A convex programming approach to H-2 control of discrete-time Markovian jump linear systems

In this paper we consider the H-2-control problem for the class of discrete-time linear systems with parameters subject to markovian jumps using a convex programming approach. We generalize the definition of the H-2 norm from the deterministic case to the markovian jump case and set a link between this norm and the observability and controllability gramians. Conditions for existence and derivation of a mean square stabilizing controller for a markovian jump linear system using convex analysis are established. The main contribution of the paper is to provide a convex programming formulation to the H-2-control problem, so that several important cases, to our knowledge not analysed in previous work, can be addressed. Regarding the transition matrix P = [p(ij)] for the Markov chain, two situations are considered: the case in which it is exactly known, and the case in which it is not exactly known but belongs to an appropriated convex set. Regarding the state variable and the jump variable, the cases in which they may or may not be available to the controller are considered. If they are not available, the H-2-control problem can be written as an optimization problem over the intersection of a convex set and a set defined by nonlinear real-valued functions. These nonlinear constraints exhibit important geometrical properties, leading to cutting-plane-like algorithms. The theory is illustrated by numerical simulations.