Section: New Results

Multiplicity-induced-dominancy

Participants : Islam Boussaada, Guilherme Mazanti, Hugues Mounier [L2S, CentraleSupélec] , Silviu-Iulian Niculescu.

The effects of multiplicity of spectral values on the exponential stability of reduced-order retarded differential equation were emphasized in recent works. In [13] the general class of second-order retarded differential equations is studied. A parametric multiplicity-induced-dominancy (MID) property is characterized, allowing a delayed stabilizing design with reduced complexity. The proposed approach is merely a delayed-output-feedback where the candidates' delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle.

The work [52] considers retarded differential equations of arbitrary order with a single delay. The existence of a real root with maximal multiplicity is characterized in terms of the equation parameters. This root is shown to be always strictly dominant, determining thus the asymptotic behavior of the system. The dominancy proof is based on improved a priori bounds on the imaginary part of roots on the complex right half-plane and a suitable factorization of the characteristic function, which is an alternative technique to the argument principle.

In [53] we extend the MID property to a given pair of complex conjugate roots for a generic second-order retarded differential equation. Necessary and sufficient conditions for the existence of such a pair are provided, and it is also shown that such a pair is always necessarily dominant. It appears also that when the frequency corresponding to this pair of roots tends to 0, then the pair of roots collapse into a real root of maximal multiplicity. The latter property is exploited in the dominancy proof together with a study of crossing imaginary roots.

In [41] a control-oriented model of torsional vibrations occurring in rotary drilling process is proposed. More precisely, a wave equations with weak damping term is considered. An appropriate stabilizing controller with a reduced number of parameters is proposed for damping such torsional vibrations. Such a controller allows to further explore the effect of multiple roots with maximal admissible multiplicity for linear neutral system with a single delay.