Finite-Time Stability And Stabilization Of Markov Jump Systems

Markov jump system is a class of stochastic systems with multiple modes. In practical applications, the industrial control systems cannot be represented by linear time-invariant equations due to the fact that the state equations of these systems are changing stochastically. As a result, the Markov jump system is adopted which accurately describe these industrial systems and the transition probabilities between different modes are governed by the stochastic Markov chain. On the other hand, researchers pay more attention to the operating trajectory of system's state in a prescribed finite-time interval. This kind of requirement is defined as finite-time stability. The objective of this paper is to investigate the finite-time stability and control problems for Markov jump systems. Details are as follows:The finite-time stochastical stability of Markov jump systems is firstly given. Based on it, by appropriately choosing Lyapunov functions, sufficient conditions for finite-time stochastical stability are derived for both continuous-time and discrete-time systems with complete knowledge on transition probabilities.Consequently, the design procedure of the stabilizing controller is presented. Secondly, due to the fact that the transition probabilities are partly unknown for most of the systems, the stability results are derived by carefully studying the properties of transition probabilities. Three techniques, including fixed-weighting matrices, free-weighting matrices and non-magnifying and reducing are suggested. The comparision between these methods has been made and the design procedure is given.Finally, the above results have been extended to systems under more general framework, that is, the finite-time control of Markov jump systems with time-varying delays. By choosing Lyapunov-Razumikhin functions, the stability and stabilization results are derived. It is noticed that all the computational results are based on linear matrix inequalities.