Characterizing the Codimension of Zero Singularities for Time-Delay Systems

Abstract

The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an algebraic multiplicity two and a geometric multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author's knowledge, the bounds of such a multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters' algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed.

Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system

“…Assertion i) follows directly from Pólya-Szegö Theorem presented in Appendix, see also [8,9]. Indeed, setting α = β = 0 gives a bound of the number of real roots for (10), which is nothing but the degree of the quasipolynomial.…”

“…Furthermore, if such a configuration holds then s 0 is the rightmost root corresponding to (8) (8) is asymptotically stable.…”

“…Recent works by the authors [7,8,9] characterized multiple Crossing Imaginary Roots (CIR) for time-delay systems using a Birkhoff/Vandermonde-based approach. In [8] it is shown that the admissible multiplicity of the zero spectral value is bounded by the generic Polya and Szegö bound denoted PS B , which is nothing but the degree of the corresponding quasipolynomial, see for instance [10].…”

“…In [8] it is shown that the admissible multiplicity of the zero spectral value is bounded by the generic Polya and Szegö bound denoted PS B , which is nothing but the degree of the corresponding quasipolynomial, see for instance [10]. In [7] it is shown that a given CIR with non vanishing frequency never reaches PS B and a sharper bound for its admissible multiplicities is established.…”

“…The present work is a natural continuation of [7,8,13,14], it aims first to improve the understanding of the correspondence between the multiple spectral value variety and the stable variety associated with the steady state solution, and secondly, to demonstrate the applicability of such a property to a mechanical engineering problem.…”