The aim of this study is to offer a comparison of the numerical procedures for an important problem, the determination of purely imaginary characteristic roots of LTITime Delayed Systems (LTI-TDS). This problem, in fact, has a crucial role in assessing the stability of the general class of vector LTI-TDS x = Ax + Bx(t -τ). There are many procedures discussed in the literature for this purpose. Those, which are exact, first determine the complete set of imaginary characteristic roots of the dynamics, as they constitute the only points where stability switching can take place. These approaches are, in fact, some variations of the five main methods, which may demand numerical procedures of different complexity and they may result in different precisions in finding the roots. There is, however, no comparative case study known to the authors to demonstrate the strengths and weaknesses of these methods. This document is prepared primarily for this purpose. We first present an overview of each of the five methods and then compare their numerical performances over an example case study.


Paper presented at the 16th IFAC World Congress (Czech Republic, 2005). DOI:10.3182/20050703-6-CZ-1902.00636

Dr. Sipahi is affiliated with Northeastern University as of the time of deposit.


linear time invariant, time-delay systems, stability, stability tests


Industrial Engineering | Mechanical Engineering



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