Detectability, Riccati Equations, and the Game-Based Control of Discrete-Time MJLSs with the Markov Chain on a Borel Space

TL;DR

This paper introduces detectability for discrete-time Markov jump linear systems with a Borel space Markov chain, establishes stability criteria via Riccati equations, and applies game-based control methods to solve complex control problems.

Contribution

It generalizes existing results to Borel space Markov chains, develops stability and Riccati equation solutions, and addresses Nash equilibrium strategies for control problems.

Findings

Established detectability for systems with Borel space Markov chains.

Derived stability criteria using generalized Lyapunov equations.

Solved Nash equilibrium strategies via coupled Riccati equations.

Abstract

In this paper, detectability is first put forward for discrete-time Markov jump linear systems with the Markov chain on a Borel space ($\Theta$, $\mathcal{B}(\Theta)$). Under the assumption that the unforced system is detectable, a stability criterion is established relying on the existence of the positive semi-definite solution to the generalized Lyapunov equation. It plays a key role in seeking the conditions that guarantee the existence and uniqueness of the maximal solution and the stabilizing solution for a class of general coupled algebraic Riccati equations (coupled-AREs). Then the nonzero-sum game-based control problem is tackled, and Nash equilibrium strategies are achieved by solving four integral coupled-AREs. As an application of the Nash game approach, the infinite horizon mixed $H_{2}/H_{\infty}$ control problem is studied, along with its solvability conditions. These works unify and generalize those set up in the case where the state space of the Markov chain is restricted to a finite or countably infinite set. Finally, some examples are included to validate the developed results, involving a practical example of the solar thermal receiver.