Stabilisation of Markov jump linear systems subject to both state and mode detection delays

This study addresses the stabilisation problem of a class of discrete-time Markov jump linear systems subject to time delays which occur in both the system state and mode detection. The mode detection delay means that there exists a lag between the switching of the system modes and the switching of the corresponding controllers. The state delays are considered to be random but with upper and lower bounds, and the mode detection delay is assumed to be constant. With an extended state space, the underlying system can be remodelled as a Markov jump linear system in which the system complexity and the number of system modes are both increased to a certain extent, but remain unchanged when the mode detection delay becomes larger. A sufficient condition guaranteeing the stochastic stability of the system is obtained via a modified stochastic Lyapunov function. The effectiveness of the proposed design approach is demonstrated by a numerical example.