Skip to main content
SpringerLink
Log in

Pinning cluster synchronization of delay-coupled Lur’e dynamical networks in a convex domain

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper devotes to investigating the cluster synchronization of coupled complex dynamical networks consisted of Lur’e systems with time-varying delay. By considering the topology structure of the complex network, an effective pinning controller is designed which not only synchronize all Lur’e systems in the same cluster, but also decrease the influence among different clusters. Firstly, based on delay interval dividing methods, expended Jensen’s inequality and Jacobian methods, sufficient conditions for local cluster synchronization of the Lur’e dynamical networks are derived by applying the specially designed pinning controller. Secondly, the convex combination theorem, S-procedure and the definition of delay rate are jointly applied in order to obtain the delay-dependent stability criteria which guarantee the global cluster synchronization of the Lur’e networks under the pinning control strategy. And finally, some numerical simulations are given to illustrate the validity of the control scheme and the theoretical analysis.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Kaluza, P., Kölzsch, A., Gastner, M.T., Blasius, B.: The complex network of global cargo ship movements. J. R. Soc. Interface 7(48), 1093–1103 (2010)

    Article  Google Scholar 

  2. Sporns, O.: The human connectome: a complex network. Ann. N. Y. Acad. Sci. 1224, 109–125 (2011)

    Article  Google Scholar 

  3. Zhang, J., Small, M.: Complex network from pseudoperiodic time series: topology versus dynamics. Phys. Rev. Lett. 96(23), 238701:1–238701:4 (2006)

  4. Boccaletti, S., Ivanchenko, M., Latora, V., Pluchino, A., Rapisarda, A.: Detecting complex network modularity by dynamical clustering. Phys. Rev. E 75(4), 045102:1–045102:4 (2007)

  5. Song, X.M., Yan, X.H.: Duality of linear estimation for multiplicative noise systems with measurement delay. IET Signal Process. 7(4), 277–284 (2013)

    Article  MathSciNet  Google Scholar 

  6. Song, X.M., Yan, X.H.: Linear quadratic Gaussian control for linear time-delay systems. IET Control Theory Appl. 8(6), 375–383 (2014)

    Article  MathSciNet  Google Scholar 

  7. Li, X., Chen, G.R.: Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50(11), 1381–1390 (2003)

  8. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.S.: Synchronization in complex dynamical networks. Phys. Rep. 469(3), 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  9. Wang, X., Fang, J.-A., Mao, H.Y., Dai, A.D.: Finite-time global synchronization for a class of Markovian jump complex networks with partially unknown transition rates under feedback control. Nonlinear Dyn. 79(1), 47–61 (2015)

    Article  MATH  Google Scholar 

  10. Wang, F., Yang, Y.Q., Hu, A.H., Xu, X.Y.: Exponential synchronization of fractional-order complex networks via pinning impulsive control. Nonlinear Dyn. 82(4), 1979–1987 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cai, G.L., Jiang, S.Q., Cai, S.M., Tian, L.X.: Cluster synchronization of overlapping uncertain complex networks with time-varying impulse disturbances. Nonlinear Dyn. 80(1), 503–513 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bouzeriba, A., Boulkroune, A., Bouden, T.: Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control. Neural Comput. Appl. 27(5), 1349–1360 (2016)

    Article  MATH  Google Scholar 

  13. Boulkroune, A., Chekireb, H., Tadjine, M., Bouatmane, S.: Observer-based adaptive feedback controller of a class of chaotic systems. Int. J. Bifurc. Chaos 16(11), 3411–3419 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Boulkroune, A., Chekireb, M., Tadjine, M., Bouatmane, S.: An adaptive feedback controller with observer for linearizable chaotic systems. Control Intell. Syst. 35(2), 162–168 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Huang, C., Ho, D.W.C., Lu, J., Kurths, J.: Pinning synchronization in T-S fuzzy complex networks with partial and discrete-time couplings. IEEE Trans. Fuzzy Syst. 23(4), 1274–1285 (2015)

    Article  Google Scholar 

  16. Gonzaga, C.A.C., Jungers, M., Daafouz, J.: Stability analysis of discrete-time Lur’e systems. Nonlinear Dyn. 82(4), 1979–1987 (2015)

    Article  MATH  Google Scholar 

  17. Khalil, H.K.: Nonlinear Dynamics, 3rd edn. Prentice Hall, New Jersey, NJ (2002). ISBN 0-13-067389-7

    Google Scholar 

  18. Park, P.G.: A revisited Popov criterion for nonlinear Lur’e systems with sector-restrictions. Int. J. Control 68(3), 461–470 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jayawardhana, B., Logemann, H., Ryan, E.P.: The circle criterion and input-to state stability. IEEE Control Syst. 31(4), 32–67 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Song, Q., Liu, F., Cao, J.D., Lu, J.Q.: Some simple criteria for pinning a Lur’e network with directed topology. IET Control Theory Appl. 8(2), 131–138 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, Z.G., Shi, P., Su, H.Y., Chu, J.: Sampled-data synchronization of chaotic Lur’e systems with time delays. IEEE Trans. Neural Netw. Learn. Syst. 24(3), 410–421 (2013)

