Averaging for a Class of Hybrid Systems

Averaging theory for ordinary differential equations is extended to a class of hybrid systems. Rapid time variations in the flow map of a hybrid system generate solutions that are also solutions of a slightly perturbed time-invariant average hybrid system. Results relating solutions of the time-varying system to solutions of the average system ensue. In the absence of finite escape times for the average system, on compact time domains each solution of the time-varying system is close to a solution of the average system. If the average system is asymptotically stable, the time-varying system exhibits semiglobal, practical asymptotic stability. These results rely on mild regularity properties for the average system. In particular, the average system is not required to exhibit unique solutions. Both periodic and non-periodic flow maps are considered. The results are partially motivated by the desire to justify a pulse-width modulated implementation of hybrid feedback control for nonlinear systems.