Robust stability criterion for uncertain stochastic systems with time-varying delay via sliding mode control☆
Introduction
Stochastic systems have received much attention since stochastic modeling has come to play an important role in many branches of science and engineering applications. Much effort has been devoted to extend many fundamental results for deterministic systems to stochastic systems [1], [2]. In the past decades, increasing attention has been devoted to the problems of stability and stabilization of stochastic systems by many researchers [3], [4], [5], [6]. On the other hand, it has been recognized that time-delay, which are inherent features of many physical processes, are often the cause for instability and poor performance of systems. Hence, stochastic differential delay equations are one of the most useful stochastic models in applications, and have attracted the attention of a considerable number of researchers, see [7], [8], [9], [10], [11], and references therein.
Sliding mode control (SMC) is an effective robust control strategy for uncertain systems, which has been successfully applied to a wide variety of complex systems and engineering [12], including uncertain systems [13], time-delay systems [14], [15], stochastic systems [16], singular systems [17], [18] and Markovian jump systems [18], [19]. Generally speaking, SMC uses a discontinuous control law (relays) to force and restrict the state trajectories to a predefined sliding surface on which the system has some desired properties such as stability, disturbance rejection capability and tracking [12], [20]. The main feature of SMC systems is its insensitive to the uncertainties and external disturbances on the sliding surface, and its applications have been extensively studied in [20], [21], [22], [23], [24]. For example, [16] developed a robust integral SMC method for uncertain stochastic systems with time-varying delay via equivalent control methods. However, to the best of the authors’ knowledge, the free weighting matrices approach has not been applied to the SMC problem of uncertain stochastic systems with time-varying delay. The existence of uncertainties, time-varying delay, and bilinear noise perturbations will make the problem more complex and challenging.
In this paper, we consider the SMC problem of uncertain stochastic systems with time-varying delay. The systems may contain time-varying parameter uncertainties and stochastic perturbations. Compared to other methods, the proposed method overcomes some of the main sources of conservatism. First, a new integral-type switching surface function is constructed. Second, some free weighting matrices are employed lead to less conservative results. Third, it does not use the inequality to estimate the bounding for cross terms, which also reduces the conservatism in the derivation of the stability condition. Fourth, a new type of Lyapunov–Krasovskii functional is constructed. This is the main advantage of our method, and is the essential difference between existing methods and ours.
The rest of the paper is organized as follows. In Section 2 gives the problem formulation and some preliminaries. The main results on designing a sliding surface and a SMC law are presented in Section 3. Numerical simulation results are shown in Section 4 and the paper ends with the concluding remarks in Section 5.
Notations. Throughout this paper, matrices, if not explicitly stated, are assumed to have compatible dimensions. denotes the n-dimensional Euclidean space, is the set of all real matrices. For the real symmetric matrices X and Y, the notation (respectively, ) means that the matrix is positive semi-definite (respectively, positive definite); I and 0 represent the identity matrix and the zero matrix with appropriate dimensions, respectively. and denote, respectively, the Euclidean norm and 1-norm of a vector or its induced matrix norm. For denotes the family of continuous functions from into with the norm . Let is a complete probability space with a naturel filtration containing all -null sets and being right continuous, where is the sample space, is a -algebra of subsets of , and the probability measure on . denotes the family of all -measurable -valued random variables such that denotes the expectation operator with respect to probability measure . We use and to denote, respectively, the transpose of and the inverse of any square matrix M. We define . Let diag denotes a block-diagonal matrix, and the symbol is used to denote the transposed elements in the symmetric positions of a matrix.
Section snippets
Preliminaries and problem formulation
Consider the following uncertain stochastic systems with time-varying delay which are described by the Itô formwhere is the state vector, is the control input, and is a one-dimensional Brownian motion defined on complete a probability space , which satisfies and . In the systems , , ,
Main results
In general, there are two steps for SMC design. The first step is to design an appropriate switching surface function so that the dynamic system restricted to the predefined switching surface (i.e., the sliding mode dynamic system) has desirable properties, such as stability, tracking capability, H-infinity and so on. The second step is to synthesize a suitable SMC law to drive the system state trajectories onto the predefined switching surface and maintain them there for all subsequent time.
Simulation example
Consider the uncertain stochastic systems with time-varying delay in (1) with the following parameters:
In addition, (so in (5) can be chosen as ), and the time-varying delay with . Note that the uncertain stochastic systems in (1)
Conclusions
This paper has presented a robust stability criterion for a class uncertain stochastic systems with time-varying delay via SMC. The criterion, which is dependent on the maximum and minimum delay bounds, was established to guarantee the existence of sliding surface by using the Lyapunov-Krasovakii functional approach combined with LMIs. A SMC law has been synthesized such that the state trajectories of the closed-loop systems are driven onto the specified switching surface by estimating the
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This work is partially supported by the National Natural Science Foundation of PR China (10971045), the Natural Science Foundation of Hebei Province (A2013208147), (F2014208042).