Model-free optimal controller for discrete-time Markovian jump linear systems: A Q-learning approach

This research paper introduces a model-free optimal controller for discrete-time Markovian jump linear systems (MJLSs), employing principles from the methodology of reinforcement learning (RL). While Q-learning methods have demonstrated efficacy in determining optimal controller gains for deterministic systems, their application to systems with Markovian switching remains unexplored. To address this research gap, we propose a Q-function involving the Markovian mode. Subsequently, a Q-learning algorithm is proposed to learn the unknown kernel matrix using raw input-state information from the system. Notably, the study proves the convergence of the proposed Q-learning optimal controller gains to the model-based optimal controller gains after proving the convergence of a value iteration algorithm as the first step. Addition of excitation noise to input which is required to ensure the leaning performance does not lead to any bias. Unlike the conventional optimal controller, the proposed method does not require any knowledge on system dynamics and eliminates the need for solving coupled algebraic Riccati equations arising in optimal control of MJLSs. Finally, the efficiency of the proposed method is demonstrated through a simulation study.