Nonlinear Dynamics

, Volume 95, Issue 3, pp 2031–2062 | Cite as

Global synchronization in fixed time for semi-Markovian switching complex dynamical networks with hybrid couplings and time-varying delays

  • Zhibo Wang
  • Huaiqin WuEmail author
Original Paper


This paper is concerned with the global synchronization in fixed time for semi-Markovian switching complex dynamical networks with hybrid couplings and time-varying delays in the presence of disturbances. Firstly, the property with respect to the global stability in fixed time is developed for semi-Markovian switching nonlinear systems. Subsequently, a novel sliding manifold with double integration is presented based on the proposed principle of convergence in fixed time. Under the designed sliding mode controller, the state trajectory of synchronization error system is driven to the prescribed sliding manifold in fixed time. In addition, the global stability in fixed time of sliding mode dynamics is proved analytically. By means of the stochastic Lyapunov–Krasovskii functional approach, the synchronization condition is established in terms of linear matrix inequalities; moreover, the stochastic fixed settling-time can be determined to any desired values in advance, via the configuration of parameters in the proposed controller. Finally, two numerical examples are provided to demonstrate the validity of the theoretical results and the feasibility of the proposed approach.


Complex dynamical networks Fixed-time synchronization Semi-Markovian switching Sliding mode control Hybrid couplings Mixed time-varying delays 



