# Observer-Based RobustH∞ Control for Switched Stochastic Systems with Time-Varying Delay

Authors: Guoxin Chen; Zhengrong Xiang; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper investigates the problem of observer-based robustH∞ control for a class of switched stochastic systems with time-varying delay. Based on the average dwell time method, an exponential stability criterion for switched stochastic delay systems is proposed. Then,  H∞ performance analysis and observer-based robust H∞  controller design for the underlying systems are developed. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Switched systems are a kind of hybrid dynamical systems composed of a set of continuous-time subsystems or discrete-time subsystems and a switching law that orchestrates the switching between them. Switched systems have attracted increasing attention during the past decades because of their wide applications in real-world systems, such as robot control systems [1], networked control systems [2, 3]. Many useful results on stability analysis and control synthesis for such systems have been reported in [4–8]. For example, H∞ control of switched linear discrete-time systems with polytopic uncertainties was investigated in [8].It is well known that the time delay phenomenon is frequently encountered in engineering and social systems, and the existence of which may cause instability or undesirable system performance. Therefore, many research efforts have been devoted to the study of switched time delay systems [9–15]. On the other hand, stochastic systems have attracted considerable attention during the past several decades. Early results can be found in [16], and the H∞ control problem of stochastic systems with time delay was investigated in [17, 18]. The study on H2/H∞ control of stochastic system was developed in [19]. Stability analysis on stochastic system with multiple delays was proposed in [20]. Moreover, some results on switched stochastic systems with and without time delay have been obtained (see [21–25] and the references cited therein).In many real-world systems, state feedback control will fail to guarantee the stabilization because the states of the systems are not all measurable [26]. One of the key approaches to solve the problem is to reconstruct the states of the systems and realize the required feedback control. Hence, the observer-based control has been an interesting topic in control theory. Some results on observer-based control for stochastic delay systems or Markovian jump systems have been presented in [27–29]. However, to the best of our knowledge, the problem of observer-based robust H∞ control for switched stochastic systems with time delay has not been fully studied, which motivates the present study.In this paper, we aim to design an observer-based robustH∞ controller for switched stochastic systems with time delay such that the closed-loop system is mean-square exponentially stable with H∞ performance. The major contributions of the work can be summarized as follows: (1) a new Lyapunov-Krasovskii functional candidate is introduced to derive the exponential stability of switched stochastic systems with time delay, and the free-weighting matrix method is employed to reduce the conservatism; (2) an observer-based robust H∞ controller design scheme for the underlying systems is proposed.The remainder of the paper is organized as follows. In Section2, problem statement and some useful lemmas are given. In Section 3, the main results are presented. 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. In this paper, the superscript “T” denotes the transpose, and the symmetric term in a matrix is denoted by ∗. The notation X>Y   (X≥Y) means that X-Y is positive definite (positive semidefinite, respectively). Rn denotes the n-dimensional Euclidean space. ∥x(t)∥ denotes the Euclidean norm. |a| denotes the absolute value of a. L2[t0,∞) is the space of square integrable functions on [t0,∞), and t0 is the initial time. λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of P, respectively. A+ denotes the Moore-Penrose pseudoinverse of A. I is the identity matrix. diag{ai} denotes a diagonal matrix with the diagonal elements ai,i=1,2,…,n.


## 2. Problem Formulation and Preliminaries

Consider the following switched stochastic system with time delay:(1a)dx(t)=[A^σ(t)x(t)+A^τσ(t)x(t-τ(t))+Bσ(t)u(t)+Gσ(t)v(t)]dt+D^σ(t)x(t)dw(t),(1b)y(t)=Cσ(t)x(t),(1c)z(t)=Jσ(t)x(t),(1d)x(t)=φ(t),t∈[t0-τ,t0],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 to belong to L2[t0,∞], y(t)∈Rr is the measurable output, z(t)∈Rq is the controlled output, and w(t) is a one-dimensional zero-mean Wiener process on a probability space (ΩFP) and satisfies
(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. τ(t) is the time delay satisfying
(3)0≤τ(t)≤τ,τ˙(t)≤τd<1,
where τ and τd are known constants.The functionσ(t):[t0,∞]→N_={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)∈N_, where t0 is the initial instant and tk denotes the kth switching instant. Moreover,  σ(t)=i means that the ith subsystem is activated. For all i∈N_, Bi, Ci, Ji, and Gi are known real-value matrices with appropriate dimensions, A^i, A^τi, and D^i are uncertain real matrices with appropriate dimensions and can be written as
(4)[A^iA^τiD^i]=[Ai+ΔAiAτi+ΔAτiDi+ΔDi],
where [ΔAiΔAτiΔDi]=HiFi(t)[E1iE2iE3i], Ai, Aτi, Di, Hi, E1i, E2i, and E3i are known real-value matrices with appropriate dimensions, and Fi(t) is an unknown time-varying matrix that satisfies
(5)FiT(t)Fi(t)≤I.The state feedback controller is designed asu(t)=Kσ(t)x(t). In actual operation, however, the states of the systems are not all measurable. The following switched system is constructed to estimate the state of system (1a), (1b), (1c), and (1d):(6a)dx^(t)=[Aσ(t)x^(t)+Aτσ(t)x^(t-τ(t))+Bσ(t)u(t)+Lσ(t)(y(t)-y^(t))]dt,(6b)y^(t)=Cσ(t)x^(t),(6c)x^(t)=ϕ(t),t∈[t0-τ,t0],where x^(t)∈Rn is the estimation of x(t), y^(t)∈Rr is the observer output, and ϕ(t)∈Rn is the initial observer state function. The real state feedback controller becomes u(t)=Kσ(t)x^(t). Li and Ki are the observer gains and controller gains to be determined, respectively.Remark 1.
It is noted that the observer-basedH∞ control for stochastic systems or Markovian jump systems was considered in [27–29]. However, the results in the aforementioned papers cannot be directly applied to the switched stochastic system considered in the paper. This motivates our study. Also, the proposed observer in (6a), (6b), and (6c) is a switching observer, which is different from the existing ones given in [27–29].From systems (1a), (1b), (1c), and (1d) and (6a), (6b), and (6c), we can obtain the following augmented closed-loop system:(7a)dξ(t)=[A-σ(t)ξ(t)+A-τσ(t)ξ(t-τ(t))+G-σ(t)v(t)]dt+D-σ(t)ξ(t)dw(t),(7b)z(t)=J-σ(t)ξ(t),(7c)ξ(t)=[φT(t)φT(t)-ϕT(t)]T,t∈[t0-τ,t0],where ξ(t)=[xT(t)eT(t)]T,  e(t)=x(t)-x^(t) denotes the state estimated error.Forσ(t)=i, the parameters of system (7a), (7b), and (7c) are given as follows:
(8)A-i=A~i+ΔA~i,A~i=[Ai+BiKi-BiKi0Ai-LiCi],ΔA~i=H~iF~iE~1i,A-τi=A~τi+ΔA~τi,A~τi=[Aτi00Aτi],ΔA~τi=H~iF~iE~2i,D-i=D~i+ΔD~i,D~i=[Di0Di0],ΔD~i=H~iF~iE~3i,G-i=G~i=[GiTGiT]T,J-i=J~i=[Ji0],H~i=[Hi00Hi],F~i=[Fi00Fi],E~gi=[Egi0Egi0],g=1,2,3.Assumption 2.
B
i is full row rank, for all i∈N_.Definition 3.
System (1a), (1b), (1c), and (1d) with v(t)=0 is said to be mean-square exponentially stable under the switching signal σ(t) if there exist scalars κ>0 and α>0, such that the solution x(t) of the system satisfies
(9)E{∥x(t)∥2}≤κe-α(t-t0)sup-τ≤θ≤0E{∥x(t0+θ)∥2},∀t≥t0.Definition 4 (see [24]).
For anyT2>T1≥t0, let Nσ(T1,T2) denote the switching number of σ(t) on an interval [T1,T2). If
(10)Nσ(T1,T2)≤N0+T2-T1Tα
holds for given constants N0≥0 and Tα>0, then the constant Tα is called the average dwell time. As commonly used in the literature, some chooses N0=0.Definition 5 (see [30]).
For anyλ>0 and γ>0, system (7a), (7b), and (7c) is said to be mean-square exponentially stable with a prescribed weighted H∞ performance level γ if the following conditions are satisfied:(1)
whenv(t)=0, system (7a), (7b), and (7c) is mean-square exponentially stable;(2)
under the zero initial condition, the outputz(t) satisfies
(11)E{∫t0∞e-λ(s-t0)zT(s)z(s)ds}≤γ2∫t0∞vT(s)v(s)ds,∀v(t)∈L2[t0,∞).Lemma 6 (see [31]).
For any positive symmetric constant matrixM∈Rn×n and a scalar r>0, if there exists a vector functiong:[0,r]→Rn  such that integrations in the following are well defined, then the following inequality holds
(12)r∫0rgT(s)Mg(s)ds≥[∫0rg(s)ds]TM[∫0rg(s)ds].Lemma 7 (see [32]).
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.The objective of this paper is to design an observer-based robustH∞ controller for switched stochastic delay system (1a), (1b), (1c), and (1d) such that the augmented closed-loop system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted H∞ performance level γ.


