This paper proposes a mechanism to fine-tune convex approximations of probabilistic reachable sets (PRS) of uncertain dynamic systems. We consider the case of unbounded uncertainties, for which it may be impossible to find a bounded reachable set of the system. Instead, we turn to find a PRS that bounds system states with high confidence. Our data-driven approach builds on a kernel density estimator (KDE) accelerated by a fast Fourier transform (FFT), which is customized to model the uncertainties and obtain the PRS efficiently. However, the non-convex shape of the PRS can make it impractical for subsequent optimal designs. Motivated by this, we formulate a mixed integer nonlinear programming (MINLP) problem whose solution result is an optimal $n$ sided convex polygon that approximates the PRS. Leveraging this formulation, we propose a heuristic algorithm to find this convex set efficiently while ensuring accuracy. The algorithm is tested on comprehensive case studies that demonstrate its near-optimality, accuracy, efficiency, and robustness. The benefits of this work pave the way for promising applications to safety-critical, real-time motion planning of uncertain dynamic systems.

翻译：本文提出了一种机制，用于微调不确定动态系统概率可达集（PRS）的凸近似。我们考虑无界不确定性的情况，此时可能无法找到系统的有界可达集。相反，我们转而寻找一个能以高置信度界定系统状态的PRS。我们的数据驱动方法基于核密度估计器（KDE），并通过快速傅里叶变换（FFT）加速，该方法被定制用于建模不确定性并高效获取PRS。然而，PRS的非凸形状可能使其在后续优化设计中不实用。基于此，我们构建了一个混合整数非线性规划（MINLP）问题，其求解结果是一个最优的$n$边凸多边形，用以近似PRS。利用这一公式，我们提出了一种启发式算法，在保证精度的同时高效找到该凸集。该算法在综合性案例研究中得到了测试，结果证明了其近最优性、准确性、高效性和鲁棒性。本文的研究成果为不确定动态系统的安全关键实时运动规划应用铺平了道路。