Robust Stability and Stabilization of a Class of Uncertain Nonlinear Discrete-Time Stochastic Systems with Interval Time-Varying Delays

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.


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][2][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][7][8][9].In [6], robust delay-dependent stability and stabilization methods for a class of nonlinear discrete-time systems with timevarying delays were proposed.In [8], the robust stabilization problem for uncertain linear systems with interval timevarying 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-distributiondependent 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 meansquare exponential stability problem for stochastic discretetime recurrent neural networks with time-varying discrete and distributed delays was investigated.In [15], the delayprobability-distribution-dependent robust stability problem for a class of uncertain stochastic neural networks with timevarying 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 2 Journal of Control Science and Engineering 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.(1) 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.(2) 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.(3) 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.(4) 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.(5) Numerical simulation examples are used to demonstrate the effectiveness and applicability of the obtained results.(6) By introducing some parameters  1 ,  2 ,  1 , and  2 , 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 R  and R × denote the -dimensional Euclidean space and the set of all  ×  real matrices, respectively.The superscripts  and −1 denote the matrix transposition and matrix inverse, respectively. min (⋅) means the smallest eigenvalue of a matrix.E{⋅} 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.

Main Results
The following result presents a sufficient condition of the robustly stochastic stability for system (1).

Journal of Control Science and Engineering
Then we get The difference of  3 () is given by From ( 17) and ( 18), we obtain From ( 14) and (19), it follows that Δ () ≤   ( + 1)  ( + 1) According to (6), we have From ( 20) and ( 21 ( Let us denote Taking the mathematical expectation, we get It is easy to see that This reduces to where with Applying Lemma 2, we get that Ω < 0 if and only if where Π =  1  2    + (  −   + 1) − .
We obtain from formula (23) that  ) . (31) From ( 4) and Lemma 3, we have ) . (33) From ( 30), (32), and (33), we get where From (10), we get that Ω < 0, which implies Ω < 0. Hence, we have Taking expected value and summing up both sides of the above equation for  ≥  0 , we have Thus, We get Obviously, ‖‖ ≤ ‖x‖ and this leads to which leads to the robust stochastic stability of (1) with () = 0 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 timedelay 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  1 ,  2 ,  1 , and  2 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 now consider the problem of robustly stochastic stability of system (1).1) is robustly stochastically stabilizable with margins  and  under the controller () = () with  =  −1 , if there exist positive scalars  1 ,  2 , and  3 , symmetric positive-definite matrices  and , and any matrix  of appropriate dimensions satisfying the following LMI: where where Π 11 = where where Figure 2 shows the simulation results for states  1 () and  2 () under initial condition (0) = (−1, 1)  .Simulation results demonstrate that our proposed design is very effective.

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 timevarying delays and nonlinear disturbances.The nonlinear disturbances are more complex with uncertainty and timevarying 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 discretetime 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.
and the gain matrix  =  −1 , we obtain