In this paper, we investigate the interval observer problem for a class of discrete-time nonlinear systems, in absence or presence of external disturbances and parametric uncertainties. The interval observers depend on the design of two preserving order observers, providing lower and upper estimations of the state. The main objective is to apply the stability radii notions and cooperativity property in the estimation error systems in order to guarantee that the lower/upper estimation is always below/above the real state trajectory at each time instant from an appropriate initialization, and the estimation errors converge asymptotically towards zero when the disturbances and/or uncertainties are vanishing. For the disturbed case, the estimation errors practically converge to a vicinity of zero, while the lower/upper estimations preserve the partial ordering with respect to the state trajectory. The design conditions, that are valid for Lipschitz nonlinearities, can be expressed as Linear Matrix Inequalities (LMIs). A numerical simulation example is provided to verify the effectiveness of the proposed method.
