Control of Networks of Coupled Dynamical Systems

  • Chai Wah WuEmail author
Part of the Understanding Complex Systems book series (UCS)


We study networks of coupled dynamical systems where an external forcing control signal is applied to the network in order to align the state of all the individual systems to the forcing signal. By considering the control signal as the state of a virtual dynamical system, this problem can be studied as a synchronization problem. The main focus of this chapter is to link the effectiveness of such control to various properties of the underlying graph. For instance, we study the relationship between control effectiveness and the network as a function of the number of nodes in the network. For vertex-balanced graphs, if the number of systems receiving control does not grow as fast as the total number of systems, then the strength of the control needed to effect control will be unbounded as the number of vertices grows. In order to achieve control in systems coupled via locally connected graphs, as the number of systems grows, both the control and the coupling among all systems need to increase. Furthermore, the algebraic connectivity of the graph is an indicator of how easy it is to control the network. We also show that for the cycle graph, the best way to achieve control is by applying control to systems that are approximately equally spaced apart. In addition, we show that when the number of controlled systems is small, it is beneficial to put the control at vertices with large degrees, whereas when the number of controlled systems is large, it is beneficial to put the control at vertices with small degrees. Finally, we give evidence to show that applying control to minimize the distances between all systems to the set of controlled systems could lead to a more effective control.



A portion of this research was sponsored by US Army Research Laboratory and the UK Ministry of Defence and was accomplished under Agreement Number W911NF-06-3-0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the US Army Research Laboratory, the US Government, the UK Ministry of Defence, or the UK Government. The US and UK Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.


