Abstract
Continuous-time dynamical networks with delays have a wide range of application fields from biology, economics to physics, and engineering sciences. Usually, two types of delays can occur in the communication network: internal delays (due to specific internal dynamics of a given node) and external delays (related to the communication process, due to the information transmission and processing). Besides, such delays can be totally different from one node to another. It is worth mentioning that the problem becomes more and more complicated when a huge number of delays has to be taken into account. In the case of constant delays , this analysis relies much on the identification and the understanding of the spectral values behavior with respect to an appropriate set of parameters when crossing the imaginary axis. There are several approaches for identifying the imaginary crossing roots, though, to the best of the authors’ knowledge, the bound of the multiplicity of such roots has not been deeply investigated so far. This chapter provides an answer for this question in the case of time-delay systems , where the corresponding quasi-polynomial function has non-spare polynomials and no coupling delays. Furthermore, we will also show the link between this multiplicity problem and Vandermonde matrices , and give the upper bound for the multiplicity of an eigenvalue at the origin for such a time-delay system modeling network dynamics in the presence of time-delay.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
L.V. Ahlfors, in Complex Analysis (McGraw-Hill Inc., 1979)
F. Atay, The consensus problem in networks with transmission delays. Philos. Trans. Royal Soc. A: Math. Phys. Eng. Sci. 371, 2013 (1999)
R. Bellman. K.L. Cooke, Differential-Difference Equations (Academic Press, 1963)
I. Boussaada, S.-I. Niculescu, Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices, in International Symposium on Mathematical Theory of Networks and Systems (MTNS) (2014)
I. Boussaada, I.-C. Morarescu, S.-I. Niculescu, Inverted pendulum stabilization: characterisation of codimension-three triple zero bifurcation via multiple delayed proportional gains. Syst. Control Lett. 82, 1–9 (2012)
I. Boussaada, D.I. Irofti, S.-I. Niculescu, Computing the codimension of the singularity at the origin for time-delay systems in the regular case: a vandermonde-based approach, in European Control Conference (ECC) (2014)
S. Campbell, Y. Yuan, Zero singularities of codimension two and three in delay differential equations. Nonlinearity 21(11), 2671–2691 (2008)
J. Carr, Application of Center Manifold Theory (Springer, 1981)
D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics (Springer, 2007)
M. Giesbrecht, D.-S. Roche, Interpolation of shifted-lacunary polynomials. Comput. Complex. 19(3), 333–354 (2010)
J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. vol. 42 (Springer, 2002)
J.K. Hale, W. Huang, Period doubling in singularly perturbed delay equations. J. Differ. Equ. 114(1), 1–23 (1994)
J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations. Applied Mathematics Sciences, vol. 99 (Springer, 1993)
Y. Kuznetsov, Elements of Applied Bifurcation Theory. Second edition of Applied Mathematics Sciences, vol. 112 (Springer, 1998)
T. Kailath, Linear Systems. Prentice-Hall information and system sciences series, vol. 1 (Prentice Hall International, 1980)
M. Marden, Geometry of Polynomials. Mathematical Surveys 3 (Providence, AMS, 1966)
M.B.S. Marquez, I. Boussaada, H. Mounier, S.-I. Niculescu, Analysis and Control of Oilwell Drilling Vibrations. Advances in Industrial Control (Springer, 2015)
W. Michiels, S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems. Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), vol. 12 (2007)
S.-I. Niculescu, W. Michiels, Stabilizing a chain of integrators using multiple delays. IEEE Trans. Autom. Control 49(5), 802–807 (2004)
G. Polya, G. Szegö, Problems and Theorems in Analysis. Vol. I: Series, Integral Calculus, Theory of Functions (Springer, 1972)
J. Sieber, B. Krauskopf, Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity. Nonlinearity 17(1), 85–103 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Irofti, DA., Boussaada, I., Niculescu, SI. (2016). On the Codimension of the Singularity at the Origin for Networked Delay Systems. In: Seuret, A., Hetel, L., Daafouz, J., Johansson, K. (eds) Delays and Networked Control Systems . Advances in Delays and Dynamics, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-32372-5_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-32372-5_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32371-8
Online ISBN: 978-3-319-32372-5
eBook Packages: EngineeringEngineering (R0)