The uncertainty and disturbance estimator (UDE)-based robust control has a two-degree-of-freedom nature through the design of the error dynamics and the design of the UDE filters. In the conventional design to handle periodic disturbances or mixed sinusoidal disturbances, high-order UDE filters incorporated with the internal model principle (IMP) or time-delay filters (TDF) are adopted to achieve the asymptotic reference tracking and the asymptotic disturbance rejection. In this paper, a new error dynamics design combined with a repetitive loop is proposed for the UDE-based robust control to achieve the asymptotic rejection of both step disturbances and periodic disturbances. The disturbance rejection performance is investigated through the two-degree-of-freedom nature, and the practical implementation of the proposed design is illustrated to eliminate the infinite bandwidth of the repetitive loop. The proposed design is validated through the simulation studies of a battery charging system with comparison to different reported designs of the conventional UDE-based robust control.