# Dynamic Output Feedback Passive Control of Uncertain Switched Stochastic Systems with Time-Varying Delay

Authors: Huimei Jia; Hamid Reza Karimi; Zhengrong Xiang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the issues of passivity analysis and dynamic output feedback (DOF) passive control for uncertain switched stochastic systems with time-varying delay via multiple storage functions (MSFs) method. Firstly, based on the MSFs method, a sufficient condition for the existence of the passivity of the underlying system is established in terms of linear matrix inequalities (LMIs). Furthermore, the problem of dynamic output feedback passive control is investigated. Based on the obtained passivity condition, a sufficient condition for the existence of the desired switched passive controller is derived. Finally, a numerical example is presented to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

Switched systems consist of a family of continuous-time or discrete-time subsystems operated by a particular type of switching rule. According to this switching rule, one of these subsystems will be activated along the system trajectory at each instant of time. Switched systems have drawn considerable attention during the past few decades because of their applicability and significance in various areas, such as power electronics, embedded systems, chemical processes, and computer-controlled systems [1–4]. Analysis and synthesis of such systems have been studied in a large number of papers, and some research results have been obtained in [5–18]. For example, stability and stabilization problems were investigated in [5–7], the sliding mode control problem was studied in [8–10], the model reduction problem was developed in [11, 12], the filtering problem was considered in [13], output feedback control was discussed in [14–17], and so on. In [18], the authors gave a summary of the recent results about the stability and stabilizability of switched systems. On the other hand, due to the presence of random disturbance, stochastic systems also play an important role in engineering applications and have received much attention during the past decades. Many results on stochastic systems and switched stochastic systems have been reported in [19–23]. Besides, time delays frequently occur in practical systems and are often the source of instability, and it is of significance to study time delay systems. Recently, many useful results on such systems have appeared in [24–27].It is well known that the Lyapunov function method [28] and the average dwell time approach [29] are widely used to study the stability of switched systems. A common Lyapunov function for all subsystems can guarantee the stability under an arbitrary switching signal [30], but sometimes the common Lyapunov function is difficult to find or even might not exist, so Lyapunov-like functions and multiple Lyapunov functions techniques are used widely to ensure the stability under some certain switching signals.Recently, passivity has been investigated by many researchers. As a special case of dissipativity, it explains the system’s internal energy relationship from the point of energy dissipation, and it means that the loss of energy inside the dynamic system is not more than the energy supplied from outside. The concept of passivity was firstly introduced by Willems [31] and was extended by Hill and Moylan [32]. Passivity has a wide range of applications, not only because of its desirable properties, but also because it is a powerful tool to analyze the stabilization of systems. Compared with the commonly used stability theory, the use of passivity presents a new tool to analyze the stability, and according to the definition of passivity, storage functions induced by passivity can provide natural candidates for Lyapunov functions to analyze the Lyapunov stability and stabilization problem of systems. The notion of passivity for a class of switched control systems was developed in [33], and the relationship between Lyapunov stability and passivity was analyzed. Because the common storage function is hard to search, to overcome this restriction, a notion of multiple storage functions (MSFs) was proposed in [34]. MSFs mean that each subsystem has its own storage function, and each storage function is nonincreasing for zero input on the “switched on” time to guarantee the stability of the system. Many scholars have done a lot of research on passivity of switched systems and have achieved fruitful research results [33–38]. A framework of dissipativity theory for switched systems using MSFs and multiple supply rates was set up in [34], a concept of passivity for switched systems using MSFs was presented in [35], and the issues of passivity, feedback equivalence, and stability for switched nonlinear systems via MSFs approach were investigated in [36, 37]. In [38], the dissipativity-based sliding mode control of switched stochastic systems was studied, and the passivity condition was established by applying the average dwell time approach. It is worth pointing out that the result presented in [38] is mainly on switched stochastic delay-free systems. However, to the best of our knowledge, the passivity analysis and control synthesis for switched stochastic systems with time-varying delay have not been fully investigated to date. This motivates the present study.In this paper, we focus on the passive analysis and DOF passive control for a class of uncertain switched stochastic systems with time-varying delay. The main contributions of this paper can be summarized as follows: (i) by utilizing the MSFs method, a passivity condition is derived in terms of LMIs; and (ii) based on the result obtained, a DOF passive controller is constructed for the system.This paper is organized as follows. In Section2, problem formulation and some necessary lemmas are given. In Section 3, the main results are presented. Section 4 gives a numerical example to illustrate the effectiveness of the proposed approach. Finally, conclusions are provided in Section 5.Notations. The notations throughout this paper are quite standard. Rn and Rn×m denote the n-dimensional Euclidean space and the set of all n×m real matrices, respectively. AT and A-1 denote the transpose and the inverse of any square matrix A. The notation A>0 means that matrix A is positive definite. I is the identity matrix with an appropriate dimension. The symmetric term in a matrix is denoted by “*.” E{·} is the expectation operator. diag(·) denotes a block diagonal matrix. Cn,τ denotes the set of all Rn-valued continuous functions defined on the interval [-τ,0]. L2[0,∞) is the space of square integrable functions on [0,∞).


