# Dissipative Delay-Feedback Control for Nonlinear Stochastic Systems with Time-Varying Delay

Authors: Guici Chen; Jianzhong Zhou; Yongchuan Zhang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The dissipative delay-feedback control problems for nonlinear stochastic delay systems (NSDSs) based on dissipativity analysis are studied in this paper. Based on the Lyapunov stability theory and stochastic analysis technique,
both delay-independent and delay-dependent dissipativity criteria are established as linear matrix inequalities- (LMIs-) based feasibility tests. The obtained results in this paper for the nominal systems include the available results onH∞ approach and passivity for stochastic delay systems as special cases. The delay-dependent feedback controller is designed by considering the relationship among the time-varying delay, its lower and upper bound, and its differential
without ignoring any terms, which effectively reduces the conservative. A numerical example is given to illustrate the theoretical developments.


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## Body

## 1. Introduction

The stochastic differential systems appear as a natural description of many observed phenomena of real world, which have been come to play an important role in many fields including population dynamics, macroeconomics, chemical reactor control, communication network, image processes, and mobile robot localization. Therefore, the stability and stabilizability of nonlinear stochastic differential systems affine in the control have been studied in the past years by means of the stochastic Lyapunov theory [1–4]. As we all know, time delay in a control loop is one of the main sources of instability, oscillation, and poor performance and naturally encountered in a number of engineering control problems and physical systems. Therefore, time-delay systems [5, 6] and stochastic time-delay systems [7–12] have attracted many researchers’ attention and have been extensively studied. The problems include stability analysis [5, 6], stabilization problems [7, 8], and robust controller design [9–12].On the other hand, studying the dissipativity analysis and synthesis problems has a strong motivation due to their simplicity and effectiveness in dealing with robust and nonlinear systems. Since the notation of dissipative dynamical system was introduced by Willems [13], dissipative systems have been of particular interest to researchers in areas of systems, circuits, networks and control, and so forth. Moreover, passivity of a certain system in a feedback interconnected system will ensure the overall stability of that feedback system if uncertainties or nonlinearities can be characterized by a strict passive system. Hence, dissipative theory has wide ranging implications and applications in control theory. For instance, dissipativity was crucially used in the stability analysis of nonlinear system [14]; the theory of dissipative systems generalizes basic tools including the passivity theory, bounded real lemma, Kalman Yakubovich lemma, and the circle criterion [15]. Among the relevant topics are the dissipativity analysis and synthesis for time-delay systems [16, 17]. These results show that the dissipativity-based methods are highly effective in design the robust controller.Due to what is above mentioned, we believe that time delay is often harmful factor of systems. However, time delay is also surprising since plenty of studies have shown that time delay can also benefit the control, such as time-delay control. As mentioned previously, many results have been published about the control of systems with state delays but without input delays, which is called memoryless controllers, or to more general, memoryless controllers only includes an instantaneous feedback term. The time-delay control is an approach which gives a small delayh in the controller design, so as to reduce the effect of instability factor and exogenous disturbance. See, for example [18–20], and the references therein. Rather than adjusting control gains or identifying model parameters, its essential idea is to use past observations regarding both the control input and system response, which is an open problem now. In this paper, dissipative delay-feedback control problems for nonlinear stochastic systems with time-varying delay are studied based on dissipativity. The delay-dependent feedback controller is designed by considering the relationship among the time-varying delay, its lower and upper bound, and its differential without ignoring any terms, which effectively reduces the conservative.


## 2. Problems Statement and Preliminaries

In this paper, we consider the following nonlinear stochastic delayed systems (NSDSs) defined on a probability space(Ω,F,P):
(1)dx(t)={F(t,x(t),x(t-τ(t)))+Bu(t,t-τ(t))hhh+Dv(t)}dt+G(t,x(t),x(t-τ(t)))dω(t),z(t)=Cx(t)+Cdx(t-τ(t)),x(t)=φ(t),t∈[-τ2,0),
where x(t)∈Rn is the state vector; u(t,t-τ(t))∈Rm is the control input, which depends on not only the real time but also the delay; we call the controller memory controller; z(t)∈Rq is the control output; v(t)∈Rp is the exogenous disturbance input, which satisfies v(t)∈L2([0,∞),Rp), where L2([0,∞),Rp) is the space of nonanticipatory square-summable stochastic process with respect to (Ft)t>0 with the following norm: ∥v(t)∥22=E∫0∞∥v(t)∥2dt. ω(t)∈Rl is a scalar Brownian motion defined on a complete probability space (Ω,F,P) with E[dω(t)]=0, E[d2ω(t)]=dt.In the sequel, we seek to study the problems of dissipative analysis and delay-feedback control for the two cases of time delay.Case 1.
Time delay is a constantτ.Case 2.
τ
(
t
) is the time-varying delay, which is a differential function satisfying
(2)0≤τ1≤τ(t)≤τ2,τ˙(t)≤τd≤1,
where τ1, τ2, and τd are nonnegative constants.Remark 1.
Obviously, whenτd=0, τ(t)=τ1=τ2, that means the time delay is a constant; this case has been extensively studied. On the other hand, the time-varying delay τ(t)≥τ1; here τ1 is equal to or greater than 0, which has less conservativeness than τ(t)>0.Assumption 2.
F
(
·
,
·
,
·
) is a nonlinear vector function which can be decomposed as follows:
(3)F(t,x(t),x(t-τ(t)))=A0x(t)+f(t,x(t))+Adx(t-τ(t))+fd(t,x(t-τ(t))),
where f, fd are vector-valued functions; we assume
(4)∥f(t,x(t))∥≤β∥x(t)∥,(5)∥fd(t,x(t-τ(t)))∥≤βd∥x(t-τ(t))∥,
where β, βd are known real positive constants.Obviously, we know that(6)f(0,0)=0,fd(0,0)=0.Equivalently stated, condition (4) implies that there exists a scalar κ>0 such that
(7)κ(β2xT(t)x(t)-fT(t,x(t))f(t,x(t)))≥0.Similarly, condition (5) implies that there exists a scalar κd>0 such that
(8)κd(βd2xT(t-τ(t))x(t-τ(t))fdT(t,x(t-τ(t)))fd(t,x(t-τ(t)))hhh-fdT(t,x(t-τ(t)))fd(t,x(t-τ(t))))≥0.Assumption 3.
G
(
·
,
·
,
·
) is a nonlinear vector function which satisfies
(9)Trace(GT(t,x(t),x(t-τ(t)))G(t,x(t),x(t-τ(t))))≤xT(t)Θ1TΘ1x(t)+xT(t-τ(t))Θ2TΘ2x(t-τ(t)),
where Θ1, Θ2 are known real matrices.Hence, nonlinear stochastic delay systems (NSDSs) (1) can be rewritten as
(10)dx(t)={Ax(t)+f(t,x(t))+Adx(t-τ(t))hhhi+fd(t,x(t-τ(t)))hhhAx(t)+f(t,x(t))+Adx(t-τ(t))+Bu(t,t-τ(t))+Dv(t)}dt+G(t,x(t),x(t-τ(t)))dω(t),z(t)=Cx(t)+Cdx(t-τ(t)),x(t)=φ(t),t∈[-τ2,0).Definition 4 (see [21]).
Given matricesQT=Q≤0, RT=R≥0, and S, nonlinear stochastic delay systems (NSDSs) (10) are called (Q,S,R)-dissipative if, for some real function η(·), η(0)=0,
(11)E∫0T‍[zT(s)Qz(s)+2vT(s)Sz(s)+vT(s)Rv(s)]ds+η(x0)≥0,∀T≥0.
Furthermore, if, for a scalarα>0,
(12)E∫0T‍[zT(s)Qz(s)+2vT(s)Sz(s)+vT(s)Rv(s)]ds+η(x0)≥E∫0T‍αvT(s)v(s)ds,∀T≥0,NSDSs (10) are called strictly (Q,S,R)-dissipative.Lemma 5 (see [22]).
Given three constant matricesS1, S2, and S3, where S3=S3T<0 and S1=S1T<0, then S1-S2TS3-1S2<0 holds if and only if (S1S2S2TS3)<0 or (S3S2S2TS1)<0.Lemma 6 (see [23]).
For given positive symmetric matrixM=MT>0, two scalars a and b satisfying a<b, and vector function x(t):[a,b]→Rn, then
(13)[∫ab‍x(s)ds]TM[∫ab‍x(s)ds]≤(b-a)∫ab‍xT(s)Mx(s)ds.


