CTCR Approach to the Complete Stability of TDS with Multiple Identical Imaginary Roots.

The dynamics we treat here is a degenerate class of linear time-invariant time-delayed systems (LTI-TDS) with several rationally independent delays, which exhibit multiple and identical imaginary characteristic roots (MIIR) for some particular values of the delays. The stability behavior of the system within the immediate proximity of this parametric setting which creates the degenerate dynamics has recently attracted a number of investigations. We bring several new perspectives to this problem. First is a limiting study in the delay space that avoids the MIIR but extracts the local stability characteristics. Second, as the main contribution of the paper, we show how a well-studied stability paradigm, the Cluster Treatment of Characteristic Roots (CTCR) can be used to resolve this mathematically degenerate problem. In addition we present several validating case studies, especially on a simpler class of systems with a single but commensurate delay.