## 3. Main Results

### 3.1. Stability Analysis

In this subsection, in order to obtain the main results, we first focus on the problem of stability analysis for the following switched stochastic systems with time delay(13a)dx(t)=[Aσ(t)x(t)+Aτσ(t)x(t-τ(t))]dt+Dσ(t)x(t)dw(t),(13b)x(t)=φ(t),t∈[t0-τ,t0].Theorem 8.
For a given scalarα>0, if there exist symmetric positive definite matrices Pi,  Qi, and Ri and any matrices Si such that
(14)[∑11iPiAτiSiTSiTDiTPi*-(1-τd)Qi-SiT-SiT0**-Si-SiT-SiT0***-τ-1Ri0****-Pi]<0,ddddddddddddddddddddddddddddddddd∀i∈N_
then system (13a) and (13b) is mean-square exponentially stable under arbitrary switching signal with the average dwell time
(15)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(16)Pi≤μPj,Qi≤μQj,Ri≤μRj,Qi≤βiPi,Ri≤βiPi,∀i,j∈N_,(17)λ+βi(1+τ)(eλτ-1)≤α,χ=maxi∈N_χi,χi=1+τβi(1+τ)(eλτ-1),∑11i=AiTPi+PiAi+Qi+τRi+αPi.Proof.
LetY(t)=Aσ(t)x(t)+Aτσ(t)x(t-τ(t)), then (13a) can be described as
(18)dx(t)=Y(t)dt+Dσ(t)x(t)dw(t).
Choose the following Lyapunov functional candidate for theith subsystem
(19)Vi(t,x(t))=V1,i(t,x(t))+V2,i(t,x(t))+V3,i(t,x(t)),
where
(20)V1,i(t,x(t))=xT(t)Pix(t),V2,i(t,x(t))=∫t-τ(t)txT(s)Qix(s)ds,V3,i(t,x(t))=∫0τ∫t-θtxT(s)Rix(s)dsdθ.
For the sake of simplicity,Vi(t,x(t)) is written as Vi(t) in this paper. According to Itô’s formula, along the trajectory of the ith subsystem, we have
(21)dVi(t)=∑g=13dVg,i,
where
(22)dV1,i(t)=ℒV1,i(t)dt+2xT(t)PiDix(t)dw(t),ℒV1,i(t)=2xT(t)Pi[Aix(t)+Aτix(t-τ(t))]+xT(t)DiTPiDix(t),dV2,i(t)=[xT(t)Qix(t)-(1-τ˙(t))xT(t-τ(t))ddd×Qix(t-τ(t))xT]dt≤[xT(t)Qix(t)-(1-τd)xT(t-τ(t))×Qix(t-τ(t))xT]dt,dV3,i(t)=[τxT(t)Rix(t)-∫t-τtxT(s)Rix(s)ds]dt.
According to Lemma6, we have(23)dV3,i(t)≤{τxT(t)Rix(t)-τ-1[∫t-τ(t)tx(s)ds]Tdfddd×Ri[∫t-τ(t)tx(s)ds]}dt.
Integrating both sides of (18) from t-τ(t) to t, we have
(24)x(t)-x(t-τ(t))=∫t-τ(t)tY(s)ds+∫t-τ(t)tDσ(s)x(s)dw(s).
Thus,(25)2[∫t-τ(t)tx(s)ds+∫t-τ(t)tY(s)ds]TSiη(t)dt=0,
where η(t)=x(t)-x(t-τ(t))-∫t-τ(t)tY(s)ds-∫t-τ(t)tDσ(s)x(s)dw(s).
Combining (21)–(25) leads to
(26)dVi(t)≤ℒVi(t)dt+Wi(t),
where
(27)ℒVi(t)=ςT(t)Θiς(t),Wi(t)=2xT(t)PiDix(t)dw(t)-2[∫t-τ(t)tx(s)ds+∫t-τ(t)tY(s)ds]T×Si[∫t-τ(t)tDσ(s)x(s)dw(s)]dt,ς(t)=[xT(t)xT(t-τ(t))∫t-τ(t)tYT(s)ds∫t-τ(t)txT(s)ds]T,Θi=[Θ11iPiAτiSiTSiT*-(1-τd)Qi-SiT-SiT**-Si-SiT-SiT***-τ-1Ri],Θ11i=AiTPi+PiAi+Qi+DiTPiDi+τRi.
By using the Schur complement, we obtain from (14) that
(28)ℒVi(t)<-αV1,i(t)<0.
According to (26), one obtains that
(29)dVi(t)≤ℒVi(t)dt+Wi(t)<-αV1,i(t)dt+Wi(t).
Then, taking mathematical expectation, we have(30)E{dVi(t)dt}≤E{ℒVi(t)}<-αE{V1,i(t)}<0.
From (16) and (19), we obtain that
(31)Vi(t)≤V1,i(t)+∫t-τtxT(s)Qix(s)ds+τ∫t-τtxT(s)Rix(s)ds≤V1,i(t)+βi(1+τ)∫t-τtV1,i(s)ds.
Letσ(tk)=i. Then, using Ito^′s formula, we can obtain that for t∈[tk,tk+1),
(32)E{eλtVi(t)}-E{eλtkVi(tk)}=E{∫tktℒ(eλsVi(s))ds}≤E{∫tkteλs[∫s-τsV1,i(ϑ)λV1,i(s)+λβi(1+τ)fffffffffffFFFFFff×∫s-τsV1,i(ϑ)dϑ-αV1,i(s)]ds∫tkteλs}.
Notice that(33)∫tkteλsds∫s-τsV1,i(ϑ)dϑ=∫tk-τtdϑ∫ϑϑ+τeλsV1,i(ϑ)ds=1λ(eλτ-1)∫tk-τteλsV1,i(s)ds.
Thus, it can be obtained that(34)E{eλtVi(t)}-E{eλtkVi(tk)}≤E{∫tkteλs[λ+βi(1+τ)(eλτ-1)-α]V1,i(s)ds}+E{Ωi(tk)},
where
(35)Ωi(tk)=βi(1+τ)(eλτ-1)∫tk-τtkeλsV1,i(s)ds≤βi(1+τ)(eλτ-1)eλtk∫tk-τtkV1,i(s)ds≤τβi(1+τ)(eλτ-1)eλtksup-τ≤θ≤0V1,i(tk+θ)≤τβi(1+τ)(eλτ-1)eλtksup-τ≤θ≤0Vi(tk+θ).
Noticing thatE{V(t)}≤sup-τ≤θ≤0E{V(t+θ)}, one gets
(36)E{Vi(t)}≤χie-λ(t-tk)sup-τ≤θ≤0E{Vi(tk+θ)}≤χe-λ(t-tk)sup-τ≤θ≤0E{Vi(tk+θ)}.
For anyt∈[tk,tk+1), from (16) and (36), it follows that
(37)E{V(t)}=E{Vi(t)}≤χe-λ(t-tk)sup-τ≤θ≤0E{Vσ(tk)(tk+θ)}≤χμe-λ(t-tk)sup-τ≤θ≤0E{Vσ(tk-)(tk+θ)}≤χ2μe-λ(t-tk)e-λ(tk+θ-tk-1)×sup-τ≤θ≤0E{Vσ(tk-1)(tk-1+θ)}≤χ2μeλτe-λ(t-tk-1)sup-τ≤θ≤0E{Vσ(tk-1)(tk-1+θ)}≤χ3(μeλτ)2e-λ(t-tk-2)sup-τ≤θ≤0E{Vσ(tk-2)(tk-2+θ)}≤⋯≤χ(χμeλτ)Nσ(t0,t)e-λ(t-t0)sup-τ≤θ≤0E{Vσ(t0)(t0+θ)}.
When (15) holds, noticing that  Nσ(t0,t)≤(t-t0)/Tα, one has
(38)E{∥x(t)∥2}≤baχe-(λ-(λτ+ln(χμ))/Tα)(t-t0)×sup-τ≤θ≤0E{∥x(t0+θ)∥2},
where a=min∀i∈N_λmin(Pi), b=max∀i∈N_λmax(Pi)+τmax∀i∈N_λmax(Qi)+(τ2/2)max∀i∈N_λmax(Ri).
The proof is completed.Remark 9.
In the derivation of Theorem8, a new Lyapunov-Krasovskii functional candidate is constructed for the stability analysis of switched stochastic systems with time delay, and it is different from the ones given in [9–15]. Also, the free-weighting matrix method is utilized to reduce the conservatism.Remark 10.
Ifμ=1 in (15), which leads to Pi=Pj, Qi=Qj, and Ri=Rj, for all i,j∈N_, then system (13a) and (13b) possesses a common Lyapunov function, and the switching signals can be arbitrary.Whenw(t)=0, system (13a) and (13b) becomes the following switched system with time delay:(39a)x˙(t)=Aσ(t)x(t)+Aτσ(t)x(t-τ(t)),(39b)x(t)=φ(t),t∈[t0-τ,t0].From Theorem8, we can readily get the exponential stability criterion for switched system (39a) and (39b).Corollary 11.
Consider system (39a) and (39b), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,  Qi, and Ri and any matrices Si such that
(40)[∑11iPiAτiSiTSiT*-(1-τd)Qi-SiT-SiT**-Si-SiT-SiT***-τ-1Ri]<0,dddddddddddddddddddddddddddd∀i∈N_
then system (39a) and (39b) is exponentially stable under arbitrary switching signal with the average dwell time scheme (15).


