Sliding mode control for uncertain discrete-time systems with Markovian jumping parameters and mixed delays

Abstract

This paper is concerned with the robust sliding mode control (SMC) problem for a class of uncertain discrete-time Markovian jump systems with mixed delays. The mixed delays consist of both the discrete time-varying delays and the infinite distributed delays. The purpose of the addressed problem is to design a sliding mode controller such that, in the simultaneous presence of parameter uncertainties, Markovian jumping parameters and mixed time-delays, the state trajectories are driven onto the pre-defined sliding surface and the resulting sliding mode dynamics is stochastically stable in the mean-square sense. A discrete-time sliding surface is firstly constructed and an SMC law is synthesized to ensure the reaching condition. Moreover, by constructing a new Lyapunov–Krasovskii functional and employing the delay-fractioning approach, a sufficient condition is established to guarantee the stochastic stability of the sliding mode dynamics. Such a condition is characterized in terms of a set of matrix inequalities that can be easily solved by using the semi-definite programming method. A simulation example is given to illustrate the effectiveness and feasibility of the proposed design scheme.

Introduction

In the past few decades, the sliding mode control (SMC) has proven to be an effective control approach in the control field through constructing an appropriate sliding surface and designing a discontinuous control law. When the state trajectories are driven onto the pre-specified sliding mode surface under the designed controller, the resulting sliding mode dynamics is insensitive to the parameter variations. It is worthwhile to mention that the SMC strategy has been successfully applied to a variety of practical systems such as robot manipulators, spacecrafts, power systems, circuits systems, etc. Consequently, the SMC design problem has received increasing research attention and there have appeared a large number of significant results in the literature, see e.g. [2], [3], [4], [5], [6], [7], [8], [9], [13], [37]. To be specific, the optimal SMC problems have been tackled in [5], [6], [7], [8] for linear/nonlinear continuous-time systems with certain performance criteria. In [2], [3], [24], [37], the SMC law has been designed for different kinds of discrete-time systems. In particular, a reaching condition has been developed in [12] for the reachability analysis in the discrete-time setting, and then has been widely applied in the design of SMC for many kinds of discrete-time systems, see [17], [18], [19], [36] for instance.

In recent years, Markovian jump systems (MJSs) have gained considerable research interest since many dynamical systems can be modeled by MJSs, such as the chemical processes, economics systems, manufacturing systems and transportation systems. The engineering significance of MJSs has been well recognized and a number of interesting yet important results have been reported on the issues of stability, control and filtering (see [1], [10], [11], [14], [28], [39], [40] and the references therein). In view of the advantages of the SMC scheme, it is not surprising that the SMC design problem for linear/nonlinear systems with Markovian jump parameters has also attracted a great deal of research attention, see e.g. [26], [29], [33], [34]. To mention a few, the SMC law has been designed in [33], [34] for singular MJSs. In [26], the SMC problem has been investigated for a class of Itô-type stochastic systems with Markovian switching, and an effective control scheme has been given where the transition rates have been involved in the design of the SMC law. It might be worth mentioning that, so far, most of the existing results concerning SMC problems with Markovian jump parameters have focused on the continuous-time systems, and the corresponding research problem for the discrete-time counterparts has not received sufficient attention due probably to the difficulties in reachability analysis.

On another research forefront, it is well known that time-delays are often encountered in practical systems such as microwave oscillators, electronics, biological systems and hydraulic systems. The existence of time-delays may be the source of instability, oscillation or significantly deteriorated performances for the controlled systems [22], [30], [35]. Therefore, the study of the time-delay systems has been a popular topic and a great number of results have been published for systems with discrete time-varying delays. Recently, another type of time-delays called distributed delays has also gained some attention since they typically occur in practice such as spatially distributed networked control systems [20]. In [32], the issue of mixed time-delays (with both discrete and distributed delays) has been thoroughly investigated for the global synchronization problem of discrete-time stochastic complex systems. Nevertheless, to the best of the authors' knowledge, the SMC problem for the discrete uncertain systems with Markovian jumping parameters and time-varying delays has not yet been dealt with so far, not to mention the case when the distributed delays are also involved. It is, therefore, our aim in this paper to shorten such a gap by applying the delay-fractioning approach [27].

Motivated by the above discussion, in this paper, we investigate the robust SMC problem for a class of uncertain discrete-time MJSs with mixed time-delays. A discrete-time integral sliding surface is firstly constructed and an SMC law is then synthesized to ensure that the trajectories of the closed-loop system are driven onto a neighborhood of the pre-specified sliding surface. By utilizing a novel Lyapunov–Krasovskii functional and combining with the delay-fractioning approach as well as the free-weighting matrix technique, a sufficient condition is established such that the resulting sliding mode dynamics is stochastically stable in the mean-square sense. Specifically, a weighting parameter is constructively introduced to fit the SMC framework and the delay-fractioning idea. The main contributions of this paper lie in the following two aspects: (i) the SMC law is, for the first time, designed for the discrete-time Markovian jump system with mixed time-delays and (ii) a novel Lyapunov–Krasovskii functional that reflects the delay-fractioning nature is constructed in order to reduce the possible conservatism introduced by the time-delays. Finally, a numerical example is given to illustrate the effectiveness of the proposed control scheme.

The rest of this paper is outlined as follows. In Section 2, the problem under consideration is formulated. In Section 3, a sliding surface is firstly constructed and then an SMC law is designed. In the same section, the stochastic stability criterion for the sliding mode dynamics is proposed that can be solved by using the semi-definite programming method. A simulation example is given in Section 4 to illustrate the effectiveness of the proposed main results. This paper is concluded in Section 5.

Notations: The notations used throughout the paper are standard except where otherwise stated. The superscript “T” stands for matrix transposition; ${\mathbb{R}}^n$ (${\mathbb{Z}}^{-}$, ${\mathbb{R}}^{n\times m}$) denote the n-dimensional Euclidean space, the set of negative integers and the set of all n×m matrices, respectively; the notation $P>0(P\ge 0)$ means that matrix P is real symmetric and positive definite (positive semi-definite); $(\Omega ,\mathcal{F},\mathcal{P})$ is a probability space, $\Omega$ is the sample space, $\mathcal{F}$ is the $\sigma \mathrm{-}\mathrm{algebra}$ of subsets of the sample space and $\mathcal{P}$ is the probability measure on $\mathcal{F}$; $\mathbb{E}\{x\}$ and $\mathbb{E}\{x|y\}$ stand for the expectation of x and the expectation of x conditional on y; I and 0 represent the identity matrix and a zero matrix with appropriate dimension, respectively; $\mathrm{diag}\{X_1,X_2,\dots ,X_n\}$ stands for a block-diagonal matrix with matrices $X_1,X_2\dots ,X_n$ on the diagonal; $\parallel ·\parallel$ denotes the Euclidean norm of a vector and its induced norm of a matrix. In symmetric block matrices or long matrix expressions, we use a star ($⁎$) to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.