In this paper a novel approach is presented for control design with guaranteed transient performance for multiple-input multiple-output discrete-time linear polytopic difference inclusions. We establish a theorem that gives necessary and sufficient conditions for the state to evolve from one polyhedral subset of the state-space to another. Then we present an algorithm which constructs a time-varying output feedback law which guarantees that the state evolves within a time-varying polyhedral target-tube specifying the system's desired transient performance. We present generalisations involving constraints on the control, and a bounded additive disturbance term. Our formulation is very general and includes reference tracking with any desired transient behaviour in the face of disturbances, as specified, for example, by the most popular step response specifications. The approach is demonstrated by an example involving the control of water levels in two coupled tanks.