The authors would like to thank the Editors and the Reviewers for their insightful and constructive comments, which help to enrich the content and improve the presentation of the results in this paper. This work was jointly supported by the Natural Science Foundation of Hebei Province of China (A2018203288), High level talent support project of Hebei province of China (C2015003054) and the Postgraduate Innovation Project of Hebei province of China (CXZZSS2018048).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Strogatz, S.H.: Exploring complex networks. Nature 410(6825), 268 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Barabási, A.L., Albert, R.: Emergence of Scaling in Random Networks. Science 86(5439), 509 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Li, C.P., Sun, W.G., Kurths, J.: Synchronization of complex dynamical networks with time delays. Phys. A Stat. Mech. Appl. 361(1), 24 (2006)CrossRefGoogle Scholar
  5. 5.
    Kim, Y., Choi, T.Y., Yan, T., Dooley, K.: Structural investigation of supply networks: a social network analysis approach. J. Oper. Manag. 29(3), 194 (2011)CrossRefGoogle Scholar
  6. 6.
    Razminia, A., Baleanu, D.: Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics 23(7), 873 (2013)CrossRefGoogle Scholar
  7. 7.
    Wang, J.L., Wu, H.N., Huang, T.: Passivity-based synchronization of a class of complex dynamical networks with time-varying delay. Automatica 56, 105 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bose, B.K.: Expert system, fuzzy logic, and neural network applications in power electronics and motion control. Proc. IEEE 82(8), 1303 (1994)CrossRefGoogle Scholar
  9. 9.
    Zhou, J., Dong, H., Feng, J.: Event-triggered communication for synchronization of Markovian jump delayed complex networks with partially unknown transition rates. Appl. Math. Comput. 293, 617 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Yogambigai, S., Cao, J.: Synchronization of master-slave Markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control. Acta Math. Sci. 37(2), 368 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Xie, Q., Si, G., Zhang, Y., Yuan, Y., Yao, R.: Finite-time synchronization and identification of complex delayed networks with Markovian jumping parameters and stochastic perturbations. Chaos Solitons Fractals 86, 35 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ainsley, C., Fu, L., Ingram, M., Novak, J., Kassaee, A., Both, S.: Exponential synchronization of complex networks with Markovian jump and mixed delays. Phys. Lett. A 372(22), 3986 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zeng, D., Zhang, R., Zhong, S., Wang, J., Shi, K.: Sampled-data synchronization control for Markovian delayed complex dynamical networks via a novel convex optimization method. Neurocomputing 266, 606–618 (2017)CrossRefGoogle Scholar
  14. 14.
    Rakkiyappan, R., Sasirekha, R., Lakshmanan, S., Lim, C.P.: Synchronization of discrete-time Markovian jump complex dynamical networks with random delays via non-fragile control. J. Frankl. Inst. 353(16), 4300 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ozcan, N., Ali, M.S., Yogambigai, J., Zhu, Q., Arik, S.: Robust synchronization of uncertain Markovian jump complex dynamical networks with time-varying delays and reaction-diffusion terms via sampled-data control. J. Frankl. Inst. 335(3), 1192–1216 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ali, M.S., Yogambigai, J.: Finite-time robust stochastic synchronization of uncertain Markovian complex dynamical networks with mixed time-varying delays and reaction-diffusion terms via impulsive control. J. Frankl. Inst. 354, 2415–2436 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, X., Yu, X., Xi, H.: Finite-time synchronization of neutral complex networks with Markovian switching based on pinning controller. Neurocomputing 153, 148 (2015)CrossRefGoogle Scholar
  18. 18.
    Dong, H., Ye, D., Feng, J., Wang, J.: Almost sure cluster synchronization of Markovian switching complex networks with stochastic noise via decentralized adaptive pinning control. Nonlinear Dyn. 87(2), 727 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, J., Shen, H.: Passivity-based fault-tolerant synchronization control of chaotic neural networks against actuator faults using the semi-Markov jump model approach. Neurocomputing 143(16), 51 (2014)CrossRefGoogle Scholar
  20. 20.
    Lee, T.H., Ma, Q., Xu, S., Ju, H.P.: Pinning control for cluster synchronisation of complex dynamical networks with semi-Markovian jump topology. Int. J. Control 88(6), 1223 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ye, D., Yang, X., Su, L.: Fault-tolerant synchronization control for complex dynamical networks with semi-Markov jump topology. Appl. Math. Comput. 312, 36 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Liang, K., Dai, M., Shen, H., Wang, J., Wang, Z., Chen, B., Simos, T.: \(L_2/L_{\infty }\) synchronization for singularly perturbed complex networks with semi-Markov jump topology. Appl. Math. Comput. 321, 450 (2018)MathSciNetGoogle Scholar
  23. 23.
    Sivaranjani, K., Rakkiyappan, R., Joo, Y.H.: Event triggered reliable synchronization of semi-Markovian jumping complex dynamical networks via generalized integral inequalities. J. Frankl. Inst. 355, 3691 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shen, H., Ju, H.P., Wu, Z.G., Zhang, Z.: Finite-time \(H_{\infty }\) synchronization for complex networks with semi-Markov jump topology. Commun. Nonlinear Sci. Numer. Simul. 24, 40–51 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sakthivel, R., Sakthivel, R., Kaviarasan, B., Wang, C., Ma, Y.K.: Finite-time nonfragile synchronization of stochastic complex dynamical networks with semi-Markov switching outer coupling. Complexity (2018)
