This paper deals with the \( H_\infty \) optimal proportional integral (PI) preview control problem for a class of continuous-time linear systems with external disturbances. A general bounded \( L_2 \) gain is applied to the \(H_\infty \) optimal PI preview tracking control problem by introducing a function with discounted performance. First, an augmented system containing a state vector, an error vector, and a PI control output vector is constructed and an optimal performance index is obtained according to the disturbance attenuation condition. Next, the optimal PI preview controller is given and game algebraic Riccati equation 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 optimisation problem. In addition, the bounded-input bounded-output stability of the closed-loop system is ensured by establishing an upper bound on the discount factor. It is further demonstrated that the system satisfies disturbance attenuation conditions in the presence of disturbances. Finally, some numerical simulation examples validate the effectiveness of the proposed method.
