On τ-Decomposition Frequency-Sweeping Test for a Class of Time-Delay Systems. Part II: Multiple Roots Case

This paper addresses the stability analysis problem of a class of linear system with commensurate delays in a frequency-domain setting. In Part I of this paper, only the simple imaginary roots (SIRs) case is considered. In Part II of the paper, the case of multiple imaginary roots (MIRs) will be studied and general results on the issues of Puiseux series expansion, invariance, and instability persistence will be presented. By the approach proposed in this two-part paper, we can study the complete stability for the time-delay systems under consideration.