### 3.2.H∞ Performance Analysis

In the sequel, we will investigate the problem ofH∞ performance analysis for switched stochastic systems with time delay. Consider the following system:(41a)dx(t)=[Aσ(t)x(t)+Aτσ(t)x(t-τ(t))+Gσ(t)v(t)]dt+Dσ(t)x(t)dw(t),(41b)z(t)=Jσ(t)x(t),(41c)x(t)=φ(t),t∈[t0-τ,t0].Theorem 12.
For a given scalarα>0, if there exist symmetric positive definite matrices Pi,  Qi, and Ri and any matrices Si such that
(42)[∏11iPiAτiSiTSiTPiGiDiTPiJiT*-(1-τd)Qi-SiT-SiT000**-Si-SiT-SiT000***-τ-1Ri000****-γ2I00*****-Pi0******-I]gggggggggggggggggggggggggggggggggggggg<0,∀i∈N_
then system (41a), (41b), and (41c) is mean-square exponentially stable with a weighted prescribed H∞ performance level γ under arbitrary switching signal with the average dwell time
(43)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(44)Pi≤μPj,Qi≤μQj,Ri≤μRj,Qi≤βiPi,Ri≤βiPi,∀i,j∈N_,(45)λ+βi(1+τ)(eλτ-1)≤α,χ=maxi∈N_χi,χi=1+τβi(1+τ)(eλτ-1),∏11i=AiTPi+PiAi+Qi+τRi+αPi.Proof.
We can easily obtain that (14) is satisfied if (42) holds. Thus, system (41a), (41b), and (41c) with v(t)=0 is mean-square exponentially stable.
Whenv(t)≠0, let
(46)Γ(t)=zT(t)z(t)-γ2vT(t)v(t).
Choosing the same Lyapunov functional candidate as (19) and following the proof line of Theorem 8, we have
(47)dVi(t)≤ℒVi(t)dt+Wi(t),
where
(48)Wi(t)=2xT(t)PiDix(t)dw(t)-2[∫t-τ(t)tx(s)ds+∫t-τ(t)tY(s)ds]T×Si[∫t-τ(t)tDσ(s)x(s)dw(s)]dt,
and ℒVi(t) satisfies
(49)ℒVi(t)+Γ(t)=ς-T(t)Θ-iς-(t),ς-(t)=[xT(t)xT(t-τ(t))∫t-τ(t)tYT(s)ds∫t-τ(t)txT(s)dsvT(t)]T,Θ-i=[Θ-11iPiAτiSiTSiTPiGi*-(1-τd)Qi-SiT-SiT0**-Si-SiT-SiT0***-τ-1Ri0****-γ2I],Θ-11i=AiTPi+PiAi+Qi+DiTPiDi+τRi+JiTJi.
Using the Schur complement, from (42), we get
(50)ℒVi(t)+Γ(t)<-αV1,i(t)<0.
It follows that(51)E{dVi(t)}≤E{ℒVi(t)dt}.
From (44), we obtain that
(52)E{Vσ(tk)(tk)}≤μE{Vσ(tk-1)(tk)}.
For anyt∈[tk,tk+1), using Ito^′s formula and taking the mathematical expectation, one has
(53)E{Vσ(t)(t)}=E{Vσ(tk)(tk)}+∫tktE{dVσ(tk)(s)}≤E{Vσ(tk)(tk)}+∫tktE{ℒVσ(tk)(s)+Γ(s)}ds-∫tktE{Γ(s)ds}<μE{Vσ(tk-1)(tk)}-∫tktE{Γ(s)ds}<μ2E{Vσ(tk-2)(tk-1)}-E{μ∫tk-1tkΓ(s)ds+∫tktΓ(s)ds}<⋯<μNσ(t0,t)E{V(t0)}-E{μNσ(t0,t)∫t0t1Γ(s)dsdddddd+μNσ(t1,t)∫t1t2Γ(s)ds+⋯+∫tktΓ(s)ds}=μNσ(t0,t)E{V(t0)}-E{∫t0teNσ(s,t)lnμΓ(s)ds}.
Under the zero initial condition, we obtain that(54)E{∫t0teNσ(s,t)lnμΓ(s)ds}<0.
According to (46), one has
(55)E{∫t0teNσ(s,t)lnμzT(s)z(s)}ds<γ2∫t0teNσ(s,t)lnμvT(s)v(s)ds.
Multiplying both sides of (55) by e-Nσ(t0,t)lnμ leads to
(56)E{∫t0te-Nσ(t0,s)lnμzT(s)z(s)ds}<γ2∫t0te-Nσ(t0,s)lnμvT(s)v(s)ds.
Noticing thatNσ(t0,s)≤(s-t0)/Tα and Tα≥τ+ln(χμ)/λ≥lnμ/λ, one obtains that
(57)E{∫t0te-λ(s-t0)zT(s)z(s)ds}<γ2∫t0tvT(s)v(s)ds.
Whent→∞, the following inequality is derived:
(58)E{∫t0∞e-λ(s-t0)zT(s)z(s)ds}<γ2∫t0∞vT(s)v(s)ds.
The proof is completed.


### 3.3. Observer-Based RobustH∞  Stabilization

Now, we are in a position to design an observer-based robustH∞ controller for system (1a), (1b), (1c), and (1d) such that the augmented closed-loop system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted H∞ performance level γ. Based on Theorem 12, a sufficient condition for the existence of such a controller is presented in the following theorem.Theorem 13.
Consider system (1a), (1b), (1c), and (1d), for a given scalar α>0, if there exist scalars εi>0, symmetric positive definite matrices Pi, Qi, and Ri, and any matrices S~i, Yi, and Zi such that, for all i∈N_,(59)[Ω11iP-iA~τiS~iTS~iTP-iG~iD~iTP-iJ~iTP-iH~i0εiE~1iTεiE~3iT*-(1-τd)Q-i-S~iT-S~iT00000εiE~2iT0**-S~i-S~iT-S~iT0000000***-τ-1R-i0000000****-γ2I000000*****-P-i00P-iH~i00******-I0000*******-εiI000********-εiI00*********-εiI0**********-εiI]<0,then there exists an observer-based controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted H∞ performance level γ under arbitrary switching signal with the average dwell time(60)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(61)Pi≤μPj,Qi≤μQj,Ri≤μRj,Qi≤βiPi,Ri≤βiPi,∀i,j∈N_,(62)λ+βi(1+τ)(eλτ-1)≤α,(63)χ=maxi∈N_χi,χi=1+τβi(1+τ)(eλτ-1),Ω11i=ΥiT+Υi+Q-i+τR-i+αP-i,Υi=[PiAi+Zi-Zi0PiAi-YiCi],P-i=diag{Pi,Pi},Q-i=diag{Qi,Qi},R-i=diag{Ri,Ri}.
