Robust stability and stabilization of discrete-time Markov jump linear systems with partly unknown transition probability matrix

An improved linear matrix inequality (LMI) approach is proposed to deal with the problems of stability, mode-dependent and mode-independent stabilization of discrete-time Markov jump linear systems (MJLS) with partly unknown transition probability matrix. As a first contribution, the uncertain parameters of the transition probability matrix are modeled in terms of the Cartesian product of simplexes, called multi-simplex. Then, convergent LMI relaxations with improved trade-off between precision and computational effort are proposed for the stability analysis of this class of MJLS. Finally, new design conditions based on LMIs with a scalar parameter are proposed for state feedback control, in both mode independent and mode dependent scenarios, providing less conservative results when compared to other conditions available in the literature, as illustrated by numerical examples.