Construction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part I: Error Estimates

TL;DR

This paper introduces a method for approximating the minimum time function in linear control systems using reachable set approximation and provides error estimates based on Hausdorff distance.

Contribution

It presents a theoretical framework for error estimation of the approximate minimum time function in linear control systems using convex analysis.

Findings

Error estimates for reachable set approximation are derived.

The approach is validated with numerical examples.

The method applies to a class of linear control systems.

Abstract

The first part of this paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Numerical examples are included in the second part.