A Model-Free Optimal Control Method with Fixed Terminal States

Optimal control of state-dependent switched systems and time-delay systems is a challenge in the control community. Classical optimality conditions-based methods rely on heavy mathematical analysis for time-delay systems or prior knowledge of switching sequences. To solve this problem, model-free algorithms are brought into the control system's research with the emergence of reinforcement learning algorithms. However, there are two practical challenges of reinforcement learning-based methods. First, learning by interacting with the environment is a highly complex process. Second, constraints on the states (boundary conditions) require additional care since the state trajectory is implicitly defined from the inputs and system dynamics. To address these problems, this paper proposes a new model-free algorithm based on basis functions, gradient estimation, and the Lagrange method. The favorable performance of the proposed algorithm is shown using several examples under state-dependent switches and/or time delays.