Nonlinear Dynamics

, Volume 67, Issue 1, pp 893–901 | Cite as

Control of a class of fractional-order chaotic systems via sliding mode

  • Di-yi Chen
  • Yu-xiao Liu
  • Xiao-yi MaEmail author
  • Run-fan Zhang
Original Paper


This paper investigates the chaos control of a class of fractional-order chaotic systems via sliding mode. First, the sliding mode control law is derived to make the states of the fractional-order chaotic systems asymptotically stable. Second, the designed control scheme guarantees asymptotical stability of the uncertain fractional-order chaotic systems in the presence of an external disturbance. Finally, simulation results are given to demonstrate the effectiveness of the proposed sliding mode control method.


Fractional order system Chaos control Sliding mode control Uncertain chaotic systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Liu, Y.J.: Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system. Nonlinear Dyn. (2011). doi: 10.1007/s11071-011-9960-2 Google Scholar
  2. 2.
    Liu, Y.J., Yang, Q.G.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Anal., Real World Appl. 11, 2563–2572 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Harb, A.M., Abdel-Jabbar, N.: Controlling Hopf bifurcation and chaos in a small power system. Chaos Solitons Fractals 18, 1055–1063 (2003) zbMATHCrossRefGoogle Scholar
  4. 4.
    Ditto, W.L.: Applications of chaos in biology and medicine. Chaos Chang. Nat. Sci. Med. 376, 175–202 (1996) Google Scholar
  5. 5.
    Ma, J., Wang, C.N., Tang, J., Xia, Y.F.: Suppression of the spiral wave and turbulence in the excitability-modulated media. Int. J. Theor. Phys. 48, 150–157 (2009) CrossRefGoogle Scholar
  6. 6.
    Lamba, P., Hudson, J.L.: Experiments on bifurcations to chaos in a forced chemical reactor. Chem. Eng. Sci. 42, 1–8 (1987) CrossRefGoogle Scholar
  7. 7.
    Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4, 75–89 (1977) zbMATHCrossRefGoogle Scholar
  8. 8.
    He, G.L., Zhou, S.P.: What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact box dimension. Chaos Solitons Fractals 26, 867–879 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Jumarie, G.: Fractional master equation: non-standard analysis and Liouville–Riemann derivative. Chaos Solitons Fractals 12, 2577–2587 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Elwakil, S.A., Zahran, M.A.: Fractional integral representation of master equation. Chaos Solitons Fractals 10, 1545–1558 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    El-Misiery, A.E.M., Ahmed, E.: On a fractional model for earthquakes. Appl. Math. Comput. 178, 207–211 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bagley, R.L., Calico, R.A.: Fractional-order state equations for the control of viscoelastically damped structures. Guid. Control Dyn. 14, 304–311 (1991) CrossRefGoogle Scholar
  13. 13.
    El-Sayed, A.M.A.: Fractional-order diffusion-wave equation. Int. J. