Stability Analysis of Quasipolynomials with Complex Coefficients: A Frequency-Sweeping Approach

For real-coefficient quasipolynomials (RCQs), which always appear as the characteristic functions of time-delay systems, the complete stability problem has been fully investi-gated within the frequency-sweeping framework. In this paper, we further study the stability property of complex-coefficient quasipolynomials (CCQs), which are a straightforward extension of RCQs and, hence, correspond to a broader class of system dynamics subject to delays. However, to the best of the authors' knowledge, the complete stability problem for CCQs still remains unsolved. Unlike for an RCQ, the spectrum for a CCQ does not possess the conjugate symmetry property. Therefore, to analyze the stability, we have to (i) exhaustively detect the critical imaginary roots, including the ones with negative imaginary parts, and (ii) systematically study their asymptotic behavior. The current paper provides a complete solution to these technical issues by generalizing the frequency-sweeping framework.