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

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.

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 {𝐹_𝑡}_𝑡≥0 satisfying the usual conditions. 𝑅^𝑛 and 𝑅^𝑚×𝑛 denote the 𝑛-dimension Euclidean space and the set of all 𝑚×𝑛 real matrices, respectively. 𝐴^𝑇 represents the transpose of the matrix 𝐴, and 𝐴⁺ represents the generalized inverse matrix of 𝐴. 𝐼 denotes the identity matrix. 𝐿^𝑝(𝑅₊;𝑅^𝑑) represents the family of 𝑅^𝑑 valued random variables 𝑋 with 𝐸|𝑋|^𝑝<∞. 𝐶^2,1(𝑅^𝑛×𝑅₊;𝑅₊) represents the family of all real-valued functions 𝑉(𝑥(𝑡),𝑡) defined on 𝑅^𝑛×𝑅₊ which are continuously twice differentiable in 𝑥∈𝑅^𝑛 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: 𝐹(𝑥)∶=𝐹₀+^𝑛_𝑖=1𝑥_𝑖𝐹_𝑖>0, (1) where 𝑥={𝑥₁,𝑥₂,…,𝑥_𝑛} is the decision vector and {𝐹_𝑖,(𝑖=0,1,2,…,𝑛)} are given symmetric matrix.

Lemma 2 (Schur complement [15]). For the given symmetric matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}, then, one has the following equivalent conditions:(a)𝑆<0;(b)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0;(c)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

Lemma 3 (see [13]). Let Ω₁(𝑥) and Ω₂(𝑥) be two arbitrary quadratic forms over 𝑅^𝑛; then Ω₁(𝑥)<0 for all 𝑥∈𝑅^𝑛−{0} satisfying Ω₂(𝑥)≤0 if and only if there exists 𝜏≥0 such that Ω₁(𝑥)−𝜏Ω₂(𝑥)<0,∀𝑥∈𝑅^𝑛−{0}. (2)

Lemma 4 (the multidimensional Itô formula [16]). Let 𝑥(𝑡) be a d-dimensional Itô process on 𝑡≥0 with the stochastic differential 𝑑𝑥(𝑡)=𝑓(𝑡)𝑑𝑡+𝑔(𝑡)𝑑𝑊(𝑡), (3) where 𝑓∈𝐿¹(𝑅₊;𝑅^𝑑),  𝑔∈𝐿²(𝑅₊;𝑅^𝑑×𝑚), and 𝑊(𝑡) is an m-dimensional Brownian motion. Let 𝑉∈𝐶^2,1(𝑅^𝑑×𝑅₊;𝑅). Then 𝑉(𝑥(𝑡),𝑡) is again an Itô process with stochastic differential given by 𝑑𝑉(𝑥(𝑡),𝑡)=𝑉_𝑡(𝑥(𝑡),𝑡)+𝑉_𝑥(𝑥(𝑡))𝑓(𝑡)+1,𝑡2trace𝑔^𝑇(𝑡)𝑉_𝑥𝑥(𝑥(𝑡),𝑡)𝑔(𝑡)𝑑𝑡+𝑉_𝑥(𝑥(𝑡),𝑡)𝑔(𝑡)𝑑𝐵(𝑡). (4)

Consider the following stochastic differential equation [16]: 𝑑𝑥(𝑡)=𝑓(𝑡)𝑑𝑡+𝑔(𝑡)𝑑𝑊(𝑡). (5) Define the differential operator 𝐿 associated with (5) by 𝜕𝐿=+𝜕𝑡^𝑛_𝑖=1𝑓_𝑖()𝜕𝑥,𝑡𝜕𝑥_𝑖+12^𝑛_𝑖,𝑗=1𝑔(𝑥,𝑡)𝑔^𝑇(𝑥,𝑡)_𝑖𝑗𝜕²𝜕𝑥_𝑖𝑥_𝑗. (6) If 𝐿 acts on a function 𝑉∈𝐶^2,1(𝑅^𝑛×𝑅₊;𝑅₊), then 𝐿𝑉(𝑥,𝑡)=𝑉_𝑡(𝑥,𝑡)+𝑉_𝑥()𝑓()+1𝑥,𝑡𝑥,𝑡2trace𝑔^𝑇(𝑥,𝑡)𝑉_𝑥𝑥(𝑥,𝑡)𝑔(𝑥,𝑡). (7) We set 𝑉_𝑡=𝜕𝑉𝜕𝑡,𝑉_𝑥=𝜕𝑉𝜕𝑥₁,𝜕𝑉𝜕𝑥₂,…,𝜕𝑉𝜕𝑥_𝑛,𝑉_𝑥𝑥=𝜕²𝑉𝜕𝑥_𝑖𝜕𝑥_{𝑗𝑛×𝑛}=𝜕²𝑉𝜕𝑥₁𝜕𝑥₁⋯𝜕²𝑉𝜕𝑥₁𝜕𝑥_𝑛𝜕⋮⋮₂𝑉𝜕𝑥_𝑛𝜕𝑥₁⋯𝜕²𝑉𝜕𝑥_𝑛𝜕𝑥_𝑛. (8) Then, we have the following lemma.

Lemma 5 (see [16]). If there exists a positive-definite decrescent function 𝑉∈𝐶^2,1(𝑅^𝑛×𝑅₊;𝑅₊) such that 𝐿𝑉(𝑥,𝑡)<0, (9) then the trivial solution of (5) is stochastically asymptotically stable.

3. Main Results

Consider the following stochastic system: 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))𝑑𝑡+𝐷𝑥(𝑡)𝑑𝑤(𝑡), (10) where 𝑥(𝑡)∈𝑅^𝑛 is the state of the system, 𝐴∈𝑅^𝑛×𝑛 and 𝐷∈𝑅^𝑛×𝑛 are constant matrix with appropriate dimensions, and 𝑤(𝑡) is one-dimensional Brownian motion which satisfies 𝐸[𝑑𝑤(𝑡)]=0,𝐸𝑑𝑤²(𝑡)=𝑑𝑡. (11)𝑓(𝑡,𝑥(𝑡)) is a known continuous vector-valued Borel measurable function and satisfies the following quadratic inequality: 𝑓^𝑇(𝑡,𝑥(𝑡))𝑓(𝑡,𝑥(𝑡))≤𝛼²𝑥^𝑇𝐹^𝑇𝐹𝑥, (12) which can be rewritten as 𝑥𝑓^𝑇−𝛼²𝐹^𝑇𝑥𝑓𝐹00𝐼≤0, (13) where 𝛼 is nonnegative constant and 𝐹 is a constant matrix with appropriate dimension.

Remark 6. For any given 𝐹, we note that inequality (12) defines a class of functions 𝐹_𝛼=𝑓∶𝑅^𝑛+1⟶𝑅^𝑛∣𝑓^𝑇𝑓≤𝛼²𝑥^𝑇𝐹^𝑇𝐹𝑥,(𝑡,𝑥)∈𝑅₊×𝑅^𝑛. (14)𝐹_𝛼 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: minimize𝛾subjectto𝑌>0𝑌𝐴^𝑇+𝐴𝑌𝐼𝑌𝐷^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌0−𝑌0𝐹𝑌00−𝛾𝐼<0, (15) where 𝛾=𝛼⁻².

Proof. In order to establish robust stability in the sense of Definition 7, the following quadratic Lyapunov function is used: 𝑉(𝑥,𝑡)=𝑥^𝑇𝑄𝑥, (16) 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 𝐿𝑉(𝑥,𝑡)=2𝑥^𝑇(𝑡)⋅𝑄⋅𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝑥^𝑇(𝑡)𝐷^𝑇⋅𝑄⋅𝐷𝑥(𝑡)=2𝑥^𝑇(𝑡)⋅𝑄⋅𝐴𝑥(𝑡)+2𝑥^𝑇(𝑡)⋅𝑄⋅𝑓(𝑡,𝑥(𝑡))+𝑥^𝑇(𝑡)𝐷^𝑇⋅𝑄⋅𝐷𝑥(𝑡)=𝑥^𝑇(𝑡)⋅𝑄𝐴⋅𝑥(𝑡)+𝑥^𝑇(𝑡)⋅𝐴^𝑇𝑄⋅𝑥(𝑡)+𝑥^𝑇(𝑡)⋅𝑄⋅𝑓(𝑡,𝑥(𝑡))+𝑓^𝑇(𝑡,𝑥(𝑡))⋅𝑄⋅𝑥(𝑡)+𝑥^𝑇(𝑡)⋅𝐷^𝑇𝑄𝐷⋅𝑥(𝑡). (17) From Lemma 5, for the stochastically asymptotically stability, we have 𝐿𝑉(𝑥,𝑡)<0. (18) We rewrite (17) as 𝑥𝑓^𝑇𝑄𝐴+𝐴^𝑇𝑄+𝐷^𝑇𝑥𝑓𝑄𝐷𝑄𝑄0<0. (19) From Lemma 3 we further rewrite (19) using (13) as the inequalities 𝑥𝑓^𝑇𝑄𝐴+𝐴^𝑇𝑄+𝐷^𝑇𝑄𝐷+𝜏𝛼²𝐹^𝑇𝑥𝑓𝐹𝑄𝑄−𝜏𝐼<0, (20) where 𝑃>0 and 𝜏≥0. Namely, we have the following inequalities:𝑄𝐴+𝐴^𝑇𝑄+𝐷^𝑇𝑄𝐷+𝜏𝛼²𝐹^𝑇𝐹𝑄𝑄−𝜏𝐼<0. (21) Further, from Lemma 2, we rewrite (21) as 𝑄𝐴+𝐴^𝑇𝑄+𝜏𝛼²𝐹^𝑇𝐹𝑄𝐷^𝑇𝑄−𝜏𝐼0𝐷0−𝑄⁻¹<0. (22) 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 𝑄⁻¹𝐴^𝑇+𝐴𝑄⁻¹+𝜏𝛼²𝑄⁻¹𝐹^𝑇𝐹𝑄⁻¹𝐼𝑄⁻¹𝐷^𝑇𝐼−𝜏𝐼0𝐷𝑄⁻¹0−𝑄⁻¹<0. (23) Multiplying the first row and the third row of matrix (23) by 𝜏, we have 𝜏𝑄⁻¹𝐴^𝑇+𝜏𝐴𝑄⁻¹+𝜏𝛼²𝑄⁻¹𝐹^𝑇𝐹𝜏𝑄⁻¹𝜏𝐼𝜏𝑄⁻¹𝐷^𝑇𝐼−𝜏𝐼0𝜏𝐷𝑄⁻¹0−𝜏𝑄⁻¹<0. (24) Let 𝑌=𝜏𝑄⁻¹; therefore, (24) is further equivalent to 𝑌𝐴^𝑇+𝐴𝑌+𝛼²𝑌𝐹^𝑇𝐹𝑌𝜏𝐼𝑌𝐷^𝑇𝐼−𝜏𝐼0𝐷𝑌0−𝑌<0. (25) Dividing the second column of the matrix (25) by 𝜏, we have 𝑌𝐴^𝑇+𝐴𝑌+𝛼²𝑌𝐹^𝑇𝐹𝑌𝐼𝑌𝐷^𝑇𝐼−𝐼0𝐷𝑌0−𝑌<0. (26) From Lemma 2, we rewrite (26) as 𝑌>0𝑌𝐴^𝑇+𝐴𝑌𝐼𝑌𝐷^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌0−𝑌0𝐹𝑌00−𝛾𝐼<0, (27) 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: minimize𝛾subjectto𝑌>0𝑌𝐴^𝑇+𝐴𝑌𝐼𝑌𝐷^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌0−𝑌0𝐹𝑌00−𝛾𝐼<0. (28) The proof is now complete.

Now, we introduce static output feedback to stabilize the system. Consider the following stochastic system: (𝑡)=(𝑡)(𝑡)((𝑡+[(𝑡)(𝑡(𝑡)𝑦(𝑡)(𝑡),𝑑𝑥𝐴𝑥+𝐵𝑢+𝑓𝑡,𝑥))𝑑𝑡𝐷𝑥+𝐶𝑢)]𝑑𝑤=𝐸𝑥 (29) where 𝑥(𝑡)∈𝑅^𝑛,𝑢(𝑡)∈𝑅^𝑚, and 𝑦(𝑡)∈𝑅^𝑝 are the system state and control input and output, respectively; 𝐴∈𝑅^𝑛×𝑛,  𝐷∈𝑅^𝑛×𝑛,  𝐵∈𝑅^𝑛×𝑚,  𝐶∈𝑅^𝑛×𝑚, and  𝐸∈𝑅^𝑝×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: 𝑢(𝑡)=𝐾𝑦(𝑡), (30) where 𝐾∈𝑅^𝑚×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:minimize𝛾subjectto𝑌>0𝑌𝐴^𝑇+𝐴𝑌+𝐿^𝑇𝐵^𝑇+𝐵𝐿𝐼𝑌𝐷^𝑇+𝐿^𝑇𝐶^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌+𝐶𝐿0−𝑌0𝐹𝑌00−𝛾𝐼<0. (31)

Proof. Applying the static output feedback (30) to the system (29), we have ()=(𝑡)(𝑡)((𝑡+[(𝑡)(𝑡(𝑡)=)𝑥(𝑡)((𝑡+()𝑥(𝑡)(𝑡)≐(𝑡)((𝑡(𝑡)(𝑡).𝑑𝑥𝑥,𝑡𝐴𝑥+𝐵𝐾𝐸𝑥+𝑓𝑡,𝑥))𝑑𝑡𝐷𝑥+𝐶𝐾𝐸𝑥)]𝑑𝑤(𝐴+𝐵𝐾𝐸+𝑓𝑡,𝑥))𝑑𝑡𝐷+𝐶𝐾𝐸𝑑𝑤𝐴𝑥+𝑓𝑡,𝑥))𝑑𝑡+𝐷𝑥𝑑𝑤 (32) Namely, 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))𝑑𝑡+𝐷𝑥(𝑡)𝑑𝑤(𝑡), (33) 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:𝑌𝐴^𝑇+𝐴𝑌+𝑌𝐸^𝑇𝐾^𝑇𝐵^𝑇+𝐵𝐾𝐸𝑌𝐼𝑌𝐷^𝑇+𝑌𝐸^𝑇𝐾^𝑇𝐶^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌+𝐶𝐾𝐸𝑌0−𝑌0𝐹𝑌00−𝛾𝐼<0, (34)which is not an LMI in 𝑌, 𝐾, and 𝐸 but can be made so by introducing the change of variable 𝐾𝐸𝑌=𝐿 (35) or 𝐾=𝐿(𝑌)⁻¹𝐸⁺. (36) Since the transformed problem is now of LMI variety, we haveminimize𝛾subjectto𝑌>0𝑌𝐴^𝑇+𝐴𝑌+𝐿^𝑇𝐵^𝑇+𝐵𝐿𝐼𝑌𝐷^𝑇+𝐿^𝑇𝐶^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌+𝐶𝐿0−𝑌0𝐹𝑌00−𝛾𝐼<0. (37)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 𝑄𝑌=¹𝑜𝑜^𝑇𝑞₂, (38) where 𝑄₁ is a (𝑛−1)×(𝑛−1) matrix, 𝑜∈𝑅^𝑛−1 is a zero vector, and 𝑞₂∈𝑅.

𝐿 is restricted as 𝑂𝑙𝐿=, (39) where 𝑂 is an (𝑛−1)×𝑛 zero matrix and 𝑙=(𝑙₁,𝑙₂,…,𝑙_𝑛) is a row vector. Now, we propose the minimization problem in order to maximize 𝛼,minimize𝛾subjectto𝑌>0𝑌𝐴^𝑇+𝐴𝑌+𝐿^𝑇𝐵^𝑇+B𝐿𝐼𝑌𝐷^𝑇+𝐿^𝑇𝐶^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌+𝐶𝐿0−𝑌0𝐹𝑌00−𝛾𝐼<0, (40)where 𝑌 and 𝐿 have the form described in (38) and (39). If the optimization problem is feasible, we can compute 𝐾𝐸 by 𝐾𝐸=𝑙⋅𝑞₂⁻¹=𝑙₁𝑞₂,𝑙₂𝑞₂𝑙,…,^𝑛𝑞₂. (41) 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 vector 𝐾 in optimization problem (40). we can restrict the size of 𝐾 by constraining 𝑞₂⁻¹ and 𝑙 (see [12]). We set 𝑙𝑙^𝑇<𝜅_𝑙,𝜅_𝑙>0, (42) which is equivalent to the LMI −𝜅_𝑙𝑙𝑙^𝑇−𝐼<0. (43) Then, we suppose that 𝑞₂⁻¹<𝜅_𝑞,𝜅_𝑞>0. (44) Inequality (44) can be rewritten as the linear matrix inequalities 𝜅_𝑞11𝑞₂>0. (45) We can get the optimized bound from constraints (42) and (44), which is shown as follows: (𝐾𝐸)(𝐾𝐸)^𝑇=𝑙𝑙^𝑇𝑞₂⁻²<𝜅_𝑙𝜅²_𝑞, (46) which has the LMI representation (43) and (45).

