Action Mapping: A Reinforcement Learning Method for Constrained-Input Systems

Existing approaches to constrained-input optimal control problems mainly focus on systems with input saturation, whereas other constraints, such as combined inequality constraints and state-dependent constraints, are seldom discussed. In this article, a reinforcement learning (RL)-based algorithm is developed for constrained-input optimal control of discrete-time (DT) systems. The deterministic policy gradient (DPG) is introduced to iteratively search the optimal solution to the Hamilton-Jacobi-Bellman (HJB) equation. To deal with input constraints, an action mapping (AM) mechanism is proposed. The objective of this mechanism is to transform the exploration space from the subspace generated by the given inequality constraints to the standard Cartesian product space, which can be searched effectively by existing algorithms. By using the proposed architecture, the learned policy can output control signals satisfying the given constraints, and the original reward function can be kept unchanged. In our study, the convergence analysis is given. It is shown that the iterative algorithm is convergent to the optimal solution of the HJB equation. In addition, the continuity of the iterative estimated Q-function is investigated. Two numerical examples are provided to demonstrate the effectiveness of our approach.