Q-Learning Methods for LQR Control of Completely Unknown Discrete-Time Linear Systems

This paper focuses on solving the linear quadratic regulator problem for discrete-time linear systems without knowing system matrices. The classical Q-learning methods for linear systems can be divided into Q-learning value iteration and Q-learning policy iteration. Q-learning value iteration converges at a linear convergence rate. Q-learning policy iteration has a second-order convergence rate but requires an initial stabilizing control policy. This paper aims to propose efficient model-free algorithms for solving the optimal control problem without requiring an initial stabilizing control policy. In this paper, we first present an equivalent problem for an auxiliary system with the same optimal control policy as the LQR problem. A Q-learning algorithm is proposed to solve the equivalent problem, which is proven to converge monotonically to the optimal solution. The convergence rate of the Q-learning algorithm is heavily dependent on the auxiliary system, so we introduce a model-free homotopy method based on Q-learning to solve the LQR problem. This homotopy method can achieve the optimal solution in a finite number of iterations by solving an LQR problem in each iteration. Additionally, we propose a Q-learning Lyapunov iteration algorithm to solve the equivalent problem for an auxiliary system and analyze its properties. Finally, two examples are provided to demonstrate our results. Note to Practitioners—This paper proposes several Q-learning methods to solve the linear quadratic regulator problem for discrete-time linear systems. On the one hand, it is difficult to know the exact system dynamics knowledge in actual engineering, so this paper is devoted to developing model-free algorithms. On the other hand, this paper focuses on the LQR problem because it is widely spread in practical applications. We propose several model-free algorithms to solve the LQR problem, which provides the basis for optimal control of actual applications. Similar to policy iteration, our algorithms need to solve the Lyapunov equation. The advantage of our methods is that all of our algorithms do not have strict constraints on initial conditions compared with policy iteration. The properties of every algorithm proposed in this paper are provided. In addition, we focus on the efficiency of algorithms to obtain the optimal control policy faster. Two practical examples are used to verify the effectiveness of our methods. Finally, the applicable situations of each algorithm are summarized in the conclusion.