Differentiability of imaginary spectra and determination of its bounds for multiple-delay LTI systems

Abstract

Several new procedures are presented to assess the stability of linear time invariant (LTI) systems with multiple time delays. The skeleton of the methods is on a paradigm named the cluster treatment of characteristic roots (CTCR). For such systems, all the imaginary characteristic roots can be determined completely on a finite set of hypersurfaces, called kernel hypersurfaces (KH), in the delay domain. The entire KH is the mere prerequisite to implement CTCR. However, as the number of delays increases, it becomes computationally prohibitive to calculate the KH in the entire delay domain. Alternatively, we explore the 2-D cross-section of these KH in the space of any two of the delays. First we investigate the bounds of the imaginary spectra as a novelty here. For this, the proof of the differentiability of the crossing-frequency variations is provided. Another novelty of the paper appears at the declaration of the KH. For this we utilize the 3-D “Building Block” and “Spectral Delay Space” concepts. The effectiveness of these procedures is shown over an example case study.