  1. 1.
    Lü, J., Yu, X., Chen, G.: Physica A 334, 281 (2004)Google Scholar
  2. 2.
    Wu, C.W.: Synchronization in systems coupled via complex networks. In: Proceedings of the 2004 International Symposium on Circuits and Systems, vol. 4, pp. IV-724-727, 23–26 May 2004Google Scholar
  3. 3.
    Wu, C.W.: On a matrix inequality and its application to the synchronization in coupled chaotic systems. In: Göknar, I.C., Sevgi, L. (eds.) Complex Computing-Networks: Brain-Like and Wave-Oriented Electrodynamic Algorithms. Springer Proceedings in Physics, vol. 104, pp. 279–287 Springer, BerlinGoogle Scholar
  4. 4.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Phys. Rep. 424, 175 (2006)Google Scholar
  5. 5.
    Arenas, A., Díaz-Guilera, D., Pérez-Vicente, C.J.: Physica D 224(1–2), 27 (2006)Google Scholar
  6. 6.
    Nishikawa, T., Motter, A.E., Lai, Y.C., Hoppensteadt, F.C.: Phys. Rev. Lett. 91(1), 014101 (2003)Google Scholar
  7. 7.
    di Bernardo, M., Garofalo, F., Sorrentino, F.: Effects of degree correlation on the synchronizability of networks of nonlinear oscillators. In: Proceedings of 44th IEEE Conference on Decision and Control and 2005 European Control Conference, pp. 4616–4621, 2005Google Scholar
  8. 8.
    Motter, A.E., Zhou, C., Kurths, J.: Phys. Rev. E 71, 016116 (2005)Google Scholar
  9. 9.
    Atay, F.M., Biyikoğlu, T., Jost, J.: IEEE Trans. Circ. Syst. I: Fundam. Theory Appl. 53(1), 92 (2006)Google Scholar
  10. 10.
    Wu, C.W.: IEEE Circ. Syst. Mag. 10, 55 (2010)Google Scholar
  11. 11.
    Wang, X.F., Chen, G.R.: Physica A 310(3–4), 521 (2002)Google Scholar
  12. 12.
    Li, X., Wang, X.F., Chen, G.R.: IEEE Trans. Circ. Syst. I 51(10), 2074 (2004)Google Scholar
  13. 13.
    Chen, T., Liu, X., Lu, W.: IEEE Trans. Circ. Syst. I 54(6), 1317 (2007)Google Scholar
  14. 14.
    Sorrentino, F., di Bernardo, M., Garofalo, F., Chen, G.: Phys. Rev. E 75, 046103 (2007)Google Scholar
  15. 15.
    Xiang, L.Y., Liu, Z.X., Chen, Z.Q., Chen, F., Yuan, Z.Z.: Physica A 379(1), 298 (2007)Google Scholar
  16. 16.
    Wu, C.W.: Localization of effective pinning control in complex networks of dynamical systems. In: Proceedings of IEEE International Symposium on Circuits and Systems, pp. 2530–2533, 18–21 May 2008Google Scholar
  17. 17.
    Wu, C.W.: CHAOS 18, 037103 (2008)Google Scholar
  18. 18.
    Wu, C.W.: On control of networks of dynamical systems. In: Proceedings of 2010 IEEE International Symposium on Circuits and Systems, pp. 3785–3788, 30 May-2 June 2010Google Scholar
  19. 19.
    Barany, E., Schaffer, S., Wedeward, K., Ball, S.: Nonlinear controllability of singularly perturbed models of power flow networks. In: Proceedings of 2004 IEEE Conference on Decision and Control, vol. 5, pp. 4826–4832, 14–17 December 2004Google Scholar
  20. 20.
    Brualdi, R.A., Ryser, H.J.: Combinatorial Matrix Theory. Cambridge University Press, Cambridge (1991)Google Scholar
  21. 21.
    Minc, H.: Nonnegative Matrices. Wiley, New York (1988)Google Scholar
  22. 22.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)Google Scholar
  23. 23.
    Heagy, J.F., Carroll, T.L., Pecora, L.M.: Phys. Rev. E 50(3), 1874 (1994)Google Scholar
  24. 24.
    Wu, C.W., Chua, L.O.: IEEE Trans. Circ. Syst. I: Fundamen. Theory Appl. 42(8), 430 (1995)Google Scholar
  25. 25.
    Pecora, L.M., Carroll, T.L.: In: Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, vol. 4, pp. IV–562–567. IEEE, New York (1998)Google Scholar
  26. 26.
    Wu, C.W.: In: Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, vol. 3, pp. III–302–305. IEEE, New York (1998)Google Scholar
  27. 27.
    Wang, X.F., Chen, G.: Int. J. Bifurcation Chaos 12(1), 187 (2002)Google Scholar
  28. 28.
    Wu, C.W.: Nonlinearity 18, 1057 (2005)Google Scholar
  29. 29.
    Wu, C.W.: Linear Algebra Appl. 402, 29 (2005)Google Scholar
  30. 30.
    Wu, C.W.: Linear Algebra Appl. 402, 207 (2005)Google Scholar
  31. 31.
    Juhász, F.: Discrete Math. 96, 59 (1991)Google Scholar
  32. 32.
    Wu, C.W.: Linear Multilinear Algebra 53(3), 203 (2005)Google Scholar
  33. 33.
    Wu, C.W.: IEEE Trans. Circ. Syst. I: Fundam. Theory Appl. 48(10), 1257 (2001)Google Scholar
  34. 34.
    Lubotzky, A., Phillips, R., Sarnak, P.: Combinatorica 8(3), 261 (1988)Google Scholar
  35. 35.
    Fiedler, M.: Czechoslovak Math. J. 23(98), 298 (1973)Google Scholar
  36. 36.
    Yueh, W.C.: Appl. Math. E-Notes 5, 66 (2005)Google Scholar
  37. 37.
    Willms, A.R.: Siam J. Matrix Anal. Appl. 30(2), 639 (2008)Google Scholar
  38. 38.
    Bapat, R.B., Pati, S.: Linear Multilinear Algebra 45, 247 (1998)Google Scholar
  39. 39.
    Grone, R., Merris, R.: Czechoslovak Math. J. 37(112), 660 (1987)Google Scholar
  40. 40.
    Kaveh, A., Rahami, H.: Asian J. Civil Eng. (Building and Housing) 7(2), 125 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.IBM T. J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations