Skip to main content
SpringerLink
Log in

Exponential synchronization for second-order nonlinear systems in complex dynamical networks with time-varying inner coupling via distributed event-triggered transmission strategy

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

Abstract

This paper focuses on the exponential synchronization problem of complex dynamical networks (CDNs) with time-varying inner coupling via distributed event-triggered transmission strategy. Information update is driven by properly defined event, which depends on the measurement error. Each node is described as a second-order nonlinear dynamic system and only exchanges information with its neighboring nodes over a directed network. Suppose that the network communication topology contains a directed spanning tree. A sufficient condition for achieving exponential synchronization of second-order nonlinear systems in CDNs with time-varying inner coupling is derived. Detailed theoretical analysis on exponential synchronization is performed by the virtues of algebraic graph theory, distributed event-triggered transmission strategy, matrix inequality and the special Lyapunov stability analysis method. Moreover, the Zeno behavior is excluded as well by the strictly positive sampling intervals based on the upper right-hand Dini derivative. It is noted that the amount of communication among network nodes and network congestion have been significantly reduced so as to avoid the waste of network resources. Finally, a simulation example is given to show the effectiveness of the proposed exponential synchronization criteria.

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
Fig. 7

Similar content being viewed by others

References

  1. Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440–442 (1998)

    Article  MATH  Google Scholar 

  2. Huberman, B.A., Adamic, L.A.: Internet: growth dynamics of the world-wide-web. Nature 401, 131–132 (1999)

    Article  Google Scholar 

  3. Yang, X.S., Cao, J.D., Lu, J.Q.: Synchronization of delayed complex dynamical networks with impulsive and stochastic effects. Nonlinear Anal. Real World Appl. 12, 2252–2266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barabási, A.L., Jeong, H., Néda, Z., Ravasz, E., Schubert, A., Vicsek, T.: Evolution of the social network of scientific collaborations. Physica A 311, 590–614 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49, 1520–1533 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ren, W., Beard, R.W.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50, 655–661 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, Z.L., Duan, Z.S., Chen, G.R., Huang, L.: Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans. Circuits Syst.I: Regul. Pap. 57, 213–224 (2010)

    Article  MathSciNet  Google Scholar 

  8. Guo, W.L.: Lag synchronization of complex networks via pinning control. Nonlinear Anal. Real World Appl. 12, 2579–2585 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, S.H., Du, H.B., Lin, X.Z.: Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica 47, 1706–1722 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, H.Q., Liao, X.F., Huang, T.W.: Second-order locally dynamical consensus of multiagent systems with arbitrarily fast switching directed topologies. IEEE Trans. Syst. Man. Cybern. 43, 1343–1353 (2013)

    Article  Google Scholar 

  11. Sun, Y.Z., Li, W., Ruan, J.: Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation. Commun. Nonlinear Sci. Numer. Simul. 18, 989–998 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yang, X.S., Cao, J.D., Lu, J.Q.: Hybrid adaptive and impulsive synchronization of uncertain complex networks with delays and general uncertain perturbations. Appl. Math. Comput. 227, 480–493 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Tang, Y., Qian, F., Gao, H.J., Kurths, J.: Synchronization in complex networks and its application-A survey of recent advances and challenges. Annu. Rev. Control 38, 184–198 (2014)

    Article  Google Scholar 

  14. Du, H.Y., Shi, P., Lü, N.: Function projective synchronization in complex dynamical networks with time delay via hybrid feedback control. Nonlinear Anal. Real World Appl. 14, 1182–1190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, C.J., Yu, W.W., Huang, T.W.: Impulsive synchronization schemes of stochastic complex networks with switching topology: average time approach. Neural Netw. 54, 85–94 (2014)