## 2. Problem Formulation and Preliminaries

Consider the following uncertain switched stochastic systems with time-varying delay:(1)dx(t)=[A-σ(t)x(t)+A-dσ(t)x(t-d(t))+Bσ(t)u(t)+Bvσ(t)v(t)A-σ(t)]dt+S-σ(t)x(t)dw(t),z(t)=C1σ(t)x(t)+D1σ(t)u(t)+D1vσ(t)v(t),dy(t)=[C-2σ(t)x(t)+C-2dσ(t)x(t-d(t))+D2vσ(t)v(t)]dt+G-σ(t)x(t)dw(t),x(t)=ϕ(t),t∈[-τ,0],
where x(t)∈Rn is the state vector, u(t)∈Rm is the control input, z(t)∈Rq is the output vector, v(t)∈Rp is the disturbance input which belongs to L2[0,∞), y(t)∈Rr is the measured output, and ϕ(t)∈Cn,τ represents the initial state function.σ
(
t
)
∈
[
0
,
∞
)
→
N
_
=
{
1,2
,
…
,
N
} is a piecewise right continuous constant function. The switching sequence can be described as Σ:{(t0,σ(t0)),(t1,σ(t1)),…,(tk,σ(tk)),…}, where t0=0 is the initial time, and tk denotes the kth switching instant. Moreover, σ(t)=i means that the ith subsystem is activated. w(t) is a zero-mean Wiener process on a probability space (Ω,F,Ρ) satisfying
(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. d(t) is the time-varying delay satisfying
(3)0≤d(t)≤τ,0≤d˙(t)≤μ<1,
where τ and μ are known constants. Consider
(4)A-i=Ai+ΔAi(t),A-di=Adi+ΔAdi(t),S-i=Si+ΔSdi(t),C-2i=C2i+ΔC2i(t),C-2di=C2di+ΔC2di(t),G-i=Gi+ΔGi(t),
where Ai, Adi, Bi, Bvi, Si, C1i, D1i, D1vi, C2i, C2di, D2vi, and Gi, i∈N_, are constant matrices with appropriate dimensions, and ΔAi(t), ΔAdi(t), ΔSi(t), ΔC2i(t), ΔC2di(t), and ΔGi(t), i∈N_, are unknown matrices representing time-varying parameter uncertainties and assumed to be of the form
(5)[ΔAi(t)ΔAdi(t)ΔSi(t)ΔCi(t)ΔCdi(t)ΔGi(t)]=[M1iM2i]Fi(t)[N1iN2iN3i],i∈N_
where M1i,M2i,N1i,N2i,andN3i are known real constant matrices, and Fi(t) is an unknown time-varying matrix function satisfying
(6)FiT(t)Fi(t)≤I.Definition 1 (see [36]).
For system (1) with u(t)=0, it is said to be stochastically passive under the switching signal σ(t) if there exists a nonnegative function Sσ(t)(x) with Sσ(t)(0)=0, such that the following inequality holds:
(7)E{Sσ(t)(x(t))-Sσ(s)(x(s))}≤∫stE{2vT(θ)z(θ)}dθ,s≤t<∞.Remark 2.
In Definition1, the nonnegative function Sσ(t)(x) is called storage function. Passivity is explained by the system internal energy relationship from the point of energy dissipation; so the stored energy function of the system is called storage function. Usually, a common storage function for all subsystems may not exist or it may be rather difficult to find, but each subsystem may have its own storage function; thus, we consider the passivity under the switching signal via MSFs method in this paper.Definition 3 (see [36]).
LetSi be a positive semidefinite function and assume that the ith subsystem is active during the interval [tk,tk+1). Si is called to be the storage-like function of the ith subsystem if the following passivity inequality holds:
(8)E{Si(x(t))-Si(x(s))}≤∫stE{2vT(θ)z(θ)}dθ,tk≤s≤t<tk+1.Remark 4.
In Definition3, inequality (8) indicates that the passivity property of the ith subsystem holds when it is active.Remark 5.
Actually, the stability-theory-based Lyapunov functions can be explained from the vision of passivity. It is often needed to construct Lyapunov functions when we consider the stability of the system; this process can be transformed to the process of constructing the storage functions to make the system passive.Lemma 6 (see [24]).
For any constant matrixR>0, scalar d>0, and vector function x(t), such that the following Jensen’s integral inequality is well defined:
(9)-d∫t-dtx(s)Rx(s)ds≤-[∫t-dtx(s)ds]TR[∫t-dtx(s)ds].Lemma 7 (see [13]).
LetU, V, W, and X be real matrices of appropriate dimensions with X satisfying X=XT, then for all VTV≤I, X+UVW+WTVTUT<0,(a)
if and only if there exists a scalarε>0 such that
(10)X+εUUT+ε-1WTW<0;(b)
for any positive definite matrixR:
(11)UW+WTUT≤UR-1UT+VTRV.Lemma 8 (see [39]).
A, M, N, and P are real matrices of appropriate dimensions. F(t) is time-varying matrix with FT(t)F(t)≤I. If there exists a scalar ε>0 such that P-εMMT>0, then we have
(12)(A+MFN)TP(A+MFN)≤AT(P-1-εMMT)-1A+ε-1NTN.


## 3. Main Results

In this section, we first investigate the passivity analysis of system (1); then based on the passive condition obtained, an output feedback controller is designed to guarantee the passivity of the resulting closed-loop system.


