The problem of periodic tracking is addressed for discrete-time linear systems. A rare-variant attracting law for the repetitive controller design is presented, and the ideal error dynamic is formed based on the proposed attracting law, by which the discrete-time repetitive controller is designed. It is shown that the control input is constrained within the pre-specified region, and the monotonic convergence of the tracking error is ensured in finite time. Consequently, the undesirable high-frequency switching between different values of the control signal is avoided. To characterize the error dynamics, the monotone convergence region, the absolute convergence boundary and the steady state error band are derived, respectively, in the presence of the bounded disturbances. The expression for the convergence steps is also given. Both numerical simulation and experiment results demonstrate the effectiveness of the proposed repetitive control scheme.