    Article  MATH  Google Scholar 

  16. Chen, W.S., Li, X.B., Ren, W., Wen, C.Y.: Adaptive consensus of multi-agent systems with unknown identical control directions based on a novel nussbaum-type function. IEEE Trans. Autom. Control 59, 1887–1892 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, Z., Chen, G.R.: Global synchronization and asymptotic stability of complex dynamical networks. IEEE Trans. IEEE Trans. Circuits Syst. II: Exp. Syst. II: Exp. Briefs 53, 28–33 (2006)

    Google Scholar 

  18. Gao, H.J., Lam, J., Chen, G.R.: New criteria for synchronization stability of general complex dynamical networks with coupling delays. Phys. Lett. A 360, 263–273 (2006)

    Article  MATH  Google Scholar 

  19. Yu, W.W., Chen, G.R., Lü, J.H.: On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tang, Y., Fang, J.A., Xia, M., Gu, X.J.: Synchronization of Takagi–Sugeno fuzzy stochastic discrete-time complex networks with mixed time-varying delays. Appl. Math. Model. 34, 843–855 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mei, J., Jiang, M.H., Xu, W.M., Wang, B.: Finite-time synchronization control of complex dynamical networks with time delay. Commun. Nonlinear Sci. Numer. Simul. 18, 2462–2478 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jin, X.Z., Yang, G.H.: Robust synchronization control for complex networks with disturbed sampling couplings. Commun. Nonlinear Sci. Numer. Simul. 19, 1985–1995 (2014)

    Article  MathSciNet  Google Scholar 

  23. Liu, X.W., Li, P., Chen, T.P.: Cluster synchronization for delayed complex networks via periodically intermittent pinning control. Neurocomputing 162, 191–200 (2015)

    Article  Google Scholar 

  24. Kaviarasan, B., Sakthivel, R., Lim, Y.: Synchronization of complex dynamical networks with uncertain inner coupling and successive delays based on passivity theory. Neurocomputing 186, 127–138 (2016)

    Article  Google Scholar 

  25. Feng, J.W., Yu, F.F., Zhao, Y.: Exponential synchronization of nonlinearly coupled complex networks with hybrid time-varying delays via impulsive control. Nonlinear Dyn. 85, 621–632 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Das, A., Lewis, F.L.: Cooperative adaptive control for synchronization of second-order systems with unknown nonlinearities. Int. J. Robust Nonlinear 21, 1509–1524 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Du, H.B., He, Y.G., Cheng, Y.Y.: Finite-time synchronization of a class of second-order nonlinear multi-agent systems using output feedback control. IEEE Trans. Circuits Syst. I: Regul. Pap. 61, 1778–1788 (2014)

    Article  Google Scholar 

  28. Liu, B., Li, S., Tang, X.H.: Adaptive second-order synchronization of two heterogeneous nonlinear coupled networks. Math. Probl. Eng. ID 578539, 1–7 (2015)

    Google Scholar 

  29. Yu, W.W., Chen, G.R., Cao, M., Kurths, J.: Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans. Syst. Man. Cybern. Part B Cybern. 40, 881–891 (2010)

    Article  Google Scholar 

  30. Qian, Y.F., Wu, X.Q., Lü, J.H., Lu, J.A.: Second-order consensus of multi-agent systems with nonlinear dynamics via impulsive control. Neurocomputing 125, 142–147 (2014)

    Article  Google Scholar 

  31. Hu, H.X., Liu, A.D., Xuan, Q., Yu, L., Xie, G.M.: Second-order consensus of multi-agent systems in the cooperationCcompetition network with switching topologies: a time-delayed impulsive control approach. Syst. Control Lett. 62, 1125–1135 (2013)

    Article  MATH  Google Scholar 

  32. Guan, Z.H., Liu, Z.W., Feng, G., Jian, M.: Impulsive consensus algorithms for second-order multi-agent networks with sampled information. Automatica 48, 1397–1404 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, W.X., Chen, Z.Q., Liu, Z.X.: Leader-following formation control for second-order multiagent systems with time-varying delay and nonlinear dynamics. Nonlinear Dyn. 72, 803–812 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, X.X., Liu, X.P.: Further results on consensus of second-order multi-agent systems with exogenous disturbance. IEEE Trans. Circuits Syst. I: Regul. Pap. 60, 3215–3226 (2013)

