Special Cases in Using the Matrix Lambert W function for the Stability Analysis of High-Order Linear Systems with Time Dela∗

A recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems is based on an assumed one to one correspondence between the branches of this multi-valued function and the characteristic roots of the system. By studying a particular, yet common, second order system, we show that in general there is no such one to one correspondence. Furthermore, it is shown that under mild conditions only two branches su?ce to find the complete spectrum of the system, and that the principal branch can be used not only the dominant root, as stated in previous works, but also some non dominant roots too. The results are presented analytically, and then verified by numerical experiments.