    Article  Google Scholar 

  22. DeLellis, P., Bernardo, M.D., Garofalo, F.: Adaptive pinning control of networks of circuits and systems in Lur’e form. IEEE Trans. Circuits Syst. I Regul. Pap. 60(11), 3033–3042 (2013)

  23. Zhang, F., Trentelman, H.L., Scherpen, J.M.A.: Fully distributed robust synchronization of networked Lur’e systems with incremental nonlinearities. Automatica 50(10), 2515–2526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, Y.L., Cao, J.D.: Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems. Nonlinear Anal. Real World Appl. 14(1), 842–851 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Su, H.S., Rong, Z.H., Chen, M.Z.Q., Wang, X.F., Chen, G.R., Wang, H.W.: Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks. IEEE Trans. Cybern. 43(1), 394–399 (2013)

    Article  Google Scholar 

  26. Feng, J.W., Tang, Z., Zhao, Y., Xu, C.: Cluster synchronization of nonlinearly coupled Lur’e networks with identical and non-identical nodes and an asymmetrical matrix. IET Control Theory Appl 7(18), 2117–2127 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wu, W., Zhou, W.J., Chen, T.P.: Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans. Circuits Syst. I Regul. Pap. 56(4), 829–839 (2009)

  28. Lu, W.L., Chen, T.P., Chen, G.R.: Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay. Phys. D 221(2), 118–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, T., Li, T., Yang, X., Fei, S.M.: Cluster synchronization for delayed Lur’e dynamical networks based on pinning control. Neurocomputing 83, 272–282 (2012)

    Google Scholar 

  30. Liu, X.W., Chen, T.P.: Cluster synchronization in directed networks via intermittent pinning control. IEEE Trans. Neural Netw. 22(7), 1009–1020 (2011)

    Article  Google Scholar 

  31. Hu, A.H., Cao, J.D., Hu, M.F., Guo, L.X.: Cluster synchronization in directed networks of non-identical systems with noises via random pinning control. Phys. A Stat. Mech. Appl. 395, 537–548 (2014)

    Article  MathSciNet  Google Scholar 

  32. Yang, X.S., Cao, J.D.: Synchronization of complex networks with coupling delay via pinning control. IMA J. Math. Control Inf. (2015). doi:10.1093/imamci/dnv065

    Google Scholar 

  33. Jiang, X.F., Han, Q.L., Liu, S.R., Xue, A.K.: A new \({\cal{H}}_\infty \) stabilization criterion for networked control systems. IEEE Trans. Autom. Control 53(4), 1025–1032 (2008)

    Article  MathSciNet  Google Scholar 

  34. Rakkiyappan, R., Sakthivel, N., Cao, J.D.: Stochastic sampled-data control for synchronization of complex dynamical networks with control packet loss and additive time-varying delays. Neural Netw. 66, 46–63 (2015)

    Article  Google Scholar 

  35. Wu, Z.G., Shi, P., Su, H.Y., Chu, J.: Local synchronization of chaotic neural networks with sampled-data and saturating actuators. IEEE Trans. Cybern. 44(12), 2635–2645 (2014)

    Article  Google Scholar 

  36. Derinkuyu, K., Pinar, M.C.: On the S-procedure and some variants. Math. Methods Oper. Res. 64(1), 55–77 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Chen, T.P., Liu, X.W., Lu, W.L.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I Regul. Pap. 54(6), 1317–1326 (2007)

  38. Li, C.G., Chen, G.R.: Synchronization in general complex dynamical networks with coupling delays. Phys. A 343(3), 263–278 (2004)

    Article  MathSciNet  Google Scholar 

  39. Gu, K. Q.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE Conference on Decision and Control, vol. 3, pp. 2805–2810. Sydney, 12–15 Dec 2000

  40. Shao, H.Y.: New delay-dependent stability criteria for systems with interval delay. Automatica 45(3), 744–749 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu, X.W., Chen, T.P.: Synchronization of complex networks via aperiodically intermittent pinning control. IEEE Trans. Autom. Control 60(12), 3316–3321 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ma, J., Qin, H.X., Song, X.L., Chu, R.T.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B 29(01), 1450239 (2015)

Download references

Acknowledgements

The work of J. H. Park and Z. Tang was supported by the BK21 Plus Program (Development of Advanced Smart Mechatronics Systems, 22A20130000136) funded by the Ministry of Education (MOE, Korea) and National Research Foundation of Korea (NRF). And the work of J. Feng is supported by the National Science Foundation of China under Grant No. 61273220.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyonsan, 38541, Republic of Korea

    Ju H. Park & Ze Tang

  2. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, People’s Republic of China

    Jianwen Feng

Authors
  1. Ju H. Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ze Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianwen Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ju H. Park or Ze Tang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Park, J.H., Tang, Z. & Feng, J. Pinning cluster synchronization of delay-coupled Lur’e dynamical networks in a convex domain. Nonlinear Dyn 89, 623–638 (2017). https://doi.org/10.1007/s11071-017-3476-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3476-3

Keywords

Search

Navigation