Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using The Matrix Lambert W function

TL;DR

This paper analyzes the stability of multi-dimensional time-delay systems using the matrix Lambert W function, revealing that only two branches are needed to find the complete spectrum and challenging previous assumptions about the principal branch.

Contribution

It demonstrates that a simplified approach using only two branches of the matrix Lambert W function suffices for stability analysis, correcting prior beliefs about the principal branch's capabilities.

Findings

No one-to-one correspondence between Lambert W branches and system roots.

Only two branches are needed to find the complete spectrum.

The principal branch can find multiple roots, not just the dominant one.

Abstract

This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments.