Minimax Q -learning design for H ∞ control of linear discrete-time systems

The H ∞ control method is an effective approach for attenuating the effect of disturbances on practical systems, but it is difficult to obtain the H ∞ controller due to the nonlinear Hamilton—Jacobi—Isaacs equation, even for linear systems. This study deals with the design of an H ∞ controller for linear discrete-time systems. To solve the related game algebraic Riccati equation (GARE), a novel model-free minimax Q -learning method is developed, on the basis of an offline policy iteration algorithm, which is shown to be Newton’s method for solving the GARE. The proposed minimax Q -learning method, which employs off-policy reinforcement learning, learns the optimal control policies for the controller and the disturbance online, using only the state samples generated by the implemented behavior policies. Different from existing Q -learning methods, a novel gradient-based policy improvement scheme is proposed. We prove that the minimax Q -learning method converges to the saddle solution under initially admissible control policies and an appropriate positive learning rate, provided that certain persistence of excitation (PE) conditions are satisfied. In addition, the PE conditions can be easily met by choosing appropriate behavior policies containing certain excitation noises, without causing any excitation noise bias. In the simulation study, we apply the proposed minimax Q -learning method to design an H ∞ load-frequency controller for an electrical power system generator that suffers from load disturbance, and the simulation results indicate that the obtained H ∞ load-frequency controller has good disturbance rejection performance.