The Cluster Treatment of Characteristic Roots and the Neutral Type Time-Delayed Systems

A new methodology is presented for assessing the stability posture of a general class of linear time-invariant—neutral time-delayed systems (LTI-NTDS). It is based on a “Cluster Treatment of Characteristic Roots CTCR” paradigm, which yields a procedure called the Direct Method (DM). The technique offers a number of unique features: It returns exact bounds of time delay for stability, as well as the number of unstable characteristic roots of the system in an explicit and nonsequentially evaluated function of time delay. As a direct consequence of the latter feature, the new methodology creates entirely, all existing stability intervals of delay, $τ$⁠. It is shown that the Direct Method inherently enforces an intriguing necessary condition for $τ$-stabilizability, which is the main contribution of this paper. This, so-called, “small delay” effect, was recognized earlier for NTDS, only through some cumbersome mathematics. Furthermore, the Direct Method is also unique in handling systems with unstable starting posture for $τ=0$⁠, which may be $τ$-stabilized for higher values of delay. Example cases are provided.