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

TL;DR

This paper develops a framework for optimal control of semi-Markov jump linear systems by approximating holding-time distributions with phase-type and matrix exponential methods, enabling efficient controller synthesis.

Contribution

It introduces a novel approximation approach for semi-Markov systems using phase-type and matrix exponential methods, facilitating optimal control design with reduced computational complexity.

Findings

Established conditions for optimal controllers in the proposed models.

Proposed matrix exponential approximation reduces computational costs.

Developed techniques to ensure non-negativity of holding-time densities.

Abstract

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 times, which is known to be consistent, and translates a 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 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 to leverage existing modeling tools, but possibly with large computational costs. Motivated by this, we propose matrix exponential approximation of holding times, resulting in pseudo-MJLSPOM representation, i.e., where 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 non-negativity of holding-time density functions. A two-step procedure consisting of a local pulling-up modification and a filtering technique is constructed to enforce non-negativity.