In this paper, linear discrete-time control systems of the formEkXk+1=AkXk+BkukYk=CkXk+Dkukare considered whereEkandAkarenot necessarily squarematrices. The problem to be solved is this. Find a system of the formÊkvk+1=Âkvk+[Bcirc]kƒk,uk=Ĉkvk+[Dcirc]kƒk, whereƒkis a partitioned column vector whose blocks are linear combinations of some of the output vectors of the original system and which possesses either or both of the following properties: (i) If the outputs of the original system areknown, then the output equation of the second system uniquely specifies the same inputs that generated the known outputs of the original system, (ii) If the output vectors of the original system aredesiredoutputs, then the equationuk=Ĉkvk+[Dcirc]kƒkyields one or more sequences of inputs for the original system that can generate the desired outputs. We shall furthermore seek the system which, among all systems having Property (i) or (ii) or both, has the smallest dimension for allk. (By dimension we mean the number of the elements ofxk) Making use of some results of Wilkinson (1978) on singular systems of linear differential equations, a procedure is developed that either yields a system having the properties listed above or provides evidence that there is no such system.