Performance Analysis And Filtering Problem For Markov Jump Systems With Time Delays

Sudden changes of parameters in the physical devices are not unusual in the practicalprocess. Such abrupt changes are well described by Markov jump systems. Due totheir extensive applications in mechanical, economics, biological, and other areas, thesesystems have been widely studied and a great number of results on estimation and controlproblems have been reported in the relevant literature. Time delays are usually includedin dynamic systems, which occurs due to signal transmission and/or inevitable defectsof control equipment. Hence, much attention has been paid to the study of time-delayMarkov jump systems.From the view of modern control theory, the contents of study for Markov jumpsystems with time delays mainly include stability analysis, feedback control and filteringproblem (i.e. state estimation problem). In general, the research method of the above con-tent is basically based on Lyapunov stability theory, i.e., construct a Lyapunov functional,and then search for the conditions so that the derivative of the Lyapunov functional isless than zero. With the development of LMI toolbox in Matlab software, LMI becomeseffective tool to deal with stability analysis, control and filtering problems. Stability isa prerequisite such that every system is up and running normally, so, stability analysisbecomes the most basic research. To the stability analysis of Markov jump systems, howto get the upper bound of delay which is close to the theoretical value becomes a keyresearch topic. The structure of singular Markov jump system is different from that ofthe state space system. The study of singular Markov jump systems may become moredifficult and complicated since the existence of singular matrix E in such systems. Theresearch results of state space methods can not be directly transplanted to that of singularsystems, especially in exponential stability analysis. In addition, from the view of systemmodeling, the demand for system modeling is increasing with the development of scienceand technology, that is, the established mathematical model is required to describe thereal-world situation as much as possible. In Markov jump systems, it is very hard andexpensive to obtain precisely all the transition probabilities of Markov chain even for asimple system. In the economic field, there exists a class of Markov jump systems withactual background, e.g., Markov chain with time-varying transition probabilities. This mathematical model has been proposed for many years, but the study of such model hasjust been started in control field. In networked control systems, the randomness of delayis a result of communication network issues, but the process itself may contain separatesources of randomness which means that the system matrices mode may be different withthe delay mode. The three types of systems mentioned above are also called Markovjump systems with complex transition probabilities, and the study of these systems hasimportant significance in both theoretical and applied aspects.The following parts are provided to show some ideas about how to derive lessconservative stability conditions and deal with the analysis and synthesis of Markov jumpsystems with time-varying delays and complex transition probabilities:1. H_∞performance analysis for Markov jump systems with time-varying delays.By using the property of convex function and free weight matrix techniques, a less con-servative sufficient condition on stochastic stability is derived. Based on the stochasticstability condition, the H_∞performance analysis of the considered systems is studied.2. Exponential admissibility in mean-square for generalized Markov jump systemswith time-varying delays and nonlinear perturbation. Based on the idea of delaydecomposition technique, by constructing an augmented Lyapunov functional, delay-dependent sufficient conditions on exponential admissibility in mean-square that are lessconservative are obtained. In addition, a new method to prove exponential stability inmean-square of generalized Markov jump systems is provided.3. The problem of L2-L_∞filtering for Markov jump systems. The systems underconsideration involve mode-dependent time-varying delays, disturbance signal and partlyunknown transition probabilities. By constructing an augmented Lyapunov functionaland by introducing some free weight matrices, we analyze the L2-L_∞performance.Further, the sufficient condition on the existence of filtering which guarantee a prescribedL2-L_∞disturbance attenuation level is obtained.4. The problem of H_∞filtering for piecewise homogeneous Markov jump systems.The systems under consideration involve mode-dependent time-varying delays, dis-tributed time-varying delays and nonlinear perturbations which are bounded by sectors.By constructing a new augmented Lyapunov functional and combining the Reciprocallyconvex technique, we analyze the H_∞performance for the considered systems. Based onthis, the the sufficient condition on the existence of filtering which guarantee a prescribed H∞disturbance attenuation level is obtained.5. Filtering-based fault detection for stochastic Markov jump system with dis-tributed time-varying delays and mixed modes. The systems under considerationinvolve mode-dependent time-varying delays, distributed time-varying delays and Ito stochastic disturbance. By constructing a new augmented Lyapunov functional andcombining free weight matrix techniques, the stochastic stability of error systems isstudied. Based on the obtained stochastic stability condition, the sufficient conditionon the existence of fault detection filter is obtained, which is formulated in terms of LMIs.