September  2017, 7(3): 289-299. doi: 10.3934/naco.2017019

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

The firrst author is supported by National Physical Laboratory

Received  January 2017 Revised  June 2017 Published  July 2017

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.

Citation: Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019
References:
[1]

N. Abaid and M. Porfiri, Consensus over numerosity-constrained random networks, IEEE Transactions on Automatic Control, 56 (2011), 649-654.  doi: 10.1109/TAC.2010.2092270.

[2]

Andries E. Brouwer and Willem H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-1939-6.

[3]

P. N. BrownG. D. Byrne and A. C. Hindmarsh, VODE: A variable coefficient ODE solver, SIAM J. Sci. Stat. Comput., 10 (1989), 1038-1051.  doi: 10.1137/0910062.

[4]

S. Chrétien and S. Darses, Perturbation bounds on the extremal singular values of a matrix after appending a column, preprint, arXiv: 1406.5441, 2014.

[5]

P. DeLellisdi Bernardo and M. Porfiri, Pinning control of complex networks via edge snapping, Chaos: An Interdisciplinary Journal of Nonlinear Science, 21 (2011), 033119.  doi: 10.1063/1.3626024.

[6]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[7]

A. Ghosh and S. Boyd, Growing well connected graphs, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 6605-6611. 

[8]

Aric A. Hagberg, Daniel A. Schult and Pieter J. Swart, Exploring network structure, dynamics, and function using network, in Proceedings of the 7th Python in Science Conference (SciPy2008)(eds. Gaël Varoquaux, Travis Vaught and Jarrod Millman), Pasadena, CA USA, (2008), 11-15.

[9] Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.  doi: 10.1017/CBO9780511810817.
[10]

J. D. Hunter, Matplotlib: A 2D graphics environment, Computing In Science & Engineering, 9 (2007), 90-95. 

[11]

G. P. JiangW. K. S Tang and G. R. Chen, A simple global synchronization criterion for coupled chaotic systems, Chaos, Solitons & Fractals, 15 (2003), 925-935.  doi: 10.1016/S0960-0779(02)00214-X.

[12]

E. Jones, E. Oliphant, P. Peterson, et al. SciPy: Open source scientific tools for Python, http://www.scipy.org/.

[13]

Chi-Kwong Li and Ren-Cang Li, A note on eigenvalues of perturbed Hermitian matrices, Linear Algebra Appl., 395 (2005), 183-190.  doi: 10.1016/j.laa.2004.08.026.

[14]

A. E. Motter, Networkcontrology, Chaos, 25 (2015), 097621.  doi: 10.1063/1.4931570.

[15]

A. E. MotterS. A. MyersM. Anghel and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nature Physics, 9 (2013), 191-197. 

[16]

T. Nishikawa and A.E. Motter, Comparative analysis of existing models for power-grid synchronization, New Journal of Physics, 17 (2015), 015012.  doi: 10.1063/1.4960617.

[17]

Maurizio Porfiri and Mario di Bernardo, Criteria for global pinning-controllability of complex networks, Automatica J. IFAC, 44 (2008), 3100-3106.  doi: 10.1016/j.automatica.2008.05.006.

[18]

L. Pecora and G. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109. 

[19]

M. Porfiri and F. Fiorilli, Node-to-node pinning control of complex networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009), 013122.  doi: 10.1063/1.3080192.

[20]

M. Porfiri and F. Fiorilli, Experiments on node-to-node pinning control of Chua's circuits, Physica D: Nonlinear Phenomena, 239 (2010), 454-464.  doi: 10.1063/1.3080192.

[21]

John W. Simpson-PorcoFlorian Dörfler and Francesco Bullo, Synchronization and power sharing for droop-controlled inverters in islanded microgrids, Automatica, 49 (2013), 2603-2611.  doi: 10.1016/j.automatica.2013.05.018.

[22]

Jean-Jacques E. Slotine and Weiping Li, Applied Nonlinear Control, NJ: Prantice-Hall, Englewood Cliffs, 1991.

[23]

F. Sorrentino, Effects of the network structural properties on its controllability, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 033101. 

[24]

Stéfan van der WaltS. Chris Colbert and Gaël Varoquaux, The NumPy array: A structure for efficient numerical computation, Computing in Science & Engineering, 13 (2011), 22-30.  doi: 10.1109/MCSE.2011.37.

