Stability and stabilisation of Markovian jump systems with time-varying delay: an input–output approach

Abstract

This study revisits the problems of stability analysis and controller synthesis of Markovian jump systems with time-varying delay via an input–output approach. First, the system under consideration is transformed into an interconnected system, and the result on stochastic scaled small-gain condition for stochastic interconnected systems is presented, which is key idea to solve the problems considered in this study. Based on the system transformation and the stochastic scaled small-gain theorem, the stochastic stability of the original system is examined via the stochastic version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time-varying delay and the stochastic scaled small-gain theorem. The proposed stability condition is demonstrated to be much less conservative than most existing results. Moreover, the problem of stabilisation is further solved with admissible controller designed via convex optimisations, whose effectiveness is also illustrated via numerical examples.