Finding the Dominant Roots of a Time Delay System without Using the Principal Branch of the Lambert W Function

TL;DR

This paper challenges existing methods for stability analysis of time delay systems by demonstrating that dominant roots can sometimes be found without the principal branch of the Lambert W function, using a numerical example.

Contribution

It shows that the dominant roots of certain time delay systems can be identified without relying on the principal branch of the Lambert W function, contradicting previous assumptions.

Findings

Dominant roots can be found without the principal Lambert W branch

The method's applicability is broader than previously thought

Numerical example illustrates the new approach

Abstract

This brief note complements some results regarding a recently developed technique for the stability analysis of linear time-invariant, time delay systems using the matrix Lambert W function. By means of a numeric example, it is shown that there are cases for which the dominant roots of the system can be found without using the principal branch of this multi-valued function, contradicting the main proposition of the methodology.