$H_{\infty}$ Control for Discrete-Time Markov Jump Linear Systems via Static Output Feedback

This paper addresses the problem of H control for discrete-time Markov jump linear systems via static output feedback. The transition probabilities of Markov chain are assumed to be partly unknown. First, by utilizing the properties that the sum of each row is one in a transition probability matrix and constructing Lyapunov function, a sufficient condition is established such that the system under consideration is stochastically stable with a pre-specified H performance index. Second, the design method of static output feedback controller is proposed to guarantee that the resultant closed-loop system is stochastically stable with the pre-specified H performance index γ. Since the conditions obtained for the existence of controllers are not expressed strictly in terms of linear matrix inequalities, a modified cone complementarity linearization algorithm is given to solve the static output feedback controller gains. Finally, a numerical example is presented to show the effectiveness of the theoretic results. Compared with the existing literature, the total number of matrix inequalities in the proposed method is less, and the optimal performance index obtained is smaller.