A New Solution Concept and Family of Relaxations for Hybrid Dynamical Systems

TL;DR

This paper presents a unified framework for analyzing, approximating, and controlling hybrid dynamical systems with discrete jumps, introducing a new solution concept and a family of smooth relaxations that converge to the hybrid solutions.

Contribution

It introduces hybrid Filippov solutions for hybrid systems and a parameterized family of smooth relaxations that approximate these solutions, unifying analysis and control approaches.

Findings

Hybrid Filippov solutions simplify hybrid system trajectories.

Smooth relaxations approximate hybrid solutions in the limit.

The framework unifies discrete and continuous dynamics analysis.

Abstract

We introduce a holistic framework for the analysis, approximation and control of the trajectories of hybrid dynamical systems which display event-triggered discrete jumps in the continuous state. We begin by demonstrating how to explicitly represent the dynamics of this class of systems using a single piecewise-smooth vector field defined on a manifold, and then employ Filippov's solution concept to describe the trajectories of the system. The resulting \emph{hybrid Filippov solutions} greatly simplify the mathematical description of hybrid executions, providing a unifying solution concept with which to work. Extending previous efforts to regularize piecewise-smooth vector fields, we then introduce a parameterized family of smooth control systems whose trajectories are used to approximate the hybrid Filippov solution numerically. The two solution concepts are shown to agree in the limit, under mild regularity conditions.