Integral sliding mode control for stochastic Markovian jump system with time-varying delay
Introduction
Being a kind of very important control system, the research on stochastic Markovian jump system with time delay has attracted considerable attention in recent years. The motivation for investigating this class of systems arises from the following three aspects. The first one is that in real-time systems, the signal transmission is usually a noisy process brought on by random fluctuations from probabilistic causes [1]. The second one is that time delay, which is a primary source of instability and performance degradation in a dynamical system, is frequently encountered in engineering, biology, economy and other areas [2]. The abrupt phenomenon, such as random failures, repairs of the components and sudden environment changes, is the last one, which is modeled by Markov chain. Many interesting results on stability analysis and controller synthesis of the stochastic time-delay systems have been presented, e.g., [3], [4], [5], [6], [7], [8], [9], [10], [11] and the references therein.
On the other hand, sliding mode control (SMC) can be seen as an effective robust control method owing to its favorable features, such as strong robustness, order reduction and fast response. To this end, much attention has been devoted to the theoretical research of SMC problems for many various uncertain systems in the past few decades, such as [12], [13], [14], [15], [16], [17], [18], [19], [20] and the references therein. Generally speaking, the basic idea of SMC is to drive the state trajectories onto a specified sliding mode manifold containing the origin in finite time by using a discontinuous control law, and then to keep the state trajectories moving along the sliding mode surface toward the origin with desired performance (see [21] for more details).
It is noticed that the SMC problem of stochastic time-delay systems with uncertainties has also been a topic of great interest in recent years. For example, by using a transformation, the design of SMC scheme is proposed for a class of linear systems with Markovian jumping parameters in [22]. Based on the singular system approach, a novel linear switching sliding surface is constructed in [23] and the sliding mode controller is developed to ensure that the states will converge to the switching surface in a finite time. When taking the stochastic perturbation term into account, the sliding mode control problem is considered for uncertain stochastic time-delay system by [24]. Recently, a further consideration of Markovian jump linear time-delay systems with generally uncertain transition rates is proposed in [25] and a robust sliding mode controller is developed. Meanwhile, under the case of containing both the discrete time-varying delays and the infinite distributed delays, the robust SMC problem for the uncertain discrete-time systems with Markovian jumping parameters is further considered in [26].
It can be clearly observed that the designed switching surfaces in the aforementioned results are linear. Different from these results, a novel SMC scheme, which is based on integral sliding mode control (ISMC), is proposed in [27], where the robustness of the system throughout the entire trajectories of the system starting from the initial time can be ensured [28]. Under the case, the closed loop systems will exhibit a better robust property and faster convergence speed. Since then, applying ISMC method to stochastic systems has been paid increasing attention and many important results have been published. In [29], [30], the robust ISMC problem is considered for uncertain stochastic time-delay systems and nonlinear stochastic system, respectively. Subsequently, the free weighting matrices approach is introduced to the research of ISMC problem for uncertain stochastic systems with time-varying delay by [31]. When the Markovian jumping parameters are included, [32] deals with the ISMC problem for nonlinear stochastic systems with Markovian switching and establishes the connections among the designed sliding surfaces corresponding to every mode by introducing some specified matrices. Based on this, the synthetic problems are further considered via sliding mode design for stochastic Markovian jump system, mainly including the fault-tolerant control [33], the output feedback control [34], etc. In addition, the ISMC problem for Markovian jump singular system is also considered in [35].
Generally speaking, the mathematical model of stochastic systems can be expressed as In many existing works, the stochastic perturbations can be well handled by assuming , where is the solution of certain LMIs. In this case, the SMC design method for deterministic system can be used for stochastic system directly. It can be observed that the aforementioned assumption is too restrictive for stochastic system. In order to remove this assumption, [24] proposes a control scheme for finite-time stabilization of stochastic delay system. Meanwhile, by designing some integral sliding surfaces, the above assumption is also removed in [21], [36], [37]. When taking the Markovian jumping parameters into account, [38] considers the sliding mode control design problem for the uncertainty Markovian neutral-type stochastic system with time delay. However, as can be seen from the existing similar results, there have two aspects which are required to be further studied. On one hand, the obtaining sufficient conditions in the existing results are mostly on asymptotically stochastic stability of sliding mode dynamics. To the best of the authors’ knowledge, the mean-square exponential stability of sliding mode dynamics has not been investigated exactly. On the other hand, the result on ISMC for stochastic Markovian jump system with time-varying delay without the above restrictive assumption is few and there has space to make further efforts.
Motivated by the above observations, the ISMC scheme for uncertain stochastic Markovian jump system with time-varying delays is studied in this paper. Firstly, by designing a modified integral sliding surface, the states can be guaranteed on the sliding surface almost surely from initial time. Then, an explicit sliding mode controller is constructed, under which the restrictive assumption mentioned above can be well removed. Furthermore, with the equivalent sliding mode controller, the resulting sliding motion can be guaranteed to be mean-square exponentially stable if a set of linear matrix inequalities (LMIs) are feasible. Finally, a numerical example is provided to verify the given control algorithm.
Notations: Throughout this paper, denotes the n dimensional Euclidean space. The superscript “T” denotes matrix transposition and “⁎” denotes transpose of the corresponding sub-matrix. The notation , where X and Y are symmetric matrices, means that is positive definite, and I denotes the identity matrix with appropriate dimensions. and denote the maximum eigenvalue and the minimum eigenvalue of a matrix, respectively. denotes the trace of a matrix. denotes Euclidean norm. is a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). is the mathematical expectation of random variable with respect to probability measure P. is the conditional mathematical expectation given Borel subfield of random variable . denote the family of continuous functions from to and the family of all bounded, -measurable and -valued random variables. If x(t) is a continuous -valued stochastic process on , we let for 0.
Section snippets
Preliminaries
Consider the following stochastic Markovian jump system with time-varying delay:where is the state, is the control input and is the time-varying delay, which satisfies and , where τ and are constants. is a q-dimensional Brownian motion. is the initial condition. is a continuous-time
Integral sliding surface and sliding mode controller design
In this paper, the integral sliding surface is designed aswhere is the solution of LMIs (14), (15).
The sliding mode controller is designed aswhere
Numerical example
Example 1 Consider the stochastic Markovian jump system (1) with two modes, that is, . The system data is shown as follows: In mode 1:In mode 2:
The density matrix is given by With the above parameters
Conclusion
In this paper, by designing a modified sliding surface and a sliding mode controller, the restrictive assumption required in the existing results can be removed. When the obtained LMIs are feasible, it is shown that the sliding motion can be achieved and maintained almost surely from the initial time and the mean-square exponential stability of the closed-loop control system can be guaranteed. It is noted that the considered stochastic systems include some linear terms, and one future research
Acknowledgments
This work is supported by Natural Science Foundation of China (61203054), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Initial Research Fund of Highly Specialized Personnel from Jiangsu University (No. 11JDG103) and Open Foundation of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education (No. MCCSE2014A01).
Li Ma was born in Anhui, China, in 1982. She received the B.S. and M.S. degrees in mathematics from Anhui Normal University, Wuhu, China, in 2004 and 2007, respectively. She received the Ph.D. degree at the School of Automation, Southeast University, Nanjing, China, 2011. Currently, she is an associate professor at School of Electrical and Information Engineering, Jiangsu University, Zhenjiang, China. Her research interests include time-delay systems, stochastic systems and Markovian jumping
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Li Ma was born in Anhui, China, in 1982. She received the B.S. and M.S. degrees in mathematics from Anhui Normal University, Wuhu, China, in 2004 and 2007, respectively. She received the Ph.D. degree at the School of Automation, Southeast University, Nanjing, China, 2011. Currently, she is an associate professor at School of Electrical and Information Engineering, Jiangsu University, Zhenjiang, China. Her research interests include time-delay systems, stochastic systems and Markovian jumping systems.