Backward reachability analysis computes the set of states that reach a target set under the competing influence of control input and disturbances. Depending on their interplay, the backward reachable set either represents all states that can be steered into the target set or all states that cannot avoid entering it -- the corresponding solutions can be used for controller synthesis and safety verification, respectively. A popular technique for backward reachable set computation solves Hamilton-Jacobi-Isaacs equations, which scales exponentially with the state dimension due to gridding the state space. In this work, we instead use set propagation techniques to design backward reachability algorithms for linear time-invariant systems. Crucially, the proposed algorithms scale only polynomially with the state dimension. Our numerical examples demonstrate the tightness of the obtained backward reachable sets and show an overwhelming improvement of our proposed algorithms over state-of-the-art methods regarding scalability, as systems with well over a hundred states can now be analyzed.

翻译：后向可达性分析旨在计算在控制输入与扰动共同影响下能够到达目标状态集的初始状态集合。根据两者交互方式的不同，后向可达集要么表征所有可被引导至目标状态集的初始状态，要么表征所有无法避免进入目标状态集的初始状态——对应的解可分别用于控制器综合与安全性验证。后向可达集计算的常用方法通过求解哈密顿-雅可比-艾萨克斯方程实现，由于需要对状态空间进行网格划分，其计算复杂度随状态维度呈指数增长。本文转而采用集传播技术为线性时不变系统设计后向可达性算法。关键创新在于，所提算法的复杂度仅随状态维度呈多项式增长。数值算例表明，本文方法所得后向可达集具有良好的紧致性，且在可扩展性方面较现有最优方法具有压倒性优势——现可对超过百维状态空间的系统进行分析。