With these modifications the optimization problem (40) becomesminimize𝛾+𝜅_𝑙+𝜅_𝑞subjectto𝑌>0𝑌𝐴^𝑇+𝐴𝑌+𝐿^𝑇𝐵^𝑇+𝐵𝐿𝐼𝑌𝐷^𝑇+𝐿^𝑇𝐶^𝑇𝑌𝐹^𝑇𝐼−𝐼00𝐷𝑌+𝐶𝐿0−𝑌0𝐹𝑌00−𝛾𝐼<0−𝜅_𝑙𝑙𝑙^𝑇𝜅−𝑙<0_𝑞11𝑞₂>0. (47)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: 𝑑𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐵₀𝑢(𝑡)((𝑡)(𝑡+𝐷+𝑓𝑡,𝑥,𝑢))𝑑𝑡₀𝑥(𝑡)+𝐶₀𝑢(𝑡(𝑡)𝑦(𝑡))𝑑𝑤=𝐸₀𝑥(𝑡), (48) where 𝑥(𝑡)∈𝑅^𝑛,  𝑢(𝑡)∈𝑅^𝑚, and 𝑦(𝑡)∈𝑅^𝑝 are the system state and control input and output, respectively; 𝐴₀∈𝑅^𝑛×𝑛,  𝐷₀∈𝑅^𝑛×𝑛,  𝐵₀∈𝑅^𝑛×𝑚,  𝐶₀∈𝑅^𝑛×𝑚, and 𝐸₀∈𝑅^𝑝×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: 𝑓^𝑇(𝑡,𝑥,𝑢)𝑓(𝑡,𝑥,𝑢)≤𝛼²𝑥^𝑇𝐻^𝑇𝐻𝑥+𝑢^𝑇𝐺^𝑇𝐺𝑢, (49) 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: 𝑢(𝑡)=𝐾₀𝑦(𝑡), (50) where 𝐾₀∈𝑅^𝑚×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∈𝑅^𝑛×𝑛, 𝑋∈𝑅^𝑚×𝑝, and 𝑍∈𝑅^𝑝×𝑝 is solvable:minimize𝛾subjectto𝑌>0𝑌𝐴^𝑇₀+𝐴₀𝑌+𝐵₀𝑋𝐸₀+𝐸^𝑇₀𝑋^𝑇𝐵^𝑇₀𝐼𝑌𝐷^𝑇+𝐸^𝑇₀𝑋^𝑇𝐶^𝑇₀𝐹^𝑇₀𝐷𝐼−𝐼00₀𝑌+𝐶₀𝑋𝐸₀𝐹0−𝑌0₀𝐸00−𝛾𝐼<0₀𝑌=𝑍𝐸₀, (51)where 𝐹₀=^𝐻𝑌_{𝐺𝑋𝐸0}.

