# Robust Quadratic Stabilizability andH∞ Control of Uncertain Linear Discrete-Time Stochastic Systems with State Delay

Authors: Xiushan Jiang; Xuemin Tian; Weihai Zhang
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper mainly discusses the robust quadratic stability and stabilization of linear discrete-time stochastic systems with state delay and uncertain parameters. By means of the linear matrix inequality (LMI) method, a sufficient condition is, respectively, obtained for the stability and stabilizability of the considered system. Moreover, we design the robustH∞ state feedback controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H∞ controllable. A sufficient condition for the existence of the desired robust H∞ controller is obtained. Finally, an example with simulations is given to verify the effectiveness of our theoretical results.


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## Body

## 1. Introduction

It is well known that stability and stabilization are very important concepts in linear system theory. Due to a great number of applications of stochastic systems in the realistic world, the studies of stability and stabilization for stochastic systems attract lots of researchers’ attention in recent years; we refer the reader to the classic book [1] and the follow-up books [2, 3], together with references [4–11] and the references therein, which include robust stochastic stability [4], exponential stabilization [6], mean-square stability, and D-stability and DR-stability [8]. The stabilization of various systems, including impulsive Markovian jump delay systems [4], stochastic singular systems [10, 12, 13], uncertain stochastic T-S fuzzy systems [14], and time-delay systems [6, 11, 15–17], has been studied extensively. H∞ control is one of the most important robust control approaches when the system is subject to the influence of external disturbance, which has been shown to be effective in attenuating the disturbance. The objective of standard H∞ control requires designing a controller to attenuate l2-gain from the external disturbance to controlled output below a given level γ>0; see [18]. The study of H∞ control of general linear discrete-time stochastic systems with multiplicative noise seems to be first initiated by [19]. Then, stochastic H∞ control and its applications have been investigated extensively; see [14, 16, 20–24].Because time-delay exists widely in practice and affects the system stability, there have been many works concerning the study in stability orH∞ control of stochastic systems [4, 6, 9, 11, 14–16, 22, 25]. Due to limitations of measurement technique and tools, it is not easy to construct exact mathematical models. Compared with the nominal stochastic systems without uncertain terms investigated in [2, 5, 24], our considered system allows the coefficient matrix to vary in a certain range.Discrete-time stochastic difference systems have attracted a great deal of attention with the development of computer technology in recent years. In our viewpoint, there are at least two motivations to study discrete-time stochastic systems, Firstly, discrete-time stochastic systems are ideal mathematical models in practical modeling such as genetic regulatory networks [23]. Secondly, discrete-time stochastic systems provide a better approach to understand extensively continuous-time stochastic Itô systems [2, 3, 26]. Therefore, it is of significance to study the stabilization and H∞ control of discrete-time stochastic time-delay uncertain systems.This paper will study quadratic stability, stabilization, and robust state feedbackH∞ control for uncertain discrete-time stochastic systems with state delay. The parameter uncertainties are time varying and norm bounded. It can be found that, up to now, many criteria for testing quadratic stabilization and H∞ control have been given in terms of LMIs and algebraic Riccati equations by applying Lyapunov function approach. One of our main contributions is to study quadratic stability and stabilization via LMIs instead of algebraic Riccati equations which is hardly solved. What we have obtained extended the work of [15] about the quadratic stability and stabilization of deterministic uncertain systems. Another contribution is to solve the state feedback H∞ control and present a state feedback H∞ controller design.The paper is organized as follows. In Section2 we give some adequate preliminaries and useful definitions. In Section 3, sufficient conditions for quadratic stability and stabilization are given in terms of LMIs which is convenient to compute by the MATLAB LMI toolbox. Section 4 designs a state feedback H∞ controller. Two numerical examples with simulations are given in Section 5 to verify the efficiency of the proposed results. Finally, we end this paper in Section 6 with a brief conclusion.For convenience, the notations in this paper are quite standard such as the following: we letRn and Rm×n represent the set of all real n-dimensional vectors and m×n real matrices. For symmetric matrices X and Y, X≥Y (resp., X>Y) stands for the idea that the matrix X-Y is positive semidefinite (resp., positive definite). I denotes the identity matrix of appropriate dimensions and XT denotes the matrix transpose of X. x=∑k=0∞‍|xk|2 represents the Euclidean norm or spectral norm of the vector x. Nk0≔{k0,k0+1,k0+2,…}, especially, N1≔1,2,…, N0≔{0,1,2,…}, and [τ1,τ2], represents the set of integers between τ1 and τ2 (inclusive). In symmetric block matrices, the symbol “∗” is used as an ellipsis for terms induced by symmetry. E(·) is the expectation operator.


