An Eigenvalue Perturbation Approach to Stability Analysis, Part II: When Will Zeros of Time-Delay Systems Cross Imaginary Axis?

This paper presents an application of the eigenvalue series developed in Part I [J. Chen et al., SIAM J. Control Optim., 48 (2010), pp. 5564-5582] to the study of linear time-invariant delay systems, focusing on the asymptotic behavior of critical characteristic zeros on the imaginary axis. We consider systems given in state-space form and as quasi-polynomials, and we develop an eigenvalue perturbation analysis approach which appears to be both conceptually appealing and computationally efficient. Our results reveal that the zero asymptotic behavior of time-delay systems can in general be characterized by solving a simple eigenvalue problem, and, additionally, when described by a quasi-polynomial, by computing the derivatives of the quasipolynomial.