H2=[0.5000.3],F1=[sin⁡t00sin⁡t],F2=[cos⁡t00cos⁡t],
the disturbance input v(t)=[50e-0.5t10e-0.5t]T.Letα=0.6, h(t)=1+0.5sint, ε1=ε2=1, γ=1; then solving the LMIs in Theorem 3.7, we have(4.2)X1=[1.05270.03920.03921.0410],Y1=[6.01160.06590.06596.5879],Z1=[-17.083730.073113.0224-37.1398],K1=[-17.329529.540113.7192-36.1921],X2=[0.9153-0.0358-0.03581.0266],Y2=[6.06510.06400.06405.4975],Z2=[-26.15264.006413.4430-5.1025],K2=[-28.45822.909914.5120-4.4641],μ=8.3261.
Then we obtain that Tα*=lnμ/α=3.5323. Thus, under the average dwell time Tα>Tα*, the designed controller can guarantee that the corresponding closed-loop system is mean-square exponentially stable with weighted H∞ performance.Simulation results are shown in Figures1–3, where the initial state x(t)=[0,0]T, t∈[-h,0), and x(0)=[2,-2]T. The switching signal with the average dwell time Tα=4 is shown in Figure 1, and Figures 2 and 3 show state x1 and x2 of the closed-loop system, respectively.Figure 1
Switching signal.Figure 2
Statex1 of the closed-loop system.Figure 3
Statex2 of the closed-loop system.


## 5. Conclusions

In this paper, the exponential stability analysis and robustH∞ control for switched stochastic time delay systems have been investigated. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criteria and H∞ performance analysis are presented. Furthermore, robust H∞ controller is designed to guarantee that the corresponding closed-loop system is mean-square exponentially stable. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. The proposed method provides a powerful tool to solve many other problems such as controller design under asynchronous switching and actuator failures. These problems are the topics of the future research.



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*Source: 202916-2012-05-13.xml*