On designing overlapping group mode-dependent H∞ controllers of discrete-time Markovian jump linear systems with incomplete mode transition probabilities

This paper is concerned with overlapping group mode-dependent H∞ control for a discrete-time Markovian jump linear system, where global modes of the system are not completely available for controller design. Firstly, a randomly overlapping decomposition method is developed to reformulate the system by a set of locally overlapping switched groups with accessible group modes. The reformulated system switches among different group modes in an overlapping manner. Secondly, an overlapping group mode-dependent state feedback controller is delicately constructed. Compared with some existing mode-dependent controllers in the literature, the proposed controller has three features: (i) it does not require all exact knowledge of global modes; (ii) it takes full advantage of group mode information of the reformulated system; and (iii) it allows overlapping local modes to exist in the formed groups. Thirdly, sufficient conditions on the existence of a desired overlapping group mode-dependent state feedback controller are derived such that the resultant closed-loop system is stochastically stable with prescribed H∞ performance. Furthermore, the proposed method is extended to design overlapping group mode-dependent state feedback controllers subject to incomplete mode transition probabilities. The proposed overlapping group mode-dependent framework is shown to be more general and includes traditional Markovian jump linear systems with completely accessible global modes as its special case. In the case of only one group in the reformulated system, it is shown that some existing result in existing literature can be retrieved. Finally, two illustrative examples are given to show the effectiveness of the obtained theoretical results.