Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees

TL;DR

This paper introduces a sampling-based method to compute finite-step invariant sets for hybrid systems, providing probabilistic guarantees and demonstrating effectiveness on various models.

Contribution

It presents a novel algorithmic framework that leverages sampling to efficiently compute invariant ellipsoids with probabilistic guarantees in hybrid systems.

Findings

Successfully computed invariant sets for low-dimensional systems.

Provided probabilistic guarantees on the invariance of the computed sets.

Demonstrated applicability to a compass-gait walking model.

Abstract

Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching behavior. The Poincare return map captures the evolution of the hybrid system on a guard surface, reducing the stability analysis of a periodic orbit to that of a discrete-time system. While linearization provides local stability information, assessing robustness to disturbances requires identifying invariant sets of the state space under the return dynamics. However, computing such invariant sets is computationally difficult, especially when system dynamics are only available through forward simulation. In this work, we propose an algorithmic framework leveraging sampling-based optimization to compute a finite-step invariant ellipsoid around a nominal periodic orbit using sampled evaluations of the return map. The resulting solution is accompanied by probabilistic guarantees on finite-step invariance satisfying a user-defined accuracy threshold. We demonstrate the approach on two low-dimensional systems and a compass-gait walking model.