# RobustH∞ Control for Discrete-Time Stochastic Interval System with Time Delay

Authors: Shengchun Yu; Guici Chen; Yi Shen
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The robustH∞ control problem for discrete-time stochastic interval system (DTSIS) with time delay is investigated in this paper. The stochastic interval system is equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing the appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robust H∞ controller for DTSIS are obtained in terms of linear matrix inequality (LMI) form, and the robust H∞ controller is designed. Finally, a numerical example with simulation is given to show the effectiveness and correctness of the designed robust H∞ controller.


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## Body

## 1. Introduction

In the past few years, much research effort has paid to the robust control problems for stochastic systems which have come to play an important role in many fields including communication network, image processes, and mobile robot localization. So far, plenty of significant results also have been published; see, for example, [1–3] and the references therein. In the meantime, we all know that time delay arises naturally in many mathematical (non) linear models of real phenomena, such as communication, circuits theory, biology, mechanics, electronics, hydraulic, rolling mill, chemical systems, and computer controlled systems, and which is frequently one of the main sources of instability, oscillation and poor performance of control systems. So the stability analysis and robust control for dynamic time-delay systems have attracted a number of researchers; see, for example, [4–8] and the references therein.Meanwhile, from Hinrichsen who presented theH∞ control problems for stochastic systems in 1998 [9], more and more experts begin to study the stochastic H∞ control problems [10–17]. In the view of dissipation, Berman develops a H∞-type theory for a large class of time-continuous stochastic nonlinear systems. In particular, it introduces the notion of stochastic dissipative systems by analogy with the familiar notion of dissipation associated with the deterministic systems. The problem of H∞ output feedback control for uncertain stochastic systems with time-varying delay is discussed in [12], and the parameter uncertainties are assumed to be time-varying norm-bounded. Xu et al. [13] investigate the problems of robust stochastic stabilization and robust H∞ control for uncertain neutral stochastic time-delay systems.On the other hand, when modeling real-time plants, the parameter uncertainties are unavoidable, which are very often the cause of instability and poor performance, such as modeling error, external perturbation, and parameter fluctuation during the physical implementation. As a result, the parameters of a system matrix are estimated only within certain closed intervals. We call this kind of system interval system or stochastic interval system (SIS) with the following form:(1)x˙(t)=Ax(t),
or
(2)dx(t)=Ax(t)dt+Hx(t)dω(t),
where A∈[A_,A¯], H∈[H_,H¯],and·_,·¯ denote the lower and upper bounds of the interval for the coefficients, respectively. Interval system and stochastic interval system have been well known for their importance in practice applications. And in recent years, the stability analysis and stabilization problems of various interval systems have received plenty of research attention; see, for example, [18–24] and the references therein. Of course, there are a lot of research works on stability analysis and stabilization problems for stochastic interval system that have been reported; see, for example, [25–29] and the references therein. However, to the best of authors’ knowledge, so far little results on robust H∞ control problem for stochastic interval system with time-delay are available in the existing literature.Inspired by the above discussions, in this paper we study the robustH∞ control problem for discrete-time stochastic interval system (DTSIS) with time delay. The stochastic interval system will be equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robust H∞ controller for DTSIS are obtained in terms of linear matrix inequality (LMI) form, and the robust H∞ controller can be designed by MATLAB LMI control toolbox.


## 2. Problems Formulation and Preliminaries

Consider the following discrete-time stochastic interval system (DTSIS) with time delay:(3)x(k+1)=Alx(k)+Adlx(k-d)+Bu(k)+Dv(k)+[Hlx(k)+Hdlx(k-d)]ω(k),z(k)=Clx(k)+Cdlx(k-d),x(k)=φ(k),k∈[-d,0],
wherex(k)∈ℝnis the state vector;d>0 is the time delay; u(k)∈ℝm is the control input;z(k)∈ℝqis the control output;v(k)∈ℝp is the exogenous disturbance input, which satisfiesv(k)∈L2([0,∞),ℝp), where L2([0,∞),ℝp) is the space of nonanticipatory square-summable stochastic process with respect to(ℱk)k∈I+ with the following norm:
(4)∥v(k)∥22=𝔼{∑k=0∞∥v(k)∥2}=∑k=0∞𝔼{∥v(k)∥2}.ω
(
k
)
∈
ℝ
lis a scalar Brownian motion defined on a complete probability space (Θ,ℱ,P) with
(5)𝔼[ω(k)]=0,𝔼[ω2(k)]=1.In the system (3), Alis an interval matrix with appropriate dimension, which means
(6)Al=[A_,A¯]={(aij)∈ℝn×n∣a_ij≤aij≤a¯ij,i,j=1,2,…,n},
where A_=(a_ij)n×n, A¯=(a¯ij)n×n are determinate matrices.Remark 1.
It is not difficult to see that the matricesAl,Adl,Hl, Hdl,Cl,Cdlcan also be time-dependent as long as the mappingsAl:R+→[A_,A¯], Adl:R+→[A_d,A¯d], Hl:R+→[H_,H¯], Hdl:R+→[H_d,H¯d], Cl:R+→[C_,C¯], and Cdl:R+→[C_d,C¯d]are continuous.Set(7)A=12[A_+A¯],Δ=12[A¯-A_]=(δij)n×n,D1=(δ11e1,…,δ1ne1,…,δn1en,…,δnnen),G1=(δ11e1,…,δ1ne1,…,δn1en,…,δnnen)T,
whereei(i=1,2,…,n)is theith column ofn×nidentification matrix, setting
(8)F¯={f∣f=diag{α11,…,α1n,…,αn1…,αnn}},
where ∥αij∥≤1. Obviously, we have F1F1T≤In2; hereIn2is then2×n2identification matrix. So the interval matrixAlcan equivalently be written as
(9)Al=[A_,A¯]=A+ΔA=A+D1F1G1,F1∈F¯.Similar to the above derivation, we can have(10)Adl=[A_d,A¯d]=Ad+ΔAd=Ad+D2F2G2,Hl=[H_,H¯]=H+ΔH=H+D3F3G3,Hdl=[H_d,H¯d]=Hd+ΔHd=Hd+D4F4G4,Cl=[C_,C¯]=C+ΔC=C+D5F5G5,(11)Cdl=[C_d,C¯d]=Cd+ΔCd=Cd+D6F6G6;
whereFi∈F¯,i=1,2,…,6; let
(12)F~=diag{F1,F2,…,F6},E1=(D1,D2,0,0,0,0),E2=(0,0,D3,D4,0,0),E3=(0,0,0,0,D5,D6),N1T=(G1T,0,G3T,0,G5T,0),N2T=(0,G2T,0,G4T,0,G6T).Then (9)–(11) can be rewritten as
(13)(ΔAΔAdΔHΔHdΔCΔCd)=(E1E2E3)F~(N1N2).So the system (3) can be readily derived as
(14)x(k+1)=(A+ΔA)x(k)+(Ad+ΔAd)x(k-d)+Bu(k)+Dv(k)+(H+ΔH)x(k)ω(k)+(Hd+ΔHd)x(k-d)ω(k),z(k)=(C+ΔC)x(k)+(Cd+ΔCd)x(k-d),x(k)=φ(k),k∈[-d,0].
For system (14), we design the following state feedback controller:
(15)u(k)=Kx(k),
where the matrix K is the controller gain, which is to be designed. Then the closed-loop stochastic control system can be rewritten as
(16)x(k+1)=A~x(k)+A~dx(k-d)+Dv(k)+[H~x(k)+H~dx(k-d)]ω(k),z(k)=C~x(k)+C~dx(k-d),
whereA~=A+BK+ΔA,A~dAd+ΔAd, H~=H+ΔH, H~d=Hd+ΔHd, C~=C+ΔC, C~d=Cd+ΔCd.Definition 2.
Discrete-time stochastic interval system (DTSIS) with time delay (3) is said to be stochastic exponentially stable in mean square, if there exist scalarsc>0,μ>1, such that
(17)𝔼|x(N)|2≤cμ-Nmax-d≤i≤0𝔼|φ(i)|2.In this paper, we aim to design the controller gain matrixKin (16) such that the requirements are simultaneously satisfied. (a)
The zero-solution of the closed-loop stochastic control system (16) withv(k)=0is stochastic exponentially stable in mean square.(b)
Under the zero-initial condition, the control outputz(k)satisfies
(18)∑k=0∞𝔼{∥z(k)∥2}≤γ2∑k=0∞𝔼{∥v(k)∥2}
for all nonzerov(k).Which is also said the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation levelγ>0.Lemma 3 (see [30]).
For given symmetric matrixΣ, and matricesH,Ewith appropriate dimension,Σ+HFE+ETFTHT<0withFsatisfying FTF≤I holds if and only if Σ+a-1HHT+aETE<0 holds for anya>0.


## 3. Main Results

Theorem 4.
If there exist two positive definite matricesPandQ, and for a given positive constantγ>0, such that the following matrix inequality holds:
(19)Σ=(Q-P00A~TH~TC~T*-Q0A~dTH~dTC~dT**-γ2IDT00***-P-100****-P-10*****-I)<0,
then the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation levelγ.Proof.
Letv(k)=0; we construct the Lyapunov-Krasovskii functional as
(20)V(k)=xT(k)Px(k)+∑i=1d‍xT(k-i)Qx(k-i).
Calculating the difference ofV(k)along with the system (16) and taking the mathematical expectation, we have
(21)𝔼{ΔV(k)}=E{V(k+1)-V(k)}=xT(k+1)Px(k+1)-xT(k)Px(k)+xT(k)Qx(k)-xT(k-d)Qx(k-d)=(x(k)x(k-d))TΠ1(x(k)x(k-d)),
where
(22)Π1=(Q-P00-Q)+(A~TA~dT)P(A~TA~dT)T+(H~TH~dT)P(H~TH~dT)T.
It is clear formΣ<0that there exists a sufficient scalara>0, such that
(23)Σ+adiag{In×n,0}<0.
Therefore, we have(24)𝔼{ΔV(k)}≤-a𝔼{|x(k)|2}.
We now in a position to analyze of the exponential stable in mean square for the stochastic interval delay system (16). According to the definition of V(k), we have
(25)𝔼{V(k)}≤ρ1𝔼{|x(k)|2}+ρ2∑k=1d‍𝔼{|x(k-i)|2},
where ρ1=λmax(P) and ρ2=λmax(Q). For any scalar μ>1, the earlier inequality, together with (23), implies that
(26)μk+1𝔼{V(k+1)}-μk𝔼{V(k)∥=μk+1𝔼∥ΔV(k)∥+μk(μ-1)𝔼{V(k)}≤β1(μ)μk𝔼{|x(k)|2}+β2(μ)μk∑i=1d‍𝔼{|x(k-i)|2},
where β1(μ)=-aμ+ρ1(μ-1),β2(μ)=(μ-1)ρ2.
Furthermore, summing up both sides of the earlier inequality from0toN-1with respect tok, we get
(27)μN𝔼{V(N)}-𝔼{V(0)}≤β1(μ)∑k=0N-1‍μk𝔼{|x(k)|2}+β2(μ)∑k=0N-1‍∑i=1d‍μk𝔼{|x(k-i)|2}.
Ford>1,
(28)∑k=0N-1‍∑i=1d‍μk𝔼{|x(k-i)|2}≤(∑i=-d-1‍∑k=0i+d‍+∑i=0N-1-d‍∑k=i+1i+d‍+∑i=N-1-dN-1‍∑k=i+1N-1‍)μk𝔼{|x(i)|2}≤∑i=-d-1‍μi+d𝔼{|x(i)|2}+d∑i=0N-1-d‍μi+d𝔼{|x(i)|2}+d∑i=N-1-dN-1‍μi+d𝔼{|x(i)|2}≤dμdmax-d≤i≤0𝔼{|φ(i)|2}+dμd∑i=0N-1‍𝔼{|x(i)|2}.
Then, from (27) and (28), it follows that
(29)μN𝔼{V(N)}≤𝔼{V(0)}+(β1(μ)+dμdβ2(μ))×∑k=0N-1‍μk𝔼{|x(k)|2}+dμdβ2(μ)max-d≤i≤0𝔼{|φ(i)|2}.
Letρ0=λmin(P) and ρ=max{ρ1,ρ2}; we have
(30)𝔼{V(N)}≥ρ0𝔼{|x(N)|2}.
It also follows that(31)𝔼{V(0)}≤ρmax-d≤i≤0𝔼{|φ(i)|2}.
We can choose appropriate scalarσ>0such that
(32)β1(σ)+dσdβ2(σ)=0,
which implies(33)𝔼{|x(N)|2}≤cμ-Nmax-d≤i≤0𝔼{|φ(i)|2},
where c=(1/ρ0)(ρ+dσdβ2(σ)). Hence the closed-loop stochastic control system (16) withv(k)=0is stochastic exponentially stable in mean square according to Definition 2.
Under the zero-initial condition, letv(k)≠0, setting
(34)J(N)=𝔼{∑k=0N‍[zT(k)z(k)-γ2vT(k)v(k)]}=𝔼{∑k=0N‍[zT(k)z(k)-γ2vT(k)v(k)+V(k+1)-V(k)zT]∑k=0N‍}-V(N+1)≤𝔼{∑k=0N‍[zT(k)z(k)-γ2vT(k)v(k)+ΔV(k)]}=∑k=0N‍(x(k)x(k-d)v(k))TΠ2(x(k)x(k-d)v(k)),
where
(35)Π2=(Q-P000-Q000-γ2I)+(A~TA~dTDT)P(A~TA~dTDT)T+(H~TH~dT0)P(H~TH~dT0)T+(C~TC~dT0)(C~TC~dT0)T.
So (19) implies that Π2<0; hence,J(N)<0. ThusN→∞, J(∞)<0, such that (18) holds. The proof is completed.To design the controller (15), matrix inequality (15) must be solvable and matrix inequality (15) in Theorem 4 must be inverted into LMI form. In the following section, we will find a way for the solution of (15).Remark 5.
