On τ-Decomposition Frequency-Sweeping Test for a Class of Time-Delay Systems. Part I: Simple Imaginary 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. First, the characteristic quasipolynominal is reformulated into a factorization form. Then, it is found that (i) the crossing direction of imaginary roots (CDIRs) can be reflected by an appropriate frequency-sweeping test, (ii) the CDIRs are determined by a certain order of the imaginary roots by magnitude and finally, (iii) a time-delay system is ultimately unstable if there is at least one crossing imaginary root. To apply the properties (i)-(iii), an easily implemented frequency-sweeping method is proposed, by which the CDIRs can be obtained without any further computation. The method can be used for the complete stability of the system. In Part II of the paper, the MIR case will be discussed.