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This paper introduces a sampling-based optimization framework to compute finite-step invariant ellipsoids for hybrid systems using Poincare return maps. The method approximates the return map through sampled evaluations and optimizes for an ellipsoid that remains invariant for a finite number of steps with probabilistic guarantees. The approach is validated on two low-dimensional systems and a compass-gait walking model, demonstrating its applicability to systems with complex hybrid dynamics.
Guaranteeing robustness of hybrid systems just got easier: this work offers a computationally tractable method to find invariant sets with probabilistic guarantees, even when dynamics are only accessible through simulation.
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.