# Analysis of Robust Stability for a Class of Stochastic Systems via Output Feedback: The LMI Approach

Authors: Xin-rong Cong; Long-suo Li
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

This paper investigates the robust stability for a class of stochastic systems with both state and control inputs. The problem of the robust stability is solved via static output feedback, and we convert the problem to a constrained convex optimization problem involving linear matrix inequality (LMI). We show how the proposed linear matrix inequality framework can be used to select a quadratic Lyapunov function. The control laws can be produced by assuming the stability of the systems. We verify that all controllers can robustly stabilize the corresponding system. Further, the numerical simulation results verify the theoretical analysis results.


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## Body

## 1. Introduction

The stochastic differential equations which play an important role in many branches of science and industry are often used as the tool of describing complex phenomena [1, 2]. The stochastic items of the stochastic differential equations lead to the solutions of the equations also with randomness, so the stability analysis is very crucial. Roughly speaking, the stability means insensitivity of the state of the system to small changes in the parameters or the initial state of the system [3]. In this paper, the robust stability for a class of stochastic systems is investigated via static output feedback by using the unified linear matrix inequalities approach.The problem of robust stability [4–6] refers to the control system under the existence of internal uncertainty and external disturbance; static or dynamic feedback controller is designed to stabilize the closed-loop system. When the uncertainty factors exist in the system, the robust stability is the indicator of the system which can still maintain normal work performance.Many important problems in stochastic differential equation theory can be converted to convex optimization problems with linear matrix inequalities, so that they will become numerically tractable. The quadratic stability theory of discrete-time systems has been paid much attention, which can be seen in [7–10]. The robust quadratic stabilization of nonlinear systems within the framework of linear matrix inequalities is investigated by Šiljak and Stipanović [11]. Further, the robust stabilization for a class of discrete-time nonlinear systems is formulated into a convex optimization via linear matrix inequalities developed by   Stipanović and Šiljak [12]. Then,   Ho and Lu  [13] who extended the work of   Stipanović and Šiljak  investigated the robust stabilization for a class of discrete-time nonlinear systems via output feedback. Based on these works, we consider a class of stochastic systems. The major objective of this paper is to show a method to design static output feedback controller for a class of stochastic systems by means of convex optimization problems via linear matrix inequalities; then, the static output feedback control laws can robustly stabilize the stochastic systems.In this paper, we suppose that(Ω,F,P)   is  a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions. Rn and Rm×n denote the n-dimension Euclidean space and the set of all m×n real matrices, respectively. AT represents the transpose of the matrix A, and A+ represents the generalized inverse matrix of A. I denotes the identity matrix. Lp(R+;Rd) represents the family of Rd valued random variables X with E|X|p<∞. C2,1(Rn×R+;R+) represents the family of all real-valued functions V(x(t),t) defined on Rn×R+ which are continuously twice differentiable in x∈Rn and once differentiable in t∈R+.