## 3. Dissipativity Analysis for NSDSs

In this section, our primary purpose is to develope delay-independent and delay-dependent stochastically stability and dissipativity criteria for NSDSs (10) based on Definition 4.


### 3.1. Delay-Independent Dissipativity

In this sequel, we consider the time delay as unknown constant pertaining to Case1 and hence the results developed hereinafter will be independent of the size of delay. Without regard to the control input, setting u(t,t-τ(t))=0, then (10) can be rewritten as
(14)dx(t)={Ax(t)+f(t,x(t))+Adx(t-τ)hhh+fd(t,x(t-τ))+Dv(t)}dt+G(t,x(t),x(t-τ))dω(t),z(t)=Cx(t)+Cdx(t-τ),x(t)=φ(t),t∈[-τ,0).Theorem 7.
Consider the NSDSs (14). Given some scalars α>0, β>0, and βd>0 and matrices Q=QT≤0, R=RT>0, and S, suppose there exist matrices P=PT>0, W=WT>0 and positive scalars κ>0, κd>0 such that the following LMI holds:
(15)(Σ1PAdΣ2PPCTQ*Σ3-CdTS00CdTQ**-Rα000***-κI00****-κdI0*****Q)<0;
then the NSDSs (14) are strictly (Q,S,R)-dissipative independent of delay, where Σ1=ATP+PA+W+Θ1TPΘ1+κβ2I, Σ2=PD-CTS, Σ3=Θ2TPΘ2-W+κdβd2I, and Rα=(R-αI).Proof.
At first we introduce the following Lyapunov-Krasovskii functional (LKF):(16)V(xt)=xT(t)Px(t)+∫t-τt‍xT(s)Wx(s)ds.
Evaluating the Itô derivative of V(xt) along the solution of NSDSs (14), we have
(17)LV(xt)=2xTP[Ax(t)+f(t,x(t))+Adx(t-τ)fd(t,x(t-τ))+Dv(t)hhhhh+fd(t,x(t-τ))+Dv(t)]+xTWx-xT(t-τ)Wx(t-τ)+GT(t,x(t),x(t-τ))PG(t,x(t),x(t-τ)).
Noting (7)–(9), we obtain
(18)LV(xt)≤LV(xt)+κ(β2xT(t)x(t)-fT(t,x(t))f(t,x(t)))+κd(βd2xT(t-τ(t))x(t-τ(t))fd(t,x(t-τ(t)))hhhhhhhhhh-fdT(t,x(t-τ(t)))hhhhhhhhhh×fd(t,x(t-τ(t)))βd2xT)≤ξTΣξ,
where
(19)ξ=(xT(t),xT(t-τ),vT(t),fT(t,x(t)),fdThhfdT(t,x(t-τ)))T,Σ=(Σ1PAdPDPP*Σ3000**000***-κI0****-kdI).
Hence, we have(20)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤ξTΣ~ξ,
where
(21)Σ~=Σ-(CTQCCTQCdTCTS00*CdTQCdCdTS00**(R-αI)00***00****0).
According to Lemma5 and applying the congruent transformation, we know that
(22)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤0.
Then, integrating (22) from 0 to T and taking mathematical expectation, we obtain that (12) holds which completes the proof.Remark 8.
WhenQ=-I, S=0, and Rα=γ2I, strictly (Q,S,R)-dissipativity reduces to the H∞ performance level. When Q=0, S=I, and Rα=γI, strictly (Q,S,R)-dissipativity reduces to the strictly passivity.So the following corollaries stand out as special cases.Corollary 9.
Consider the NSDSs (14). Given some scalars γ>0, β>0, and βd>0, suppose there exist matrices P=PT>0, W=WT>0 and positive scalars κ>0, κd>0 such that the following LMI holds:
(23)(Σ1PAPDPP-CT*Σ3000-CdT**-γ2I000***-κI00****-κdI0*****-I)<0;
then the NSDSs (14) are stochastically asymptotically stable and independent of delay with disturbance level γ.Corollary 10.
Consider the NSDSs (14). Given some scalars γ>0, β>0, and βd>0, suppose there exist matrices P=PT>0, W=WT>0 and positive scalars κ>0, κd>0 such that the following LMI holds:
(24)(Σ1PAPD-CTPP*Σ3-CdT00**-γI00***-κI0****-κdI)<0;
then the NSDSs (14) are strictly passive independent of delay.


