State-Feedback Control of Positive Switching Systems with Markovian Jumps

Abstract

This chapter deals with positive linear systems in continuous-time affected by a switching signal representing a disturbance driven by a Markov chain. A state-feedback control law has to be designed in order to ensure mean stability and input–output \(\mathcal{L}_{\infty }\)-induced or \(\mathcal{L}_{1}\)-induced mean performance. The chapter is divided into two parts. In the first, the control action is based on the knowledge of both the state of the system and the sample path of the Markovian process (mode-dependent control). In the second, instead, only the state-variable is known (mode-independent control). In the mode-dependent case, as well as in the single-input mode-independent case, necessary and sufficient conditions for the existence of feasible feedback gains are provided based on linear programming tools, also yielding a full parametrization of feasible solutions. In the multi-input mode-independent case, sufficient conditions are worked out in terms of convex programming. Some numerical examples illustrate the theory.