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.

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., [

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 [

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 [

The remainder of this paper is organized as follows. In Section

Consider the uncertain nonlinear discrete stochastic system with time-varying delay described by

At the end of this section, we introduce a definition and some lemmas for the development of our results.

System (

Given constant matrices

Let

The following result presents a sufficient condition of the robustly stochastic stability for system (

For given integers

Consider the following Lyapunov-Krasovskii functional for system (

Applying Lemma

We obtain from formula (

From (

In [

In this paper, scalars

We have reformulated this theorem as an optimization problem which is given below as a separated theorem.

Let

We now consider the problem of robustly stochastic stability of system (

System (

Substituting

Similar to the proof of Theorem

We have

Using (

The proposed feedback controller can ensure stochastic stability of the closed-loop system in Theorem

We have reformulated this theorem as an optimization problem which is given below as a separated theorem.

Let

Unlike robust control results available in the literature [

In [

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 [

In this section, two numerical examples are provided to illustrate the usefulness of the proposed criteria.

Consider system (

By using Matlab LMI Toolbox, we solve LMI (

The simulation of the state response of

State trajectories of the open-loop system.

We consider the uncertain nonlinear discrete stochastic system (

Figure

State trajectories of the closed-loop system.

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.

The authors declare that there is no conflict of interests regarding the publication of this paper