Under-approximations of reachable sets and tubes have been receiving growing research attention due to their important roles in control synthesis and verification. Available under-approximation methods applicable to continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general, and/or suffer from high computational costs. In this note, we attempt to overcome these drawbacks for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes, utilizing approximations of the matrix exponential and its integral. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes, when implemented using zonotopes, with first-order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, we implement our approach in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.

翻译：可达集与可达管的欠近似因在控制综合与验证中的重要作用而日益受到关注。现有适用于连续时间线性系统的欠近似方法通常假设能够精确计算转移矩阵及其积分，这在一般情况下难以实现，且/或存在计算成本过高的问题。本文试图针对一类线性时不变（LTI）系统克服上述缺陷，提出一种基于矩阵指数及其积分近似来欠近似有限时间前向可达集与可达管的新方法。具体而言，我们考虑输入矩阵为单位阵、且初始值与输入值属于闭单位球仿射变换所得满维集合的连续时间LTI系统。当采用多面体（zonotope）实现时，所提方法可高效计算可达集与可达管的欠近似，并在豪斯多夫距离意义下具备一阶收敛保证。为展示算法性能，我们在三个数值算例中实施该方法，系统维度介于2至200之间。