Robustness analysis of a class of discrete-time systems with applications to neural networks

We study the robust stability properties of a large class of nonlinear discrete-time systems by addressing the following question: given a nonlinear discrete-time system with specified exponentially stable equilibria, under what conditions will a perturbed model of the discrete-time system possess exponentially stable equilibria that are close (in distance) to the exponentially stable equilibria of the unperturbed discrete-time system? In arriving at our results, we establish robust stability results for the perturbed discrete-time systems considered herein and we determine conditions which ensure the existence of exponentially stable equilibria of perturbed discrete-time systems which are near the exponentially stable equilibria of the original unperturbed discrete-time systems. These results involve quantitative estimates of the distance between the corresponding equilibrium points of the unperturbed and perturbed discrete-time systems. We apply the above results to the robustness analysis of a large class of discrete-time recurrent neural networks