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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.
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 . 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 . 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 . Then, Ho and Lu  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 is a complete probability space with a filtration satisfying the usual conditions. and denote the -dimension Euclidean space and the set of all real matrices, respectively. represents the transpose of the matrix , and represents the generalized inverse matrix of . denotes the identity matrix. represents the family of valued random variables with . represents the family of all real-valued functions defined on which are continuously twice differentiable in and once differentiable in .
Let us introduce some definitions and lemmas which are needed to prove our main results of this paper.
Definition 1 (see ). A linear matrix inequality can be represented as the following form: where is the decision vector and are given symmetric matrix.
Lemma 2 (Schur complement ). For the given symmetric matrix , then, one has the following equivalent conditions:(a);(b);(c).
Lemma 3 (see ). Let and be two arbitrary quadratic forms over ; then for all satisfying if and only if there exists such that
Lemma 4 (the multidimensional Itô formula ). Let be a d-dimensional Itô process on with the stochastic differential where , , and is an m-dimensional Brownian motion. Let . Then is again an Itô process with stochastic differential given by
3. 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 , and corresponding to the initial value is a trivial solution of system (10). For the convenience of the discussion, let .
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 .
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 . 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 and . Namely, we have the following inequalities: Further, from Lemma 2, we rewrite (21) as Because , the inequalities (22) represent nonstrict linear matrix inequalities. From the known result of , if there is a solution for , there is a solution for some and sufficiently small ; then, we can replace by . Multiplying the first row and the first column of matrix (22) by , we have Multiplying the first row and the third row of matrix (23) by , we have Let ; 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 . 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).
Proof. Applying the static output feedback (30) to the system (29), we have Namely, where and . 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 haveThe 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 is a matrix, is a zero vector, and .
is restricted as where is an zero matrix and 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 Then, we have the following corollary.
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 and (see ). 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: which has the LMI representation (43) and (45).
With these modifications the optimization problem (40) becomesThen, we have the following corollary.
Further, by expanding the system (29), 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, we suppose that rank ; is known vector-valued function and satisfies the following constraint condition: where and are constant matrices with appropriate dimensions and is nonnegative constant.
Consider the following form of linear static output feedback controller: where 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 in the sense of Definition 7, then, we say that system (48) is robustly stabilized with degree by the control law (50).
Proof. Applying the static output feedback (50) to the system (48), we have
where can be transformed to
where and . That is, system (54) is equivalent to (10) with constraint (12) if
Noting that the full row rank of and is a positive-definite matrix from LMI (51), then matrix implies that
then, is nonsingular. If the gain of control law in the form of (50) can be chosen as
then, in LMI (26), we have
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.
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
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 then, by Theorem 10 the control law for linear static output feedback can be chosen as We conclude that the stochastic system (29) is robustly stabilizable by means of static output feedback (30) if we choose .
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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Copyright © 2013 Xin-rong Cong and Long-suo Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.