Zero-sum game-based optimal control for discrete-time Markov jump systems: A parallel off-policy Q-learning method

In this paper, the zero-sum game problem for linear discrete-time Markov jump systems is solved by two novel model-free reinforcement Q-learning algorithms, on-policy Q-learning and off-policy Q-learning. Firstly, under the framework of the zero-sum game, the game-coupled algebraic Riccati equation is derived. On this basis, subsystem transformation technology is employed to decouple the jumping modes. Then, a model-free on-policy Q-learning algorithm is introduced in the zero-sum game architecture to obtain the optimal control gain by measured system data. However, the probing noise will produce biases in on-policy algorithm. Thus, an off-policy Q-learning algorithm is proposed to eliminate the effect of probing noise. Subsequently, convergence is discussed for the proposed methods. Finally, an inverted pendulum system is employed to verify the validity of the proposed methods.