Moreover, if the above conditions have a feasible solution, the controller gain matrices and the observer gain matrices can be obtained byKi=Bi+Pi-1Zi and  Li=Pi-1Yi.Proof.
According to Theorem12, we get that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted H∞ performance level γ if the following inequality is satisfied:
(64)Ψ-i=[Ψ-11iP~iA-τiS~iTS~iTP~iG-iD-iTP~iJ-iT*-(1-τd)Q~i-S~iT-S~iT000**-S~i-S~iT-S~iT000***-τ-1Ri000****-γ2I00*****-P~i0******-I]<0,
where Ψ-11i=A~iTP~i+P~iA~i+Q~i+τR~i+αP~i, and P~i, Q~i, and R~i are symmetric positive definite matrices with appropriate dimensions.
Then, we have(65)Ψ-i=Ψi+ΔΨi<0,
where
(66)Ψi=[Ψ11iP~iA~τiS~iTS~iTP~iG~iD~iTP~iJ~iT*-(1-τd)Q~i-S~iT-S~iT000**-S~i-S~iT-S~iT000***-τ-1Ri000****-γ2I00*****-P~i0******-I],ΔΨi=[E~1iTF~iTH~iTP~i+P~iH~iF~iE~1iP~iH~iF~iE~2i000E~3iTF~iTH~iTP~i0E~2iTF~iTH~iTP~i000000000000000000000000000P~iH~iF~iE~3i0000000000000]=[P~iH~i0000000000P~iH~i00][F~i00F~i][E~1iTE~3iTE~2iT00000000000]T+[E~1iTE~3iTE~2iT00000000000][F~i00F~i]T[P~iH~i0000000000P~iH~i00]T,Ψ11i=AiTP~i+P~iAi+Q~i+τR~i+αP~i.
From Lemma7, we have
(67)Ψ-i≤Ψi+εi-1[P~iH~i0000000000P~iH~i00][P~iH~i0000000000P~iH~i00]T+εi[E~1iTE~3iTE~2iT00000000000][E~1iTE~3iTE~2iT00000000000]T<0.Choose P~i=P-i=diag{Pi,Pi}, Q~i=Q-i=diag{Qi,Qi}, and R~i=R-i=diag{Ri,Ri}, and let Zi=PiBiKi and Yi=PiLi. By using the Schur complement, we can obtain that (67) is equivalent to (59).
Thus, according to Theorem12, we obtain from (59)–(62) that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted H∞ performance level γ. Moreover, we can obtain the controller gain matrices Ki=Bi+Pi-1Zi and the observer gain matrices Li=Pi-1Yi.
The proof is completed.Remark 14.
An observer-basedH∞ controller design scheme is proposed in the paper. Compared with the existing results presented in [27–29], a remark advantage of the work is that the proposed observer is mode-dependent, which means that each subsystem has its individual observer. Moreover, the proposed observer not only ensures the convergence of the estimated error of each subsystem, but also guarantees that the estimated error of the whole system converges to zero exponentially.In Theorem13, when we choose that Qi=Ri=Pi, it is not difficult to get that βi=1 and χ=χi=1+τ(1+τ)(eλτ-1),  for all i∈N_. Then, the following corollary is derived.Corollary 15.
Consider system (1a), (1b), (1c), and (1d), for a given scalar α>0, if there exist scalars εi>0, symmetric positive definite matrices Pi>0, and matrices S~i, Yi, and Zi such that, for all i∈N_,(68)[Ξ11iP-iA~τiS~iTS~iTP-iG~iD~iTP-iJ~iTP-iH~i0εiE~1iTεiE~3iT*-(1-τd)P-i-S~iT-S~iT00000εiE~2iT0**-S~i-S~iT-S~iT0000000***-τ-1P-i0000000****-γ2I000000*****-P-i00P-iH~i00******-I0000*******-εiI000********-εiI00*********-εiI0**********-εiI]<0, then there exists an observer-based robust H∞ controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted H∞ performance level γ under arbitrary switching signal with the average dwell time
(69)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(70)χ=1+τ(1+τ)(eλτ-1),λ+(1+τ)(eλτ-1)≤α,Pi≤μPj,∀i,j∈N_,Ξ11i=ΛiT+Λi+P-i+τP-i+αP-i,Λi=[PiAi+Zi-Zi0PiAi-YiCi],P-i=diag{Pi,Pi}.
Moreover, the controller gain matrices areKi=Bi+Pi-1Zi, and the observer gain matrices are Li=Pi-1Yi.


