Technical communiqueSliding mode control of singular stochastic hybrid systems☆
Introduction
Singular systems, also referred to as descriptor systems, generalized state-space systems, differential-algebraic systems or semi-state systems, provide convenient and natural representations in the description of economic systems, power systems and circuits systems (Xu & Lam, 2006). Many results have been reported on control of singular systems, e.g., Dai (1989), Uezato and Ikeda (1999), Xu and Lam (2006) and Xu, Van Dooren, Stefan, and Lam (2002). On the other hand, in the past few decades, considerable attention has been devoted to stochastic systems governed by Itô stochastic differential equations, due to their extensive applications in mechanical systems, economics, systems with human operators, and other areas (Mao & Yuan, 2006). A variety of works have been published with respect to the stability, stabilization and filtering problems of Itô stochastic systems, e.g., Wang, Lam, and Liu (2003) and Wang, Qiao, and Burnham (2002) and the references therein.
SMC has been proven to be an effective robust control strategy. It has been successfully applied to a wide variety of practical engineering systems such as robot manipulators, aircrafts, underwater vehicles, spacecrafts, flexible space structures, electrical motors, power systems, and automotive engines. The main idea of SMC is to utilize a discontinuous control to force the system state trajectories to some predefined sliding surfaces on which the system has desired properties such as stability, disturbance rejection capability, and tracking ability. The SMC strategy has been successfully applied to many kinds of systems, such as, uncertain time-delay systems (Fridman et al., 2003, Seuret et al., 2007, Wu et al., 2008, Xia and Jia, 2003), stochastic systems (Niu, Ho, & Wang, 2007), and Markovian jump systems (Shi, Xia, Liu, & Rees, 2006). However, to the authors’ knowledge, there is little related results reported on SMC of singular systems (or singular hybrid systems). Research in this area should be theoretically interesting and challenging.
In this paper, we will investigate the SMC for nonlinear singular Itô stochastic systems with Markovian switching parameters. The key questions to be addressed are stated as follows:
- Q1.
How to design a suitable sliding surface function such that the resulting sliding mode dynamics exists with an easily available stability condition?
- Q2.
How to propose a strict LMI condition (easy to check) for the stochastic stability of the Markovian jump singular stochastic hybrid systems?
- Q3.
How to analyze and synthesize a SMC law for singular stochastic systems with an disturbance?
SMC of singular systems may become complicated due to the singular matrix of in the systems. Since the rank of may not equal to the rank of in a simple singular system of , one cannot obtain the ‘regular’ form (see Eq. (2.5) in Shi et al. (2006)) through model transformation. Thus, the approaches proposed in Shi et al. (2006) are not workable for singular systems. For this reason, the linear sliding surface function is not applicable for singular systems either. In this work, we will design an appropriate integral sliding surface function and take the singular matrix into account. In this case, the order of the resulted sliding mode dynamics equals to that of the original system, which is convenient to analyze the disturbance attenuation performance for the singular systems with an external disturbance. In addition, motivated by Uezato and Ikeda (1999), we shall propose some strict LMI conditions of the stochastic stability and optimal performance for the singular Itô stochastic systems with Markovian jump parameters.
Section snippets
System description and preliminaries
Let be a continuous-time Markov process on the probability space which takes values in a finite state-space , and the generator matrix , with transition probability from mode at time to mode at time is given by where and ; and for each .
In this work, let us consider a nonlinear singular stochastic hybrid system with Markovian switching. This system is
Sliding surface design
We design the following integral sliding surface function: where and are real matrices to be designed. Matrix is designed to satisfy that is nonsingular and . A necessary condition for the existence of such matrix is that there does not exist a common column vector in both and .
We give the solution of as following: It follows from (7), (8) subject to
SMC with performance
In this section, we shall answer the third question (Q3). Specifically, we will propose a sufficient condition by which the sliding mode dynamics of the controlled system is guaranteed to be stochastically stable with an performance.
Illustrative example
Consider the Markovian jump singular stochastic hybrid system () in (2) with two operating modes, that is, and the following parameters: In addition, (thus, in (3) can be chosen as
Conclusions
SMC of Markovian jump singular stochastic hybrid systems has been investigated in this paper. An integral sliding surface has been designed and some sufficient conditions have been proposed for the stochastic stability of sliding mode dynamics in terms of strict LMI. A SMC law has been synthesized to guarantee the reachability of the system’s trajectories to the sliding surface. Moreover, we have further analyzed the stochastic stability and disturbance attenuation performance for the
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This work was supported in part by the National Natural Science Foundation of China under Grants 60804002 and 60834003, the Natural Science Foundation of Heilongjiang Province of China (QC2009C58), Program for New Century Excellent Talents in University, 973 Project (2009CB320600), and in part by a research grant from GRF (CityU 101109). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associated editor James Lam under the direction of André L. Tits.