Stability and stabilization of piecewise homogeneous partially known hidden Markov jump linear systems

This brief deals with the asynchronous stabilization problem for discrete-time piecewise homogeneous Markov jump linear systems (MJLSs) with stochastic switching transition probabilities (TPs) that are governed by a higher-level Markov chain. Two practical scenarios are considered: (i) the modes of the system and its TP matrix cannot always be accurately detected, leading to the mismatched modes phenomenon; and (ii) the information of probability matrices cannot be fully obtained. First, we adopt a dual hidden Markov model (HMM) to characterize the mismatched modes phenomenon. It is expected that the feasibility of the established stability conditions for the resulting closed-loop system is enhanced because both observed modes are used in the design. Second, we allow all transition and observation probability (T/OP) matrices to be partially known, considering that some T/OPs cannot be obtained. We propose a novel nonconservative probability decoupling principle to deal with the simultaneous existence of unknown T/OPs, and we then obtain several novel stability conditions based on the probability decoupling principle. We also show that the obtained stability conditions recover some existing results in the literature as special cases without sacrifice of any conservatism. In the end, we use an example of an economic model to demonstrate the effectiveness of the proposed controller design method.