Proof. Applying the static output feedback (50) to the system (48), we have 𝑑𝑥(𝑥,𝑡)=𝐴₀𝑥(𝑡)+𝐵₀𝐾₀𝐸₀𝑥(𝑡)+𝑓𝑡,𝑥(𝑡),𝐾₀𝐸₀𝑥(𝑡)+𝐷𝑑𝑡₀𝑥(𝑡)+𝐶₀𝐾₀𝐸₀𝑥(𝑡(𝑡)=𝐴)𝑑𝑤₀+𝐵₀𝐾₀𝐸₀𝑥(𝑡)+𝑓𝑡,𝑥(𝑡),𝐾₀𝐸₀𝑥(𝑡)+𝐷𝑑𝑡₀+𝐶₀𝐾₀𝐸₀𝑥(𝑡)(𝑡)≐(𝑡)(𝑡)𝑑𝑤𝐴𝑥+𝑓𝑡,𝑥,𝐾₀𝐸₀𝑥(𝑡)+(𝑡)(𝑡),𝑑𝑡𝐷𝑥𝑑𝑤 (52) where 𝑓(𝑡,𝑥(𝑡),𝐾₀𝐸₀𝑥(𝑡)) can be transformed to 𝑓^𝑇𝑡,𝑥(𝑡),𝐾₀𝐸₀𝑥(𝑡)𝑓𝑡,𝑥(𝑡),𝐾₀𝐸₀𝑥(𝑡)≤𝛼²𝑥^𝑇(𝑡)𝐻^𝑇𝐻𝑥(𝑡)+𝐾₀𝐸₀𝑥(𝑡)^𝑇𝐺^𝑇𝐺𝐾₀𝐸₀𝑥(𝑡)=𝛼²𝑥^𝑇(𝑡)𝐻^𝑇𝐻𝑥(𝑡)+𝑥^𝑇(𝑡)𝐸^𝑇₀𝐾^𝑇₀𝐺^𝑇𝐺𝐾₀𝐸₀𝑥(𝑡)=𝛼²𝑥^𝑇(𝑡)𝐻𝐺𝐾₀𝐸₀^𝑇𝐻𝐺𝐾₀𝐸₀𝑥(𝑡). (53) Namely, 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓𝑡,𝑥(𝑡),𝐾₀𝐸₀𝑥(𝑡)𝑑𝑡+𝐷𝑥(𝑡)𝑑𝑤(𝑡), (54) where 𝐴=𝐴₀+𝐵₀𝐾₀𝐸₀ and 𝐷=𝐷₀+𝐶₀𝐾₀𝐸₀. That is, system (54) is equivalent to (10) with constraint (12) if 𝐴=𝐴=𝐴₀+𝐵₀𝐾₀𝐸₀,𝐷=𝐷=𝐷₀+𝐶₀𝐾₀𝐸₀,𝐻𝐹=𝐺𝐾₀𝐸₀. (55) Noting that the full row rank of 𝐸 and 𝑌 is a positive-definite matrix from LMI (51), then matrix 𝐸₀𝑌=𝑍𝐸₀ implies that 𝑝≥rank(𝑍)≥rank𝑍𝐸₀=rank𝐸₀𝑌≥rank𝐸₀𝑌⋅𝑌⁻¹=rank𝐸₀=𝑝, (56) then, 𝑍 is nonsingular. If the gain of control law in the form of (50) can be chosen as 𝐾₀=𝑋𝑍⁻¹, (57) then, in LMI (26), we have 𝐴𝑌=𝐴₀+𝐵₀𝐾₀𝐸₀𝑌=𝐴₀𝑌+𝐵₀𝐾₀𝐸₀𝑌=𝐴₀𝑌+𝐵₀𝐾₀𝑍𝐸₀=𝐴₀𝑌+𝐵₀𝑋𝐸₀𝐷𝑌=𝐷₀+𝐶₀𝐾₀𝐸₀𝑌=𝐷₀𝑌+𝐶₀𝐾₀𝐸₀𝑌=𝐷₀𝑌+𝐶₀𝐾₀𝑍𝐸₀=𝐷₀𝑌+𝐶₀𝑋𝐸₀𝐻𝐹𝑌=𝐺𝐾₀𝐸₀𝑌=𝐻𝑌𝐺𝐾₀𝐸₀𝑌=𝐻𝑌𝐺𝐾₀𝑍𝐸₀=𝐻𝑌𝐺𝑋𝐸₀, (58) then, for systems (54), the following LMI is equivalent to the LMI in (15):𝑌𝐴^𝑇₀+𝐴₀𝑌+𝐵₀𝑋𝐸₀+𝐸^𝑇₀𝑋^𝑇𝐵^𝑇₀𝐼𝑌𝐷^𝑇+𝐸^𝑇₀𝑋^𝑇𝐶^𝑇₀𝑌𝐻^𝑇,𝐸^𝑇₀𝑋^𝑇𝐺^𝑇𝐷𝐼−𝐼00₀𝑌+𝐶₀𝑋𝐸₀0−𝑌0𝐻𝑌𝐺𝑋𝐸₀00−𝛾𝐼<0. (59)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 01,12.𝐴=−51−2−3,𝐵=,𝐷=10.1−0.2−0.5𝐶=,𝐹=1−2,𝐸=01 (60)

In this example, Theorem 10 is applied to construct a static output feedback law.

By using the Matlab control toolbox to solve the convex optimization problem (31) and obtain 𝑌=,9.33101.75991.75991.8743𝐿=2.92930.6304,𝛾=2.0008, (61) then, by Theorem 10 the control law for linear static output feedback can be chosen as 𝐾=0.0505. (62) 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.