用采样优化计算周期轨道的有限步不变椭球，带概率保证。
Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees
- 基于采样优化，从返回映射中构造不变椭球。
- 在用户设定精度下，保证有限步内状态不跳出椭球。
- 适用于足式机器人等切换系统的稳定性分析。
庞加莱返回映射是分析混合动力系统周期轨道的核心工具，涵盖足式运动、电力电子等具有切换行为的网络物理系统。该映射刻画系统在守卫面上的演化，将周期轨道的稳定性分析转化为离散时间系统的分析。虽然线性化可提供局部稳定性信息，但评估扰动鲁棒性需识别返回动力学下的状态空间不变集。然而，当系统动态仅可通过前向仿真获取时，计算此类不变集极为困难。本文提出一种算法框架，利用采样优化方法，基于返回映射的采样评估，在名义周期轨道周围构造有限步不变椭球。所得解附带概率保证，满足用户定义的精度阈值。我们在两个低维系统及一个指南针式步行模型上验证了该方法的有效性。
原文摘要 · Abstract (English)
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.
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