  26. 26.
    Polyakov, A.: Fixed-time stabilization of linear systems via sliding mode control. In: 12th IEEE Workshop on Variable Structure Systems, pp. 1–6 (2012)Google Scholar
  27. 27.
    Liu, X., Chen, T.: Fixed-time cluster synchronization for complex networks via pinning control (2015). arXiv:1509.03350v1
  28. 28.
    Zhou, Y., Sun, C.: Fixed Time Synchronization of Complex Dynamical Networks. In: Proceedings of the 2015 Chinese Intelligent Automation Conference, pp. 163–170 (2015)Google Scholar
  29. 29.
    Yang, X., Lam, J., Ho, D.W.C., Feng, Z.: Fixed-time synchronization of complex networks with impulsive effects via non-chattering control. IEEE Trans. Autom.Control 62(11), 5511 (2017)CrossRefzbMATHGoogle Scholar
  30. 30.
    Jiang, S., Lu, X., Cai, G., Cai, S.: Adaptive fixed-time control for cluster synchronisation of coupled complex networks with uncertain disturbances. Int. J. Syst. Sci. 48, 1 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Khanzadeh, A., Pourgholi, M.: Fixed-time sliding mode controller design for synchronization of complex dynamical networks. Nonlinear Dyn. 88, 2637–2649 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhang, W., Li, C., Huang, T., Huang, J.: Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations. Phys. A 492, 1531 (2018)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wang, Z., Wu, H.: Projective synchronization in fixed time for complex dynamical networks with nonidentical nodes via second-order sliding mode control strategy. J. Frankl. Inst. 355, 7306–7334 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wei, Y., Ju, H.P., Qin, J., Wu, L., Jung, H.W.: Sliding mode control for semi-Markovian jump systems via output feedback. Automatica 81, 133 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wei, Y., Park, J.H., Karimi, H.R., Tian, Y.C., Jung, H.: Improved stability and stabilization results for stochastic synchronization of continuous-time semi-Markovian jump neural networks with time-varying delay. IEEE Trans Neural Netw. Learn. Syst. 29(6), 2488 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Schwartz, C.: Control of Semi-Markov Jump Linear Systems with Application to the Bunch-Train Cavity Interaction. Northwestern University, Evanston (2003)Google Scholar
  37. 37.
    Foucher, Y., Mathieu, E., Saint-Pierre, P., Durand, J.F., Daurès, J.P.: A semi-Markov model based on generalized Weibull distribution with an illustration for HIV disease. Biomet. J. 47(6), 825 (2005)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Huang, J., Shi, Y.: Stochastic stability of semi-Markov jump linear systems: an LMI approach 413(1), p. 4668 (2012)Google Scholar
  40. 40.
    Liu, X., Yu, X., Zhou, X., Xi, H.: Finite-time \(H_{\infty }\) control for linear systems with semi-Markovian switching. Nonlinear Dyn. 85(4), 2297 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Gu, K., Chen, J., Kharitonov, V.: Stability of Time-Delay Systems. Birkhäuser Boston, Boston (2003)CrossRefzbMATHGoogle Scholar
  42. 42.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, London (1951)zbMATHGoogle Scholar
  43. 43.
    Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequality in Systems and Control Theory. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  44. 44.
    Liu, Y., Wang, Z., Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 19(5), 667 (2006)CrossRefzbMATHGoogle Scholar
  45. 45.
    Wang, Z., Shu, H., Liu, Y., Ho, D.W.C., Liu, X.: Robust stability analysis of generalized neural networks with discrete and distributed time delays. Chaos Solitons Fractals 30(4), 886 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Liu, Y., Wang, Z., Liang, J., Liu, X.: Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays. IEEE Trans. Neural Netw. 20(7), 1102 (2009)CrossRefGoogle Scholar
  47. 47.
    Xu, Y., Lu, R., Shi, P., Tao, J., Xie, S.: Robust estimation for neural networks with randomly occurring distributed delays and markovian jump coupling. IEEE Trans. Neural Netw. Learn. Syst. 29(4), 845 (2018)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Zhang, H., Wang, Z., Liu, D.: A comprehensive review of stability analysis of continuous-time recurrent neural networks. IEEE Trans. Neural Netw. Learn. Syst. 25(7), 1229 (2017)CrossRefGoogle Scholar
  49. 49.
    Zhou, Q., Yao, D., Wang, J., Wu, C.: Robust control of uncertain semi-Markovian jump systems using sliding mode control method. Appl. Math. Comput. 286(C), 72 (2016)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Chen, X., Ju, H.P., Cao, J., Qiu, J.: Sliding mode synchronization of multiple chaotic systems with uncertainties and disturbances. Appl. Math. Comput. 308, 161 (2017)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Chen, W.H., Yang, J., Guo, L., Li, S.: Disturbance-observer-based control and related methods—an overview. IEEE Trans. Ind. Electron. 63(2), 1083 (2016)CrossRefGoogle Scholar
  52. 52.
    Perruquetti, W., Barbot, J.P.: Sliding Mode Control in Engineering. Marcel Dekker, New York (2002)CrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of ScienceYanshan UniversityQinhuangdaoChina

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