    Article  MathSciNet  Google Scholar 

  35. Xu, C.J., Zheng, Y., Su, H.S., Chen, M.Z.Q., Zhang, C.: Cluster consensus for second-order mobile multi-agent systems via distributed adaptive pinning control under directed topology. Nonlinear Dyn. 83, 1975–1985 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Han, G.S., Guan, Z.H., Li, J., He, D.X., Zheng, D.F.: Multi-tracking of second-order multi-agent systems using impulsive control. Nonlinear Dyn. 84, 1771–1781 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yu, M., Shang, W.P., Chen, Z.Q.: Exponential synchronization for second-order nodes in complex dynamical network with communication time delays and switching topologies. J. Control Sci. Eng. 7836316, 1–10 (2017)

    MathSciNet  Google Scholar 

  38. Xie, D.S., Xu, S.Y., Li, Z., Zou, Y.: Event-triggered consensus control for second-order multi-agent systems. IET Control Theory Appl. 9, 667–680 (2015)

    Article  MathSciNet  Google Scholar 

  39. Li, H.Q., Liao, X.F., Huang, T.W., Zhu, W.: Event-triggering sampling based leader-following consensus in second-order multi-agent systems. IEEE Trans. Autom. Control 60, 1998–2003 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mu, N.K., Liao, X.F., Huang, T.W.: Consensus of second-order multi-agent systems with random sampling via event-triggered control. J. Frankl. Inst. 353, 1423–1435 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xie, T.T., Liao, X.F., Li, H.Q.: Leader-following consensus in second-order multi-agent systems with input time delay: an event-triggered sampling approach. Neurocomputing 177, 130–135 (2016)

    Article  Google Scholar 

  42. Li, H.Q., Chen, G., Dong, Z.Y., Xia, D.W.: Consensus analysis of multiagent systems with second-order nonlinear dynamics and general directed topology: an event-triggered scheme. Inform. Sci. 370–371, 598–622 (2016)

    Article  Google Scholar 

  43. Dai, H., Chen, W.S., Jia, J.P., Liu, J.Y., Zhang, Z.Q.: Exponential synchronization of complex dynamical networks with time-varying inner coupling via event-triggered communication. Neurocomputing 245, 124–132 (2017)

    Article  Google Scholar 

  44. Seyboth, G.S., Dimarogonas, D.V., Johansson, K.H.: Event-based broadcasting for multi-agent average consensus. Automatica 49, 245–252 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wu, X.J., Lu, H.T.: Generalized projective synchronization between two different general complex dynamical networks with delayed coupling. Phys. Lett. A 374, 3932–3941 (2010)

    Article  MATH  Google Scholar 

  46. Tuna, S.E.: Sufficient conditions on observability grammian for synchronization in arrays of coupled linear time-varying systems. IEEE Trans. Autom. Control 55, 2586–2590 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA (1994)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Numbers 61503292, 61673308 and 61502373)and Fundamental Research Funds for the Central Universities(Grant Numbers JB181305 and XJS15030).

Author information

Authors and Affiliations

  1. School of Aerospace Science and Technology, Xidian University, Xi’an, 710071, People’s Republic of China

    Hao Dai & Weisheng Chen

  2. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, People’s Republic of China

    Jin Xie & Jinping Jia

Authors
  1. Hao Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weisheng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jin Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jinping Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weisheng Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, H., Chen, W., Xie, J. et al. Exponential synchronization for second-order nonlinear systems in complex dynamical networks with time-varying inner coupling via distributed event-triggered transmission strategy. Nonlinear Dyn 92, 853–867 (2018). https://doi.org/10.1007/s11071-018-4096-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4096-2

Keywords

Search

Navigation