### 3.2. Delay-Dependent Dissipativity

We now direct attention to the type of dissipativity which depends on the time-varying delay, which pertains to Case2. Without consideration of the control input, and defining a new state variable
(25)y(t)=Ax(t)+f(t,x(t))+Adx(t-τ(t))+fd(t,x(t-τ(t)))+Dv(t),
then, the NSDSs (10) can be rewritten as
(26)dx(t)=y(t)dt+G(t,x(t),x(t-τ(t)))dω(t),z(t)=Cx(t)+Cdx(t-τ(t)),x(t)=φ(t),t∈[-τ2,0).Theorem 11.
Consider the NSDSs (26). For the given scalars α>0, τ2>τ1≥0, τd>0, β>0, and βd>0, the NSDSs (26) are strictly (Q,S,R)-dissipative for all time-varying delays if there exist symmetric positive-definite matrices P, V1, V2, V3, W1, W2, Z1, and Z2; any appropriately dimensioned matrices N, M, F, and H; and two positive scalars κ>0, κd>0 such that the following LMIs hold:
(27)(Ξ-τ2M^*-τ2W1)<0,(Ξ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))<0,(Ξ-(τ2-τ1)F^*-(τ2-τ1)W2)<0,
where
(28)Ξ=(Ξ11Ξ12Ξ13Ξ14ATHΞ16*Ξ22-CdTS0AdTHΞ26**-Rα0DTH0***Ξ44Ξ450****Ξ550*****Ξ66),Ξ11=PA+ATP+V1+V2+V3+τ2Z1+(τ2-τ1)Z2+M+MT+κβ2I,Ξ12=N-F-M+PAd,Ξ13=PD-CTS,Ξ14=(F-NPP),Ξ16=(CTQΘ1P0),Ξ22=-(1-τd)V2+κdβd2I,Ξ26=(CdTQ0Θ2P),Ξ44=diag⁡{-V1,-V3,-κI,-κdI},Ξ55=τ2W1+(τ2-τ1)W2-H-HT,Ξ66=diag⁡{Q,-P,-P},Ξ45=(00HTHT)T,N^=(NT,01×10)T,M^=(MT,01×10)T,F^=(FT,01×10)T.Proof.
We construct a LKF as follows:(29)V(xt)=V1(xt)+V2(xt)+V3(xt)+V4(xt),
where
(30)V1(xt)=xT(t)Px(t),V2(xt)=∫t-τ1t‍xT(s)V1x(s)ds+∫t-τ(t)t‍xT(s)V2x(s)ds+∫t-τ2t‍xT(s)V3x(s)ds,V3(xt)=∫-τ20‍∫t+θt‍yT(s)W1y(s)ds+∫-τ2-τ1‍∫t+θt‍yT(s)W2y(s)ds,V4(xt)=∫-τ20‍∫t+θt‍xT(s)Z1x(s)ds+∫-τ2-τ1‍∫t+θt‍xT(s)Z2x(s)ds.
Then, the weak infinitesimal operatorL of the stochastic process xt along the evolution of V(xt) is given by
(31)LV(xt)=LV1(xt)+LV2(xt)+LV3(xt)+LV4(xt),
where
(32)LV1(xt)≤2xT(t)P[Ax(t)+f(t,x(t))fdhhhhhhhh+Adx(t-τ(t))hhhhhhhh+fd(t,x(t-τ(t)))+Dv(t)]+xT(t)Θ1TPΘ1x(t)+x(t-τ(t))Θ2TPΘ2x(t-τ(t)),(33)LV2(xt)≤xT(t)(V1+V2+V3)x(t)-xT(t-τ1)V1x(t-τ1)-xT(t-τ2)V3x(t-τ2)-xT(t-τ(t))(1-τd)V2x(t-τ(t)),(34)LV3(xt)=τ2yT(t)W1y(t)-∫t-τ2t‍yT(s)W1y(s)ds+(τ2-τ1)yT(t)W2y(t)-∫t-τ2t-τ1‍yT(s)W2y(s)ds.
By using Lemma6, we get the following inequality:
(35)LV4(xt)≤τ2xT(t)Z1x(t)+(τ2-τ1)xT(t)Z2x(t)-1τ2(∫t-τ(t)t‍x(s)ds)TZ1(∫t-τ(t)t‍x(s)ds)-1τ2-τ1(∫t-τ2t-τ(t)‍x(s)ds)TZ2(∫t-τ2t-τ(t)‍x(s)ds)-1τ2-τ1(∫t-τ(t)t-τ1‍x(s)ds)TZ2(∫t-τ(t)t-τ1‍x(s)ds),
settingξ(t)=(xT(t),xT(t-τ(t)),vT(t),xT(t-τ1),xT(t-τ2),fT(x(t)),fdT(x(t-τ(t))),yT(t),(∫t-τ(t)t‍x(s)ds)T,(∫t-τ2t-τ(t)‍x(s)ds)T,(∫t-τ(t)t-τ1‍x(s)ds)T)T. And we introduce the following four zero equations:
(36)2xT(t)M[x(t)-x(t-τ(t))-∫t-τ(t)t‍dx(s)]=0,(37)2xT(t)N[x(t-τ(t))-x(t-τ2)-∫t-τ2t-τ(t)‍dx(s)]=0,(38)2xT(t)F[x(t-τ1)-x(t-τ(t))-∫t-τ(t)t-τ1‍dx(s)]=0,(39)2yT(t)H[Ax(t)+f(t,x(t))+Adx(t-τ(t))hhhhhhhh+fd(t,x(t-τ(t)))+Dv(t)-y(t)]=0.
Summing up (31)–(39), we obtain
(40)LV(xt)≤LV(xt)+κ(β2xT(t)x(t)-fT(t,x(t))f(t,x(t)))+κd(βd2xT(t-τ(t))x(t-τ(t))hhhhh-fdT(t,x(t-τ(t)))fd(t,x(t-τ(t))))+2xT(t)M[x(t)-x(t-τ(t))∫t-τ(t)t‍hhhhhhhhhhh-∫t-τ(t)t‍y(s)ds]-2xT(t)M∫t-τ(t)t‍G(xs)dω(s)+2xT(t)N[x(t-τ(t))-x(t-τ2)∫t-τ2t-τ(t)‍hhhhhhhhhhh-∫t-τ2t-τ(t)‍y(s)ds]-2xT(t)N∫t-τ2t-τ(t)‍G(xs)dω(s)+2xT(t)F[x(t-τ1)-x(t-τ(t))∫t-τ(t)t-τ1‍hhhhhhhhhhh-∫t-τ(t)t-τ1‍y(s)ds]-2xT(t)F∫t-τ(t)t-τ1‍G(xs)dω(s)+2yT(t)H[Ax(t)+f(t,x(t))hhhhhhhhhh+Adx(t-τ(t))hhhhhhhhhh+fd(t,x(t-τ(t)))hhhhhhhhhh+Dv(t)-y(t)],
where M, N, F, and H are matrices with appropriate dimensions. Hence,
(41)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤ξT(t)Φξ(t)-2xT(t)M×[∫t-τ(t)t‍y(s)ds+∫t-τ(t)t‍G(xs)dω(s)]-2xT(t)N[∫t-τ2t-τ(t)‍y(s)ds+∫t-τ2t-τ(t)‍G(xs)dω(s)]-2xT(t)F[∫t-τ(t)t-τ1‍y(s)ds+∫t-τ(t)t-τ1‍G(xs)dω(s)]-∫t-τ2t‍yT(s)W1y(s)ds-∫t-τ2t-τ1‍yT(s)W2y(s)ds,
where
(42)Φ=(Φ11Φ12Φ13Φ14ATH0*Φ22-CdTS0AdTH0**-Rα0DTH0***Φ44Φ450****Φ550*****Φ66),Φ11=PA+ATP+Θ1TPΘ1+V1+V2+V3+τ2Z1+(τ2-τ1)Z2+M+MT-CTQC+κβ2I,Φ12=N-F-M+PAd-CTQCd,Φ13=PD-CTS,Φ14=(F-NPP),Φ22=Θ2TPΘ2-CdTQCd-(1-τd)V2+κdβd2I,Φ44=diag⁡{-V1,-V3,-κI,-κdI},Φ45=(00HTHT)T,Φ55=τ2W1+(τ2-τ1)W2-H-HT,Φ66=diag⁡{-1τ2Z1,-1τ2-τ1Z2,-1τ2-τ1Z2}.
So it easy to obtain that
(43)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤1τ2∫t-τ(t)t‍ηT(t,s)(Φ-τ2M^*-τ2W1)η(t,s)ds+1τ2-τ1∫t-τ(t)t‍ηT(t,s)hhhhhhhhhhhhh×(Φ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))η(t,s)ds+1τ2-τ1∫t-τ(t)t‍ηT(t,s)(Φ-(τ2-τ1)F^*-(τ2-τ1)W2)η(t,s)ds-2xT(t)M∫t-τ(t)t‍G(xs)dω(s)-2xT(t)N∫t-τ2t-τ(t)‍G(xs)dω(s)-2xT(t)F∫t-τ(t)t-τ1‍G(xs)dω(s),
where ηT(t,s)=[ξT(t),yT(s)].
By Lemma5 and applying the congruent transformation to (27), it follows that
(44)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤-2xT(t)M∫t-τ(t)t‍G(xs)dω(s)-2xT(t)N∫t-τ2t-τ(t)‍G(xs)dω(s)-2xT(t)F∫t-τ(t)t-τ1‍G(xs)dω(s).
Then, integrating both sides of (44) from 0 to T and taking mathematical expectation, we obtain that (12) holds, which completes the proof.Similarly, as the special case, we can easily obtain the following corollaries.Corollary 12.
Consider the NSDSs (26). For the given scalars γ>0, τ2>τ1≥0, τd>0, β>0, and βd>0, the NSDSs (26) are stochastically asymptotically stable with H∞ performance level for all time-varying delays if there exist symmetric positive-definite matrices P, V1, V2, V3, W1, W2, Z1, and Z2; any appropriately dimensioned matrices N, M, F, and H; and two positive scalars κ>0, κd>0 such that the following LMIs hold:
(45)(Ξ-τ2M^*-τ2W1)<0,(Ξ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))<0,(Ξ-(τ2-τ1)F^*-(τ2-τ1)W2)<0,
where
(46)Ξ=(Ξ11Ξ12PDΞ14ATHΞ16*Ξ2200AdTHΞ26**-γ2I0DTH0***Ξ44Ξ450****Ξ550*****Ξ66),Ξ16=(-CTΘ1P0),Ξ26=(-CdT0Θ2P),Ξ66=diag⁡{-I,-P,-P},Ξ11,Ξ12,Ξ14,Ξ22,Ξ44,Ξ45,Ξ55,
defined in Theorem 11.Corollary 13.
Consider the NSDSs (26). For the given scalars γ>0, τ2>τ1≥0, τd>0, β>0, and βd>0, the NSDSs (26) are strictly (Q,S,R)-passive for all time-varying delays if there exist symmetric positive-definite matrices P, V1, V2, V3, W1, W2, Z1, Z2; any appropriately dimensioned matrices N, M, F, and H; and two positive scalars κ>0, κd>0 such that the following LMIs hold:
(47)(Ξ-τ2M^*-τ2W1)<0,(Ξ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))<0,(Ξ-(τ2-τ1)F^*-(τ2-τ1)W2)<0,
where
(48)Ξ=(Ξ11Ξ12PD-CTΞ14ATHΞ16*Ξ22-CdT0AdTHΞ26**-γI0DTH0***Ξ44Ξ450****Ξ550*****Ξ66),Ξ16=(Θ1P0),Ξ26=(0Θ2P),Ξ66=diag⁡{-P,-P},N^=(NT,01×9)T,M^=(MT,01×9)T,F^=(FT,01×9)T,Ξ11,Ξ12,Ξ14,Ξ22,Ξ44,Ξ45,Ξ55
defined in Theorem 11.


