Technical communique: The Lambert W function and the spectrum of some multidimensional time-delay systems

Authors:
Elias Jarlebring;Tobias Damm
Affiliations:
TU Braunschweig, Institut Computational Mathematics, Postfach 3329, 38023 Braunschweig, Germany;TU Kaiserslautern, Technomathematics Group, Postfach 3049, 67653 Kaiserslautern, Germany
Venue:
Automatica (Journal of IFAC)
Year:
2007

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Abstract

In this note we find an explicit expression for the eigenvalues of a retarded time-delay system with one delay, for the special case that the system matrices are simultaneously triangularizable, which includes the case where they commute. Using matrix function definitions we define a matrix version of the Lambert W function, from which we form the expression. We prove by counter-example that some expressions in other publications on Lambert W for time-delay systems do not always hold.