In this paper, a constructive design procedure is presented that solves the problem of realizing stabilizing discrete time min-max controllers via measured outputs for non-square systems with matched uncertainties. The conditions for the existence of such a controller will be given and a parameterization of the available design freedom will be proposed. An explicit procedure explains how a Lyapunov matrix, which satisfies both the discrete Riccati inequality and a structural constraint, is obtained. A numerical linear matrix inequality-based method, to optimally select the Lyapunov matrix, so that the computable upper bound on the allowable uncertainty is maximized, is suggested. The results of this paper will be illustrated by an aircraft example.