The results obtained in the paper are based on the Lyapunov-Krasovskii functional and the corresponding technique used in the proof of Theorem4, and it is easy to extend the results to stochastic system with time-varying delay or multiple delays. Details are omitted here due to page length consideration.Theorem 6.
If there exist two positive definite matricesX,Q~, a matrix Y with appropriate dimension, and a positive constant ε, for a given positive constant γ>0, such that the following linear matrix inequality (LMI) holds
(36)(Q~-X00ΩXHTXCTXN1T0*-Q~0XAdTXHdTXCdTXN2T0**-γ2IDT0000***-X000εE1****-X00εE2*****-I0εE3******-εI0*******-εI)<0,
where Ω=YTBT+XAT, then the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation level γ with the controller designed as K=YX-1.Proof.
By (13) and (19), we have
(37)Π+Θ1F~TΘ2T+Θ2F~Θ1T<0,
where
(38)Π=(Q-P00(A+BK)THTCT*-Q0AdTHdTCdT**-γ2IDT00***-P-100****-P-10*****-I)Θ1T=(N1,N2,01×4),Θ2T=(01×3,E1T,E2T,E3T).
By Lemma3, we know that
(39)Π+ε-1Θ1Θ1T+εΘ2Θ2T<0.
Using Schur complement lemma and contragradient transformation, we can get the LMI (36) holds.  The proof is completed.Without considering of interval matrices, the system (3) changes into the following discrete-time stochastic time-delay system:
(40)x(k+1)=Ax(k)+Adx(k-d)+Bu(k)+Dv(k)+[Hx(k)+Hdx(k-d)]ω(k),z(k)=Cx(k)+Cdx(k-d),x(k)=φ(k),k∈[-d,0].We can design the controller (15) by the following the corollary without proof.Corollary 7.
If there exist two positive definite matricesX,Q~ and a matrix Y with appropriate dimension, for a given positive constant γ>0, such that the following linear matrix inequality (LMI) holds
(41)(Q~-X00ΩXHTXCT*-Q~0XAdTXHdTXCdT**-γ2IDT00***-X00****-X0*****-I)<0,
where Ω is defined in (36), then the closed-loop stochastic control system (40) is stochastic exponentially stable in mean square with disturbance attenuation levelγwith the controller designed asK=YX-1.If we do not consider the stochastic disturbance in system (3), the system (3) can be written as
(42)x(k+1)=Alx(k)+Adlx(k-d)+Bu(k)+Dv(k),z(k)=Clx(k)+Cdlx(k-d),x(k)=φ(k),k∈[-d,0].Then the robustH∞ controller can be designed by the following corollary.Corollary 8.
If there exist two positive definite matricesX,Q~, a matrixYwith appropriate dimension, and a positive constant ε, for a given positive constantγ>0, such that the following linear matrix inequality (LMI) holds
(43)(Q~-X00ΩXCTXN1T0*-Q~0XAdTXCdTXN2T0**-γ2IDT000***-X00εE1****-I0εE3*****-εI0******-εI)<0,
where Ω is defined in (36), then the closed-loop stochastic control system (42) is exponentially stable with disturbance attenuation level γ with the controller designed as K=YX-1.