## 3.1. Stability Analysis

In this subsection, in order to obtain the main results, we first focus on the problem of stability analysis for the following switched stochastic systems with time delay(13a)dx(t)=[Aσ(t)x(t)+Aτσ(t)x(t-τ(t))]dt+Dσ(t)x(t)dw(t),(13b)x(t)=φ(t),t∈[t0-τ,t0].Theorem 8.
For a given scalarα>0, if there exist symmetric positive definite matrices Pi,  Qi, and Ri and any matrices Si such that
(14)[∑11iPiAτiSiTSiTDiTPi*-(1-τd)Qi-SiT-SiT0**-Si-SiT-SiT0***-τ-1Ri0****-Pi]<0,ddddddddddddddddddddddddddddddddd∀i∈N_
then system (13a) and (13b) is mean-square exponentially stable under arbitrary switching signal with the average dwell time
(15)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(16)Pi≤μPj,Qi≤μQj,Ri≤μRj,Qi≤βiPi,Ri≤βiPi,∀i,j∈N_,(17)λ+βi(1+τ)(eλτ-1)≤α,χ=maxi∈N_χi,χi=1+τβi(1+τ)(eλτ-1),∑11i=AiTPi+PiAi+Qi+τRi+αPi.Proof.
LetY(t)=Aσ(t)x(t)+Aτσ(t)x(t-τ(t)), then (13a) can be described as
(18)dx(t)=Y(t)dt+Dσ(t)x(t)dw(t).
Choose the following Lyapunov functional candidate for theith subsystem
(19)Vi(t,x(t))=V1,i(t,x(t))+V2,i(t,x(t))+V3,i(t,x(t)),
where
(20)V1,i(t,x(t))=xT(t)Pix(t),V2,i(t,x(t))=∫t-τ(t)txT(s)Qix(s)ds,V3,i(t,x(t))=∫0τ∫t-θtxT(s)Rix(s)dsdθ.
For the sake of simplicity,Vi(t,x(t)) is written as Vi(t) in this paper. According to Itô’s formula, along the trajectory of the ith subsystem, we have
(21)dVi(t)=∑g=13dVg,i,
where
(22)dV1,i(t)=ℒV1,i(t)dt+2xT(t)PiDix(t)dw(t),ℒV1,i(t)=2xT(t)Pi[Aix(t)+Aτix(t-τ(t))]+xT(t)DiTPiDix(t),dV2,i(t)=[xT(t)Qix(t)-(1-τ˙(t))xT(t-τ(t))ddd×Qix(t-τ(t))xT]dt≤[xT(t)Qix(t)-(1-τd)xT(t-τ(t))×Qix(t-τ(t))xT]dt,dV3,i(t)=[τxT(t)Rix(t)-∫t-τtxT(s)Rix(s)ds]dt.
According to Lemma6, we have(23)dV3,i(t)≤{τxT(t)Rix(t)-τ-1[∫t-τ(t)tx(s)ds]Tdfddd×Ri[∫t-τ(t)tx(s)ds]}dt.
Integrating both sides of (18) from t-τ(t) to t, we have
(24)x(t)-x(t-τ(t))=∫t-τ(t)tY(s)ds+∫t-τ(t)tDσ(s)x(s)dw(s).
Thus,(25)2[∫t-τ(t)tx(s)ds+∫t-τ(t)tY(s)ds]TSiη(t)dt=0,
where η(t)=x(t)-x(t-τ(t))-∫t-τ(t)tY(s)ds-∫t-τ(t)tDσ(s)x(s)dw(s).
Combining (21)–(25) leads to
(26)dVi(t)≤ℒVi(t)dt+Wi(t),
where
(27)ℒVi(t)=ςT(t)Θiς(t),Wi(t)=2xT(t)PiDix(t)dw(t)-2[∫t-τ(t)tx(s)ds+∫t-τ(t)tY(s)ds]T×Si[∫t-τ(t)tDσ(s)x(s)dw(s)]dt,ς(t)=[xT(t)xT(t-τ(t))∫t-τ(t)tYT(s)ds∫t-τ(t)txT(s)ds]T,Θi=[Θ11iPiAτiSiTSiT*-(1-τd)Qi-SiT-SiT**-Si-SiT-SiT***-τ-1Ri],Θ11i=AiTPi+PiAi+Qi+DiTPiDi+τRi.
By using the Schur complement, we obtain from (14) that
(28)ℒVi(t)<-αV1,i(t)<0.
According to (26), one obtains that
(29)dVi(t)≤ℒVi(t)dt+Wi(t)<-αV1,i(t)dt+Wi(t).
Then, taking mathematical expectation, we have(30)E{dVi(t)dt}≤E{ℒVi(t)}<-αE{V1,i(t)}<0.
From (16) and (19), we obtain that
(31)Vi(t)≤V1,i(t)+∫t-τtxT(s)Qix(s)ds+τ∫t-τtxT(s)Rix(s)ds≤V1,i(t)+βi(1+τ)∫t-τtV1,i(s)ds.
Letσ(tk)=i. Then, using Ito^′s formula, we can obtain that for t∈[tk,tk+1),
(32)E{eλtVi(t)}-E{eλtkVi(tk)}=E{∫tktℒ(eλsVi(s))ds}≤E{∫tkteλs[∫s-τsV1,i(ϑ)λV1,i(s)+λβi(1+τ)fffffffffffFFFFFff×∫s-τsV1,i(ϑ)dϑ-αV1,i(s)]ds∫tkteλs}.
Notice that(33)∫tkteλsds∫s-τsV1,i(ϑ)dϑ=∫tk-τtdϑ∫ϑϑ+τeλsV1,i(ϑ)ds=1λ(eλτ-1)∫tk-τteλsV1,i(s)ds.
Thus, it can be obtained that(34)E{eλtVi(t)}-E{eλtkVi(tk)}≤E{∫tkteλs[λ+βi(1+τ)(eλτ-1)-α]V1,i(s)ds}+E{Ωi(tk)},
where
(35)Ωi(tk)=βi(1+τ)(eλτ-1)∫tk-τtkeλsV1,i(s)ds≤βi(1+τ)(eλτ-1)eλtk∫tk-τtkV1,i(s)ds≤τβi(1+τ)(eλτ-1)eλtksup-τ≤θ≤0V1,i(tk+θ)≤τβi(1+τ)(eλτ-1)eλtksup-τ≤θ≤0Vi(tk+θ).
Noticing thatE{V(t)}≤sup-τ≤θ≤0E{V(t+θ)}, one gets
(36)E{Vi(t)}≤χie-λ(t-tk)sup-τ≤θ≤0E{Vi(tk+θ)}≤χe-λ(t-tk)sup-τ≤θ≤0E{Vi(tk+θ)}.
For anyt∈[tk,tk+1), from (16) and (36), it follows that
(37)E{V(t)}=E{Vi(t)}≤χe-λ(t-tk)sup-τ≤θ≤0E{Vσ(tk)(tk+θ)}≤χμe-λ(t-tk)sup-τ≤θ≤0E{Vσ(tk-)(tk+θ)}≤χ2μe-λ(t-tk)e-λ(tk+θ-tk-1)×sup-τ≤θ≤0E{Vσ(tk-1)(tk-1+θ)}≤χ2μeλτe-λ(t-tk-1)sup-τ≤θ≤0E{Vσ(tk-1)(tk-1+θ)}≤χ3(μeλτ)2e-λ(t-tk-2)sup-τ≤θ≤0E{Vσ(tk-2)(tk-2+θ)}≤⋯≤χ(χμeλτ)Nσ(t0,t)e-λ(t-t0)sup-τ≤θ≤0E{Vσ(t0)(t0+θ)}.
When (15) holds, noticing that  Nσ(t0,t)≤(t-t0)/Tα, one has
(38)E{∥x(t)∥2}≤baχe-(λ-(λτ+ln(χμ))/Tα)(t-t0)×sup-τ≤θ≤0E{∥x(t0+θ)∥2},
where a=min∀i∈N_λmin(Pi), b=max∀i∈N_λmax(Pi)+τmax∀i∈N_λmax(Qi)+(τ2/2)max∀i∈N_λmax(Ri).
The proof is completed.Remark 9.
In the derivation of Theorem8, a new Lyapunov-Krasovskii functional candidate is constructed for the stability analysis of switched stochastic systems with time delay, and it is different from the ones given in [9–15]. Also, the free-weighting matrix method is utilized to reduce the conservatism.Remark 10.
Ifμ=1 in (15), which leads to Pi=Pj, Qi=Qj, and Ri=Rj, for all i,j∈N_, then system (13a) and (13b) possesses a common Lyapunov function, and the switching signals can be arbitrary.Whenw(t)=0, system (13a) and (13b) becomes the following switched system with time delay:(39a)x˙(t)=Aσ(t)x(t)+Aτσ(t)x(t-τ(t)),(39b)x(t)=φ(t),t∈[t0-τ,t0].From Theorem8, we can readily get the exponential stability criterion for switched system (39a) and (39b).Corollary 11.
Consider system (39a) and (39b), for a given scalar α>0, if there exist symmetric positive definite matrices Pi,  Qi, and Ri and any matrices Si such that
(40)[∑11iPiAτiSiTSiT*-(1-τd)Qi-SiT-SiT**-Si-SiT-SiT***-τ-1Ri]<0,dddddddddddddddddddddddddddd∀i∈N_
then system (39a) and (39b) is exponentially stable under arbitrary switching signal with the average dwell time scheme (15).