## 3.1. Delay-Independent Dissipativity

In this sequel, we consider the time delay as unknown constant pertaining to Case1 and hence the results developed hereinafter will be independent of the size of delay. Without regard to the control input, setting u(t,t-τ(t))=0, then (10) can be rewritten as
(14)dx(t)={Ax(t)+f(t,x(t))+Adx(t-τ)hhh+fd(t,x(t-τ))+Dv(t)}dt+G(t,x(t),x(t-τ))dω(t),z(t)=Cx(t)+Cdx(t-τ),x(t)=φ(t),t∈[-τ,0).Theorem 7.
Consider the NSDSs (14). Given some scalars α>0, β>0, and βd>0 and matrices Q=QT≤0, R=RT>0, and S, suppose there exist matrices P=PT>0, W=WT>0 and positive scalars κ>0, κd>0 such that the following LMI holds:
(15)(Σ1PAdΣ2PPCTQ*Σ3-CdTS00CdTQ**-Rα000***-κI00****-κdI0*****Q)<0;
then the NSDSs (14) are strictly (Q,S,R)-dissipative independent of delay, where Σ1=ATP+PA+W+Θ1TPΘ1+κβ2I, Σ2=PD-CTS, Σ3=Θ2TPΘ2-W+κdβd2I, and Rα=(R-αI).Proof.
At first we introduce the following Lyapunov-Krasovskii functional (LKF):(16)V(xt)=xT(t)Px(t)+∫t-τt‍xT(s)Wx(s)ds.
Evaluating the Itô derivative of V(xt) along the solution of NSDSs (14), we have
(17)LV(xt)=2xTP[Ax(t)+f(t,x(t))+Adx(t-τ)fd(t,x(t-τ))+Dv(t)hhhhh+fd(t,x(t-τ))+Dv(t)]+xTWx-xT(t-τ)Wx(t-τ)+GT(t,x(t),x(t-τ))PG(t,x(t),x(t-τ)).
Noting (7)–(9), we obtain
(18)LV(xt)≤LV(xt)+κ(β2xT(t)x(t)-fT(t,x(t))f(t,x(t)))+κd(βd2xT(t-τ(t))x(t-τ(t))fd(t,x(t-τ(t)))hhhhhhhhhh-fdT(t,x(t-τ(t)))hhhhhhhhhh×fd(t,x(t-τ(t)))βd2xT)≤ξTΣξ,
where
(19)ξ=(xT(t),xT(t-τ),vT(t),fT(t,x(t)),fdThhfdT(t,x(t-τ)))T,Σ=(Σ1PAdPDPP*Σ3000**000***-κI0****-kdI).
Hence, we have(20)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤ξTΣ~ξ,
where
(21)Σ~=Σ-(CTQCCTQCdTCTS00*CdTQCdCdTS00**(R-αI)00***00****0).
According to Lemma5 and applying the congruent transformation, we know that
(22)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤0.
Then, integrating (22) from 0 to T and taking mathematical expectation, we obtain that (12) holds which completes the proof.Remark 8.
WhenQ=-I, S=0, and Rα=γ2I, strictly (Q,S,R)-dissipativity reduces to the H∞ performance level. When Q=0, S=I, and Rα=γI, strictly (Q,S,R)-dissipativity reduces to the strictly passivity.So the following corollaries stand out as special cases.Corollary 9.
Consider the NSDSs (14). Given some scalars γ>0, β>0, and βd>0, suppose there exist matrices P=PT>0, W=WT>0 and positive scalars κ>0, κd>0 such that the following LMI holds:
(23)(Σ1PAPDPP-CT*Σ3000-CdT**-γ2I000***-κI00****-κdI0*****-I)<0;
then the NSDSs (14) are stochastically asymptotically stable and independent of delay with disturbance level γ.Corollary 10.
Consider the NSDSs (14). Given some scalars γ>0, β>0, and βd>0, suppose there exist matrices P=PT>0, W=WT>0 and positive scalars κ>0, κd>0 such that the following LMI holds:
(24)(Σ1PAPD-CTPP*Σ3-CdT00**-γI00***-κI0****-κdI)<0;
then the NSDSs (14) are strictly passive independent of delay.


