On the relations between stability optimization of linear time-delay systems and multiple rightmost characteristic roots

Several recent results on spectrum-based analysis and control of linear time-invariant time-delay system concern the characterization and exploitation of situations where the so-called multiplicity-induced dominancy property holds, that is, the higher multiplicity of a characteristic roots implies that it is a rightmost root. This direction of research is inspired by observed multiple roots after minimizing the spectral abscissa as a function of controller parameters. However, unlike the relation between multiple roots and rightmost roots, barely theoretical results about the relation of the former with minimizers of the spectral abscissa are available. Consequently, in the first part of the paper the characterization of rightmost roots in such minimizers is briefly revisited for all second-order systems with input delay, controlled with state feedback. As the main theoretical results, the governing multiple root configurations are proved to correspond not only to rightmost roots, but also to global minimizers of the spectrum abscissa function. The proofs rely on perturbation theory of nonlinear eigenvalue problems and exploit the quasi-convexity of the spectral abscissa function. In the second part, a computational characterization of minima of the spectral abscissa is made for output feedback, yielding a more complex picture, which includes configurations with both multiple and simple rightmost roots. In the analysis, the pivotal role of the invariant zeros is highlighted, which translate into restrictions on the tunable parameters in the closed-loop quasi-polynomial.