Conservative Time Discretization: A Comparative Study

TL;DR

This paper reviews and compares various methods for overapproximating reachable states of linear systems over short time horizons, highlighting their trade-offs in accuracy and computational efficiency.

Contribution

It provides the first isolated comparison of discretization methods for reachability analysis, including numerical experiments and implementation strategies.

Findings

Different discretization methods have unique accuracy and cost trade-offs.

Some methods are better suited for specific system characteristics.

Preprocessing and implementation strategies can enhance results.

Abstract

We present the first review of methods to overapproximate the set of reachable states of linear time-invariant systems subject to uncertain initial states and input signals for short time horizons. These methods are fundamental to state-of-the-art reachability algorithms for long time horizons, which proceed in two steps: First they use such a method to discretize the system for a short time horizon, and then they efficiently obtain a solution of the new discrete system for the long time horizon. Traditionally, both qualitative and quantitative comparison between different reachability algorithms has only considered the combination of both steps. In this paper we study the first step in isolation. We perform a variety of numerical experiments for six fundamental discretization methods from the literature. As we show, these methods have different trade-offs regarding accuracy and computational cost and, depending on the characteristics of the system, some methods may be preferred over others. We also discuss preprocessing steps to improve the results and efficient implementation strategies.