In this paper, linear discrete-time control systems of the form E k X k+1 = A k X k + B k u k Y k = C k X k + D k u k are considered where E k and A k are not necessarily square matrices. The problem to be solved is this. Find a system of the form Ê k v k+1 = Â k v k + [Bcirc] k ƒ k , u k = Ĉkvk + [Dcirc]kƒk , where ƒk is 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 are known, 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 are desired outputs, then the equation uk = Ĉkvk + [Dcirc]kƒk yields 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 all k. (By dimension we mean the number of the elements of xk ) 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.