Abstract
This paper deals with the problems of the robust stochastic stability and stabilization for a class of uncertain discrete-time stochastic systems with interval time-varying delays and nonlinear disturbances. By utilizing a new Lyapunov-Krasovskii functional and some well-known inequalities, some new delay-dependent criteria are developed to guarantee the robust stochastic stability of a class of uncertain discrete-time stochastic systems in terms of the linear matrix inequality (LMI). Then based on the state feedback controller, the delay-dependent sufficient conditions of robust stochastic stabilization for a class of uncertain discrete-time stochastic systems with interval time-varying delays are established. The controller gain is designed to ensure the robust stochastic stability of the closed-loop system. Finally, illustrative examples are given to demonstrate the effectiveness of the proposed method.
1. Introduction
In the past decade, the stability analyses (see, e.g., feedback stabilization for discrete-time nonlinear systems, robustness of exponential stability, and optimal stabilizing compensator) and discrete-time stochastic systems have been extensively studied because of their potential applications (see, e.g., [1–3] and the references therein). On the other hand, time delays, both time-varying and constant, are frequently encountered in various biological, engineering, and economic systems [4, 5]. The stability analysis and control of time-delay systems have been widely studied during the past years [6–9]. In [6], robust delay-dependent stability and stabilization methods for a class of nonlinear discrete-time systems with time-varying delays were proposed. In [8], the robust stabilization problem for uncertain linear systems with interval time-varying delays was investigated. Some delay-dependent stability criteria were derived based on an improved Wirtinger’s inequality.
On the other hand, the research on stochastic systems has aroused much interest in the past few years, because stochastic modeling has come to play an important role in many real systems [10]. In [11], a robust delay-distribution-dependent stochastic stability analysis was conducted for a class of discrete-time stochastic delayed neural networks with parameter uncertainties. The robust stability and stabilization of a class of nonlinear discrete stochastic systems were reported in [7]. In [12], the global exponential stability of switched stochastic neural networks with time-varying delays was considered. Authors in [13] studied the robust stability of discrete-time stochastic neural networks with time-varying delays, and the stability analysis problem for stochastic neural networks becomes increasingly significant. In [14], the mean-square exponential stability problem for stochastic discrete-time recurrent neural networks with time-varying discrete and distributed delays was investigated. In [15], the delay-probability-distribution-dependent robust stability problem for a class of uncertain stochastic neural networks with time-varying delay was investigated, and some stability criteria were proposed.
In this paper, we contribute to the further development of robust stability and feedback stabilization methods for a class of uncertain nonlinear discrete-time stochastic systems with interval time-varying delays. The parameter uncertainties are time-varying matrices which are norm-bounded, and the unknown nonlinear time-varying perturbations with time-varying delay are quadratically bounded. Comparing with [3, 7, 8], the stochastic nonlinearity and parameter uncertainties and unknown nonlinearities with time-varying delays are considered for discrete-time stochastic systems and therefore the model in this paper may be more general. The main contributions of this paper can be summarized as follows. The system model is comprehensive that covers stochastic nonlinearity, parameter uncertainties, and the unknown nonlinearities that are time-varying perturbations with time-varying delay, thereby better reflecting the reality. An appropriate Lyapunov-Krasovskii functional is constructed to exhibit the delay-dependent dynamics, and delay-dependent robustly stochastic stability analysis is performed to characterize linear matrix inequalities- (LMI-) based conditions under which the discrete-time nonlinear stochastic delay system which does not contain control is robustly stochastically stable. Robust feedback stabilization methods are provided based on state feedback control. The new sufficient conditions are established under which the closed-loop system is robustly stochastically stable and the calculation method of the control gain is given. The result presented in this paper designs a state feedback control law that stabilizes the closed-loop system and is maximally robust with respect to considered nonlinear perturbations. Numerical simulation examples are used to demonstrate the effectiveness and applicability of the obtained results. By introducing some parameters , and , our method leads to less conservatism compared with the existing ones.
The remainder of this paper is organized as follows. In Section 2, the problem description and preliminaries are stated and some lemmas and a definition are given. In Section 3, by using Lyapunov-Krasovskii functional, novel LMI sufficient conditions for the robust stochastic stability of a class of uncertain discrete-time stochastic systems with interval time-varying delays and nonlinear disturbances are derived. Furthermore, the robust stochastic stabilizable criteria for uncertain nonlinear discrete-time stochastic delayed systems are presented. In Section 4, two numeric examples are given to illustrate the results. Finally, some conclusions are drawn in Section 5.
Notations. denotes the set of all real nonnegative integers and and denote the -dimensional Euclidean space and the set of all real matrices, respectively. The superscripts and denote the matrix transposition and matrix inverse, respectively. means the smallest eigenvalue of a matrix. stands for the mathematical expectation operator with respect to the given probability measure . The asterisk in a matrix is used to denote term that is induced by symmetry. is the identity matrix with compatible dimension.
2. Problem Formulation
Consider the uncertain nonlinear discrete stochastic system with time-varying delay described by where is the state vector and . is a sequence of identically, independently normally distributed random function with and is the control input. The positive integer denotes the time-varying delay satisfying where and are known positive integers, respectively, and , and , whereas matrices , , and represent the time-varying parameter uncertainties and are assumed to satisfy the following condition: where and are known constant matrices and is the unknown time-varying matrix-valued function satisfying the following condition: The crucial assumptions about the nonlinear functions and are that they are uncertain and satisfy the following quadratic inequalities for all :
At the end of this section, we introduce a definition and some lemmas for the development of our results.
