Stability and Fuzzy Optimal Control for Nonlinear Itô Stochastic Markov Jump Systems via Hybrid Reinforcement Learning

This paper addresses the stability and optimal control problem (OCP) for nonlinear Itô stochastic Markov jump systems (NISMJSs) by employing the Takagi-Sugeno (T-S) fuzzy model. First, the NISMJSs undergo transformation through T-S fuzzy model into a class of Itô stochastic Markov jump fuzzy systems (ISMJFSs), which is a composite of a set of linear subsystems. Then, based on the Lyapunov stability theorem, a sufficient condition is provided for establishing the mean-square stability of ISMJFSs. For solving the OCP, the most commonly used method is policy iteration (PI), but it requires an initial stabilizing control policy to be given, which relies on precise information about the system dynamics. To remove this limitation, we introduce hybrid iteration (HI) to design the optimal controller for the Itô stochastic systems for the first time without requiring the initial stabilizing control gain, and then the stability and convergence of the offline HI algorithm for the ISMJFSs are proven. Moreover, by incorporating reinforcement learning (RL) techniques, we develop an online HI algorithm to successfully solve the OCP for ISMJFSs with unknown system dynamics matrices, which initializes the algorithm without the need for a set of initial stabilizing control gain matrices. Finally, the efficiency of the proposed methods is illustrated through simulation results using the RLC electric circuit.