## 2. Preliminaries

Consider a class of uncertain linear discrete-time stochastic systems with state delay described by(1)xk+1=A0+ΔA0kxk+A0d+ΔA0dkxk-d+B0+ΔB0kuk+∑i=1sC0+ΔC0kxk+C0d+ΔC0dkxk-d+D0+ΔD0kukwik,xj=ϕj∈Rn,j∈-d,-d+1,…,0,k∈N0,where x(k)∈Rn is the system state and u(k)∈Rm is the control input, and {w(k)}k≥0 are independent white noise process satisfying the following assumptions:(H1)
E[wk]=0, E[wkwj]=δkj, where δkj is a Kronecker function defined by δkj=0 for k≠j while δkj=1 for k=j.(H2)
{w(k)}k≥0 are defined on the filtered probability space (Ω,F,Fk,P) with Fk=σ{w(0),…,w(k)}. In addition, {Fk}k∈N0 is an increasing sequence of σ-algebras with Ft⊂F.A 0, A0d, B0, C0, C0d, D0 are known real constant matrices with compatible dimensions. ΔA0(k), ΔA0d(k), ΔB0(k), ΔC0(k), ΔD0(k), ΔC0d(k) are norm bounded and time-varying uncertain parameter which are assumed to have the following form:(2)ΔA0kΔA0dkΔB0kΔC0kΔC0dkΔD0k=EFkGA0GA0dGB0GC0GC0dGD0,where E, GA0, GA0d, GB0, GC0, GC0d, GD0 are constant matrices and F(k)∈Rm×n is the uncertain matrix satisfying(3)FkTFk≤I,k∈N0.For the purpose of simplicity, throughout this paper, we write system (1) in the following form:(4)xk+1=A0Δxk+A0dΔxk-d+B0Δuk+∑i=1sC0Δxk+C0dΔxk-d+D0Δukwik,xj=ϕj∈Rn,j∈-d,0,k∈N0,where A0Δ, A0dΔ, B0Δ, C0Δ, C0dΔ are bounded uncertain system matrices with(5)A0Δ=A0+ΔA0k=A0+EFkGA0,A0dΔ=A0d+ΔA0dk=A0d+EFkGA0d,B0Δ=B0+ΔB0k=B0+EFkGB0,C0Δ=C0+ΔC0k=C0+EFkGC0,C0dΔ=C0d+ΔC0dk=C0d+EFkGC0d,D0Δ=D0+ΔD0k=D0+EFkGD0.Below, we define robust quadratic stability and robust quadratic stabilizability for the uncertain time-delay discrete-time system (1), which generalize Definition 1 of [15] to stochastic systems.Definition 1.
Uncertain discrete time-delay system (1) is said to be robustly quadratically stable, if there exist matrices P>0, Q>0 and a scalar ω>0 such that, for all admissible uncertain terms and given initial condition x(j)=ϕ(j)∈Rn for j=0,-1,…,-d, the unforced system of (1) (with u(k)≡0) satisfies(6)EΔVk=EVk+1-EVk≤-ωEx⌢k2for x⌢k∈R2n with x⌢k=(x(k)T,x(k-d)T)T and(7)Vk=xkTPxk+∑j=1dxk-jTQxk-j.Definition 2.
Uncertain discrete time-delay system (1) is said to be robustly quadratically stabilizable if there exists a matrix K∈Rm×n such that closed-loop system (1) with u(k)=Kx(k), that is,(8)xk+1=A0Δ+B0ΔKxk+A0dΔkxk-d+∑i=1sC0Δ+D0ΔKxk+C0dΔxk-dwik,is robustly quadratically stable for given x(j)=ϕ(j)∈Rn for j=0,-1,…,-d.


