A numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systems

Authors:
André Ricardo Fioravanti;Catherine Bonnet;Hitay ÖZbay;Silviu-Iulian Niculescu
Affiliations:
INRIA Saclay - Ilê-de-France, Supélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France;INRIA Saclay - Ilê-de-France, Supélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France;Bilkent University, Department of Electrical and Electronics Engineering, Ankara 06800, Turkey;L2S (UMR CNRS 8506), CNRS-Supélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France
Venue:
Automatica (Journal of IFAC)
Year:
2012

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Abstract

This paper aims to provide a numerical algorithm able to locate all unstable poles, and therefore the characterization of the stability as a function of the delay, for a class of linear fractional-order neutral systems with multiple commensurate delays. We start by giving the asymptotic position of the chains of poles and the conditions for their stability for a small delay. When these conditions are met, the root continuity argument and some simple substitutions allow us to determine the locations where some roots cross the imaginary axis, providing therefore the complete characterization of the stability windows. The same method can be extended to provide the position of all unstable poles as a function of the delay.