Novel results in averaging analysis of singularly perturbed hybrid systems

We investigate stability of a class of singularly perturbed systems whose slow system is a set-valued average defined via an appropriate averaging procedure of the solutions of the continuous-time boundary layer system. An approximate hybrid system consisting of this average, the projection of the jump map in the direction of the slow states and flow and jump sets from the original dynamics is shown to approximate the actual singularly perturbed hybrid system. In particular, using forward pre-completeness of the average system we show that solutions of the actual and approximate systems are close in an appropriate sense on compact time intervals. It is also shown that global asymptotic stability of the average system implies semi-global practical asymptotic stability of the actual system. Several examples are presented to illustrate our results and relate them to previously published results in the literature.