Use of Lambert W function to stability analysis of time-delay systems

The Lambert W function has been used in an extremely wide variety of applications, including the stability analysis of fractional-order as well as integer-order time-delay systems. In this paper, we re-examine an application of using the Lambert W function through actually constructing the root distributions of the derived TCEs of some chosen orders. It is found that the rightmost root of the original TCE is not necessarily a principal branch Lambert W function solution, and that a derived TCE obtained by taking the nth power of the original TCE introduces superfluous roots to the system. With these observations, some deficiencies displayed in the literature are pointed out. Moreover, we clarify the correct use of Lambert W function to stability analysis of a class of time-delay systems. This actually enlarges the application scope of the Lambert W function, which is becoming a standard library function for various commercial symbolic software packages, to time-delay systems.