On the Relaxation of Hybrid Dynamical Systems

TL;DR

This paper introduces a new relaxation framework for hybrid dynamical systems that ensures fundamental properties like trajectory existence and uniqueness, even beyond Zeno points, and enables reliable numerical simulation and sensitivity analysis.

Contribution

The authors develop a novel relaxation approach that recovers key system properties and provides convergent numerical methods for hybrid systems, including through Zeno phenomena.

Findings

Ensures existence and uniqueness of trajectories beyond Zeno

Provides convergent numerical approximation methods

Enables sensitivity analysis of hybrid trajectories

Abstract

Hybrid dynamical systems have proven to be a powerful modeling abstraction, yet fundamental questions regarding the dynamical properties of these systems remain. In this paper, we develop a novel class of relaxations which we use to recover a number of classic systems theoretic properties for hybrid systems, such as existence and uniqueness of trajectories, even past the point of Zeno. Our relaxations also naturally give rise to a class of provably convergent numerical approximations, capable of simulating through Zeno. Using our methods, we are also able to perform sensitivity analysis about nominal trajectories undergoing a discrete transition -- a technique with many practical applications, such as assessing the stability of periodic orbits.