Abstract: The design of minimum settling-time and ripple-free digital control systems are investigated in this paper. The basic specifications for the system are that with a reference signal within a pre-specified class the system's response can have no ripple and no error after the settling-time, and the settling-time for the transient response must be as short as possible. To improve robustness and flexibility to input, it is necessary to design the non-minimum settlingtime system and introduce a staleness-weighting factor. This is a compromise design between the limited settling-time and the overshoot. Some fundamental algebraic concepts are introduced. Then the ripple-free tracking problem for admissible input and its CAD with MATLAB are considered. Simulative examples are reported.