Stability of linear switching systems and Markov-Bernstein inequalities for exponents

TL;DR

This paper investigates the stability of continuous-time linear switching systems by estimating discretization step sizes using Markov-Bernstein inequalities, providing a method with guaranteed accuracy and initial computational cost estimates.

Contribution

It introduces a novel approach to estimate discretization step size for stability analysis of LSS using Markov-Bernstein inequalities, including universal bounds and initial cost estimates.

Findings

Established a method to estimate discretization step size with guaranteed accuracy.

Derived universal bounds for constants in Markov-Bernstein inequalities for exponents.

Provided the first computational cost estimate for stability analysis of LSS with real spectra.

Abstract

We analyse the problem of stability of a continuous time linear switching system (LSS) versus the stability of its Euler discretization. It is well-known that the existence of a positive {\tau} for which the corresponding discrete time system with step size {\tau} is stable implies the stability of LSS. Our main goal is to obtain a converse statement, that is, to estimate the discretization step size {\tau} > 0 up to a given accuracy {\epsilon} > 0. This leads to a method of deciding the stability of continuous time LSS with a guaranteed accuracy. As the first step, we solve this problem for matrices with real spectrum and conjecture that our method stays valid for the general case. Our approach is based on Markov-Bernstein type inequalities for systems of exponents. We obtain universal estimates for sharp constants in those inequalities. Our work provides the first estimate of the computational cost of the stability problem for continuous-time LSS (though restricted to the real-spectrum case).