We study the problem of computing robust controllable sets for discrete-time linear systems with additive uncertainty. We propose a tractable and scalable approach to inner- and outer-approximate robust controllable sets using constrained zonotopes, when the additive uncertainty set is a symmetric, convex, and compact set. Our least-squares-based approach uses novel closed-form approximations of the Pontryagin difference between a constrained zonotopic minuend and a symmetric, convex, and compact subtrahend. Unlike existing approaches, our approach does not rely on convex optimization solvers, and is projection-free for ellipsoidal and zonotopic uncertainty sets. We also propose a least-squares-based approach to compute a convex, polyhedral outer-approximation to constrained zonotopes, and characterize sufficient conditions under which all these approximations are exact. We demonstrate the computational efficiency and scalability of our approach in several case studies, including the design of abort-safe rendezvous trajectories for a spacecraft in near-rectilinear halo orbit under uncertainty. Our approach can inner-approximate a 20-step robust controllable set for a 100-dimensional linear system in under 15 seconds on a standard computer.

翻译：我们研究了离散时间线性系统在加性不确定性下的鲁棒可控集计算问题。当加性不确定集为对称凸紧集时，我们提出了一种可行且可扩展的方法，利用约束zonotopes对鲁棒可控集进行内逼近和外逼近。该基于最小二乘法的方法采用新型闭式逼近，计算约束zonotopic被减集与对称凸紧减集之间的Pontryagin差。与现有方法不同，本方法不依赖凸优化求解器，且对于椭球和zonotopic不确定集无需投影操作。我们还提出了一种基于最小二乘法的凸多面体外逼近方法，用于逼近约束zonotopes，并刻画了这些逼近均为精确逼近的充分条件。通过多个案例研究（包括航天器在近直线晕轨道不确定性下设计避撞安全交会轨迹）证明了本方法的计算效率和可扩展性。在标准计算机上，本方法可在15秒内对100维线性系统完成20步鲁棒可控集的内逼近。