Under-Approximate Reachability Analysis for a Class of Linear Systems with Inputs

TL;DR

This paper introduces a computationally efficient method for under-approximating reachable sets of certain linear systems using matrix exponential approximations, with convergence guarantees and practical demonstrations.

Contribution

It proposes a novel under-approximation technique for LTI systems with identity input matrices, leveraging matrix exponential approximations and zonotopes, improving efficiency and accuracy.

Findings

Method provides first-order convergence guarantees.

Applicable to systems with dimensions up to 200.

Demonstrated effectiveness through three numerical examples.

Abstract

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.