Tropical Matrix Exponential

TL;DR

This paper introduces a tropical matrix exponential and explores its role in stability analysis of discrete event systems, including generalized eigenvectors and robustness conditions.

Contribution

It presents a novel exponential function for tropical matrices and characterizes generalized eigenvectors related to stability in discrete event systems.

Findings

The tropical matrix exponential is essential for stability analysis.

Generalized eigenvectors exist up to a certain order related to matrix period.

A sufficient condition for robustness of the matrix exponential is established.

Abstract

In this article, we introduce an exponential for tropical matrices and show that this series is essential for the analysis of certain kinds of stability in discrete event dynamic systems. A notion of a generalised eigenvector is introduced to discuss this kind of stability and prove it exists at most in the order of $1,p/2,p$, where $p$ is the period of the corresponding matrix. Thus characterizing the generalised eigenvectors of all powers of the matrix. Also, a sufficient condition is proved for the exponential of a matrix to be robust.