## 3.2. Delay-Dependent Dissipativity

We now direct attention to the type of dissipativity which depends on the time-varying delay, which pertains to Case2. Without consideration of the control input, and defining a new state variable
(25)y(t)=Ax(t)+f(t,x(t))+Adx(t-τ(t))+fd(t,x(t-τ(t)))+Dv(t),
then, the NSDSs (10) can be rewritten as
(26)dx(t)=y(t)dt+G(t,x(t),x(t-τ(t)))dω(t),z(t)=Cx(t)+Cdx(t-τ(t)),x(t)=φ(t),t∈[-τ2,0).Theorem 11.
Consider the NSDSs (26). For the given scalars α>0, τ2>τ1≥0, τd>0, β>0, and βd>0, the NSDSs (26) are strictly (Q,S,R)-dissipative for all time-varying delays if there exist symmetric positive-definite matrices P, V1, V2, V3, W1, W2, Z1, and Z2; any appropriately dimensioned matrices N, M, F, and H; and two positive scalars κ>0, κd>0 such that the following LMIs hold:
(27)(Ξ-τ2M^*-τ2W1)<0,(Ξ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))<0,(Ξ-(τ2-τ1)F^*-(τ2-τ1)W2)<0,
where
(28)Ξ=(Ξ11Ξ12Ξ13Ξ14ATHΞ16*Ξ22-CdTS0AdTHΞ26**-Rα0DTH0***Ξ44Ξ450****Ξ550*****Ξ66),Ξ11=PA+ATP+V1+V2+V3+τ2Z1+(τ2-τ1)Z2+M+MT+κβ2I,Ξ12=N-F-M+PAd,Ξ13=PD-CTS,Ξ14=(F-NPP),Ξ16=(CTQΘ1P0),Ξ22=-(1-τd)V2+κdβd2I,Ξ26=(CdTQ0Θ2P),Ξ44=diag⁡{-V1,-V3,-κI,-κdI},Ξ55=τ2W1+(τ2-τ1)W2-H-HT,Ξ66=diag⁡{Q,-P,-P},Ξ45=(00HTHT)T,N^=(NT,01×10)T,M^=(MT,01×10)T,F^=(FT,01×10)T.Proof.
We construct a LKF as follows:(29)V(xt)=V1(xt)+V2(xt)+V3(xt)+V4(xt),
where
(30)V1(xt)=xT(t)Px(t),V2(xt)=∫t-τ1t‍xT(s)V1x(s)ds+∫t-τ(t)t‍xT(s)V2x(s)ds+∫t-τ2t‍xT(s)V3x(s)ds,V3(xt)=∫-τ20‍∫t+θt‍yT(s)W1y(s)ds+∫-τ2-τ1‍∫t+θt‍yT(s)W2y(s)ds,V4(xt)=∫-τ20‍∫t+θt‍xT(s)Z1x(s)ds+∫-τ2-τ1‍∫t+θt‍xT(s)Z2x(s)ds.
Then, the weak infinitesimal operatorL of the stochastic process xt along the evolution of V(xt) is given by
(31)LV(xt)=LV1(xt)+LV2(xt)+LV3(xt)+LV4(xt),
where
(32)LV1(xt)≤2xT(t)P[Ax(t)+f(t,x(t))fdhhhhhhhh+Adx(t-τ(t))hhhhhhhh+fd(t,x(t-τ(t)))+Dv(t)]+xT(t)Θ1TPΘ1x(t)+x(t-τ(t))Θ2TPΘ2x(t-τ(t)),(33)LV2(xt)≤xT(t)(V1+V2+V3)x(t)-xT(t-τ1)V1x(t-τ1)-xT(t-τ2)V3x(t-τ2)-xT(t-τ(t))(1-τd)V2x(t-τ(t)),(34)LV3(xt)=τ2yT(t)W1y(t)-∫t-τ2t‍yT(s)W1y(s)ds+(τ2-τ1)yT(t)W2y(t)-∫t-τ2t-τ1‍yT(s)W2y(s)ds.
By using Lemma6, we get the following inequality:
(35)LV4(xt)≤τ2xT(t)Z1x(t)+(τ2-τ1)xT(t)Z2x(t)-1τ2(∫t-τ(t)t‍x(s)ds)TZ1(∫t-τ(t)t‍x(s)ds)-1τ2-τ1(∫t-τ2t-τ(t)‍x(s)ds)TZ2(∫t-τ2t-τ(t)‍x(s)ds)-1τ2-τ1(∫t-τ(t)t-τ1‍x(s)ds)TZ2(∫t-τ(t)t-τ1‍x(s)ds),
settingξ(t)=(xT(t),xT(t-τ(t)),vT(t),xT(t-τ1),xT(t-τ2),fT(x(t)),fdT(x(t-τ(t))),yT(t),(∫t-τ(t)t‍x(s)ds)T,(∫t-τ2t-τ(t)‍x(s)ds)T,(∫t-τ(t)t-τ1‍x(s)ds)T)T. And we introduce the following four zero equations:
(36)2xT(t)M[x(t)-x(t-τ(t))-∫t-τ(t)t‍dx(s)]=0,(37)2xT(t)N[x(t-τ(t))-x(t-τ2)-∫t-τ2t-τ(t)‍dx(s)]=0,(38)2xT(t)F[x(t-τ1)-x(t-τ(t))-∫t-τ(t)t-τ1‍dx(s)]=0,(39)2yT(t)H[Ax(t)+f(t,x(t))+Adx(t-τ(t))hhhhhhhh+fd(t,x(t-τ(t)))+Dv(t)-y(t)]=0.
Summing up (31)–(39), we obtain
(40)LV(xt)≤LV(xt)+κ(β2xT(t)x(t)-fT(t,x(t))f(t,x(t)))+κd(βd2xT(t-τ(t))x(t-τ(t))hhhhh-fdT(t,x(t-τ(t)))fd(t,x(t-τ(t))))+2xT(t)M[x(t)-x(t-τ(t))∫t-τ(t)t‍hhhhhhhhhhh-∫t-τ(t)t‍y(s)ds]-2xT(t)M∫t-τ(t)t‍G(xs)dω(s)+2xT(t)N[x(t-τ(t))-x(t-τ2)∫t-τ2t-τ(t)‍hhhhhhhhhhh-∫t-τ2t-τ(t)‍y(s)ds]-2xT(t)N∫t-τ2t-τ(t)‍G(xs)dω(s)+2xT(t)F[x(t-τ1)-x(t-τ(t))∫t-τ(t)t-τ1‍hhhhhhhhhhh-∫t-τ(t)t-τ1‍y(s)ds]-2xT(t)F∫t-τ(t)t-τ1‍G(xs)dω(s)+2yT(t)H[Ax(t)+f(t,x(t))hhhhhhhhhh+Adx(t-τ(t))hhhhhhhhhh+fd(t,x(t-τ(t)))hhhhhhhhhh+Dv(t)-y(t)],
