Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems

TL;DR

This paper demonstrates that near exponentially stable periodic orbits, hybrid dynamical systems can be approximated by smooth systems through contraction and topological reduction, facilitating analysis and control design.

Contribution

It introduces a method to reduce hybrid systems near stable periodic orbits to smooth systems, enabling easier analysis and control synthesis.

Findings

Hybrid models contract superexponentially near stable orbits

Finite-time contraction under rank conditions

Reduction of high-dimensional models to smooth systems

Abstract

We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constant-dimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincare map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high-dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior.