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

TL;DR

This work develops detectability and stability theory for discrete-time Markov jump linear systems with a Markov chain on a Borel space, linking EMSS to Lyapunov and spectral criteria and establishing existence and uniqueness of maximal and stabilizing solutions to integral coupled Riccati equations. It then builds a comprehensive infinite-horizon control framework based on a four-equation integral Riccati system to realize Nash equilibrium strategies in a two-player game, and applies this to mixed $H_2/H_{\infty}$ control under solvability conditions. The results extend previous finite/countable-state analyses to uncountable Markov state spaces, providing a unified approach for LQ control and game-based design in this broader setting. A solar-thermal receiver example demonstrates practical applicability and highlights performance gains of the $H_2/H_{\infty}$ approach. The work also discusses key mathematical challenges (boundedness, measurability, integrability) and potential generalizations to multiplicative noise scenarios.

Abstract

In this paper, detectability is first put forward for discrete-time Markov jump linear systems with the Markov chain on a Borel space ($Θ$, $\mathcal{B}(Θ)$). 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.

Figures (8)

• Figure 1: $P_1(1,t)$$(P_1(2,t))$ is plotted on the left (right), and $P_1(i,t)_{hl}$ is the $(h,l)$-th entry of $P_1(i,t)$, $h, l\in\{1,2\}$.
• Figure 2: $K_1(1,t)$$(K_1(2,t))$ is plotted on the left (right), and $K_1(i,t)_{h}$ is the $h$-th entry of $K_1(i,t)$, $h\in\{1,2\}$.
• Figure 3: $P_2(1,t)$$(P_2(2,t))$ is plotted on the left (right).
• Figure 4: $K_2(1,t)$$(K_2(2,t))$ is plotted on the left (right).
• Figure 5: 1000 possible trajectories of $x_{1}(k)$$(x_{2}(k))$ for $(A+BK_{2}|\mathbb{G})$ with the initial conditions $(x_{0},\vartheta_{0})$ are plotted on the left (right).

Theorems & Definitions (46)

• Proposition 1
• Proposition 2
• proof
• Lemma 1
• proof
• Definition 1
• Remark 1
• Theorem 1
• proof
• Remark 2