We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances and uncertain initial conditions. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing over-approximation tools tend to be conservative or computationally expensive. In this work, we characterize the convex hulls of reachable sets as the convex hulls of solutions of an ordinary differential equation with initial conditions on the sphere. This finite-dimensional characterization unlocks an efficient sampling-based estimation algorithm to accurately over-approximate reachable sets. We also study the structure of the boundary of the reachable convex hulls and derive error bounds for the estimation algorithm. We give applications to neural feedback loop analysis and robust MPC.

翻译：我们研究了存在有界扰动和不确定初始条件的非线性系统可达集的凸包。可达集在控制领域至关重要，但其计算一直以来极富挑战性，现有的过近似方法往往过于保守或计算成本高昂。在本工作中，我们将可达集的凸包刻画为以球面上的点为初始条件的常微分方程解的凸包。这种有限维刻画使我们能够提出一种高效的基于采样的估计算法，从而精确地过近似可达集。我们还研究了可达凸包边界的结构，并推导了该估计算法的误差界。最后，我们将该方法应用于神经反馈回路分析与鲁棒模型预测控制。