Static Output Feedback Variable Structure Control For Markov Jump Systems

The relevant researches of time-delay systems, Markov jump systems and variable structure control theory are introduced, and the research background and research significance of this paper is pointed out briefly. Then, based on Lyapunov stability theory, variable structure control theory and the singular Markov jump system theory, the problem of static output feedback variable structure control for Markov jump systems with nonlinear terms and time-delay Markov jump systems with nonlinear terms are discussed, respsctively. The nonlinear terms are norm bounded and the time-delay is constant. In this paper, the stochastic stability of sliding modes and the designing of static output feedback variable structure controllers are discussed based on singular system approach. The following results are proposed:(1) In chapter2, the static output feedback variable structure control for Markov jump systems is discussed. First, the sliding mode dynamics and the switching surface are treated as a singular Markov jump system, based on stochastic stability theory of singular Markov jump systems, a linear matrix inequality (LMI) sufficient condition which guarantees that the sliding mode dynamics is stochastically stable and the switching surface is existent is given. Then, by using LMI, the design method of a static output feedback variable structure controller is given, which guarantees that the closed-loop system is stochastically stable, and the state trajectories of the closed-loop system are convergent to the switching surface in finite time. Last, a numerical example is given to show the correctness and effectiveness of the proposed method. This part has been published in32nd Chinese Control Conference (CCC2013).(2) In chapter3, the static output feedback variable structure control for Markov jump systems with time-delay is discussed. First, based on the research methods in chapter2, a singular time-delay Markov jump system can be constructed by the sliding mode of system and the linear switching surface. By constructing Lyapunov-Krasovskii functional, an LMI sufficient condition is given to guarantee the stochastic stability of the sliding mode. Then, the static output feedback variable structure controller is obtained by using LMI, which guarantees that the closed-loop system is stochastically stable, and the state trajectories of the closed-loop system reach the switching surface in finite time.The first innovation of this paper is that the methods for design switch-ing surfaces and variable structure controllers do not need to decompose the coefficient matrices of the system and switching surface. The given condition is indicated by the original coefficient matrix of the system and switching sur-face, and the gain matrix of switching surface is directly designed. It can avoid the complicated numerical problem caused by decomposing the systems. The given condition can be used to design multiple switching surface and single switching surface at the same time.The second innovation of this paper is that the conditions for stochastic stability of the sliding mode and stochastic stability of the closed-loop systems are all strict LMIs. In the design of the controllers, there are no equality con-straints, and the LMIs are easy to solve. And free matrices in LMI conditions can increase the degrees of freedom of the solution to get a better result.