Exponential stability of a class of nonlinear singularly perturbed systems with delayed impulses

A class of nonlinear singularly perturbed systems with delayed impulses is considered. By delayed impulses we mean that the impulse maps describing the state's jumping at impulsive moments are dependent on delayed state variables. Assuming that each of two lower order subsystems possesses a Lyapunov function, exponential stability criteria for all small enough values of singular perturbation parameter are obtained. It turns out that the achieved exponential stability is robust with respect to small impulse input delays. A stability bound on perturbation parameter is also derived through using those Lyapunov functions. Additionally, for a class of singularly perturbed Lur'e systems with delayed impulses, an LMI-based method to determine stability and an upper bound of the singular perturbation parameter is presented. The results are illustrated by an example for the position control of a dc-motor with unmodelled dynamics.