## 3.2.H∞ Performance Analysis

In the sequel, we will investigate the problem ofH∞ performance analysis for switched stochastic systems with time delay. Consider the following system:(41a)dx(t)=[Aσ(t)x(t)+Aτσ(t)x(t-τ(t))+Gσ(t)v(t)]dt+Dσ(t)x(t)dw(t),(41b)z(t)=Jσ(t)x(t),(41c)x(t)=φ(t),t∈[t0-τ,t0].Theorem 12.
For a given scalarα>0, if there exist symmetric positive definite matrices Pi,  Qi, and Ri and any matrices Si such that
(42)[∏11iPiAτiSiTSiTPiGiDiTPiJiT*-(1-τd)Qi-SiT-SiT000**-Si-SiT-SiT000***-τ-1Ri000****-γ2I00*****-Pi0******-I]gggggggggggggggggggggggggggggggggggggg<0,∀i∈N_
then system (41a), (41b), and (41c) is mean-square exponentially stable with a weighted prescribed H∞ performance level γ under arbitrary switching signal with the average dwell time
(43)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(44)Pi≤μPj,Qi≤μQj,Ri≤μRj,Qi≤βiPi,Ri≤βiPi,∀i,j∈N_,(45)λ+βi(1+τ)(eλτ-1)≤α,χ=maxi∈N_χi,χi=1+τβi(1+τ)(eλτ-1),∏11i=AiTPi+PiAi+Qi+τRi+αPi.Proof.
We can easily obtain that (14) is satisfied if (42) holds. Thus, system (41a), (41b), and (41c) with v(t)=0 is mean-square exponentially stable.
Whenv(t)≠0, let
(46)Γ(t)=zT(t)z(t)-γ2vT(t)v(t).
Choosing the same Lyapunov functional candidate as (19) and following the proof line of Theorem 8, we have
(47)dVi(t)≤ℒVi(t)dt+Wi(t),
where
(48)Wi(t)=2xT(t)PiDix(t)dw(t)-2[∫t-τ(t)tx(s)ds+∫t-τ(t)tY(s)ds]T×Si[∫t-τ(t)tDσ(s)x(s)dw(s)]dt,
and ℒVi(t) satisfies
(49)ℒVi(t)+Γ(t)=ς-T(t)Θ-iς-(t),ς-(t)=[xT(t)xT(t-τ(t))∫t-τ(t)tYT(s)ds∫t-τ(t)txT(s)dsvT(t)]T,Θ-i=[Θ-11iPiAτiSiTSiTPiGi*-(1-τd)Qi-SiT-SiT0**-Si-SiT-SiT0***-τ-1Ri0****-γ2I],Θ-11i=AiTPi+PiAi+Qi+DiTPiDi+τRi+JiTJi.
Using the Schur complement, from (42), we get
(50)ℒVi(t)+Γ(t)<-αV1,i(t)<0.
It follows that(51)E{dVi(t)}≤E{ℒVi(t)dt}.
From (44), we obtain that
(52)E{Vσ(tk)(tk)}≤μE{Vσ(tk-1)(tk)}.
For anyt∈[tk,tk+1), using Ito^′s formula and taking the mathematical expectation, one has
(53)E{Vσ(t)(t)}=E{Vσ(tk)(tk)}+∫tktE{dVσ(tk)(s)}≤E{Vσ(tk)(tk)}+∫tktE{ℒVσ(tk)(s)+Γ(s)}ds-∫tktE{Γ(s)ds}<μE{Vσ(tk-1)(tk)}-∫tktE{Γ(s)ds}<μ2E{Vσ(tk-2)(tk-1)}-E{μ∫tk-1tkΓ(s)ds+∫tktΓ(s)ds}<⋯<μNσ(t0,t)E{V(t0)}-E{μNσ(t0,t)∫t0t1Γ(s)dsdddddd+μNσ(t1,t)∫t1t2Γ(s)ds+⋯+∫tktΓ(s)ds}=μNσ(t0,t)E{V(t0)}-E{∫t0teNσ(s,t)lnμΓ(s)ds}.
Under the zero initial condition, we obtain that(54)E{∫t0teNσ(s,t)lnμΓ(s)ds}<0.
According to (46), one has
(55)E{∫t0teNσ(s,t)lnμzT(s)z(s)}ds<γ2∫t0teNσ(s,t)lnμvT(s)v(s)ds.
Multiplying both sides of (55) by e-Nσ(t0,t)lnμ leads to
(56)E{∫t0te-Nσ(t0,s)lnμzT(s)z(s)ds}<γ2∫t0te-Nσ(t0,s)lnμvT(s)v(s)ds.
Noticing thatNσ(t0,s)≤(s-t0)/Tα and Tα≥τ+ln(χμ)/λ≥lnμ/λ, one obtains that
(57)E{∫t0te-λ(s-t0)zT(s)z(s)ds}<γ2∫t0tvT(s)v(s)ds.
Whent→∞, the following inequality is derived:
(58)E{∫t0∞e-λ(s-t0)zT(s)z(s)ds}<γ2∫t0∞vT(s)v(s)ds.
The proof is completed.