## 2. Preliminaries

Let us introduce some definitions and lemmas which are needed to prove our main results of this paper.Definition 1 (see [14]).
A linear matrix inequality can be represented as the following form:(1)F(x):=F0+∑i=1n‍xiFi>0,
where x={x1,x2,…,xn} is the decision vector and {Fi,(i=0,1,2,…,n)} are given symmetric matrix.Lemma 2 (Schur complement [15]).
For the given symmetric matrixS=[S11S12S12TS22], then, one has the following equivalent conditions:(a)
S<0;(b)
S11<0,S22-S12TS11-1S12<0;(c)
S22<0,S11-S12S22-1S12T<0.Lemma 3 (see [13]).
LetΩ1(x) and Ω2(x) be two arbitrary quadratic forms over Rn; then Ω1(x)<0 for all x∈Rn-{0} satisfying Ω2(x)≤0 if and only if there   exists  τ≥0 such that
(2)Ω1(x)-τΩ2(x)<0,∀x∈Rn-{0}.Lemma 4 (the multidimensional Itô formula [16]).
Letx(t) be a d-dimensional Itô process on t≥0 with the stochastic differential
(3)dx(t)=f(t)dt+g(t)dW(t),
where f∈L1(R+;Rd),  g∈L2(R+;Rd×m), and W(t) is an m-dimensional Brownian motion. Let V∈C2,1(Rd×R+;R). Then V(x(t),t) is again an Itô process with stochastic differential given by
(4)dV(x(t),t)=[12Vt(x(t),t)+Vx(x(t),t)f(t)f+12trace(gT(t)Vxx(x(t),t)g(t))]dt+Vx(x(t),t)g(t)dB(t).Consider the following stochastic differential equation [16]:
(5)dx(t)=f(t)dt+g(t)dW(t).
Define the differential operator L associated with (5) by
(6)L=∂∂t+∑i=1n‍fi(x,t)∂∂xi+12∑i,j=1n‍[g(x,t)gT(x,t)]ij∂2∂xixj.
If L acts on a function V∈C2,1(Rn×R+;R+), then
(7)LV(x,t)=Vt(x,t)+Vx(x,t)f(x,t)+12trace[gT(x,t)Vxx(x,t)g(x,t)].
We set
(8)Vt=∂V∂t,Vx=(∂V∂x1,∂V∂x2,…,∂V∂xn),Vxx=(∂2V∂xi∂xj)n×n=(∂2V∂x1∂x1⋯∂2V∂x1∂xn⋮⋮∂2V∂xn∂x1⋯∂2V∂xn∂xn).
Then, we have the following lemma.Lemma 5 (see [16]).
If there exists a positive-definite decrescent functionV∈C2,1(Rn×R+;R+) such that
(9)LV(x,t)<0,
then the trivial solution of (5) is stochastically asymptotically stable.


