ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

Contributed talks on Dynamical Systems and Control

Tue 14:30–15:00, Auditorium Max Weber, Chair: Pieter Lietaert
Further remarks on the effect of multiple spectral values on the dynamics of Time-delay systems: A Birkhoff-based approach
Hakki U. Unal (Centrale Supélec)
Joint work with Islam Boussaada (L2S, CNRS-Centrale), Silviu-Iulian Niculescu (L2S, CNRS-CentraleSupelec-U PSUD), Hakki Unal (L2S, CNRS-CentraleSupelec-U PSUD), Tomas Vyhlidal (CTU in Prague)

A standard approach in analyzing the linear stability of Time-delay dynamical systems consists in characterizing the corresponding spectrum. Furthermore, symmetries in networked dynamical systems often induce multiple spectral values. Such multiple roots are often at the origin of complex behaviors as well as unstable solutions. For instance, it is shown in the literature that the Hopf points near a $1:1$ resonant double Hopf point in a general three-dimensional parameter space form a Whitney umbrella, a phenomenon which cannot occur near the standard Hopf points. Moreover, it is well known the crucial effect of multiplicities (algebraic/geometric) of a given spectral values on the local stability of the steady-state of the corresponding dynamical system. In recent works [1,2], multiple imaginary spectral values for time-delay systems are characterized using an approach based on a class of structured matrices, namely, the class of functional Birkhoff/Vandermonde matrices. In [1], it is shown that the admissible multiplicity of the zero spectral value is generically bounded by the degree of the corresponding quasipolynomial. In [2], in the case of non vanishing frequency, it is shown that such a bound in never reached and a sharper bound for the multiplicity is established. Also an interesting observation is formulated in [2], "if a non zero spectral value of a given scalar time-delay system reaches its maximal (admissible) multiplicity, then, the equilibrium point is locally asymptotically stable". In [3], it is shown that such a surprising property can be extended to a class of planar time-delay dynamical systems. In addition, it is shown that under appropriate assumptions such multiple spectral value is nothing but the rightmost root. Motivated by the potential implication of such a property in control systems application, in this work, we would like to further explore the existing link between the multiplicity and stable manifolds of Time-delay systems. Concret example from mechanical engineering application illustrate this work.

  1. I. Boussaada and S-I. Niculescu. Characterizing the codimension of zero singularities for time-delay systems: A link with Vandermonde and Birkhoff incidence matrices. To appear in: Acta Applicandae Mathematicae, pages 1–46, 2016.
  2. I. Boussaada and S-I. Niculescu. Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A vandermonde-based approach. To appear in: IEEE Transactions on Automatic Control, page 6pp, 2016.
  3. S-I. Niculescu I. Boussaada, H. Unal. Multiplicity and stable varieties of time-delay systems: A missing link. In Submitted to: The 22nd International Symposium on Mathematical Theory of Networks and Systems, pages 1–6, 2016.