Definition 1 (see [7]). System (1) with is said to be robustly stochastically stable with margins and , if there exists a constant such that
Lemma 2 (Schur complements). Given constant matrices , , and of appropriate dimensions, where and , then if and only if
Lemma 3 (see [16]). Let , , and be real matrices of appropriate dimensions with satisfying . Then one has the following inequality. For any scalar ,
3. Main Results
The following result presents a sufficient condition of the robustly stochastic stability for system (1).
Theorem 4. For given integers and , system (1) with is robustly stochastically stable with margins and , if there exist positive scalars , , , and and symmetric positive-definite matrices and of appropriate dimensions satisfying the following LMI: where
Proof. Consider the following Lyapunov-Krasovskii functional for system (1):where Then, the difference of , along the solution of system (1) is given by Since , we have Then we get The difference of is given by From (17) and (18), we obtain From (14) and (19), it follows that According to (6), we haveFrom (20) and (21), we get Let us denote Taking the mathematical expectation, we get It is easy to see thatThis reduces to where with
Applying Lemma 2, we get that if and only if where .
We obtain from formula (23) that where From (4) and Lemma 3, we have Similarly, it is not difficult to verify that From (30), (32), and (33), we get where From (10), we get that , which implies Hence, we have Taking expected value and summing up both sides of the above equation for , we have Thus, We get Obviously, and this leads to which leads to the robust stochastic stability of (1) with with margins and . This completes the proof of the theorem.
Remark 5. In [7], the stochastic stability analysis problem had been studied for discrete-time system with stochastic disturbance. But the time-delay and parameter uncertainties and unknown nonlinearities with time-varying delays were not considered in [7]. In this paper, we consider the time-delay and parameter uncertainties and the unknown nonlinear time-varying perturbations with time-varying delay. Comparing with [7], the model that is given by uncertain nonlinear discrete-time stochastic system (1) is a more general one.
Remark 6. In this paper, scalars , and are introduced with the aim to obtain a tractable matrix condition, while the conservatism does not increase much. Compared to [7], by choosing these scalars appropriately, the conservatism can be further reduced.
We have reformulated this theorem as an optimization problem which is given below as a separated theorem.
Theorem 7. Let and be the optimal solutions of the following optimization problem:whereThen, for any and , system (1) with is robustly stochastically stable with margins and .
We now consider the problem of robustly stochastic stability of system (1).
Theorem 8. System (1) is robustly stochastically stabilizable with margins and under the controller with , if there exist positive scalars , , and , symmetric positive-definite matrices and , and any matrix of appropriate dimensions satisfying the following LMI: where
Proof. Substituting into (1) yields the dynamics of the closed-loop system described by Denote
Similar to the proof of Theorem 4, we get that the closed-loop system (45) is stochastically stable if there exist symmetric positive-definite matrices and satisfying the following LMI: where
We have where Let , , and pre- and postmultiplying (47) by yield where Using the well-known relationships we can get Using (52) and the gain matrix , we obtain where We have known that is equivalent to By Lemma 2, we get LMI (43) implying that , which concludes the proof of the theorem.
Remark 9. The proposed feedback controller can ensure stochastic stability of the closed-loop system in Theorem 8. If and are given, the feasibility problem of LMI can be solved to get a suitable stabilization controller gain.
We have reformulated this theorem as an optimization problem which is given below as a separated theorem.
Theorem 10. Let and be the optimal solutions of the following optimization problem:withThen, for any and , system (1) with , where , is robustly stochastically stable with margins and .
Remark 11. Unlike robust control results available in the literature [17, 18], the result presented in this paper designs a linear control law that stabilizes the closed-loop system and is maximally robust with respect to considered nonlinear perturbations.
Remark 12. In [19], by using a linear controller, delay-dependent sufficient conditions of stabilization for a class of nonlinear discrete-time systems with varying time delay were given. However, the system in [19] did not involve stochastic disturbance. In [7], authors considered the robust state feedback stability and stabilization of nonlinear discrete-time stochastic system, but the stochastic system in [7] did not include time delay. Compared with [7, 19], the results obtained in this paper have a greater range of applications.
Remark 13. In this paper, we use the linear state feedback control law which has many applications in stochastic stability analysis and control synthesizing. For example, in [20], for the robust stabilization problem, a linear state feedback controller was designed, which ensured that the closed-loop system was robustly stochastically stable with maximal decay rate. In [7], a linear state feedback controller was used to explore the stabilization of a class of nonlinear discrete-time stochastic systems. In [21], asymptotic stabilization of a discrete-time switched stochastic system was investigated based on a linear state feedback controller.
4. Numerical Examples
In this section, two numerical examples are provided to illustrate the usefulness of the proposed criteria.
Example 1. Consider system (1) with and the following parameters: By using Matlab LMI Toolbox, we solve LMI (10) and obtain the feasible solutions as follows: The simulation of the state response of under initial condition is given in Figure 1.
Example 2. We consider the uncertain nonlinear discrete stochastic system (1) with the following parameters: Given that , and , then the solution of LMI (40) is By the formula , we get the controller gain
Figure 2 shows the simulation results for states and under initial condition . Simulation results demonstrate that our proposed design is very effective.
5. Conclusions
In this paper, we have investigated the robust stochastic stability and stabilization for a class of uncertain nonlinear discrete-time stochastic systems with interval time-varying delays and nonlinear disturbances. The nonlinear disturbances are more complex with uncertainty and time-varying delays. By constructing a new Lyapunov-Krasovskii functional and utilizing some well-known inequalities, we present novel delay-dependent criteria which guarantee the robust stochastic stability of a class of uncertain discrete-time stochastic systems. Then based on a state feedback control law, we give the delay-dependent sufficient conditions of robust stochastic stabilization for a class of uncertain discrete-time stochastic systems with interval time-varying delays, and the controller gain is designed. In this paper, we convert the complex stability analysis problem into the resolvable LMI problem. The results of this paper can be easily extended to the global exponential stability problem.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper