Nonlinear Dynamics

, Volume 71, Issue 1–2, pp 269–278 | Cite as

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

Original Paper


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.


Adaptive sliding mode control Chaos synchronization Fractional-order chaotic system Unknown parameter 



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


  1. 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) CrossRefGoogle Scholar
  2. 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) MATHCrossRefGoogle Scholar
  3. 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) CrossRefGoogle Scholar
  4. 4.
    Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971) Google Scholar
  5. 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) CrossRefGoogle Scholar
  6. 6.
    Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003) CrossRefGoogle Scholar
  7. 7.
    Zhang, W., Zhou, S., Li, H., Zhu, H.: Chaos in a fractional-order Rössler system. Chaos Solitons Fractals 42, 1684–1691 (2009) MATHCrossRefGoogle Scholar
  8. 8.
    Li, C.G., Chen, G.R.: Chaos in the fractional order Chen system and its control. Chaos Solitons Fractals 22, 549–554 (2004) MATHCrossRefGoogle Scholar
  9. 9.
    Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005) CrossRefGoogle Scholar
  10. 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) MathSciNetMATHCrossRefGoogle Scholar
  11. 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) MATHCrossRefGoogle Scholar
  12. 12.
    Wu, X.J., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009) MATHCrossRefGoogle Scholar
  13. 13.
    Li, C.P., Deng, W.H.: Chaos synchronization of fractional-order differential systems.Int. J. Mod. Phys. B 20, 791–803 (2006) MathSciNetMATHCrossRefGoogle Scholar
  14. 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) MathSciNetMATHCrossRefGoogle Scholar
  15. 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) MathSciNetMATHCrossRefGoogle Scholar
  16. 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) MATHCrossRefGoogle Scholar
  17. 17.
    Odibat, Z.M.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn. 60, 479–487 (2010) MathSciNetMATHCrossRefGoogle Scholar
  18. 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) MATHCrossRefGoogle Scholar
  19. 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) MATHCrossRefGoogle Scholar
  20. 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) MATHCrossRefGoogle Scholar
  21. 21.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) MATHGoogle Scholar
  22. 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. 23.
    Qi, G., Chen, Z., Yuan, Z.: Model-free control of affine chaotic systems. Phys. Lett. A 344, 189–202 (2005) MATHCrossRefGoogle Scholar
  24. 24.
    Itkis, U.: Control System of Variable Structure. Wiley, New York (1976) Google Scholar
  25. 25.
    Utkin, V.I.: Sliding mode and their application in variable structure systems. Mir, Moscow (1978) Google Scholar
  26. 26.
    Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991) MATHGoogle Scholar
  27. 27.
    Deng, W.H.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1522 (2007) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.College of Elementary EducationXingtai UniversityXingtaiP.R. China
  2. 2.College of Physics Science and Information EngineeringHebei Normal UniversityShijiazhuangP.R. China

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