where M, N, F, and H are matrices with appropriate dimensions. Hence,
(41)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤ξT(t)Φξ(t)-2xT(t)M×[∫t-τ(t)t‍y(s)ds+∫t-τ(t)t‍G(xs)dω(s)]-2xT(t)N[∫t-τ2t-τ(t)‍y(s)ds+∫t-τ2t-τ(t)‍G(xs)dω(s)]-2xT(t)F[∫t-τ(t)t-τ1‍y(s)ds+∫t-τ(t)t-τ1‍G(xs)dω(s)]-∫t-τ2t‍yT(s)W1y(s)ds-∫t-τ2t-τ1‍yT(s)W2y(s)ds,
where
(42)Φ=(Φ11Φ12Φ13Φ14ATH0*Φ22-CdTS0AdTH0**-Rα0DTH0***Φ44Φ450****Φ550*****Φ66),Φ11=PA+ATP+Θ1TPΘ1+V1+V2+V3+τ2Z1+(τ2-τ1)Z2+M+MT-CTQC+κβ2I,Φ12=N-F-M+PAd-CTQCd,Φ13=PD-CTS,Φ14=(F-NPP),Φ22=Θ2TPΘ2-CdTQCd-(1-τd)V2+κdβd2I,Φ44=diag⁡{-V1,-V3,-κI,-κdI},Φ45=(00HTHT)T,Φ55=τ2W1+(τ2-τ1)W2-H-HT,Φ66=diag⁡{-1τ2Z1,-1τ2-τ1Z2,-1τ2-τ1Z2}.
So it easy to obtain that
(43)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤1τ2∫t-τ(t)t‍ηT(t,s)(Φ-τ2M^*-τ2W1)η(t,s)ds+1τ2-τ1∫t-τ(t)t‍ηT(t,s)hhhhhhhhhhhhh×(Φ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))η(t,s)ds+1τ2-τ1∫t-τ(t)t‍ηT(t,s)(Φ-(τ2-τ1)F^*-(τ2-τ1)W2)η(t,s)ds-2xT(t)M∫t-τ(t)t‍G(xs)dω(s)-2xT(t)N∫t-τ2t-τ(t)‍G(xs)dω(s)-2xT(t)F∫t-τ(t)t-τ1‍G(xs)dω(s),
where ηT(t,s)=[ξT(t),yT(s)].
By Lemma5 and applying the congruent transformation to (27), it follows that
(44)LV(xt)-zT(s)Qz(s)-2vT(s)Sz(s)-vT(s)(R-αI)v(s)≤-2xT(t)M∫t-τ(t)t‍G(xs)dω(s)-2xT(t)N∫t-τ2t-τ(t)‍G(xs)dω(s)-2xT(t)F∫t-τ(t)t-τ1‍G(xs)dω(s).
Then, integrating both sides of (44) from 0 to T and taking mathematical expectation, we obtain that (12) holds, which completes the proof.Similarly, as the special case, we can easily obtain the following corollaries.Corollary 12.
Consider the NSDSs (26). For the given scalars γ>0, τ2>τ1≥0, τd>0, β>0, and βd>0, the NSDSs (26) are stochastically asymptotically stable with H∞ performance level for all time-varying delays if there exist symmetric positive-definite matrices P, V1, V2, V3, W1, W2, Z1, and Z2; any appropriately dimensioned matrices N, M, F, and H; and two positive scalars κ>0, κd>0 such that the following LMIs hold:
(45)(Ξ-τ2M^*-τ2W1)<0,(Ξ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))<0,(Ξ-(τ2-τ1)F^*-(τ2-τ1)W2)<0,
where
(46)Ξ=(Ξ11Ξ12PDΞ14ATHΞ16*Ξ2200AdTHΞ26**-γ2I0DTH0***Ξ44Ξ450****Ξ550*****Ξ66),Ξ16=(-CTΘ1P0),Ξ26=(-CdT0Θ2P),Ξ66=diag⁡{-I,-P,-P},Ξ11,Ξ12,Ξ14,Ξ22,Ξ44,Ξ45,Ξ55,
defined in Theorem 11.Corollary 13.
Consider the NSDSs (26). For the given scalars γ>0, τ2>τ1≥0, τd>0, β>0, and βd>0, the NSDSs (26) are strictly (Q,S,R)-passive for all time-varying delays if there exist symmetric positive-definite matrices P, V1, V2, V3, W1, W2, Z1, Z2; any appropriately dimensioned matrices N, M, F, and H; and two positive scalars κ>0, κd>0 such that the following LMIs hold:
(47)(Ξ-τ2M^*-τ2W1)<0,(Ξ-(τ2-τ1)N^*-(τ2-τ1)(W1+W2))<0,(Ξ-(τ2-τ1)F^*-(τ2-τ1)W2)<0,
where
(48)Ξ=(Ξ11Ξ12PD-CTΞ14ATHΞ16*Ξ22-CdT0AdTHΞ26**-γI0DTH0***Ξ44Ξ450****Ξ550*****Ξ66),Ξ16=(Θ1P0),Ξ26=(0Θ2P),Ξ66=diag⁡{-P,-P},N^=(NT,01×9)T,M^=(MT,01×9)T,F^=(FT,01×9)T,Ξ11,Ξ12,Ξ14,Ξ22,Ξ44,Ξ45,Ξ55
defined in Theorem 11.