## 4. Numerical Example with Simulation

In this section, an example is given to show the usefulness of the designed controller. Consider the following discrete-time stochastic interval system with time delay:(44)x(k+1)=(1[0.5,0.7]0.2[0.1,0.3])x(k)+(0.20.3)u(k)+([-0.1,0.1]00.15[-0.5,0.2])x(k-10)+(-0.10.1-0.10.2)(e-0.2ksink-e-0.1kcos2k)+{([0.1,0.3]00.3[-0.2,0.1])x(k)+(-0.1[-0.1,0.1]0[-0.1,0.2])x(k-10)}ω(k),z(k)=([-0.1,0.1]0.100.2)x(k)+([0,0.1][0.1,0.3]-0.10)x(k-10),x(k)=φ(k),k∈[-10,0].Without considering the state-feedback control, we see that the open-loop stochastic interval system is unstable from Figure1.Figure 1
Response of open-loop DTSIS (44) to initial state(5,-3).We aim at designing a state-feedback controller for stochastic interval system (44), such that the closed-loop stochastic interval system is stochastic exponentially stable in mean square with disturbance attenuation levelγ.By using the method discussed in the previous section, we can calculate the matricesA,Ad,H,Hd,C,Cd,Di,Gi, i=1,2,…,6, setting γ=0.9, solving the LMI (36) in Theorem 6 and then obtain the matrices as follows:
(45)X=(0.71470.07060.07061.3303),Q~=(0.24420.02850.02850.9091),Y=(-2.8706-0.5642),ε=2.2710.Note that in expression of Theorem6, we can design the controller as
(46)K=YX-1=(-3.9954-0.2119).The response of the closed-loop stochastic interval system (44) is shown in Figures 2 and 3, which demonstrate that the designed controller for the stochastic interval system (44) is effective.Figure 2
Response of closed-loop DTSIS (44) to initial state(5,-3).Figure 3
Response of control outputz(k)of closed-loop DTSIS (44).


## 5. Conclusions

In this paper, we have studied the robustH∞control problem for discrete-time stochastic interval system (DTSIS) with time delay. The stochastic interval system was equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robustH∞controller for DTSIS have been obtained in terms of linear matrix inequality (LMI) form, and the robustH∞controller has been designed easily by MATLAB LMI control toolbox. A numerical example with simulation has been given to demonstrate the feasibility and effectiveness of the proposed method.Of course, this method proposed in this paper also can be used to the the state estimation problem and the sampled-data synchronization control problem for the complex networks, which has been studied in [31, 32] and is one of our future research topics.



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