## 3. Robust Quadratic Stabilization

In this section, a sufficient condition about robust quadratic stability and robust quadratic stabilization will be presented via LMIs, respectively. First, we cite the following lemma which is essential in proving our main results.Lemma 3 (see [27]).
Suppose thatW=WT, F(k) satisfies (2), and then for any real matrices W, M, and N of suitable dimensions we have(9)W+MFkN+NTFkTMT<0if and only if (iff), for some α>0,(10)W+αMMT+α-1NTN<0.Theorem 4.
Consider uncertain discrete-time stochastic delay system (1) with u(k)=0. This system is robustly quadratically stable if there exist positive definite matrices X>0, Y>0 such that the following LMI holds.(11)Δ11Δ12A0TXs1/2C0TX00∗Δ22A0dTXs1/2C0dTX00∗∗-X0XE0∗∗∗-X0XE∗∗∗∗-I0∗∗∗∗∗-I<0,where(12)Δ11=Y-X+GA0TGA0+sGC0TGC0,Δ12=GA0dTGA0d+sGC0dTGC0d,Δ22=-Y+GA0dTGA0d+sGC0dTGC0d.Proof.
From Definition1, taking a Lyapunov function Vk as in the form of (7), if uncertain discrete time-delay stochastic system (1) is quadratically stable, then, for all admissible uncertainties of (1), there exist matrices P>0, Q>0 and a scalar α>0 such that E(ΔVk) associated with unforced system (8) satisfies (6). In view of the assumption (H1), it is easy to compute(13)EVk+1-EVk=ExkTA0ΔkTPA0Δk+sC0ΔkTPC0Δk+Q-Pxk+xkTA0ΔkTPA0dΔk+sC0ΔkTPC0dΔkxk-d+xk-dTA0dΔkTPA0Δk+sC0dΔkTPC0Δkxk+xk-dTA0dΔkTPA0dΔk+sC0dΔkTPC0dΔk-Qxk-d=xkxk-dTΠxkxk-d,where A0Δ, A0dΔ, C0Δ, and C0dΔ are given in (5) and Π is shown as (14)Π=A0ΔkTPA0Δk+sC0ΔkTPC0Δk+Q-PA0ΔkTPA0dΔk+sC0ΔkTPC0dΔkA0dΔkTPA0Δk+sC0dΔkTPC0ΔkA0dΔkTPA0dΔk+sC0dΔkTPC0dΔk-Q.By Definition 1, system (1) with u(k)=0 is robustly quadratically stable, only if(15)Π<0which is equivalent to (16)Π=Π1+Π2=sC0ΔkTPC0Δk+Q-PsC0ΔkTPC0dΔksC0dΔkTPC0ΔksC0dΔkTPC0dΔk-Q+A0ΔkTPA0ΔkA0ΔkTPA0dΔkA0dΔkTPA0ΔkA0dΔkTPA0dΔk<0.Note that Π2 can be rewritten as(17)Π2=A0ΔkTPA0dΔkTPP-1PA0ΔkPA0dΔk. By Schur’s complement, it is easy to derive that Π<0 is equivalent to(18)Π^=π11sC0ΔkTPC0dΔkA0ΔkTP∗π22A0dΔkTP∗∗-P,where(19)π11=sC0ΔkTPC0Δk+Q-P,π22=sC0dΔkTPC0dΔk-Q.Then, using the same way as in (16)–(19) yields(20)Π^=Q-P0A0ΔkTPs1/2C0ΔkTP∗-QA0dΔkTPs1/2C0dΔkTP∗∗-P0∗∗∗-P<0.The above inequality can be rewritten as(21)Π^=Π3+0000PE00PEdiag⁡Fk,FkGA0GA0d00s1/2GC0s1/2GC0d00+GA0Ts1/2GC0TGA0dTs1/2GC0dT0000diag⁡FkT,FkT00ETP0000ETP<0,where(22)Π3=Q-P0A0TPs1/2C0TP∗-QA0dTPs1/2C0dTP∗∗-P0∗∗∗-P.Because Π3 is a symmetric matrix, applying