Design of zero-sum game-based H ∞ $$ {H}_{\infty } $$ optimal preview repetitive control systems with external disturbance and input delay

In this article, an $H_{\infty }$ optimal preview repetitive control (OPRC) scheme is proposed to deal with the disturbance attenuation problem for continuous-time linear systems with external unknown disturbance and input delay. A general bounded $L_2$ gain is applied to the $H_{\infty }$ OPRC tracking control problem by introducing a function with discounted performance. First, an augmented system containing system state equation, tracking error dynamics, and modified repetitive control output equation is constructed, which is then transformed into a non-delayed one by state transformation. Next, the OPRC controller is given and the game algebraic Riccati equation (GARE) is derived by transforming the $H_{\infty }$ tracking problem into a 2-player zero-sum game problem to give a Nash equilibrium solution of the associated min–max optimization problem. Besides, a value iteration (VI) algorithm is introduced to optimize the solution of continuous time GARE and ensure its convergence. Furthermore, the bounded-input bounded-output stability of the closed-loop system is obtained by giving an upper bound on the discount factor. Finally, the numerical simulation example is provided to illustrate the effectiveness of the proposed method.