Cluster treatment of characteristic roots, CTCR, a unique methodology for the complete robustness analysis of linear time invariant multiple time delay systems against delay uncertainties

Date of Completion

January 2005


Engineering, Electronics and Electrical|Engineering, Mechanical|Engineering, System Science




Today, many high-tech applications such as simultaneous metal machining, neural networks, internet data congestion, population dynamics, chemical processes, formation flight problems, etc., appear in the form of multiple time delay systems (MTDS). Due to the presence of delays, the stability of the dynamics is under question and needs to be addressed. ^ In the last four decades, the stability analysis of the general class, Linear Time Invariant MTDS (LTI-MTDS), has been one of the challenging problems of the systems and control community, which originates from the complex stability analysis due to their infinite dimensionality. For tackling this, a unique procedure is presented, in this Ph.D. thesis, for the complete stability robustness of the general class LTI-MTDS. The uniqueness of the treatment is simply due to the fact that there is no comparable methodology, presently, in the literature. The backbone of the method is a novel framework called the Cluster Treatment of Characteristic Roots, CTCR. CTCR is an efficient, exact and exhaustive methodology, which is based on outstanding characteristics that originate from two fundamental propositions. ^ Proposition I suggests that there exists only an upper-bounded number of hyperplanes in a p-dimensional delay domain where all the purely imaginary characteristic roots of the dynamics reside. These hyperplanes are called the Kernel Hyperplanes and they are exhaustively determined by CTCR. This property considerably alleviates the problem, which is thus known to be notoriously complex to handle. ^ Using a unique mapping, Kernel Hyperplanes give rise to their Offspring Hyperplanes, together which they from all possible time delay combinations where the dynamics may change its stability posture. ^ With the exhaustive determination of Kernel Hyperplanes and their corresponding imaginary roots, Proposition 2 follows an interesting invariance property on a crossing tendency of these imaginary roots across the Kernel and Offspring Hyperplanes so long one moves parallel to any one of the p delay-axis. ^ CTCR's unique stability results are demonstrated by way of challenging examples in Mechatronics, Manufacturing Engineering and Mathematics, all of which are prohibitively difficult, if not impossible to solve using any other peer methodology. ^