Lemma 3, (21) holds iff the following inequality holds:(23)Π3+α0000PE00PE00ETP0000ETP+α-1GA0Ts1/2GC0TGA0dTs1/2GC0dT0000GA0GA0d00s1/2GC0s1/2GC0d00=Λ11Λ12A0TPs1/2C0TP∗Λ22A0dTPs1/2C0dTP∗∗Λ330∗∗∗Λ44<0,where(24)Λ11=Q-P+α-1GA0TGA0+sα-1GC0TGC0,Λ12=α-1GA0TGA0d+sα-1GC0TGC0d,Λ22=-Q+α-1GA0dTGA0d+sα-1GC0dTGC0d,Λ33=Λ44=-P+αPEETP.Take(25)P=α-1X,Q=α-1Yand then by substituting (25) into (23), for α>0, we get(26)Δ11Δ12A0TXs1/2C0TX∗Δ22A0dTXs1/2C0dTX∗∗-X+XEETX0∗∗∗-X+XEETX<0,where Δ11, Δ12, Δ22 are shown in (12).
Using the same method as in (16)–(20), (11)-(12) follow immediately from the above inequality.Theorem 5.
System (1) is robustly quadratically stabilizable if there exist positive matrices X>0, Y>0, K∈Rm×n and a scalar ε>0 with εI-X-1<0 such that the following LMI holds.(27)Θ11Θ12A0+B0KTs1/2C0+D0KT00GA0+GB0KTs1/2GC0+GD0KT∗Θ22A0dTs1/2C0dT0000∗∗-εI0E000∗∗∗-εI0E00∗∗∗∗-I000∗∗∗∗∗-I00∗∗∗∗∗∗-I0∗∗∗∗∗∗∗-I<0,where(28)Θ11=Y-X,Θ12=GA0+GB0KTGA0d+GC0+GD0KTGC0d,Θ22=-Y+GA0dTGA0d+GC0dTGC0d.Moreover, a quadratically stabilizing state feedback controller is given by(29)uk=Kxk.Proof.
By Definition2, using the same way as in the proof of Theorem 4, the following inequality which has a similar form to (11)-(12) can be obtained by taking u(k)=Kx(k)(30)Θ^11Θ12A0+B0KTXs1/2C0+D0KTX00∗Θ22A0dTXs1/2C0dTX00∗∗-X0XE0∗∗∗-X0XE∗∗∗∗-I0∗∗∗∗∗-I<0,where(31)Θ^11=Y-X+GA0+GB0KTGA0+GB0K+GC0+GD0KTGC0+GD0K.In order to eliminate the nonlinear quadratic terms(32)A0+B0KTX,C0+D0KTX,pre- and postmultiplying(33)diag⁡I,I,X-1,I,I,Ion both sides of (30) and considering X-1>εI, (27)-(28) can be obtained easily. This theorem is proved.Remark 6.
Compared with the results about quadratic stability and quadratic stabilizability of deterministic systems given in [14], our two theorems not only extend the results of [14] to stochastic systems, but also provide the corresponding LMI criteria which can be easily tested by MATLAB LMI toolbox.Remark 7.
From these two theorems, we also can get the result about quadratic stability with the given decay rate. Take the function(34)xλk=xkekλ,and then, substituting (34) into (8), we obtain the following new system:(35)xλk+1=A~0Δ+B~0ΔKxλk+A~0dΔxλk-d+∑i=1sC~0Δ+D~0ΔKxλk+C~0dΔxλk-dwik,where (36)A~0Δ=eλA0Δ,B~0Δ=eλB0Δ,C~0Δ=eλC0Δ,D~0Δ=eλD0Δ,A~0dΔ=ed+1λA0dΔ,C~0dΔ=ed+1λC0dΔ.So system (1) is quadratically stabilizable with decay rate λ if system (35) is quadratically stabilizable.