## 3.3. Observer-Based RobustH∞  Stabilization

Now, we are in a position to design an observer-based robustH∞ controller for system (1a), (1b), (1c), and (1d) such that the augmented closed-loop system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted H∞ performance level γ. Based on Theorem 12, a sufficient condition for the existence of such a controller is presented in the following theorem.Theorem 13.
Consider system (1a), (1b), (1c), and (1d), for a given scalar α>0, if there exist scalars εi>0, symmetric positive definite matrices Pi, Qi, and Ri, and any matrices S~i, Yi, and Zi such that, for all i∈N_,(59)[Ω11iP-iA~τiS~iTS~iTP-iG~iD~iTP-iJ~iTP-iH~i0εiE~1iTεiE~3iT*-(1-τd)Q-i-S~iT-S~iT00000εiE~2iT0**-S~i-S~iT-S~iT0000000***-τ-1R-i0000000****-γ2I000000*****-P-i00P-iH~i00******-I0000*******-εiI000********-εiI00*********-εiI0**********-εiI]<0,then there exists an observer-based controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted H∞ performance level γ under arbitrary switching signal with the average dwell time(60)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(61)Pi≤μPj,Qi≤μQj,Ri≤μRj,Qi≤βiPi,Ri≤βiPi,∀i,j∈N_,(62)λ+βi(1+τ)(eλτ-1)≤α,(63)χ=maxi∈N_χi,χi=1+τβi(1+τ)(eλτ-1),Ω11i=ΥiT+Υi+Q-i+τR-i+αP-i,Υi=[PiAi+Zi-Zi0PiAi-YiCi],P-i=diag{Pi,Pi},Q-i=diag{Qi,Qi},R-i=diag{Ri,Ri}.
Moreover, if the above conditions have a feasible solution, the controller gain matrices and the observer gain matrices can be obtained byKi=Bi+Pi-1Zi and  Li=Pi-1Yi.Proof.
According to Theorem12, we get that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted H∞ performance level γ if the following inequality is satisfied:
(64)Ψ-i=[Ψ-11iP~iA-τiS~iTS~iTP~iG-iD-iTP~iJ-iT*-(1-τd)Q~i-S~iT-S~iT000**-S~i-S~iT-S~iT000***-τ-1Ri000****-γ2I00*****-P~i0******-I]<0,
where Ψ-11i=A~iTP~i+P~iA~i+Q~i+τR~i+αP~i, and P~i, Q~i, and R~i are symmetric positive definite matrices with appropriate dimensions.
Then, we have(65)Ψ-i=Ψi+ΔΨi<0,
where
(66)Ψi=[Ψ11iP~iA~τiS~iTS~iTP~iG~iD~iTP~iJ~iT*-(1-τd)Q~i-S~iT-S~iT000**-S~i-S~iT-S~iT000***-τ-1Ri000****-γ2I00*****-P~i0******-I],ΔΨi=[E~1iTF~iTH~iTP~i+P~iH~iF~iE~1iP~iH~iF~iE~2i000E~3iTF~iTH~iTP~i0E~2iTF~iTH~iTP~i000000000000000000000000000P~iH~iF~iE~3i0000000000000]=[P~iH~i0000000000P~iH~i00][F~i00F~i][E~1iTE~3iTE~2iT00000000000]T+[E~1iTE~3iTE~2iT00000000000][F~i00F~i]T[P~iH~i0000000000P~iH~i00]T,Ψ11i=AiTP~i+P~iAi+Q~i+τR~i+αP~i.
From Lemma7, we have
(67)Ψ-i≤Ψi+εi-1[P~iH~i0000000000P~iH~i00][P~iH~i0000000000P~iH~i00]T+εi[E~1iTE~3iTE~2iT00000000000][E~1iTE~3iTE~2iT00000000000]T<0.Choose P~i=P-i=diag{Pi,Pi}, Q~i=Q-i=diag{Qi,Qi}, and R~i=R-i=diag{Ri,Ri}, and let Zi=PiBiKi and Yi=PiLi. By using the Schur complement, we can obtain that (67) is equivalent to (59).
Thus, according to Theorem12, we obtain from (59)–(62) that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted H∞ performance level γ. Moreover, we can obtain the controller gain matrices Ki=Bi+Pi-1Zi and the observer gain matrices Li=Pi-1Yi.
The proof is completed.Remark 14.
An observer-basedH∞ controller design scheme is proposed in the paper. Compared with the existing results presented in [27–29], a remark advantage of the work is that the proposed observer is mode-dependent, which means that each subsystem has its individual observer. Moreover, the proposed observer not only ensures the convergence of the estimated error of each subsystem, but also guarantees that the estimated error of the whole system converges to zero exponentially.In Theorem13, when we choose that Qi=Ri=Pi, it is not difficult to get that βi=1 and χ=χi=1+τ(1+τ)(eλτ-1),  for all i∈N_. Then, the following corollary is derived.Corollary 15.
