A note on the use of the Lambert W function in the stability analysis of time-delay systems

The Lambert W function is defined to be the multivalued inverse of the function w -> we(w) = z. This 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. The latter application is based on taking the mth power and/or nth root of the transcendental characteristic equation (TCE) and representing the roots of the derived TCE(s) in terms of W functions. In this note, we re-examine such an application of using the Lambert W function through actually computing 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 mth power of the original TCE introduces superfluous roots to the system. With these observations, some deficiencies displayed in the literature (Chen, Y. Q., & Moore, K. L. (2002a). Analytical stability bound for delayed second-order systems with repeating poles using Lambert W function. Automatica, 38(5), 891-895, Chen, Y. Q., & Moore, K. L. (2002b). Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dynamics, 29(1-4), 191-200) are pointed out. Moreover, we clarify the correct use of Lambert W function to stability analysis of a class of time-delay systems. This will actually enlarge 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. (c) 2005 Elsevier Ltd. All rights reserved.