### 3.1. Passivity Analysis

Theorem 9.
Consider system (1) with u(t)=0, the system is stochastically passive for the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist symmetric positive matrices Pi, Qi, Zi, Ri and matrices Li, i∈N_, with appropriate dimensions such that
(13)[Ξ11PiA-di0PiBvi-C1iTS-iTPiA-iTPi0*-(1-μ)Qi000A-diTPi0**Ξ33000Li***-D1viT-D1vi0BviTPi0****-Pi00*****-Pi0******-Pi]<0,
where Ξ11=Qi+τZi+A-iTPi+PiA-i, Ξ33=τRi-LiT-Li.Proof.
Letf(t):=A-σ(t)x(t)+A-dσ(t)x(t-d(t))+Bvσ(t)v(t), and η(t)dt=dx(t), when the ith subsystem is activated, we have
(14)2ηT(t)Li[(f(t)-η(t))dt+S-ix(t)dw(t)]=0.
Choose the following Lyapunov functional candidate
(15)Vi(x(t),t)=V1i(x(t),t)+V2i(x(t),t)+V3i(x(t),t)+V4i(x(t),t),
where
(16)V1i(x(t),t)=xT(t)Pix(t),V2i(x(t),t)=∫t-d(t)txT(s)Qix(s)ds,V3i(x(t),t)=∫-τ0∫t+θtxT(s)Zix(s)ds,V4i(x(t),t)=∫-τ0∫t+θtηT(s)Riη(s)ds.
According to the Itô formula, along the trajectory of the system, we have
(17)dVi(x(t),t)=LVi(x(t),t)dt+2xT(t)PiS-ix(t)dw(t).
Substituting (14) into (17), we get
(18)dVi(x(t),t)=LV~i(x(t),t)dt+2[xT(t)PiS-ix(t)+ηT(t)LiS-ix(t)]dw(t),
where LV~i(x(t),t)=LVi(x(t),t)+2ηT(t)Li[(f(t)-η(t))].
Taking the expectation, we have(19)E{dVi(x(t),t)dt}=E{LV~i(x(t),t)}.
For convenience, we write LV~i=LV~i(x(t),t); then it can be obtained from (17) that
(20)LV~i≤2xT(t)Pif(t)+xT(t)S-iTPiS-ix(t)+xT(t)Qix(t)-(1-μ)xT(t-d(t))Qix(t-d(t))+τxT(t)Zix(t)+τηT(t)Riη(t)-∫t-d(t)txT(s)Zix(s)ds-∫t-d(t)tηT(s)Riη(s)ds+2ηT(t)Li[(f(t)-η(t))].
It follows that
(21)LV~i(x(t))-2vT(t)z(t)≤ξT(t)(Ξ+Ξ1+Ξ1T+Ξ2)ξ(t)-∫t-d(t)t[xT(s)ηT(s)][Zi00Ri][x(s)η(s)]ds,
where
(22)ξT(t)=[xT(t)xT(t-d(t))ηT(t)vT(t)],Ξ1=L~iA~,L~iT=[00LiT0],A~=[A-iA-di0Bvi],Ξ2=[S-i000]TPi[S-i000],Ξ=[Ξ11PiA-di0PiBvi-C1iT*-(1-μ)Qi00**Ξ330***-D1viT-D1vi].
By Lemma 7 and Schur complement lemma, we can obtain from (13) that
(23)Ξ+Ξ1+Ξ1T+Ξ2<0;
then the following inequality can be obtained:
(24)E{dVi(x(t),t)dt}=E{LVi(x(t),t)}<2E{vT(t)z(t)}.
For tk≤s<t<tk+1, the following inequality holds:
(25)E{Vσ(tk)(x(t),t)-Vσ(tk)(x(s),s)}<∫stE{2vT(s)z(s)}ds.
When the switching signal is chosen as σ(t)=min{i∣i=argmini∈N_Vi(x(t))}, for tk≤t<tk+1, one obtains
(26)E{Vσ(tk)(x(t),t)-Vσ(t0)(x(0),0)}=E{∫0t1LVσ(t0)(x(s),s)ds+∫t1t2LVσ(t1)(x(s),s)ds+⋯∫tktVσ(tk)(x(t),t)ds}+E{∑g=1k(Vσ(tg)(x(tg),tg)-Vσ(tg-1)(x(tg),tg))}<∫0tE{2vT(s)z(s)}dt.
This completes the proof.Remark 10.
The passive condition presented in Theorem9 is delay dependent, and it is obtained by constructing an appropriate multiple Lyapunov functions. Some other methods dealing with time delay systems, such as delay divisioning method and input-output method, can also be utilized to investigate the systems under consideration, and this will be our further work.
When the matrix parameters in system (1) are determined, according to Theorem 9, we have the following result.Corollary 11.
Consider the nominal system in (1), it is stochastically passive under the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist symmetric positive matrices Pi, Qi, Ri and matrices Li, i∈N_, with appropriate dimensions such that
(27)[Γ11PiAdi0PiBvi-C1iTSiTPiAiTPi0*-(1-μ)Qi000AdiTPi0**Γ33000Li***-D1viT-D1vi0BviTPi0****-Pi00*****-Pi0******-Pi]<0,
where Γ11=Qi+τZi+AiTPi+PiAi, Γ33=τRi-LiT-Li.


### 3.2. DOF Passivity Analysis

In this subsection, we focus on designing a DOF passive controller for system (1) such that the corresponding closed-loop system is stochastically passive.Consider a full-order DOF passive controller with the following form:(28)Σk:{dx^(t)=Akσ(t)x^(t)dt+Bkσ(t)dy(t)u(t)=Ckσ(t)x^(t),
where x^∈Rn is the controller state; Aki, Bki, and Cki, i∈N_, are matrices to be determined.Applying this controller to system (1), we obtain the following closed-loop system:
(29)Σc:{dξ(t)=[A-ciξ(t)+A-cdiHξ(t-d(t))+Bcviv(t)]dt+S-ciξ(t)dw(t)z(t)=Cciξ(t)+Dcviv(t),
where
(30)ξT(t)=[xT(t)x^T(t)],A-ci=Aci+ΔAci(t),A-cdi=Acdi+ΔAcdi(t),S-ci=Sci+ΔSci(t),H=[I0],A-ci=[A-iBiCkiBkiC-2iAki],A-cdi=[A-diBkiC-2di],Bcvi=[BviBkiD2vi],S-ci=[S-iBkiG-i]H,Cci=[C1iD1iCki],Dcvi=D1vi,M~1i=[M1iBkiM2i],N~1i=N1i,N~2i=N2i,N~3i=N3i.
Then, the output feedback passive controller design problem can be converted into the problem of passivity analysis of the closed-loop system (29).Theorem 12.
Consider system (29), it is stochastically passive for the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist symmetric positive matrices Pi, Qi, Ri and matrices Lci, i∈N_, with appropriate dimensions such that the following matrix inequality holds:
(31)[Θ11PiA-cdi0PiBcvi-CciTS-ciTPiA-ciTPi0*-(1-μ)Qi000A-cdiTPi0**Θ33000Lci***-DcviT-Dcvi0BcviTPi0****-Pi00*****-Pi0******-Pi]<0,
where Θ11=HTQiH+τHTZiH+A-ciTPi+PiA-ci, Θ33=τ2HTRiH-LciT-Lci.Proof.
Letg(t):=A-ciξ(t)+A-cdiξ(t-d(t))+Bcviv(t) and ς(t)dt=dξ(t), then we have
(32)2ςT(t)Lci[(g(t)-ς(t))dt+S-ciξ(t)dw(t)]=0.
Choose the following Lyapunov functional candidate:
(33)Vi(ξ(t),t)=ξT(t)Piξ(t)+∫t-d(t)tξT(s)HTQiHξ(s)ds+∫-τ0∫t+θtξT(s)HTZiHξ(s)ds+∫-τ0∫t+θtςT(s)HTRiHς(s)ds.
From the relation between ξ(t) and x(t), we obtain
(34)Hξ(t)=x(t)Hξ(t-d(t))=x(t-d(t)).
Denote ζT(t)=[ξT(t)xT(t-d(t))ςT(t)vT(t)], then following the proof line of Theorem 9, it can be obtained form (31) that
(35)LVi(ξ(t),t)-2vT(t)z(t)<0.
It follows that
(36)E{dVi(ξ(t),t)dt}=E{LVi(ξ(t),t)}<2E{vT(t)z(t)}.
For tk≤s<t<tk+1, one obtains
(37)E{Vσ(k)(ξ(t),t)-Vσ(k)(ξ(s),s)}<∫stE{2vT(s)z(s)}ds.
Under the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))}, for tk≤t<tk+1, we obtain
(38)E{Vσ(tk)(ξ(t),t)-Vσ(t0)(ξ(0),0)}=E{∫0t1LVσ(t0)(ξ(s),s)ds+∫t1t2LVσ(t1)(ξ(s),s)ds+⋯∫tktVσ(tk)(ξ(t),t)ds}+E{∑g=1k(Vσ(tg)(ξ(tg),tg)-Vσ(tg-1)(ξ(tg),tg))}<∫0tE{2vT(s)z(s)}dt.
This completes the proof.