[25]

S. YamaguchiH. IsejimaT. MatsuoR. OkuraK. YagitaM. Kobayashi and H. Okamura, Synchronization of cellular clocks in the suprachiasmatic nucleus, Science, 302 (2003), 1408-1412. 

show all references

References:
[1]

N. Abaid and M. Porfiri, Consensus over numerosity-constrained random networks, IEEE Transactions on Automatic Control, 56 (2011), 649-654.  doi: 10.1109/TAC.2010.2092270.

[2]

Andries E. Brouwer and Willem H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-1939-6.

[3]

P. N. BrownG. D. Byrne and A. C. Hindmarsh, VODE: A variable coefficient ODE solver, SIAM J. Sci. Stat. Comput., 10 (1989), 1038-1051.  doi: 10.1137/0910062.

[4]

S. Chrétien and S. Darses, Perturbation bounds on the extremal singular values of a matrix after appending a column, preprint, arXiv: 1406.5441, 2014.

[5]

P. DeLellisdi Bernardo and M. Porfiri, Pinning control of complex networks via edge snapping, Chaos: An Interdisciplinary Journal of Nonlinear Science, 21 (2011), 033119.  doi: 10.1063/1.3626024.

[6]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[7]

A. Ghosh and S. Boyd, Growing well connected graphs, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 6605-6611. 

[8]

Aric A. Hagberg, Daniel A. Schult and Pieter J. Swart, Exploring network structure, dynamics, and function using network, in Proceedings of the 7th Python in Science Conference (SciPy2008)(eds. Gaël Varoquaux, Travis Vaught and Jarrod Millman), Pasadena, CA USA, (2008), 11-15.

[9] Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.  doi: 10.1017/CBO9780511810817.
[10]

J. D. Hunter, Matplotlib: A 2D graphics environment, Computing In Science & Engineering, 9 (2007), 90-95. 

[11]

G. P. JiangW. K. S Tang and G. R. Chen, A simple global synchronization criterion for coupled chaotic systems, Chaos, Solitons & Fractals, 15 (2003), 925-935.  doi: 10.1016/S0960-0779(02)00214-X.

[12]

E. Jones, E. Oliphant, P. Peterson, et al. SciPy: Open source scientific tools for Python, http://www.scipy.org/.

[13]

Chi-Kwong Li and Ren-Cang Li, A note on eigenvalues of perturbed Hermitian matrices, Linear Algebra Appl., 395 (2005), 183-190.  doi: 10.1016/j.laa.2004.08.026.

[14]

A. E. Motter, Networkcontrology, Chaos, 25 (2015), 097621.  doi: 10.1063/1.4931570.

[15]

A. E. MotterS. A. MyersM. Anghel and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nature Physics, 9 (2013), 191-197. 

[16]

T. Nishikawa and A.E. Motter, Comparative analysis of existing models for power-grid synchronization, New Journal of Physics, 17 (2015), 015012.  doi: 10.1063/1.4960617.

[17]

Maurizio Porfiri and Mario di Bernardo, Criteria for global pinning-controllability of complex networks, Automatica J. IFAC, 44 (2008), 3100-3106.  doi: 10.1016/j.automatica.2008.05.006.

[18]

L. Pecora and G. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109. 

[19]

M. Porfiri and F. Fiorilli, Node-to-node pinning control of complex networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009), 013122.  doi: 10.1063/1.3080192.

[20]

M. Porfiri and F. Fiorilli, Experiments on node-to-node pinning control of Chua's circuits, Physica D: Nonlinear Phenomena, 239 (2010), 454-464.  doi: 10.1063/1.3080192.

[21]

John W. Simpson-PorcoFlorian Dörfler and Francesco Bullo, Synchronization and power sharing for droop-controlled inverters in islanded microgrids, Automatica, 49 (2013), 2603-2611.  doi: 10.1016/j.automatica.2013.05.018.

[22]

Jean-Jacques E. Slotine and Weiping Li, Applied Nonlinear Control, NJ: Prantice-Hall, Englewood Cliffs, 1991.

[23]

F. Sorrentino, Effects of the network structural properties on its controllability, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 033101. 

[24]

Stéfan van der WaltS. Chris Colbert and Gaël Varoquaux, The NumPy array: A structure for efficient numerical computation, Computing in Science & Engineering, 13 (2011), 22-30.  doi: 10.1109/MCSE.2011.37.

[25]

S. YamaguchiH. IsejimaT. MatsuoR. OkuraK. YagitaM. Kobayashi and H. Okamura, Synchronization of cellular clocks in the suprachiasmatic nucleus, Science, 302 (2003), 1408-1412. 

Figure 1.  Evolution of the phase at each bus as a function of time for a random network
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
Figure 3.  A last evolution of the phase at each bus, same configuration
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