H2 dynamic output feedback control of phase-type semi-Markov jump linear systems

Abstract

This paper deals with the design of $H_2$ dynamic output feedback controllers for phase-type distributed semi-Markov jump linear systems. It is assumed that the state-space of the semi-Markov jump process can be written as the union of disjoint sets, called clusters, and that the only information available to the controller regarding the jumping process is which cluster it belongs to. We provide two sets of design conditions for the $H_2$ control problem, written in terms of bilinear matrix inequalities, which are associated with the observability and controllability Grammians (referred to as the “primal” and “dual” approaches, respectively). An iterative separation procedure, formulated as a sequence of linear matrix inequalities optimization problems, is proposed to reduce an upper bound of the $H_2$ norm of the system for both the primal and dual design conditions. We show that our conditions are not conservative in the sense that, for the Markov mode-dependent case, they also become necessary. Finally, we study the robust case, considering that the system matrices and transition rate matrix have polytopic uncertainties, and the observer-based control case, for which the conditions can be simplified and written directly as linear matrix inequalities. The paper concludes with an illustrative example in the context of systems subject to actuator and sensor faults.