Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees
This work addresses robustness analysis in cyber-physical systems like legged locomotion, but it is incremental as it builds on existing Poincare map methods with a new computational approach.
The authors tackled the problem of computing invariant sets for hybrid systems with switching behavior, which is computationally difficult when only forward simulation is available, by proposing a sampling-based optimization framework that computes finite-step invariant ellipsoids with probabilistic guarantees, demonstrating it on low-dimensional systems and a walking model.
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.