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

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.


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][5][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][8][9][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.

Main Results
Consider the following stochastic system: where () ∈   is the state of the system,  ∈  × and  ∈  × are constant matrix with appropriate dimensions, and () is one-dimensional Brownian motion which satisfies (, ()) is a known continuous vector-valued Borel measurable function and satisfies the following quadratic inequality: which can be rewritten as where  is nonnegative constant and  is a constant matrix with appropriate dimension.
Remark 6.For any given , we note that inequality (12) defines a class of functions includes function (, 0) = 0, and  = 0 corresponding to the initial value ( 0 ) = 0 is a trivial solution of system (10).
For the convenience of the discussion, let  =  −2 .
Theorem 8. System (10) is robustly stable with degree  if there exists positive-definite matrix  such that the following convex optimization problem is solvable: where  =  −2 .
Proof.In order to establish robust stability in the sense of Definition 7, the following quadratic Lyapunov function is used: where  is a symmetric positive-definite matrix ( > 0).According to (7), when we compute (, ) with respect to system (10), we have From Lemma 5, for the stochastically asymptotically stability, we have We rewrite (17) as From Lemma 3 we further rewrite (19) using ( 13) as the inequalities where  > 0 and  ≥ 0. Namely, we have the following inequalities: Further, from Lemma 2, we rewrite (21) as Because  ≥ 0, the inequalities (22) represent nonstrict linear matrix inequalities.From the known result of [14], if there is a solution for  = 0, there is a solution for some  > 0 and sufficiently small ; then, we can replace  ≥ 0 by  > 0.
Multiplying the first row and the first column of matrix ( 22) by  −1 , we have Multiplying the first row and the third row of matrix (23) by , we have Let  =  −1 ; therefore, (24) is further equivalent to Dividing the second column of the matrix (25) by , we have From Lemma 2, we rewrite (26) as where  =  −1 .We assume that the matrix  is selected and maximize parameter  by solving the following LMI problem in  and : The proof is now complete.Now, we introduce static output feedback to stabilize the system.Consider the following stochastic system: where () ∈   , () ∈   , and () ∈   are the system state and control input and output, respectively;  ∈  × ,  ∈  × ,  ∈  × ,  ∈  × , and  ∈  × are constant matrices; without loss of generality, suppose that rank () = ; (, ()) satisfies the constraint condition (12).
Consider the following form of linear static output feedback controller: where  ∈  × 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  in the sense of Definition 7, then, we say that system (29) is robustly stabilized with degree  by the control law (30).where Â =  +  and D =  + .Using quadratic function () and computing (, ) with respect to system (33) which is similar to Theorem 8, we have the following inequalities: which is not an LMI in , , and  but can be made so by introducing the change of variable  = or Since the transformed problem is now of LMI variety, we have The proof is now complete.
To make the outcomes of problem (31) practical, we have to impose some restrictions on  and . is restricted to the structure where Q1 is a ( − 1) × ( − 1) matrix,  ∈  −1 is a zero vector, and q2 ∈ . is restricted as where  is an ( − 1) ×  zero matrix and  = ( 1 ,  2 , . . .,   ) is a row vector.Now, we propose the minimization problem in order to maximize , where  and  have the form described in (38) and (39).If the optimization problem is feasible, we can compute  by We note that it is possible to incorporate constraints on the size of the vector  in optimization problem (40).we can restrict the size of  by constraining q−1 2 and  (see [12]).We set which is equivalent to the LMI Then, we suppose that Inequality (44) can be rewritten as the linear matrix inequalities We can get the optimized bound from constraints (42) and (44), which is shown as follows: Further, by expanding the system (29), consider the following stochastic system: where () ∈   , () ∈   , and () ∈   are the system state and control input and output, respectively;  0 ∈  × ,  0 ∈  × ,  0 ∈  × ,  0 ∈  × , and  0 ∈  × are constant matrices; without loss of generality, we suppose that rank ( 0 ) = ; (, (), ()) is known vector-valued function and satisfies the following constraint condition:   (, , )  (, , ) ≤  2 (     +     ) , (49) where  and  are constant matrices with appropriate dimensions and  is nonnegative constant.then, for systems (54), the following LMI is equivalent to the LMI in (15): From Theorem 8, the system (54) are robustly stable with degree .
The proof is now complete.

Numerical Simulation
To further verify the flexibility and correctness of the theory in this paper, we present a simple numerical example.
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 We conclude that the stochastic system (29) is robustly stabilizable by means of static output feedback (30) if we choose  = 0.0505.

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.

Example 15 .
We consider the stochastic systems (29) with Theorem 10 the control law for linear static output feedback can be chosen as  = 0.0505.