The simultaneous ripple-free tracking problem of a prespecified class of inputs is considered. The purpose is that of providing a general and complete characterisation of all ripple-free controllers for SISO discrete systems. In the control problem treated in the paper, references are assumed to be generated by an ensemble of linear time-variant systems. Both the plant and the inputs may be unstable. By adoptintg a polynomial algebra approach, a necessary and sufficient condition is established for the existence of a causal stabilising deadbeat controller, so that the ripple free performance is attained, independent of the initial state of the plant, and for each admissible input. The proposed design procedure proves to be very simple and efficient. A detailed analysis of the least degree solution of the control problem is also presented.