通过可调松弛变量，实现对动态系统可达集的精准估计与鲁棒性保障。
Data-driven Reachable Set Estimation with Tunable Adversarial and Wasserstein Distributional Guarantees
- 引入带松弛变量的宽松场景规划，平衡集合大小与未来轨迹包容性。
- 在对抗扰动和分布偏移下仍能提供事后概率保证，且保证随Wasserstein距离衰减。
- 适用于需要高可靠性轨迹预测的系统安全验证场景。
我们研究仅基于采样状态轨迹的未知离散时间动力系统的有限时域可达集估计问题。不同于将场景优化视为黑箱工具，本文展示如何将其定制用于可达集估计——需基于完整轨迹学习一组集合，并保持对未来轨迹在整个时域内包含的概率保证。为此，我们提出一种带松弛变量的松弛场景规划，实现可达集大小与跨时域外样本轨迹包含概率之间的可调权衡，从而降低对异常值的敏感性。结合最近的对抗鲁棒场景优化成果，进一步扩展该框架以考虑观测轨迹的有界对抗扰动，并推导出未来轨迹包含的事后概率保证。当概率分布发生以Wasserstein距离度量的偏移时，获得理论保证退化程度的显式边界。针对不同几何形状（p-范数球、椭球、拟柱体），我们推导出可计算的凸重构形式，并在仿真中验证了理论结果。
原文摘要 · Abstract (English)
We study finite horizon reachable set estimation for unknown discrete-time dynamical systems using only sampled state trajectories. Rather than treating scenario optimization as a black-box tool, we show how it can be tailored to reachable set estimation, where one must learn a family of sets based on whole trajectories, while preserving probabilistic guarantees on future trajectory inclusion for the entire horizon. To this end, we formulate a relaxed scenario program with slack variables that yields a tunable trade-off between reachable set size and out-of-sample trajectory inclusion over the horizon, thereby reducing sensitivity to outliers. Leveraging the recent results in adversarially robust scenario optimization, we then extend this formulation to account for bounded adversarial perturbations of the observed trajectories and derive a posteriori probabilistic guarantees on future trajectory inclusion. When probability distribution shifts in the Wasserstein distance occur, we obtain an explicit bound on how gracefully the theoretical probabilistic guarantees degrade. For different geometries, i.e., $p$-norm balls, ellipsoids, and zonotopes, we derive tractable convex reformulations and corroborate our theoretical results in simulation.
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