### 3.3. DOF Passivity Controller Design

In Theorem12, we have obtained a sufficient condition under which the closed-loop system ∑c is stochastically passive, but the designed parameters have not been given. The following theorem presents a controller design method and gives all parameters of the controller.Theorem 13.
Consider system (1), there exists a controller (28) such that the closed-loop system ∑c is stochastically passive for the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist scalars ε>0, β>0, and positive definite symmetric matrices Ωi, Υi, Qi, Ri, Zi and matrices Xi, Yi, Ti, i∈N_, with appropriate dimensions such that, for all i∈N_,
(39)[Ψ11Ψ120Ψ14Ψ15Ψ160U1*Ψ22000Ψ260U2**2βψ000-βψU3***Ψ440Ψ4600****ψ00U1*****ψ00******ψU4*******V]<0,
where
(40)Ψ11=[AiΩi+BiYi+ΩiAiT+YiTBiTAi+TiT*ΥiAi+XiC2i+AiTΥi+C2iTXiT],Ψ12=[AdiΥiAdi+XiC2di],Ψ14=[Bvi-ΩiC1iT-YiTD1iTΥiBvi+XiD2vi-C1iT],Ψ15=[ΩiSiT0SiTSiTΥi+GiTXi],Ψ16=[ΩiAiT+YiTBiTTiTAiTAiTΥi+C2iTXiT],Ψ22=-(1-μ)Qi,Ψ44=-D1viT-D1vi,Ψ26=[AdiTAdiTΥi+C2diTXiT],Ψ46=[BviTBviTΥi+D2viTXiT],ψ=[-Ωi-I-I-Υi],U1=[M1i00ΩiΩiN1iTΩiN3iTΩiΩi0ΥiM1i+XiM2i000N1iTN3iTII0],U2=[0000N2iT0000],U3=[00000000Ωi00000000I],U4=[M1i00000000ΥiM1i+XiM2i0SiTΥi+GiTXiT000000],U5=[0M1i00000000ΥiM1i+XiM2i0000000],V=diag{-εI,-εI,-I,-I,-ε-1I,-ε-1I,-Qi-1,-τ-1Zi-1,-Ri-1}.
Moreover, if the above condition is feasible, then a DOF passive controller realization is given by
(41)Aki=Pi12-1(Ti-ΥiAiΩi-XiC2iΩi-ΥiBiYi)Wi12-TBki=Pi12-1Xi,Cki=YiWi12-T,
where Pi12 and Wi12 are any nonsingular matrices and satisfy
(42)ΩiΥi+Wi12Pi12T=I.Proof.
According to Theorem12, let the matrices Pi and Pi-1 be partitioned as
(43)Pi:=[Pi11Pi12*Pi22],Pi-1:=[Wi11Wi12*Wi22],
where Pi11∈Rn×Rn, Wi11∈Rn×Rn, and Pi12, Wi12 are nonsingular matrices. Then, define the following matrices:
(44)Ji:=[Wi11IWi12T0],J~i:=[IPi110Pi12T].
From Pi-1Pi=I, we have
(45)Wi11Pi11+Wi12Pi12T=I,Wi12TPi11+Wi22Pi12T=0,
and PiJi=J~i, Pi-1J~i=Ji.
By using Lemmas7 and 8 and Schur complement lemma, we can get that (31) is equivalent to the following inequality:
(46)[Σ11Σ120PiBcvi-CciTSciTPiAciTPi0PiM~1i0*Σ22000AcdiTPi000**Σ33000Lci00***-DcviT-Dcvi0BcviTPi000****-Pi00PiM~1i0*****-Pi000******-Pi0PiM~1i*******-εI0********-εI]<0,
where
(47)Σ11=HTQiH+τHTZiH+AciTPi+PiAci+εHT×(N1iTN1i+N3iTN3i)H,Σ12=PiAcdi+εHTN1iTN2i,Σ22=-(1-μ)Qi+εN2iTN2i,Σ33=τHTRiH-LiT-Li.
Pre- and post-multiplying (46) by matrices diag{JiT,I,JiT,I,JiT,JiT,JiT,I,I,I,I} and diag{Ji,I,Ji,I,Ji,Ji,Ji,I,I,I,I}, respectively, and denoting Li:=βPi, Ωi:=Wi11, Υi:=Pi11, Xi:=Pi12Bki, Yi:=CkiWi12T, and
(48)Ti:=(Pi11Ai+Pi12BkiC2i)Wi11+Pi11BiCkiWi12T+Pi12AkiWi12T,
we have
(49)[Ψ~11Ψ~120Ψ14Ψ~15Ψ160Ψ180*Ψ~22000Ψ26000**Ψ33000-βψ00***Ψ440Ψ46000****ψ00Ψ580*****ψ000******ψ0Ψ79*******-εI0********-εI]<0,
where
(50)Ψ~11=Ψ11+ε[ΩiI](Qi+τZi+N1iTN1i+N3iTN3i)[ΩiI],Ψ~12=Ψ12+ε[ΩiI]N1iTN2i,Ψ~22=Ψ22+εN2iTN2i,Ψ~15=[ΩiSiTΩiSiTΥi+ΩiGiTXiTSiTSiTΥi+GiTXiT],Ψ33=βψ+ε[ΩiI]Ri[ΩiI],Ψ18=Ψ58=Ψ79=[M1iΥiM1i+XiM2i].
Applying Schur complement lemma, one obtains(51)[Ψ~11Ψ~120Ψ14Ψ~15Ψ160Ψ180*Ψ~22000Ψ26000**Ψ33000-βψ00***Ψ440Ψ46000****ψ00Ψ580*****ψ000******ψ0Ψ79*******-εI0********-εI]+υTϑ+ϑTυ<0,
where
(52)υ=[Ωi0000000000000],ϑ=[0000000SiTΥi+GiTXiT000000].
Then by Lemma 7 and Schur complement lemma, we can obtain (39) and the parameters of the designed passive controller.Remark 14.
It should be noted that the dissipativity-based sliding mode control of switched stochastic delay-free systems was studied in [38]. However, the focus of our work is on DOF passive controller design for switched stochastic delay systems, and this is also the major contribution of the paper.Remark 15.
Theorem13 proposes a design method of the DOF passive controller for system (1). For the prescribed positive definite symmetric matrices Qi, Ri, and Zi, we can solve the inequality (39) easily by LMI Toolbox in MATLAB. However, when these matrices are unknown, the inequality is no longer a LMI. In this case, we can use an iterative algorithm involving a nonlinear optimization problem with LMI constraint in [20] to solve it.The procedure of the passive controller design can be concluded as follows:(1)
by solving LMI (39), we can get the feasible solution of the positive definite symmetric matrices Ωi, Υi, Qi, Ri, and Zi and any matrices Xi, Yi, and Ti.(2)
Applying singular value decomposition to (42), we can obtain the nonsingular matrices Pi12, Wi12.(3)
According to (41), we can calculate the controller parameters Aki, Bki, and Cki.


