Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees

2026-04-06

AI Summary

This work addresses the challenge of analyzing robustness of periodic orbits in hybrid dynamical systems when system dynamics are accessible only through forward simulation, rendering traditional computation of invariant sets under Poincaré return maps intractable. The authors propose a novel framework that integrates sampling-based optimization with probabilistic verification to efficiently estimate finite-step invariant ellipsoids for simulation-driven return maps. For the first time, this approach provides rigorous probabilistic guarantees on the accuracy and confidence level specified by the user. The method is validated on two low-dimensional benchmark systems and the compass-gait walking model, demonstrating significantly enhanced scalability and reliability in computing invariant sets for complex hybrid systems.

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.

Problem

• hybrid systems
• Poincare return maps
• invariant sets
• robustness
• periodic orbits

Innovation

• finite-step invariance
• Poincare return map
• sampling-based optimization
• probabilistic guarantees
• hybrid dynamical systems