On the averaging of a class of hybrid systems

Abstract

Abstract-Modeling abstraction and time-scale separation in the design of complex systems often leads to hybrid dynamics. Discontinuities in the continuous evolution of a hybrid system may however create difficulties in the formal analysis, as well as in numerical simulation and verification. Here we study a class of hybrid systems that are excited by high-frequency external signals. These systems arise in the modeling of switched power converters, mechanical systems with friction and quantized systems. For a quite general class of excitation signals, an averaging result is shown stating that the hybrid system can be approximated by a Lipschitzcontinuous system. The approximation is in the order of the maximal repetition interval of the excitation signal.

Dynamics and bifurcations of nonsmooth systems: A survey

“…Versions of the first Bogolyubov's theorem for differential equations with bounded variation right-hand-sides are developed in Iannelli-Johansson-Jonsson-Vasca [174,175,176] in the context of control systems subject to a dither noise. The response of a piecewise-linear FitzHugh-Nagumo model to a white noise is investigated in Simpson-Kuske [320].…”

Averaging of nonsmooth systems using dither

“…However, for most cases in applications the existence of a Carathéodory solution is reasonable. Existence and uniqueness of solutions for dithered relay systems are discussed in Iannelli, Johansson, Jönsson, and Vasca (2004).…”

Fuzzy Model and Control for Hybrid Systems Using Averaging Techniques

“…1. According to the results in [15], we have the following theorem. According to Theorem 1, for a sufficiently small switching period T, x(t) will be very close to xp(t).…”