## 3. Main Results

Consider the following stochastic system:(10)dx(t)=[Ax(t)+f(t,x(t))]dt+Dx(t)dw(t),
where x(t)∈Rn is the state of the system, A∈Rn×n and D∈Rn×n are constant matrix with appropriate dimensions, and w(t) is one-dimensional Brownian motion which satisfies
(11)E[dw(t)]=0,E[dw2(t)]=dt.f(t,x(t)) is a known continuous vector-valued Borel measurable function and satisfies the following quadratic inequality:
(12)fT(t,x(t))f(t,x(t))≤α2xTFTFx,
which can be rewritten as
(13)[xf]T[-α2FTF00I][xf]≤0,
where α is nonnegative constant and F is a constant matrix with appropriate dimension.Remark 6.
For any givenF, we note that inequality (12) defines a class of functions
(14)Fα={f:Rn+1⟶Rn∣fTff≤α2xTFTFx,(t,x)∈R+×Rn}.Fα includes function F(t,0)=0, and x=0 corresponding to the initial value x(t0)=0 is a trivial solution of system (10). For the convenience of the discussion, let γ=α-2.Definition 7.
System (10) is robustly stable with degree α if the trivial solution x=0 is stochastically asymptotically stable for all f(t,x(t)) satisfying constraint (12).Theorem 8.
System (10) is robustly stable with degree α if there exists positive-definite matrix Y such that the following convex optimization problem is solvable:
(15)minimizeγsubjecttoY>0fffffffff[YAT+AYIYDTYFTI-I00DY0-Y0FY00-γI]<0,
where γ=α-2.Proof.
In order to establish robust stability in the sense of Definition7, the following quadratic Lyapunov function is used:
(16)V(x,t)=xTQx,
where Q is a symmetric positive-definite matrix (Q>0). According to (7), when we compute LV(x,t) with respect to system (10), we have
(17)LV(x,t)=2xT(t)·Q·[Ax(t)+f(t,x(t))]+xT(t)DT·Q·Dx(t)=2xT(t)·Q·Ax(t)+2xT(t)·Q·f(t,x(t))+xT(t)DT·Q·Dx(t)=xT(t)·QA·x(t)+xT(t)·ATQ·x(t)+xT(t)·Q·f(t,x(t))+fT(t,x(t))·Q·x(t)+xT(t)·DTQD·x(t).
From Lemma 5, for the stochastically asymptotically stability, we have
(18)LV(x,t)<0.
We rewrite (17) as
(19)[xf]T[QA+ATQ+DTQDQQ0][xf]<0.
From Lemma 3 we further rewrite (19) using (13) as the inequalities
(20)[xf]T[QA+ATQ+DTQD+τα2FTFQQ-τI][xf]<0,
where P>0 and τ≥0. Namely, we have the following inequalities:(21)[QA+ATQ+DTQD+τα2FTFQQ-τI]<0.
Further, from Lemma 2, we rewrite (21) as
(22)[QA+ATQ+τα2FTFQDTQ-τI0D0-Q-1]<0.
Because τ≥0, the inequalities (22) represent nonstrict linear matrix inequalities. From the known result of [14], if there is a solution for   τ=0,  there is a solution for some τ>0 and sufficiently small   α;  then, we can replace τ≥0 by τ>0.   Multiplying  the first row and the first column of matrix (22) by Q-1, we have
(23)[Q-1AT+AQ-1+τα2Q-1FTFQ-1IQ-1DTI-τI0DQ-10-Q-1]<0.
Multiplying the first row and the third row of matrix (23) by τ, we have
(24)[τQ-1AT+τAQ-1+τα2Q-1FTFτQ-1τIτQ-1DTI-τI0τDQ-10-τQ-1]<0.
Let Y=τQ-1; therefore, (24) is further equivalent to
(25)[YAT+AY+α2YFTFYτIYDTI-τI0DY0-Y]<0.
Dividing  the second column of the matrix (25) by τ, we have
(26)[YAT+AY+α2YFTFYIYDTI-I0DY0-Y]<0.
From Lemma 2, we rewrite (26) as
(27)Y>0[YAT+AYIYDTYFTI-I00DY0-Y0FY00-γI]<0,
where Y=τQ-1. We assume that the matrix F is selected and maximize parameter α by solving the following LMI problem in Y and γ:
(28)minimizeγsubjecttoY>0ffffffffff[YAT+AYIYDTYFTI-I00DY0-Y0FY00-γI]<0.
The proof is now complete.Now, we introduce static output feedback to stabilize the system. Consider the following stochastic system:(29)dx(t)=[Ax(t)+Bu(t)+f(t,x(t))]dt+[Dx(t)+Cu(t)]dw(t)y(t)=Ex(t),
where x(t)∈Rn,u(t)∈Rm, and y(t)∈Rp are the system state and control input and output, respectively; A∈Rn×n,  D∈Rn×n,  B∈Rn×m,  C∈Rn×m, and  E∈Rp×n are constant matrices; without loss of generality, suppose that rank (E)=p; f(t,x(t))   satisfies  the constraint condition (12).Consider the following form of linear static output feedback controller:(30)u(t)=Ky(t),
