Nonlinear Dynamics

, Volume 93, Issue 4, pp 1809–1821 | Cite as

Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay

  • Ardashir Mohammadzadeh
  • Sehraneh GhaemiEmail author
Original Paper


This paper presents a new technique using a recurrent non-singleton type-2 sequential fuzzy neural network (RNT2SFNN) for synchronization of the fractional-order chaotic systems with time-varying delay and uncertain dynamics. The consequent parameters of the proposed RNT2SFNN are learned based on the Lyapunov–Krasovskii stability analysis. The proposed control method is used to synchronize two non-identical and identical fractional-order chaotic systems, with time-varying delay. Also, to demonstrate the performance of the proposed control method, in the other practical applications, the proposed controller is applied to synchronize the master–slave bilateral teleoperation problem with time-varying delay. Simulation results show that the proposed control scenario results in good performance in the presence of external disturbance, unknown functions in the dynamics of the system and also time-varying delay in the control signal and the dynamics of system. Finally, the effectiveness of proposed RNT2SFNN is verified by a nonlinear identification problem and its performance is compared with other well-known neural networks.


Time-varying delay Fractional-order chaotic systems Robust stability analysis Lyapunov–Krasovskii 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Lee, S., Wong, S.: Group-based approach to predictive delay model based on incremental queue accumulations for adaptive traffic control systems. Transp. Res. Part B Methodol. 98, 1–20 (2017)CrossRefGoogle Scholar
  2. 2.
    Banks, H.T., Banks, J.E., Bommarco, R., Laubmeier, A., Myers, N., Rundlöf, M., Tillman, K.: Modeling bumble bee population dynamics with delay differential equations. Ecol. Model. 351, 14–23 (2017)CrossRefGoogle Scholar
  3. 3.
    Balas, M.J., Frost, S.A.: Normal form for linear infinite-dimensional systems in Hilbert space and its role in direct adaptive control of distributed parameter systems. In: AIAA Guidance, Navigation, and Control Conference, p. 1501 (2017)Google Scholar
  4. 4.
    Zhou, B., Egorov, A.V.: Razumikhin and Krasovskii stability theorems for time-varying time-delay systems. Automatica 71, 281–291 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Medvedeva, I.V., Zhabko, A.P.: Synthesis of razumikhin and Lyapunov–Krasovskii approaches to stability analysis of time-delay systems. Automatica 51, 372–377 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sanz, R., García, P., Zhong, Q.-C., Albertos, P.: Predictor-based control of a class of time-delay systems and its application to quadrotors. IEEE Trans. Ind. Electron. 64(1), 459–469 (2017)CrossRefGoogle Scholar
  7. 7.
    Hamamci, S.E.: An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers. IEEE Trans. Autom. Control 52(10), 1964–1969 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lazarević, M.P., Spasić, A.M.: Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Math. Comput. Model. 49(3), 475–481 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhang, X.: Some results of linear fractional order time-delay system. Appl. Math. Comput. 197(1), 407–411 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, Y., Moore, K.L.: Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dyn. 29(1), 191–200 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Deng, W., Li, C., Lü, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48(4), 409–416 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gao, Z.: A computing method on stability intervals of time-delay for fractional-order retarded systems with commensurate time-delays. Automatica 50(6), 1611–1616 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, F., Li, X., Liu, X., Tang, Y.: Parameter identification of fractional-order chaotic system with time delay via multi-selection differential evolution. Syst. Sci. Control Eng. 5(1), 42–48 (2017)CrossRefGoogle Scholar
  14. 14.
    Stamov, G., Stamova, I.: Impulsive fractional-order neural networks with time-varying delays: almost periodic solutions. Neural Comput. Appl. 28(11), 3307–3316 (2017)CrossRefGoogle Scholar
  15. 15.
    Song, X., Song, S., Li, B., Tejado Balsera, I.: Adaptive projective synchronization for time-delayed fractional-order neural networks with uncertain parameters and its application in secure communications. Trans. Inst. Meas. Control 0142331217714523 (2017)Google Scholar
  16. 16.
    Hu, W., Ding, D., Wang, N.: Nonlinear dynamic analysis of a simplest fractional-order delayed memristive chaotic system. J. Comput. Nonlinear Dyn. 12(4), 041003 (2017)CrossRefGoogle Scholar
  17. 17.
    Rakkiyappan, R., Udhayakumar, K., Velmurugan, G., Cao, J., Alsaedi, A.: Stability and hopf bifurcation analysis of fractional-order complex-valued neural networks with time delays. Adv. Differ. Equ. 2017(1), 225 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fei-Fei, L., Zhe-Zhao, Z.: Synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive neural network control. Acta Phys. Sin. 66(9) (2017).
  19. 19.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  20. 20.
    Rong, H.-J., Sundararajan, N., Huang, G.-B., Saratchandran, P.: Sequential adaptive fuzzy inference system (SAFIS) for nonlinear system identification and prediction. Fuzzy Sets Syst. 157(9), 1260–1275 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hokayem, P.F., Spong, M.W.: Bilateral teleoperation: an historical survey. Automatica 42(12), 2035–2057 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sadeghi, M.S., Momeni, H., Amirifar, R.: \({H_\infty }\) and \({L_1}\) control of a teleoperation system via LMIs. Appl. Math. Comput. 206(2), 669–677 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yingwei, L., Sundararajan, N., Saratchandran, P.: A sequential learning scheme for function approximation using minimal radial basis function neural networks. Neural Comput. 9(2), 461–478 (1997)CrossRefzbMATHGoogle Scholar
  24. 24.
    Angelov, P.P., Filev, D.P.: An approach to online identification of Takagi–Sugeno fuzzy models. IEEE Trans. Syst. Man Cybern. Part B Cybern. 34(1), 484–498 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Control Engineering Department, Faculty of Electrical and Computer EngineeringUniversity of TabrizTabrizIran

Personalised recommendations