Skip to main content
SpringerLink
Log in

Number estimation of controllers for pinning a complex dynamical network

  • Published:
Journal of Zhejiang University SCIENCE C Aims and scope Submit manuscript

Abstract

Number estimation of controllers is a fundamental question in pinning synchronization of complex networks. This paper studies the problem of controller number in synchronizing a complex network of coupled dynamical systems by means of pinning. For a complex network with a symmetric coupling matrix and full coupling between the nodes, we formulate network synchronization via pinning as a linear matrix inequality criterion, and provide a lower bound and an upper bound of the controller number for a given complex network with fixed architecture. Several numerical examples with Barabási-Albert network topologies are provided to verify our theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Albert, R., Barabási, A.L., 2002. Statistical mechanics of complex networks. Rev. Modern Phys., 74(1):47–97. [doi:10.1103/RevModPhys.74.47]

    Article  MathSciNet  MATH  Google Scholar 

  • Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C., 2008. Synchronization in complex networks. Phys. Rep., 469(3):95–153. [doi:10.1016/j.physrep.2008.09.002]

    Article  MathSciNet  Google Scholar 

  • Barabási, A.L., Albert, R., 1999. Emergence of scaling in random networks. Science, 286(5439):509–512. [doi:10. 1126/science.286.5439.509]

    Article  MathSciNet  Google Scholar 

  • Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U., 2006. Complex networks: structure and dynamics. Phys. Rep., 424(4–5):178–308. [doi:10.1016/j.physrep.2005.10.009]

    MathSciNet  Google Scholar 

  • Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., 1994. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, USA.

    MATH  Google Scholar 

  • Chen, M.Y., Zhou, D.H., 2006. Synchronization in uncertain complex networks. Chaos, 16(1):013101. [doi:10.1063/1.2126581]

    Article  MathSciNet  Google Scholar 

  • Chen, T., Liu, X., Lu, W., 2007. Pinning complex networks by a single controller. IEEE Trans. Circ. Syst. I, 54(6):1317–1326. [doi:10.1109/TCSI.2007.895383]

    Article  MathSciNet  Google Scholar 

  • Duan, Z., Wang, J., Chen, G., Huang, L., 2008. Stability analysis and decentralized control of a class of complex dynamical networks. Automatica, 44(4):1028–1035. [doi:10.1016/j.automatica.2007.08.005]

    MathSciNet  Google Scholar 

  • Grigoriev, R.O., Cross, M.C., Schuster, H.G., 1997. Pinning control of spatiotemporal chaos. Phys. Rev. Lett., 79(15): 2795–2798. [doi:10.1103/PhysRevLett.79.2795]

    Article  Google Scholar 

  • Hu, G., Yang, J., Liu, W., 1998. Instability and controllability of linearly coupled oscillators: eigenvalue analysis. Phys. Rev. E, 58(4):4440–4453. [doi:10.1103/PhysRevE.58.4440]

    Article  Google Scholar 

  • Kurths, J., Maraun, D., Zhou, C.S., Zamora-Lopez, G., Zou, Y., 2009. S dynamics in complex systems. Eur. Rev., 17(2): 357–370. [doi:10.1017/S1062798709000726]

    Article  Google Scholar 

  • Li, D., Lu, J., Wu, X., Chen, G., 2006. Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system. J. Math. Anal. Appl., 323(2): 844–853. [doi:10.1016/j.jmaa.2005.11.008]

    Article  MathSciNet  MATH  Google Scholar 

  • Li, X., Wang, X., Chen, G., 2004. Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circ. Syst. I, 51(10):2074–2087. [doi:10.1109/TCSI.2004.835655]

    Article  MathSciNet  Google Scholar 

  • Li, Z., Chen, G., 2004. Robust adaptive synchronization of uncertain dynamical networks. Phys. Lett. A, 324(2–3): 166–178. [doi:10.1016/j.physleta.2004.02.058]

    Article  MathSciNet  MATH  Google Scholar 

  • Lü, J., Chen, G., 2002. A new chaotic attractor coined. Int. J. Bifurc. Chaos, 12(3):659–661. [doi:10.1142/S0218127402004620]

    Article  MATH  Google Scholar 

  • Lü, J., Chen, G., 2005. A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans. Automat. Control, 50(6):841–846. [doi:10.1109/TAC.2005.849233]

    Article  MathSciNet  Google Scholar 

  • Newman, M.E.J., 2003. The structure and function of complex networks. SIAM Rev., 45(2):167–256. [doi:10.1137/S003614450342480]