## 4. Dissipative Delay-Feedback Control for NSDSs

Extending on the results of the foregoing section, our aim is to develope an LMIs-based solution to the problem of designing a delay-feedback controller as(49)u(t,t-τ(t))=K0x(t)+K1x(t-τ(t)),
which will render the NSDSs (10) strictly (Q,S,R)-dissipative. The closed-loop systems is now described by
(50)dx(t)={A~x(t)+f(t,x(t))+A~dx(t-τ(t))hhhA~d+fd(t,x(t-τ(t)))+Dv(t)}dt+G(t,x(t),x(t-τ(t)))dω(t),z(t)=Cx(t)+Cdx(t-τ(t)),x(t)=φ(t),t∈[-τ2,0),
where A~=A+BK0, A~d=Ad+BK1.Applying Theorem7, together with Lemma 5 and congruent transformation, we can get the following theorem without detailed proofs.Theorem 14.
Consider the NSDSs (50). Given some scalars α>0, β>0, βd>0, κ>0, and κd>0 and matrices Q=QT≤0, R=RT>0, and S, suppose there exist matrices X=XT>0, W=WT>0, Y, and Yd such that the following LMI holds:
(51)(Γ1AdX+BYdD-XCTSΓ2*-W~-XCdTSΓ3**-Rα0***Γ4)<0;
then the NSDSs (50) are strictly (Q,S,R)-dissipative independent of delay; the feedback gain is K0=YX-1, K1=YdX-1, where Γ1=XAT+AX+YTBT+BY+W~, Γ2=(IIXCTQXΘ10κβX0), Γ3=(00XCdTQ0XΘ20κdβdX), and Γ4=diag⁡(-κI,-κdI,Q,-X,-X,-κI,-κdI).Similarly, applying Theorem11, together with Lemma 5 and congruent transformation, we can obtain the following theorem without detailed proofs.Theorem 15.
Consider the NSDSs (50). For the given scalars α>0, τ2>τ1≥0, τd>0, κ>0, κd>0, β>0, and βd>0, the NSDSs (50) are strictly (Q,S,R)-dissipative for all time-varying delays and the feedback gain is K0=YX-1, K1=YdX-1, if there exist symmetric positive-definite matrices X, V~1, V~2, V~3, W~1, W~2, Z~1, and Z~2 and any appropriately dimensioned matrices N~, M~, F~, Y, and Yd such that the following LMIs hold:
(52)(Δ-τ2Mˇ*-τ2W~1)<0,(Δ-(τ2-τ1)Nˇ*-(τ2-τ1)(W~1+W~2))<0,(Δ-(τ2-τ1)Fˇ*-(τ2-τ1)W~2)<0,
where
(53)Δ=(Δ11Δ12Δ13Δ14Δ15Δ16*Δ22-XCdTS0Δ25Δ26**-Rα0DT0***Δ44Δ450****Δ550*****Δ66),Δ11=AX+BY+XAT+YTBT+V~1+V~2+V~3+τ2Z~1+(τ2-τ1)Z~2+M~+M~T,Δ12=N~-F~-M~+AdX+BYd,Δ13=D-XCTS,Δ14=(F~-N~II),Δ15=YTBT+XAT,Δ16=(XCTQXΘ10κβX0),Δ22=-(1-τd)V~2,Δ25=YdTBT+XAdT,Δ26=(XCdTQ0XΘ20κdβdX),Δ44=diag⁡{-V~1,-V~3,-κI,-κdI},Δ45=(00II)T,Δ55=τ2W~1+(τ2-τ1)W~2-X-XT,Δ66=diag⁡{Q,-X,-X,-κI,-κdI},Nˇ=(N~T,01×12)T,Mˇ=(M~T,01×12)T,Fˇ=(F~T,01×12)T.


## 5. Numerical Example with Simulation

In this section, we will give an example to show the correctness of the derived results and the effectiveness of the designed controller. Consider the following nonlinear stochastic delay systems:(54)dx(t)=[(-0.11012-12.50-1-1.5-50002-5)x(t)h+(0.1x1sin⁡(x3)0.4x2cos⁡⁡(x4)0.3x3sin⁡(x1x4)0.5x4cos⁡⁡(x2x3))h+(-11012-32.50-1-2-300023)x(t-τ)h+(0.1x1(t-τ)sin(x3(t-τ))0.2x2(t-τ)cos⁡(x4(t-τ))0.3x3(t-τ)sin(x1(t-τ)x4(t-τ))0.3x4(t-τ)cos⁡(x2(t-τ)x3(t-τ)))h+(0-1-0.50-0.200-0.2)u(t,t-τ)h+(0.50.10.100.10.10.50.200.1010010.4)v(t)]dt+(0.35x1sin⁡(x1)+0.35x1(t-τ)×cos⁡⁡(x2(t-τ))0.71t1+t(x2+x2(t-τ))0.35x3sin⁡(x1x3)-0.35x3(t-τ)×sin⁡(x32(t-τ))14t7+100t(x4-x4(t-τ)))dω(t),z(t)=(120000.5000010000.21)x(t)+(110-10.50.5-100012002-1)x(t-τ).For Case1, when the initial condition x(0)=[-2,2,3,-4]T is used and v(t) is a random vector of zero mean and 0.3 standard deviation, we can see that the uncontrolled NSDSs (54) are not stable from Figure 1. Hence the design of dissipative delay-feedback controller is necessary. From (54), we can get Θ1=Θ2=diag⁡(0.5,1,0.5,2), and β=0.5, βd=0.3, for given S=I, R=5I, and Q=-0.2I; applying Theorem 14 to this example, we have
(55)X=(38.2649-9.3804-3.69247.8006-9.380425.989610.5274-1.9189-3.692410.52744.5464-0.73377.8006-1.9189-0.73372.2634),Y=1.0e+004*(-0.02541.55970.6249-0.00440.8881-0.0450-0.01750.1783),Yd=1.0e+003*(1.19510.50870.21000.23130.03080.24030.0994-0.0020).Figure 1
The states curves of open-loop NSDSs in (54) without the control with initial state (-2,2,3,-4)T and time delay τ=1.So the delay-feedback controller parameters can be calculated as follows:(56)K=(149.4677780.3725-306.768927.4064258.0621100.2942-67.4539-38.6640),Kd=(43.186333.70780.3173-17.95045.89537.65506.9022-12.4934).The states curves and the output curves of closed-loop NSDSs in (54) can be seen in Figures 2, 3, and 4; from Figure 5, we can see that (12) holds. Hence, the closed-loop NSDSs are strictly (Q,S,R)-dissipative; we can also see that the delay-feedback controller is delay-independent.Figure 2
The states curves of closed-loop NSDSs in (54) under the delay-feedback control with initial state (-2,2,3,-4)T and time delay τ=1.Figure 3
The states curves of closed-loop NSDSs in (54) under the delay-feedback control with initial state (-2,2,3,-4)T and time delay τ=2.Figure 4
The output curves of closed-loop NSDSs in (54) under the delay-feedback control with initial state (-2,2,3,-4)T and time delay τ=1.Figure 5
The dissipativity level of closed-loop NSDSs in (54) under the delay-feedback control with initial state (-2,2,3,-4)T and time delay τ=1.


## 6. Conclusions

The dissipative delay-feedback control problems for nonlinear stochastic delay systems (NSDSs) have been investigated based on dissipativity analysis. The systems are subjected to stochastic disturbance, nonlinear disturbance, and two cases time-delay effects, which often exist in a wide variety of industrial processes and are the main sources of instability. Based on the Lyapunov stability theory and stochastic analysis technique, both delay-independent and delay-dependent dissipativity criteria have been established in terms of linear matrix inequalities (LMIs). The available results onH∞ approach and passivity for stochastic delay systems as special cases of the developed results also have been given in this paper. The delay-dependent feedback controller has been designed by considering the relationship among the time-varying delay, its lower and upper bound, and its difference without ignoring any terms, which effectively reduces the conservative. A numerical example also has been given to verify the effectiveness of the proposed methods.



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*Source: 249860-2014-05-21.xml*