Linear quadratic regulation of polytopic time-inhomogeneous Markov jump linear systems (extended version)

TL;DR

This paper develops a control strategy for polytopic time-inhomogeneous Markov jump linear systems, providing a solution to the infinite-horizon optimal control problem using coupled Riccati equations, with demonstrated numerical efficiency.

Contribution

It introduces a novel approach to optimal control of systems with time-varying, polytopic transition probabilities using coupled Riccati equations, extending existing methods.

Findings

Optimal controller derived from coupled algebraic Riccati equations.

Finite-horizon cost converges exponentially to infinite-horizon cost.

Numerical example confirms the effectiveness of the proposed method.

Abstract

In most real cases transition probabilities between operational modes of Markov jump linear systems cannot be computed exactly and are time-varying. We take into account this aspect by considering Markov jump linear systems where the underlying Markov chain is polytopic and time-inhomogeneous, i.e. its transition probability matrix is varying over time, with variations that are arbitrary within a polytopic set of stochastic matrices. We address and solve for this class of systems the infinite-horizon optimal control problem. In particular, we show that the optimal controller can be obtained from a set of coupled algebraic Riccati equations, and that for mean square stabilizable systems the optimal finite-horizon cost corresponding to the solution to a parsimonious set of coupled difference Riccati equations converges exponentially fast to the optimal infinite-horizon cost related to the set of coupled algebraic Riccati equations. All the presented concepts are illustrated on a numerical example showing the efficiency of the provided solution.