    Article  MathSciNet  MATH  Google Scholar 

  • Parekh, N., Parthasarathy, S., Sinha, S., 1998. S global and local control of spatiotemporal chaos in coupled map lattices. Phys. Rev. Lett., 81(7):1401–1404. [doi:10.1103/PhysRevLett.81.1401]

    Article  Google Scholar 

  • Pecora, L.M., Carroll, T.L., 1998. Master stability functions for synchronized coupled systems. Phys. Rev. Lett., 80(10): 2109–2112. [doi:10.1103/PhysRevLett.80.2109]

    Article  Google Scholar 

  • Pikovsky, A., Rosenblum, M., Kurths, J., 2001. Synchronization, a Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge, UK.

    Book  MATH  Google Scholar 

  • Sorrentino, F., di Benardo, M., Garofalo, F., Chen, G., 2007. Controllability of complex networks via pinning. Phys. Rev. E, 75(4):046103. [doi:10.1103/PhysRevE.75.046103]

    Article  Google Scholar 

  • Wang, L., Dai, H.P., Dong, H., Cao, Y.Y., Sun, Y.X., 2008a. Adaptive synchronization of weighted complex dynamical networks through pinning. Eur. Phys. J. B, 61(3):335–342. [doi:10.1140/epjb/e2008-00081-5]

    Article  Google Scholar 

  • Wang, L., Kong, X.J., Shi, H., Dai, H.P., Sun, Y.X., 2008b. LMI-based criteria for synchronization of complex dynamical networks. J. Phys. A: Math. Theor., 41(28): 285102. [doi:10.1088/1751-8113/41/28/285102]

    Article  MathSciNet  Google Scholar 

  • Wang, L., Dai, H.P., Kong, X.J., Sun, Y.X., 2009. Synchronization of uncertain complex dynamical networks via adaptive control. Int. J. Robust Nonlinear Control, 19(5):495–511. [doi:10.1002/rnc.1326]

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, X., Chen, G., 2002. Pinning control of scale-free dynamical networks. Phys. A, 310(3–4):521–531. [doi:10.1016/S0378-4371(02)00772-0]

    MathSciNet  MATH  Google Scholar 

  • Wu, C.W., 2008. On the relationship between pinning control effectiveness and graph topology in complex networks of dynamical systems. Chaos, 18(3):037103. [doi:10.1063/1.2944235]

    Article  MathSciNet  Google Scholar 

  • Wu, C.W., Chua, L.O., 1995. Synchronization in an array of linearly coupled dynamical systems. IEEE Trans. Circ. Syst. I, 42(8):430–447. [doi:10.1109/81.404047]

    Article  MathSciNet  MATH  Google Scholar 

  • Xiang, J., Chen, G., 2007. On the V-stability of complex dynamical networks. Automatica, 43(6):1049–1057. [doi:10.1016/j.automatica.2006.11.014]

    Article  MathSciNet  MATH  Google Scholar 

  • Xiang, L.Y., Liu, Z.X., Chen, Z.Q., Chen, F., Yuan, Z.Z., 2007. Pinning control of complex dynamical networks with general topology. Phys. A, 379(1):298–306. [doi:10.1016/j.physa.2006.12.037]

    Article  Google Scholar 

  • Zhou, J., Lu, J., Lü, J., 2006. Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans. Automat. Control, 51(4):652–656. [doi:10.1109/TAC.2006.872760]

    Article  MathSciNet  Google Scholar 

  • Zhou, J., Lu, J., Lü, J., 2008. Pinning adaptive synchronization of a general complex dynamical network. Automatica, 44(4):996–1003. [doi:10.1016/j.automatica.2007.08.016]

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Systems Science and LMIB, Beihang University, Beijing, 100191, China

    Lei Wang

  2. State Key Lab of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou, 310027, China

    Huan Shi & You-xian Sun

Authors
  1. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huan Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. You-xian Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Wang.

Additional information

Project supported by the National Natural Science Foundation of China (No. 61004106) and the Fundamental Research Funds for the Central Universities, China

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, L., Shi, H. & Sun, Yx. Number estimation of controllers for pinning a complex dynamical network. J. Zhejiang Univ. - Sci. C 12, 470–477 (2011). https://doi.org/10.1631/jzus.C1010247

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1631/jzus.C1010247

Key words

CLC number

Search

Navigation