A novel discrete-time repetitive control (RC) scheme is proposed to achieve both the fast convergence rate and the perfect tracking/rejection by updating the input period by period and in the internal of the periods simultaneously. By using the internal model (IM) of the external periodic signals, the classic RC achieves perfect tracking/rejection. Furthermore, a polynomial IM (PIM) is introduced, which only incorporates the dominant frequencies of the external signals. The PIM-based RC is faster than the classic RC because of the fact that the PIM makes the input update in the internal of the periods. However, the PIM-based RC cannot achieve perfect tracking/rejection. Motivated by the fast convergence rate of the PIM-based RC and the perfect tracking/rejection of the classic RC, a new RC scheme is proposed whose IM is the product of the IMs of the classic and PIM-based RCs. As such, the new RC scheme is named as a dual IM (DIM)-based RC. By using the 2-D H-infinity theory, it is verified that the DIM-based RC can achieve both the fast convergence rate and perfect tracking/rejection. In addition, by further comparing the classic, PIM-based, and DIM-based RCs in frequency domain, the merits and demerits of the three RCs on the convergence rate and the steady-state performance are deeply discussed. Finally, an experimental apparatus about the rotational system with two dc motors is given to illustrate the advantage of the proposed DIM-based RC.