Skip to main content
Log in

Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach

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

Abstract

In this paper, an adaptive sliding mode control method is introduced to ensure robust synchronization of two different fractional-order chaotic systems with fully unknown parameters and external disturbances. For this purpose, a fractional integral sliding surface is defined and an adaptive sliding mode controller is designed. In this method, no knowledge of the bounds of parameters and perturbation is required in advance and the parameters are updated through an adaptive control process. The proposed scheme is global and theoretically rigorous. Two examples are given to illustrate effectiveness of the scheme, in which the synchronizations between fractional-order chaotic Chen system and fractional-order chaotic Rössler system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.

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

Access this article

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
Fig. 8

Similar content being viewed by others

References

  1. Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304–311 (1991)

    Article  Google Scholar 

  2. Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional power.IEEE. IEEE Trans. Autom. Control 29, 441–444 (1984)

    Article  MATH  Google Scholar 

  3. Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of non-integer order transfer functions for analysis of electrode processes. J. Electroanal. Chem. Interfacial Electrochem. 33, 253–265 (1971)

    Article  Google Scholar 

  4. Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971)

    Google Scholar 

  5. Hartly, T.T., Lorenzo, C.F., Qamme, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. 42(8), 485–490 (1995)

    Article  Google Scholar 

  6. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003)

    Article  Google Scholar 

  7. Zhang, W., Zhou, S., Li, H., Zhu, H.: Chaos in a fractional-order Rössler system. Chaos Solitons Fractals 42, 1684–1691 (2009)

    Article  MATH  Google Scholar 

  8. Li, C.G., Chen, G.R.: Chaos in the fractional order Chen system and its control. Chaos Solitons Fractals 22, 549–554 (2004)

    Article  MATH  Google Scholar 

  9. Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005)

    Article  Google Scholar 

  10. Chang, C.M., Chen, H.K.: Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems. Nonlinear Dyn. 62, 851–858 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yuan Wang, X., Mei Song, J.: Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Commun. Nonlinear Sci. Numer. Simul. 14, 3351–3357 (2009)

    Article  MATH  Google Scholar 

  12. Wu, X.J., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009)

    Article  MATH  Google Scholar 

  13. Li, C.P., Deng, W.H.: Chaos synchronization of fractional-order differential systems.Int. J. Mod. Phys. B 20, 791–803 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bhalekar, S., Daftardar-Gejji, V.: Synchronization of different fractional order chaotic systems using active control. Commun. Nonlinear Sci. Numer. Simul. 15, 3536–3546 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Taghvafard, H., Erjaee, G.H.: Phase and anti-phase synchronization of fractional order chaotic systems via active control. Commun. Nonlinear Sci. Numer. Simul. 16, 4079–4088 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, R.X., Yang, S.P.: Adaptive synchronization of fractional-order chaotic systems via a single driving variable. Nonlinear Dyn. 66, 831–837 (2011)

    Article  MATH  Google Scholar 

  17. Odibat, Z.M.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn. 60, 479–487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, R.X., Yang, S.P.: Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller. Nonlinear Dyn. 68, 45–51 (2012)

    Article  MATH  Google Scholar 

  19. Chen, D.Y., Liu, Y.X., Ma, X.Y., Zhang, R.F.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 67, 893–901 (2012)

    Article  MATH  Google Scholar 

  20. Lu, J.G.: Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Chaos Solitons Fractals 27, 519–525 (2006)

    Article  MATH  Google Scholar 

  21. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  22. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus. Springer, New York (1997)

    Google Scholar 

  23. Qi, G., Chen, Z., Yuan, Z.: Model-free control of affine chaotic systems. Phys. Lett. A 344, 189–202 (2005)

    Article  MATH  Google Scholar 

  24. Itkis, U.: Control System of Variable Structure. Wiley, New York (1976)

    Google Scholar 

  25. Utkin, V.I.: Sliding mode and their application in variable structure systems. Mir, Moscow (1978)

    Google Scholar 

  26. Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)

    MATH  Google Scholar 

  27. Deng, W.H.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

The present work is supported by Natural Science Foundation of Hebei Province under Grant No. 2010000343.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shiping Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, R., Yang, S. Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach. Nonlinear Dyn 71, 269–278 (2013). https://doi.org/10.1007/s11071-012-0659-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0659-9

Keywords

Navigation