where K∈Rm×p is a constant matrix to be determined.Definition 9.
If there exists a controller in the form of (30) such that the system (29) and (30) is robustly stable with degree α in the sense of Definition 7, then, we say that system (29) is robustly stabilized with degree α by the control law (30).Theorem 10.
System (29) is robustly stabilizable with degree α by means of static output feedback (30) if the following optimization problem is solvable:(31)minimizeγsubjecttoY>0ffffffffff[YAT+AY+LTBT+BLIYDT+LTCTYFTI-I00DY+CL0-Y0FY00-γI]<0.Proof.
Applying the static output feedback (30) to the system (29), we have
(32)dx(x,t)=[Ax(t)+BKEx(t)+f(t,x(t))]dt+[Dx(t)+CKEx(t)]dw(t)=[(A+BKE)x(t)+f(t,x(t))]dt+(D+CKE)x(t)dw(t)≐[A^x(t)+f(t,x(t))]dt+D^x(t)dw(t).
Namely,
(33)dx(t)=[A^x(t)+f(t,x(t))]dt+D^x(t)dw(t),
where A^=A+BKE and D^=D+CKE. Using quadratic function V(x) and computing LV(x,t) with respect to system (33) which is similar to Theorem 8, we have the following inequalities:(34)[YAT+AY+YETKTBT+BKEYIYDT+YETKTCTYFTI-I00DY+CKEY0-Y0FY00-γI]<0,which is not an LMI in Y, K, and E but can be made so by introducing the change of variable
(35)KEY=L
or
(36)K=L(Y)-1E+.
Since the transformed problem is now of LMI variety, we have(37)minimizeγsubjecttoY>0ffffffffff[YAT+AY+LTBT+BLIYDT+LTCTYFTI-I00DY+CL0-Y0FY00-γI]<0.The proof is now complete.To make the outcomes of problem (31) practical, we have to impose some restrictions on Y and L. Y is restricted to the structure
(38)Y=[Q~1ooTq~2],
where Q~1 is a (n-1)×(n-1) matrix, o∈Rn-1 is a zero vector, and q~2∈R.L is restricted as
(39)L=[Ol],
where O is an (n-1)×n zero matrix and l=(l1,l2,…,ln) is a row vector. Now, we propose the minimization problem in order to maximize α,(40)minimizeγsubjecttoY>0ffffffffff[YAT+AY+LTBT+BLIYDT+LTCTYFTI-I00DY+CL0-Y0FY00-γI]<0,where Y and L have the form described in (38) and (39). If the optimization problem is feasible, we can compute KE by
(41)KE=l·q~2-1=(l1q~2,l2q~2,…,lnq~2).
Then, we have the following corollary.Corollary 11.
System (29) is robustly stabilizable with degree α by means of static output feedback (30) if the problem (41) is feasible.We note that it is possible to incorporate constraints on the size of the vectorK in optimization problem (40). we can restrict the size of K by constraining q~2-1 and l (see [12]). We set
(42)llT<κl,κl>0,
which is equivalent to the LMI
(43)[-κlllT-I]<0.
Then, we suppose that
(44)q~2-1<κq,κq>0.
Inequality  (44) can be rewritten as the linear matrix inequalities
(45)[κq11q~2]>0.
We can get the optimized bound from constraints (42) and (44), which is shown as follows:
(46)(KE)(KE)T=llTq~2-2<κlκq2,
which has the LMI representation (43) and (45).With these modifications the optimization problem (40) becomes(47)minimizeγ+κl+κqsubjecttoY>0ffffffffff[YAT+AY+LTBT+BLIYDT+LTCTYFTI-I00DY+CL0-Y0FY00-γI]<0ffffffffff[-κlllT-l]<0ffffffffff[κq11q~2]>0.Then, we have the following corollary.Corollary 12.
System (29) is robustly stabilizable with degree α by means of static output feedback (30) if the problem (47) is feasible.Further,   by expanding  the system (29), consider the following stochastic system:
(48)dx(t)=[A0x(t)+B0u(t)+f(t,x(t),u(t))]dt+[D0x(t)+C0u(t)]dw(t)y(t)=E0x(t),
where x(t)∈Rn,  u(t)∈Rm, and y(t)∈Rp are the system state   and  control input and output, respectively; A0∈Rn×n,  D0∈Rn×n,  B0∈Rn×m,  C0∈Rn×m, and E0∈Rp×n are constant matrices; without loss of generality, we suppose that rank (E0)=p; f(t,x(t),u(t)) is known vector-valued function and satisfies the following constraint condition:
