A Principled Approximation Framework for Optimal Control of Semi-Markov Jump Linear Systems

We consider continuous-time, finite-horizon, optimal quadratic control of semi-Markov jump linear systems (S-MJLS), and develop principled approximations through Markov-like representations for the holding-time distributions. We adopt a phase-type approximation for holding-time distributions, which is known to be consistent, and translates an S-MJLS into a specific MJLS with partially observable modes (MJLSPOM), where the modes in a cluster have the same dynamic, the same cost weighting matrices, and the same control policy. For a general MJLSPOM, we give necessary and sufficient conditions for optimal (switched) linear controllers. When specialized to our particular MJLSPOM, we additionally establish the existence of an optimal linear controller, as well as its optimality within the class of general controllers satisfying standard smoothness conditions. The known equivalence between phase-type distributions and positive linear systems allows us to leverage existing modeling tools, but possibly with large computational costs. Motivated by this, we propose a matrix-exponential approximation of holding-time distributions, resulting in pseudo-MJLSPOM representation, wherein the transition rates could be negative. Such a representation is of relatively low order, and maintains the same optimality conditions as for the MJLSPOM representation, but could violate nonnegativity of holding-time density functions. A two-step procedure consisting of a local pulling-up modification and a filtering technique is constructed to enforce nonnegativity.