The problem of designing asymptotic observers along with observer-based feedbacks for a class of discrete-time non- linear systems is considered. We assume that the system non-linearity is globally Lipschitz and the system is supposed to be stabilizable by a linear controller. Sufficient linear matrix inequality condition is derived to ensure the stability of the considered system under the action of feedback control based on the reconstructed states. A numerical example of a single-link flexible joint robot is presented to illustrate the efficacy of the theoretical developments.