(49)fT(t,x,u)f(t,x,u)≤α2(xTHTHx+uTGTGu),
where H and G are constant matrices with appropriate dimensions and α is nonnegative constant.Consider the following form of linear static output feedback controller:(50)u(t)=K0y(t),
where K0∈Rm×p is a constant matrix to be determined.Definition 13.
If there exists a controller in the form of (50) such that the system (48) and (50) is robustly stable with degree α in the sense of Definition 7, then, we say that system (48) is robustly stabilized with degree α by the control law (50).Theorem 14.
System (48) is robustly stabilizable with degree α by means of static output feedback (50) if the following optimization problem on Y∈Rn×n, X∈Rm×p, and Z∈Rp×p is solvable:(51)minimizeγsubjecttoY>0fffffffff[YA0T+A0Y+B0XE0+E0TXTB0TIYDT+E0TXTC0TF0TI-I00D0Y+C0XE00-Y0F000-γI]<0fffffffffE0Y=ZE0,where F0=(HYGXE0).Proof.
Applying the static output feedback (50) to the system (48), we have
(52)dx(x,t)=[A0x(t)+B0K0E0x(t)+f(t,x(t),K0E0x(t))]dt+[D0x(t)+C0K0E0x(t)]dw(t)=[(A0+B0K0E0)x(t)+f(t,x(t),K0E0x(t))]dt+[D0+C0K0E0]x(t)dw(t)≐[A^x(t)+f(t,x(t),K0E0x(t))]dt+D^x(t)dw(t),
where f(t,x(t),K0E0x(t)) can be transformed to
(53)fT(t,x(t),K0E0x(t))f(t,x(t),K0E0x(t))≤α2[xT(t)HTHx(t)+(K0E0x(t))TGTG(K0E0x(t))]=α2[xT(t)HTHx(t)+xT(t)E0TK0TGTGK0E0x(t)]=α2xT(t)(HGK0E0)T(HGK0E0)x(t).
Namely,
(54)dx(t)=[A^x(t)+f(t,x(t),K0E0x(t))]dt+D^x(t)dw(t),
where A^=A0+B0K0E0 and D^=D0+C0K0E0. That is, system (54) is equivalent to (10) with constraint (12) if
(55)A=A^=A0+B0K0E0,D=D^=D0+C0K0E0,F=(HGK0E0).
Noting that the full row rank of E and Y is a positive-definite matrix from LMI (51), then matrix E0Y=ZE0 implies that
(56)p≥rank(Z)≥rank(ZE0)=rank(E0Y)≥rank[(E0Y)·Y-1]=rank(E0)=p,
then, Z is nonsingular. If the gain of control law in the form of (50) can be chosen as
(57)K0=XZ-1,
then, in LMI (26), we have
(58)AY=(A0+B0K0E0)Y=A0Y+B0K0E0Y=A0Y+B0K0ZE0=A0Y+B0XE0DY=(D0+C0K0E0)Y=D0Y+C0K0E0Y=D0Y+C0K0ZE0=D0Y+C0XE0FY=(HGK0E0)Y=(HYGK0E0Y)=(HYGK0ZE0)=(HYGXE0),
then, for systems (54), the following LMI is equivalent to the LMI in (15):(59)[YA0T+A0Y+B0XE0+E0TXTB0TIYDT+E0TXTC0T(YHT,E0TXTGT)I-I00D0Y+C0XE00-Y0(HYGXE0)00-γI]<0.From Theorem 8, the system (54) are robustly stable with degree α.
The proof is now complete.


## 4. Numerical Simulation

To further verify the flexibility and correctness of the theory in this paper, we present a simple numerical example.Example 15.
We consider the stochastic systems (29) with
(60)A=[-51-2-3],B=[01],D=[10.1-0.2-0.5],C=[12],F=[1-2],E=[01].In this example, Theorem10 is applied to construct a static output feedback law.By using the Matlab control toolbox to solve the convex optimization problem (31) and obtain
(61)Y=[9.33101.75991.75991.8743],L=[2.92930.6304],γ=2.0008,
then, by Theorem 10 the control law for linear static output feedback can be chosen as
(62)K=0.0505.
We conclude that the stochastic system (29) is robustly stabilizable by means of static output feedback (30) if we choose K=0.0505.


## 5. Conclusion

The problems of robust stability by means of output feedback law for a class of stochastic systems are investigated in this paper. It is shown that the problems can be transformed to the convex optimization problems in the form of linear matrix inequalities. The unified approach is presented in this paper, and how the linear matrix inequalities methods can be used to stabilize the stochastic system is given. The sufficient conditions for the existence of output feedback laws are obtained. Further, we give the numerical example by using the computer simulation to verify the correctness of the theoretical analysis results.



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