## 4. State FeedbackH∞ Control

In this section we consider the state feedback discrete-timeH∞ control problem for the following uncertain linear stochastic system with state delay:(37)xk+1=A0+ΔA0kxk+A0d+ΔA0dkxk-d+Bξk+B0+ΔB0kuk+∑i=1sC0+ΔC0kxk+C0d+ΔC0dkxk-d+D0+ΔD0kukwik,xj=ϕj∈Rn,j∈-d,-d+1,…,0,k∈N0zk=Cxk+Duk,where z(k)∈Rnz and ξ(k)∈Rq are called the controlled output and external disturbance, respectively. In addition, the effect of the disturbance ξ(k) on the controlled output z(k) is described by a perturbation operator Gzξ:ξ↦z, which maps any finite energy disturbance signal ξ into the corresponding finite energy output signal z of the closed-loop system. The size of this linear operator, that is, Gzξ, measures the influence of the disturbances in the worst case. We denote by lw2(N0,Rl) the set of all nonanticipative square summable Rl-valued stochastic processes (38)y=yk:yk∈L2Ω,Rl,ykisFk-1measurablek∈N0.l2-norm of y∈lw2(N0,Rl) is defined by (39)ylw2N0,Rl=∑k=0∞Eyk21/2.Firstly, for system (37), we define the perturbed operator Gzξ and its norm as follows.Definition 8.
The perturbed operator of system (37), Gzξ:lw2(N0,Rq)↦lw2(N0,Rnz), is defined as (40)Gzξ:ξk∈lw2N0,Rq⟼Cxk+Duk,xj=0,j=0,-1,-2,…,-dwith its norm (41)Gzξ=supξk∈lw2N0,Rq,ξk≠0,xj=0,j∈-d,0⁡zklw2N0,Rnzξklw2N0,Rq=supξk∈lw2N0,Rq,ξk≠0,xj=0,j∈-d,0⁡∑k=0∞ECxk+Duk21/2∑k=0∞Eξk21/2.Next, we present the definition about stochastic robust H∞ control.Definition 9.
For a certain levelγ>0, u∗(k)=Kx(k) is the H∞ control of the system (37), if (i)
system (37) is internally stabilizable when ξ(k)≡0;(ii)
the norm of the perturbed operator of system (37) satisfies Gzξ<γ for all external disturbance ξ(k)∈lw2(N0,Rq).Besides, ifu∗(k) exists, then system (37) is called H∞ controllable in the disturbance attenuation. Furthermore, it is called strongly robust H∞ controllable if γ=1.Theorem 10.
Consider system (37). For the given γ>0 and some β>0 with P<β-1I and α>0 if there exist P>0, Q>0, and K∈Rm×n satisfying the following LMI(42)-2P+Q+CTCħ120s1/2C0+D0KTA0+B0KKTDTħ17ħ18∗ħ220s1/2C0dTA0dT000∗∗-γ2I0BT000∗∗∗-βI+αEET0000∗∗∗∗-βI+αEET000∗∗∗∗∗-I00∗∗∗∗∗∗-I0∗∗∗∗∗∗∗-I<0,where(43)ħ12=α-1s1/4GC0+GD0KTGC0d+GA0+GB0KTGA0d,ħ22=-Q+α-1s1/4GC0dTGC0d+GA0dTGA0d,ħ17=α-1s1/2GC0+GD0KT,ħ18=α-1GA0+GB0KT,then system (37) is robustly H∞ controllable with a control law u(k)=Kx(k).Proof.
By Theorem5, when disturbance ξ(k)=0, it is easy to test that system (37) is internally stabilizable with u∗(k)=Kx(k). Now we only need to show Gzξ<γ. By Definition 1, choose the Lyapunov function Vk=x(k)TPx(k)+∑j=1d‍x(k-j)TQx(k-j) with P>0 and Q>0 to be determined, and then (44)
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,where (46)Ξ=Ξ11A0Δ+B0ΔKTPA0dΔA0Δ+B0ΔKTPBs1/2C0Δ+D0ΔKTP∗A0dΔTPA0dΔ-QA0dΔTPBs1/2C0dΔTP∗∗BTPB-γ2I0∗∗∗-P,Ξ11=A0Δ+B0ΔKTPA0Δ+B0ΔK-2P+Q+CTC+KTDTDK.Obviously, it is easy to get that Gzξ<γ if Ξ<0. Then, we need to eliminate the uncertainties. Using the same method as in the proof of Theorem 4, we know that, for some α>0, a sufficient condition for Ξ<0 can be got from the following matrix inequality.(47)Γ11Γ120s1/2C0+D0KTPA0+B0KTP∗Γ220s1/2C0dTPA0dTP∗∗-γ2I0BTP∗∗∗-P+αPEETP0∗∗∗∗-P+αPEETP<0,where (48)Γ11=-2P+Q+CTC+KTDTDK+α-1s1/4GC0+GD0KTGC0+GD0K+α-1GA0+GB0KTGA0+GB0K,Γ12=α-1s1/4GC0+GD0KTGC0d+GA0+GB0KTGA0d,Γ22=-Q+α-1s1/4GC0dTGC0d+GA0dTGA0d.Then, by pre- and postmultiplying (49)diag⁡IIIP-1P-1on both sides of (47), we have(50)Γ11Γ120s1/2C0+D0KTA0+B0KT∗Γ220s1/2C0dTA0dT∗∗-γ2I0BT∗∗∗-P-1+αEET0∗∗∗∗-P-1+αEET<0.For some constant β>0 with P-1>βI, Theorem 10 is concluded; that is, an H∞ control of system (37) is obtained by solving LMIs (42)-(43). This completes the proof.