Consider system (1a), (1b), (1c), and (1d), for a given scalar α>0, if there exist scalars εi>0, symmetric positive definite matrices Pi>0, and matrices S~i, Yi, and Zi such that, for all i∈N_,(68)[Ξ11iP-iA~τiS~iTS~iTP-iG~iD~iTP-iJ~iTP-iH~i0εiE~1iTεiE~3iT*-(1-τd)P-i-S~iT-S~iT00000εiE~2iT0**-S~i-S~iT-S~iT0000000***-τ-1P-i0000000****-γ2I000000*****-P-i00P-iH~i00******-I0000*******-εiI000********-εiI00*********-εiI0**********-εiI]<0, then there exists an observer-based robust H∞ controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted H∞ performance level γ under arbitrary switching signal with the average dwell time
(69)Tα>Tα*=τ+ln(χμ)λ,
where μ, χ, and λ satisfy
(70)χ=1+τ(1+τ)(eλτ-1),λ+(1+τ)(eλτ-1)≤α,Pi≤μPj,∀i,j∈N_,Ξ11i=ΛiT+Λi+P-i+τP-i+αP-i,Λi=[PiAi+Zi-Zi0PiAi-YiCi],P-i=diag{Pi,Pi}.
Moreover, the controller gain matrices areKi=Bi+Pi-1Zi, and the observer gain matrices are Li=Pi-1Yi.


## 4. Numerical Example

Consider system (1a), (1b), (1c), and (1d) with the following parameters
(71)A1=[310-3],Aτ1=[-0.100.10.1],B1=[1211],C1=[2003],J1=[3014],D1=[0.100.10.6],G1=[1.500.10.4],H1=[0.1000],E11=[0.1000],E21=[00.100],E31=[000.10],A2=[400-2],Aτ2=[-0.10.10-0.2],B2=[1121],C2=[3021],J2=[2114],D2=[0.100.30.5],G2=[0.50.200.6],H2=[0000.1],E12=[0000.1],E22=[000.10],E32=[00.100],F1(t)=[sint00sint],F2(t)=[cost00cost].The disturbance inputv(t)=[15e-0.2t12e-0.3t]T, and τ(t)=0.3+0.2sint; by calculation, we can obtain that τ˙(t)≤τd=0.2 and τ(t)≤τ=0.5. Choosing α=1.4,  γ=1.0 and solving the LMIs in Corollary 15, we have
(72)P1=[4.0319-0.7058-0.705813.0192],Y1=[82.457919.7441-15.992625.9295],Z1=[-98.8224-4.9793-5.0165-48.5838],P2=[4.1782-0.5687-0.568711.9156],Y2=[91.2715-45.8464-48.736499.0875],Z2=[-105.7969-5.8641-5.1805-64.8133],S~1=[1.9716-0.2408-0.08090.0342-0.23015.22800.0555-0.4124-0.08220.05532.0559-0.30490.0286-0.4044-0.30176.0017],S~2=[1.5102-0.0781-0.07300.1131-0.07964.65230.0254-0.6518-0.07920.01192.1184-0.23290.1161-0.6486-0.24135.2970],ε1=54.6307,ε2=51.0439.Then, we can obtain the following observer gain matrices:(73)L1=[20.43015.2958-0.12092.2787],L2=[21.4270-9.9052-3.06757.8431]
and the controller gain matrices
(74)K1=[21.3520-5.7638-23.08251.9288],K2=[23.8921-3.3845-49.43821.2267].Moreover, we haveμ=4.3051, λ=0.7, χ=1.3143, and Tα*=τ+ln(χμ)/λ=2.9759. Taking Tα=3>Tα*, and letting x(t)=[00]T, t∈[-0.5,0), x(0)=[2-2]T, and e(0)=[11]T, and simulation results are shown in Figures 1–5.Figure 1
Switching signal.Figure 2
Statex1 of the closed-loop system.Figure 3
Statex2 of the closed-loop system.Figure 4
The estimated errore1.Figure 5
The estimated errore2.Figure1 shows the switching signal of the switched system with the average dwell time Tα=3. Figures 2 and 3 illustrate the state trajectories of the closed-loop system. The estimated errors are plotted in Figures 4 and 5, respectively. We can see from Figures 2–5 that the proposed observer can guarantee the convergence of the estimated error and the designed controller can guarantee the stability of the corresponding closed-loop system. This demonstrates the effectiveness of the proposed method.In addition, some observer-based controller design approaches proposed in the existing literature [27–29] are only applicable to stochastic systems or Markovian jump systems and they cannot be used to stabilize the system considered in this section, which also shows the advantage of the proposed method.


## 5. Conclusions

In this paper, the problem of observer-based robustH∞  stabilization for stochastic switched systems with time delay has been investigated. By using the average dwell time method, sufficient conditions which guarantee the mean-square exponential stability of switched stochastic systems with time delay are derived. Then, H∞ performance analysis and observer-based H∞ control for such systems are developed. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.



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*Source: 320703-2013-09-08.xml*