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A note on the use of the Lambert W function in the stability analysis of time-delay systems. (English) Zbl 1125.93440

Summary: The Lambert \(W\) function is defined to be the multivalued inverse of the function \(w\to we^w=z\). 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 \(m\)th power and/or \(n\)th 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 \(m\)th power of the original TCE introduces superfluous roots to the system. With these observations, some deficiencies displayed in the literature [Y. Q. Chen and K. L. Moore, Automatica 38, No. 5, 891–895 (2002; Zbl 1020.93019), Nonlinear Dyn. 29, No. 1–4, 191–200 (2002; Zbl 1020.34064)] 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.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34K20 Stability theory of functional-differential equations
93C23 Control/observation systems governed by functional-differential equations

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References:

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