## 5. Simulation Example

In this section, we consider two simple examples with simulations to illustrate the effectiveness of the proposed approach.Example 11.
Consider discrete-time stochastic system (1) with the following parameters:(51)A0=1000.8,A0d=0.02000.1,B0=31,C0=0.2000.4,C0d=0.1000.2,D0=0.20.7,E=0.2000.4,GA0=0.01000.03,GA0d=0.04000.05,GB0=0.20.1,GC0=0.3000.1,GD0=0.10.7,GC0d=0.1000.01,Fk=cos⁡wk00sin⁡wk,s=1.Using LMI toolbox to solve (11)-(12) in Theorem 4, we find out that tmin=0.0086>0 which means that there is no feasible solution and indicates that system (1) with u≡0 is unstable. Figure 1 verifies the result. By solving LMI (27), a group of feasible solutions with tmin=-0.9649<0 are shown as ε=16.7436 and (52)X=22.10260.65190.651920.0007,Y=11.3608-0.6710-0.671013.2231.
By Theorem5, the system is mean-square stabilizable which is verified by Figure 2. A robust stabilizing controller is given by(53)uk=Kxk=-0.092-0.1344xk.Figure 1
State trajectories of the autonomous system.Figure 2
State trajectories of the closed-loop system.Example 12.
Consider system (42) with the following parameters:(54)A0=1.2001.1,A0d=0.2000.1,B0=1.31,C0=0.2000.4,B=10.65,C=0.40.20.10.8,D=0.81,C0d=0.1000.2,D0=0.20.7,E=0.2000.4,GA0=0.1000.3,GA0d=0.4000.5,GB0=0.20.1,GC0=0.3000.1,GD0=0.10.7,GC0d=0.1000.1,Fk=cos⁡wk00sin⁡wk,s=1.
For perturbed system (42), we take the external disturbance as ξ(k)=e-k and the certain level as γ=0.8. In addition, according to Lemma 3, an appropriate α is given as α=4.9. Then, by the result of Theorem 10, using LMI toolbox to solve (43) and (47), we find that tmin=-0.1046, which means we have got a group of feasible solutions with(55)P=1.72580.03200.03201.8314,Q=1.6581-0.0480-0.04801.5180,K=-0.3281-0.2836,β=3.0531.The simulation results of state trajectories and controlled output trajectories of system (42) are given in Figures 3 and 4 with the H∞ controller (56)uk=Kxk=-0.3281-0.2836xk.This further verifies the effectiveness of Theorem 10.Figure 3
State trajectories of the closed-loop system.Figure 4
Controlled output trajectories of the closed-loop system.


## 6. Conclusion

In this paper, we have studied the robust quadratic stability, quadratic stabilization, and robustH∞ state feedback control of discrete-time stochastic systems with state delay and uncertain parameters. Based on LMI technique, a sufficient condition about quadratic stability and quadratic stabilization of our considered system is, respectively, given. Moreover, an H∞ state feedback controller is obtained by solving two LMIs. Finally, we supply two simulation examples to show the validity of the proposed results. It is expected to solve the H∞ output feedback control and H∞ filtering in our forthcoming work.



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*Source: 7843940-2016-11-13.xml*