## 3.1. Passivity Analysis

Theorem 9.
Consider system (1) with u(t)=0, the system is stochastically passive for the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist symmetric positive matrices Pi, Qi, Zi, Ri and matrices Li, i∈N_, with appropriate dimensions such that
(13)[Ξ11PiA-di0PiBvi-C1iTS-iTPiA-iTPi0*-(1-μ)Qi000A-diTPi0**Ξ33000Li***-D1viT-D1vi0BviTPi0****-Pi00*****-Pi0******-Pi]<0,
where Ξ11=Qi+τZi+A-iTPi+PiA-i, Ξ33=τRi-LiT-Li.Proof.
Letf(t):=A-σ(t)x(t)+A-dσ(t)x(t-d(t))+Bvσ(t)v(t), and η(t)dt=dx(t), when the ith subsystem is activated, we have
(14)2ηT(t)Li[(f(t)-η(t))dt+S-ix(t)dw(t)]=0.
Choose the following Lyapunov functional candidate
(15)Vi(x(t),t)=V1i(x(t),t)+V2i(x(t),t)+V3i(x(t),t)+V4i(x(t),t),
where
(16)V1i(x(t),t)=xT(t)Pix(t),V2i(x(t),t)=∫t-d(t)txT(s)Qix(s)ds,V3i(x(t),t)=∫-τ0∫t+θtxT(s)Zix(s)ds,V4i(x(t),t)=∫-τ0∫t+θtηT(s)Riη(s)ds.
According to the Itô formula, along the trajectory of the system, we have
(17)dVi(x(t),t)=LVi(x(t),t)dt+2xT(t)PiS-ix(t)dw(t).
Substituting (14) into (17), we get
(18)dVi(x(t),t)=LV~i(x(t),t)dt+2[xT(t)PiS-ix(t)+ηT(t)LiS-ix(t)]dw(t),
where LV~i(x(t),t)=LVi(x(t),t)+2ηT(t)Li[(f(t)-η(t))].
Taking the expectation, we have(19)E{dVi(x(t),t)dt}=E{LV~i(x(t),t)}.
For convenience, we write LV~i=LV~i(x(t),t); then it can be obtained from (17) that
(20)LV~i≤2xT(t)Pif(t)+xT(t)S-iTPiS-ix(t)+xT(t)Qix(t)-(1-μ)xT(t-d(t))Qix(t-d(t))+τxT(t)Zix(t)+τηT(t)Riη(t)-∫t-d(t)txT(s)Zix(s)ds-∫t-d(t)tηT(s)Riη(s)ds+2ηT(t)Li[(f(t)-η(t))].
It follows that
(21)LV~i(x(t))-2vT(t)z(t)≤ξT(t)(Ξ+Ξ1+Ξ1T+Ξ2)ξ(t)-∫t-d(t)t[xT(s)ηT(s)][Zi00Ri][x(s)η(s)]ds,
where
(22)ξT(t)=[xT(t)xT(t-d(t))ηT(t)vT(t)],Ξ1=L~iA~,L~iT=[00LiT0],A~=[A-iA-di0Bvi],Ξ2=[S-i000]TPi[S-i000],Ξ=[Ξ11PiA-di0PiBvi-C1iT*-(1-μ)Qi00**Ξ330***-D1viT-D1vi].
By Lemma 7 and Schur complement lemma, we can obtain from (13) that
(23)Ξ+Ξ1+Ξ1T+Ξ2<0;
then the following inequality can be obtained:
(24)E{dVi(x(t),t)dt}=E{LVi(x(t),t)}<2E{vT(t)z(t)}.
For tk≤s<t<tk+1, the following inequality holds:
(25)E{Vσ(tk)(x(t),t)-Vσ(tk)(x(s),s)}<∫stE{2vT(s)z(s)}ds.
When the switching signal is chosen as σ(t)=min{i∣i=argmini∈N_Vi(x(t))}, for tk≤t<tk+1, one obtains
(26)E{Vσ(tk)(x(t),t)-Vσ(t0)(x(0),0)}=E{∫0t1LVσ(t0)(x(s),s)ds+∫t1t2LVσ(t1)(x(s),s)ds+⋯∫tktVσ(tk)(x(t),t)ds}+E{∑g=1k(Vσ(tg)(x(tg),tg)-Vσ(tg-1)(x(tg),tg))}<∫0tE{2vT(s)z(s)}dt.
This completes the proof.Remark 10.
The passive condition presented in Theorem9 is delay dependent, and it is obtained by constructing an appropriate multiple Lyapunov functions. Some other methods dealing with time delay systems, such as delay divisioning method and input-output method, can also be utilized to investigate the systems under consideration, and this will be our further work.
When the matrix parameters in system (1) are determined, according to Theorem 9, we have the following result.Corollary 11.
Consider the nominal system in (1), it is stochastically passive under the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist symmetric positive matrices Pi, Qi, Ri and matrices Li, i∈N_, with appropriate dimensions such that
(27)[Γ11PiAdi0PiBvi-C1iTSiTPiAiTPi0*-(1-μ)Qi000AdiTPi0**Γ33000Li***-D1viT-D1vi0BviTPi0****-Pi00*****-Pi0******-Pi]<0,
where Γ11=Qi+τZi+AiTPi+PiAi, Γ33=τRi-LiT-Li.


## 3.2. DOF Passivity Analysis

In this subsection, we focus on designing a DOF passive controller for system (1) such that the corresponding closed-loop system is stochastically passive.Consider a full-order DOF passive controller with the following form:(28)Σk:{dx^(t)=Akσ(t)x^(t)dt+Bkσ(t)dy(t)u(t)=Ckσ(t)x^(t),
where x^∈Rn is the controller state; Aki, Bki, and Cki, i∈N_, are matrices to be determined.Applying this controller to system (1), we obtain the following closed-loop system:
(29)Σc:{dξ(t)=[A-ciξ(t)+A-cdiHξ(t-d(t))+Bcviv(t)]dt+S-ciξ(t)dw(t)z(t)=Cciξ(t)+Dcviv(t),
where
(30)ξT(t)=[xT(t)x^T(t)],A-ci=Aci+ΔAci(t),A-cdi=Acdi+ΔAcdi(t),S-ci=Sci+ΔSci(t),H=[I0],A-ci=[A-iBiCkiBkiC-2iAki],A-cdi=[A-diBkiC-2di],Bcvi=[BviBkiD2vi],S-ci=[S-iBkiG-i]H,Cci=[C1iD1iCki],Dcvi=D1vi,M~1i=[M1iBkiM2i],N~1i=N1i,N~2i=N2i,N~3i=N3i.
Then, the output feedback passive controller design problem can be converted into the problem of passivity analysis of the closed-loop system (29).Theorem 12.
Consider system (29), it is stochastically passive for the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist symmetric positive matrices Pi, Qi, Ri and matrices Lci, i∈N_, with appropriate dimensions such that the following matrix inequality holds:
(31)[Θ11PiA-cdi0PiBcvi-CciTS-ciTPiA-ciTPi0*-(1-μ)Qi000A-cdiTPi0**Θ33000Lci***-DcviT-Dcvi0BcviTPi0****-Pi00*****-Pi0******-Pi]<0,
where Θ11=HTQiH+τHTZiH+A-ciTPi+PiA-ci, Θ33=τ2HTRiH-LciT-Lci.Proof.
Letg(t):=A-ciξ(t)+A-cdiξ(t-d(t))+Bcviv(t) and ς(t)dt=dξ(t), then we have
(32)2ςT(t)Lci[(g(t)-ς(t))dt+S-ciξ(t)dw(t)]=0.
Choose the following Lyapunov functional candidate:
(33)Vi(ξ(t),t)=ξT(t)Piξ(t)+∫t-d(t)tξT(s)HTQiHξ(s)ds+∫-τ0∫t+θtξT(s)HTZiHξ(s)ds+∫-τ0∫t+θtςT(s)HTRiHς(s)ds.
From the relation between ξ(t) and x(t), we obtain
(34)Hξ(t)=x(t)Hξ(t-d(t))=x(t-d(t)).
Denote ζT(t)=[ξT(t)xT(t-d(t))ςT(t)vT(t)], then following the proof line of Theorem 9, it can be obtained form (31) that
(35)LVi(ξ(t),t)-2vT(t)z(t)<0.
It follows that
(36)E{dVi(ξ(t),t)dt}=E{LVi(ξ(t),t)}<2E{vT(t)z(t)}.
For tk≤s<t<tk+1, one obtains
(37)E{Vσ(k)(ξ(t),t)-Vσ(k)(ξ(s),s)}<∫stE{2vT(s)z(s)}ds.
Under the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))}, for tk≤t<tk+1, we obtain
(38)E{Vσ(tk)(ξ(t),t)-Vσ(t0)(ξ(0),0)}=E{∫0t1LVσ(t0)(ξ(s),s)ds+∫t1t2LVσ(t1)(ξ(s),s)ds+⋯∫tktVσ(tk)(ξ(t),t)ds}+E{∑g=1k(Vσ(tg)(ξ(tg),tg)-Vσ(tg-1)(ξ(tg),tg))}<∫0tE{2vT(s)z(s)}dt.
This completes the proof.


## 3.3. DOF Passivity Controller Design

In Theorem12, we have obtained a sufficient condition under which the closed-loop system ∑c is stochastically passive, but the designed parameters have not been given. The following theorem presents a controller design method and gives all parameters of the controller.Theorem 13.
Consider system (1), there exists a controller (28) such that the closed-loop system ∑c is stochastically passive for the switching signal σ(t)=min{i∣i=argmini∈N_Vi(x(t))} if there exist scalars ε>0, β>0, and positive definite symmetric matrices Ωi, Υi, Qi, Ri, Zi and matrices Xi, Yi, Ti, i∈N_, with appropriate dimensions such that, for all i∈N_,
(39)[Ψ11Ψ120Ψ14Ψ15Ψ160U1*Ψ22000Ψ260U2**2βψ000-βψU3***Ψ440Ψ4600****ψ00U1*****ψ00******ψU4*******V]<0,
where
(40)Ψ11=[AiΩi+BiYi+ΩiAiT+YiTBiTAi+TiT*ΥiAi+XiC2i+AiTΥi+C2iTXiT],Ψ12=[AdiΥiAdi+XiC2di],Ψ14=[Bvi-ΩiC1iT-YiTD1iTΥiBvi+XiD2vi-C1iT],Ψ15=[ΩiSiT0SiTSiTΥi+GiTXi],Ψ16=[ΩiAiT+YiTBiTTiTAiTAiTΥi+C2iTXiT],Ψ22=-(1-μ)Qi,Ψ44=-D1viT-D1vi,Ψ26=[AdiTAdiTΥi+C2diTXiT],Ψ46=[BviTBviTΥi+D2viTXiT],ψ=[-Ωi-I-I-Υi],U1=[M1i00ΩiΩiN1iTΩiN3iTΩiΩi0ΥiM1i+XiM2i000N1iTN3iTII0],U2=[0000N2iT0000],U3=[00000000Ωi00000000I],U4=[M1i00000000ΥiM1i+XiM2i0SiTΥi+GiTXiT000000],U5=[0M1i00000000ΥiM1i+XiM2i0000000],V=diag{-εI,-εI,-I,-I,-ε-1I,-ε-1I,-Qi-1,-τ-1Zi-1,-Ri-1}.
Moreover, if the above condition is feasible, then a DOF passive controller realization is given by
(41)Aki=Pi12-1(Ti-ΥiAiΩi-XiC2iΩi-ΥiBiYi)Wi12-TBki=Pi12-1Xi,Cki=YiWi12-T,
where Pi12 and Wi12 are any nonsingular matrices and satisfy
(42)ΩiΥi+Wi12Pi12T=I.Proof.
According to Theorem12, let the matrices Pi and Pi-1 be partitioned as
(43)Pi:=[Pi11Pi12*Pi22],Pi-1:=[Wi11Wi12*Wi22],
where Pi11∈Rn×Rn, Wi11∈Rn×Rn, and Pi12, Wi12 are nonsingular matrices. Then, define the following matrices:
(44)Ji:=[Wi11IWi12T0],J~i:=[IPi110Pi12T].
From Pi-1Pi=I, we have
(45)Wi11Pi11+Wi12Pi12T=I,Wi12TPi11+Wi22Pi12T=0,
and PiJi=J~i, Pi-1J~i=Ji.
By using Lemmas7 and 8 and Schur complement lemma, we can get that (31) is equivalent to the following inequality:
(46)[Σ11Σ120PiBcvi-CciTSciTPiAciTPi0PiM~1i0*Σ22000AcdiTPi000**Σ33000Lci00***-DcviT-Dcvi0BcviTPi000****-Pi00PiM~1i0*****-Pi000******-Pi0PiM~1i*******-εI0********-εI]<0,
where
(47)Σ11=HTQiH+τHTZiH+AciTPi+PiAci+εHT×(N1iTN1i+N3iTN3i)H,Σ12=PiAcdi+εHTN1iTN2i,Σ22=-(1-μ)Qi+εN2iTN2i,Σ33=τHTRiH-LiT-Li.
Pre- and post-multiplying (46) by matrices diag{JiT,I,JiT,I,JiT,JiT,JiT,I,I,I,I} and diag{Ji,I,Ji,I,Ji,Ji,Ji,I,I,I,I}, respectively, and denoting Li:=βPi, Ωi:=Wi11, Υi:=Pi11, Xi:=Pi12Bki, Yi:=CkiWi12T, and
(48)Ti:=(Pi11Ai+Pi12BkiC2i)Wi11+Pi11BiCkiWi12T+Pi12AkiWi12T,
we have
(49)[Ψ~11Ψ~120Ψ14Ψ~15Ψ160Ψ180*Ψ~22000Ψ26000**Ψ33000-βψ00***Ψ440Ψ46000****ψ00Ψ580*****ψ000******ψ0Ψ79*******-εI0********-εI]<0,
where
(50)Ψ~11=Ψ11+ε[ΩiI](Qi+τZi+N1iTN1i+N3iTN3i)[ΩiI],Ψ~12=Ψ12+ε[ΩiI]N1iTN2i,Ψ~22=Ψ22+εN2iTN2i,Ψ~15=[ΩiSiTΩiSiTΥi+ΩiGiTXiTSiTSiTΥi+GiTXiT],Ψ33=βψ+ε[ΩiI]Ri[ΩiI],Ψ18=Ψ58=Ψ79=[M1iΥiM1i+XiM2i].
Applying Schur complement lemma, one obtains(51)[Ψ~11Ψ~120Ψ14Ψ~15Ψ160Ψ180*Ψ~22000Ψ26000**Ψ33000-βψ00***Ψ440Ψ46000****ψ00Ψ580*****ψ000******ψ0Ψ79*******-εI0********-εI]+υTϑ+ϑTυ<0,
where
(52)υ=[Ωi0000000000000],ϑ=[0000000SiTΥi+GiTXiT000000].
Then by Lemma 7 and Schur complement lemma, we can obtain (39) and the parameters of the designed passive controller.Remark 14.
It should be noted that the dissipativity-based sliding mode control of switched stochastic delay-free systems was studied in [38]. However, the focus of our work is on DOF passive controller design for switched stochastic delay systems, and this is also the major contribution of the paper.Remark 15.
Theorem13 proposes a design method of the DOF passive controller for system (1). For the prescribed positive definite symmetric matrices Qi, Ri, and Zi, we can solve the inequality (39) easily by LMI Toolbox in MATLAB. However, when these matrices are unknown, the inequality is no longer a LMI. In this case, we can use an iterative algorithm involving a nonlinear optimization problem with LMI constraint in [20] to solve it.The procedure of the passive controller design can be concluded as follows:(1)
by solving LMI (39), we can get the feasible solution of the positive definite symmetric matrices Ωi, Υi, Qi, Ri, and Zi and any matrices Xi, Yi, and Ti.(2)
Applying singular value decomposition to (42), we can obtain the nonsingular matrices Pi12, Wi12.(3)
According to (41), we can calculate the controller parameters Aki, Bki, and Cki.


## 4. Numerical Example

In this section, we present a numerical example to show the effectiveness of the proposed design method in the previous section. Consider system (1) with parameters as follows:Subsystem 1.
(53)
A
1
=
[
0.2
0.3
0.1
0.5
]
,
A
d
1
=
[
0.1
0
0
0.4
]
,
B
1
=
[
0.2
0.6
]
,
B
v
1
=
[
0.3
-
0.1
]
,
S
1
=
[
-
0.4
-
1
-
1
0.2
]
,
C
21
=
[
0.1
-
0.4
-
0.2
0.6
]
,
C
2
d
1
=
[
0.3
-
0.2
-
0.4
0.5
]
,
D
2
v
1
=
[
0.1
-
0.5
]
,
G
d
1
=
[
-
0.2
0.5
-
0.1
0.3
]
,
C
11
=
[
0.1
0.8
]
,
D
11
=
0.5
,
D
1
v
1
=
0.4
,
M
11
=
M
21
=
diag
{
0.1
,
0.1
}
,
N
11
=
N
21
=
N
31
=
diag
{
0.2
,
0.2
}
,
F
1
=
diag
{
sin
(
0.01
t
)

,
sin
(
0.01
t
)
}
.Subsystem 2.
(54)
A
2
=
[
-
0.1
0.5
-
0.8
0.6
]
,
A
d
2
=
[
0.3
0
0
0.9
]
,
B
2
=
[
0.5
-
0.1
]
,
B
v
2
=
[
-
0.1
0.2
]
,
S
2
=
[
-
0.2
0
0
0.1
]
,
C
22
=
[
0.1
0
0.1
0.5
]
,
C
2
d
2
=
[
-
0.3
0
0
-
0.4
]
,
D
2
v
2
=
[
-
0.1
0.6
]
,
G
2
=
[
0.6
0.2
-
0.3
0.5
]
,
C
12
=
[
0.2
0.3
]
,
D
12
=
0.8
,
D
1
v
2
=
0.6
,
M
12
=
M
22
=
diag
{
0.2
,
0.2
}
,
N
12
=
N
22
=
N
32
=
diag
{
0.1
,
0.1
}
,
τ
=
0.5
.Chooseε1=ε2=1 and matrices Q1, Q2, R1, R2, Z1, and Z2 as follows:
(55)Q1=diag{0.5,0.5},Q2=diag{0.2,0.2},R1=diag{0.1,0.1},R2=diag{0.2,0.2},Z1=diag{0.1,0.1},Z2=diag{0.4,0.4}.In this case, solving (39) in Theorem 13, we can obtain
(56)Ω1=[0.25370.04070.04070.7178],Ω2=[0.53730.20540.20540.2056],Υ1=[0.2688-0.2717-0.27170.4615],Υ2=[1.6648-0.2134-0.21340.3725],X1=[0.34670.6160-0.1359-1.8058],X2=[-0.2034-0.2568-0.0305-0.6650],Y1=[-0.7717-3.4812],Y2=[-1.96390.2288],T1=[-0.50480.2014-0.3514-1.0718],T2=[-0.40010.6342-0.3454-1.4366].Then from (42), matrices P112, W112, P212, and W212 can be obtained:
(57)P112=[-0.9484-0.3170-0.31700.9484],W112=[-0.9101-0.2513-0.39010.5864],P212=[-0.29560.95530.95530.2956],W212=[-0.00770.15391.01210.0012].From (41), we can get the DOF passive controller parameters as follows:
(58)Ak1=[-0.0413-2.33492.1534-9.1815],Bk1=[-0.2857-0.0119-0.2388-1.9080],Ck1=[2.1013-4.5387],Ak2=[-1.3064-4.8026-0.07336.5615],Bk2=[0.0310-0.5594-0.2034-0.4419],Ck2=[0.2407-12.7494].Let the disturbance inputv(t)=0.02e-0.1tsin(t); simulation results are shown in Figures 1–3, where the initial states x(0)=[1-2]T and x^(0)=[00]T. The switching signal is depicted in Figure 1. Under the obtained passivity-based controller, trajectories of states x1 and x2 are shown in Figures 2 and 3, respectively. From simulation results, we can see that the designed passivity-based controller can guarantee that the corresponding closed-loop system is stable; this shows the effectiveness of the proposed method.Figure 1
Switching signal.Figure 2
The trajectory of statex1.Figure 3
The trajectory of statex2.


## 5. Conclusions

This paper has investigated the problem of DOF passive control for a class of switched stochastic systems with time-varying delay. By utilizing MSFs method, a delay-dependent sufficient condition for the existence of stochastic passivity was derived in terms of LMIs. In addition, a DOF passive controller design method was proposed. A numerical example was presented to illustrate the effectiveness of the proposed method.



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*Source: 281747-2013-04-11.xml*