A general TD-Q learning control approach for discrete-time Markov jump systems

This paper develops a novel temporal difference Q (TD-Q) learning approach, designed to address the robust control challenge in discrete-time Markov jump systems (MJSs) which are characterized by entirely unknown dynamics and transition probabilities (TPs). The model-free TD-Q learning method is uniquely comprehensive, including two special cases: Q learning for MJSs with unknown dynamics, and TD learning for MJSs with undetermined TPs. We propose an innovative ternary policy iteration framework, which iteratively refines the control policies through a dynamic loop of alternating updates. This loop consists of three synergistic processes: firstly, aligning TD value functions with current policies; secondly, enhancing Q-function's matrix kernels (QFMKs) using these TD value functions; and thirdly, generating greedy policies based on the enhanced QFMKs. We demonstrate that, with a sufficient number of episodes, the TD value functions, QFMKs, and control policies converge optimally within this iterative loop. To illustrate efficiency of the developed approach, we introduce a numerical example that highlights its substantial benefits through a thorough comparison with current learning control methods for MJSs. Moreover, a structured population dynamics model for pests is utilized to validate the practical applicability.