Complete Stability Map of Third Order LTI, Multiple Time-Delay Systems

摘要

A unique procedure is presented in this paper, for a complete stability map of the third order LTI multiple time-delay systems (LTI-MTDS). For simplicity the number of time delays is taken as two. The uniqueness of the treatment is simply due to the fact that there is no comparable methodology, presently, in the literature. The end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays. The backbone of the method is a novel framework called "the cluster treatment of characteristic roots, CTCR". CTCR is constructed over two fundamental propositions. The first proposition claims the existence of a bounded number of so called "kernel curves" where the only imaginary characteristic roots of the system occur. The second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots. The new methodology is expandable to higher order dynamics with more time delays than two, as the authors intend to demonstrate in future publications.