Stochastic stability and stabilization of Markov jump linear systems with instantly time-varying transition rates: A unified framework

This paper investigates the stochastic stability and stabilization problems of non-homogeneous Markov jump linear systems (NHMJLSs) characterized by instantly unconditionally time-varying transition rates (TRs). The novelty of the study lies in proposing a systematic method for achieving finite dimensional conditions with an acceptable degree of conservativeness for the stability and the stabilization problems of the system. In this framework, by first processing the time-varying TRs, a finite number of uncertain but time-constant TR matrices are obtained. Then, a high-level switching signal is constructed for the system, which models the contribution of each possible time-constant TR matrix. Based on the results, the NHMJLS is reformed into an uncertain switching structure referred to as the associated switched Markov jump linear system (AS-MJLS). Finally, by taking advantage of the new representation, sufficient conditions are obtained to ensure the stability and stabilizability of the system, also the controller gains are designed. The proposed framework provides a realistic representation as well as practically solvable analysis and synthesis conditions for the NHMJLS. It also leads to less conservative results compared with the existing well-known techniques. Comparative simulation studies for a single-machine infinite-bus (SMIB) power system subject to stochastically varying load demonstrate the efficiency and superiority of the method.