On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids

Stéphane Chrétien; Sébastien Darses; Christophe Guyeux; Paul Clarkson

doi:10.3934/naco.2017019

Article Contents

2017, Volume 7, Issue 3: 289-299. Doi: 10.3934/naco.2017019

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On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids

1.
National Physical Laboratory, Hampton Road, Teddington, UK
2.
Aix Marseille Univ, CNRS, Centrale Marseille, I2M. Technopôle Château-Gombert, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France
3.
Femto-ST Institute, UMR 6174 CNRS, Université de Bourgogne Franche-Comté, 16 route de Gray 25000, Besançon, France

* Corresponding author: Stephane Chretien
* Corresponding author: Stephane Chretien

The firrst author is supported by National Physical Laboratory

Received: December 31, 2016

Revised: May 31, 2017

Published: October 2017

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Abstract

Pinning control on complex dynamical networks has emerged as a very important topic in recent trends of control theory due to the extensive study of collective coupled behaviors and their role in physics, engineering and biology. In practice, real-world networks consist of a large number of vertices and one may only be able to perform a control on a fraction of them only. Controllability of such systems has been addressed in [17], where it was reformulated as a global asymptotic stability problem. The goal of this short note is to refine the analysis proposed in [17] using recent results in singular value perturbation theory.

Keywords:

Pinned control; eigenvalue perturbation; sum of rank one matrices; singular value perturbation; control theory.

Mathematics Subject Classification: Primary: 93Bxx; Secondary: 15A18.

Citation:

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Figure 1. Evolution of the phase at each bus as a function of time for a random network

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Figure 2. A second example of evolution of the phase at each bus as a function of time for a random network, with same setup

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Figure 3. A last evolution of the phase at each bus, same configuration

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