Over-approximating reachable tubes of linear time-varying systems

TL;DR

This paper introduces a method for over-approximating the reachable states of linear time-varying systems with uncertainties, using numerical and zonotopic techniques, to ensure safety and robustness analysis.

Contribution

It provides a novel, convergent approach for over-approximating reachable tubes in linear time-varying systems with convex uncertainties, including a zonotopic variant using only linear algebra.

Findings

The method converges with first-order accuracy.

The zonotopic variant efficiently computes over-approximations.

Demonstrated effectiveness on a practical example.

Abstract

We present a method to over-approximate reachable tubes over compact time-intervals, for linear continuous-time, time-varying control systems whose initial states and inputs are subject to compact convex uncertainty. The method uses numerical approximations of transition matrices, is convergent of first order, and assumes the ability to compute with compact convex sets in finite dimension. We also present a variant that applies to the case of zonotopic uncertainties, uses only linear algebraic operations, and yields zonotopic over-approximations. The performance of the latter variant is demonstrated on an example.