Numerical methods for solving minimum-time problem for linear systems

TL;DR

This paper reviews classical algorithms for the minimum-time problem in linear systems, introduces a new algorithm with proven convergence, and demonstrates its superior performance through numerical experiments.

Contribution

The paper presents a novel algorithm for MTPLS with a convergence proof, outperforming existing methods in high-precision scenarios.

Findings

The new algorithm outperforms existing algorithms by factors of tens or hundreds in high-precision computations.

It exhibits the lowest failure rate among compared algorithms.

Numerical experiments validate the effectiveness of the proposed method.

Abstract

This paper offers a contemporary and comprehensive perspective on the classical algorithms utilized for the solution of minimum-time problem for linear systems (MTPLS). The use of unified notations supported by visual geometric representations serves to highlight the differences between the Neustadt-Eaton and Barr-Gilbert algorithms. Furthermore, these notations assist in the interpretation of the distance-finding algorithms utilized in the Barr-Gilbert algorithm. Additionally, we present a novel algorithm for solving MTPLS and provide a constructive proof of its convergence. Similar to the Barr-Gilbert algorithm, the novel algorithm employs distance search algorithms. The design of the novel algorithm is oriented towards solving such MTPLS for which the analytic description of the reachable set is available. To illustrate the advantages of the novel algorithm, we utilize the isotropic rocket benchmark. Numerical experiments demonstrate that, for high-precision computations, the novel algorithm outperforms others by factors of tens or hundreds and exhibits the lowest failure rate.