A sliding mode approach to H∞ synchronization of master–slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties
Introduction
The sliding mode method has been recognized as one of the efficient tools to design robust controllers for the complex high-order nonlinear dynamic system operating under uncertainty conditions. The research in this area were initiated in the former Soviet Union about 40 years ago, and then the sliding mode control methodology has been receiving much more attention from the international control community within the last two decades. The major advantage of sliding mode is low sensitivity to plant parameter variations and disturbances, which eliminates the necessity of exact modeling. Sliding mode control enables the decoupling of the overall system motion into independent partial components of lower dimension and, as a result, reduces the complexity of feedback design [1], [2], [3], [4], [5].
In recent years, more attention has been devoted to the study of stochastic hybrid systems, where the so-called Markov jump systems. These systems represent an important class of stochastic systems that is popular in modeling practical systems like manufacturing systems, power systems, aerospace systems and networked control systems that may experience random abrupt changes in their structures and parameters [6], [7], [8], [9]. Random parameter changes may result from random component failures, repairs or shut down, or abrupt changes of the operating point. Many such events can be modeled using a continuous time finite-state Markov chain, which leads to the hybrid description of system dynamics known as a Markov jump parameter system [10], [11], [12], [13], [14], [15], [16]. Furthermore, the delay effects problem on the stability of systems is a problem of recurring interest since the delay presence may induce complex behaviors for the schemes, see for instance [17], [18]. The problem of filtering for state delayed systems with Markovian switching is proposed in [19], [20], [21], [22], [23]. The problem of robust mode-dependent delayed state feedback control is investigated for a class of uncertain time-delay systems with Markovian switching parameters and mixed discrete, neutral and distributed delays in [24]. Moreover, the sliding mode control problem for uncertain systems with time delays and stochastic jump systems are also investigated in [25], [26], [27], [28], [29], [30], [31], [32], [33], respectively. Recently, the problem of sliding mode control for a class of nonlinear uncertain stochastic systems with Markovian switching is studied in [34]. More recently, in [35], sliding mode control of nonlinear singular stochastic systems with Markovian switching is proposed.
On another research front line, synchronization is a basic motion in nature that has been studied for a long time, ever since the discovery of Christian Huygens in 1665 on the synchronization of two pendulum clocks. The results of chaos synchronization are utilized in biology, chemistry, secret communication and cryptography, nonlinear oscillation synchronization and some other nonlinear fields. The first idea of synchronizing two identical chaotic systems with different initial conditions was introduced by Pecora and Carroll [36], and the method was realized in electronic circuits. The methods for synchronization of the chaotic systems have been widely studied in recent years, and many different methods have been applied theoretically and experimentally to synchronize chaotic systems; see for instance [37], [38], [39], [40]. On the synchronization problems of systems with time-delays and nonlinear perturbation terms, we see that there have been some research works; see for instance [41], [42], [43], [44], [45], [46] and the references therein. So the development of synchronization methods for master–slave systems with Markovian switching parameters and time-varying delays is important and has not been fully investigated in the past and remains to be important and challenging. This motivates the present study.
In this paper, the problem of exponential H∞ synchronization is studied for a class of master–slave systems with both discrete and distributed time-delays, norm-bounded nonlinear uncertainties and Markovian switching parameters. Using an appropriate Lyapunov–Krasovskii functional, some delay-dependent sufficient conditions and a synchronization law, which include the master–slave parameters are established for designing a delay-dependent mode-dependent sliding mode exponential H∞ synchronization control law in terms of linear matrix inequalities (LMIs). The controller guarantees the H∞ synchronization of the two coupled master and slave systems regardless of their initial states. Two numerical examples are given to show the effectiveness of the method. The contribution of this paper is three-fold: first, this paper extends previous works on synchronization problem to time-delay systems with Markovian jumping parameters and nonlinear uncertainties and derives some new theoretical results; second, this paper shows how the synchronization problem can be reduced to a convex problem with additional degrees of freedom to design a synchronization law; third, using a Lyapunov–Krasovskii functional and a suitable change of variables, we establish new required sufficient conditions in terms of delay-dependent mode-dependent LMIs under which the desired sliding mode synchronization law exists, and derive the explicit expression of these salve systems to satisfy both stochastically exponential stability and an H∞ performance condition.
The rest of this paper is organized as follows. Section 2 formulates the exponential H∞ synchronization problem of the master and slave systems with Markovian switching parameters and mixed discrete and distributed time-varying delays and nonlinear perturbations. In Section 3, both H∞ performance analysis and sliding model control design are presented for the system under consideration. In Section 4, computer simulations are provided to demonstrate the effectiveness of the proposed synchronization scheme. Finally, conclusions are presented in Section 5.
Notation: The notations used throughout the paper are fairly standard. and 0 represent identity matrix and zero matrix, respectively; the superscript ‘T’ stands for matrix transposition. ‖⋅‖ refers to the Euclidean vector norm or the induced matrix 2-norm. represents a block diagonal matrix and the operator represents . Let and denotes the expectation operator with respect to some probability measure . If is a continuous -valued stochastic process on , we let for , which is regarded as a -valued stochastic process. The notations stand for . The notation means that is real symmetric and positive definite; the symbol denotes the elements below the main diagonal of a symmetric block matrix.
Section snippets
Problem description
Consider a model of master and slave systems with Markovian switching parameters and mixed discrete and distributed time-varying delays and nonlinear perturbations in the form of
Main results
In this section, we propose sufficient conditions for the stochastic stability of the sliding error motion (10) using the Lyapunov method.
Simulation results
In this section, with the aid of MATLAB LMI Toolbox [49], we use two numerical examples to illustrate the effectiveness and advantage of our design methods. Example 1 We give an example for the application of the theoretical results to a realistic master–slave time-delay synchronization problem without the switching modes, i.e., . One of the master–slave single flexible link model used in the literature (see, e.g. [50]) is given by
Conclusion
In this paper, the problem of exponential H∞ synchronization was studied for a class of master–slave systems with both discrete and distributed time-delays, norm-bounded nonlinear uncertainties and Markovian switching parameters. Using an appropriate Lyapunov–Krasovskii functional, some delay-dependent sufficient conditions and a synchronization law which include the master–slave parameters were established for designing a delay-dependent mode-dependent sliding mode exponential H∞
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