Critical simple imaginary characteristic roots and invariance properties for quasipolynomials: A missing link

This paper addresses some properties of simple characteristic roots of quasipolynomials including commensurate delays. Although such a problem seems easy and was largely treated in the literature, to the best of the authors' knowledge, the invariance and the ultimate stability properties have not been fully investigated. In this paper, we propose a new frequency-sweeping framework, which simultaneously considers the Taylor series of the critical imaginary roots as well as the Puiseux series of singular points of the frequency-sweeping curves. Through analyzing the algebraic properties of the corresponding frequency-sweeping curves, we are able to completely characterize the invariance property of the simple imaginary roots with respect to the corresponding critical delay values. As a consequence of the invariance property, the ultimate stability property can be easily derived. Finally, as a byproduct of the approach proposed in the paper, the complete stability for time-delay systems with only simple imaginary roots can be systematically derived. Some illustrative examples complete the presentation.