For a class of discrete linear systems, this paper deals with the problem of designing an observer-based iterative learning control scheme in the finite frequency range. First, the controller is constructed by combining an observer-based state feedback with a PID-type feedforward learning term on the basis of the two-dimensional system theory. Then, by means of the Kalman–Yakubovich–Popov(KYP) lemma, the finite frequency domain specifications of the resulting closed-loop system are transformed into corresponding linear matrix inequalities(LMI), furthermore, the sufficient conditions for the existence of the controller and the observer are also obtained. Simultaneously, these conditions guarantee the stability of the close-loop controlled system and the monotonic convergence of the tracking error. Finally, the effectiveness of the proposed method is verified by the simulation of a gantry robot.