The paper deals with the problem of designing an unknown input observer for discrete-time non-linear systems. In particular, with the use of the Lyapunov method, it is shown that the proposed observer is convergent under certain, non-restrictive conditions. Based on the achieved results, a general solution for increasing the convergence rate is proposed and implemented with the use of stochastic robustness techniques. In particular, it is shown that the problem of increasing the convergence rate of the observer can be formulated as a stochastic robustness analysis task that can be transformed into a structure selection and parameter estimation problem of a non-linear function, which can be solved with the B-spline approximation and evolutionary algorithms. The final part of the paper shows an illustrative example based on a two phase induction motor. The presented results clearly exhibit the performance of the proposed observer.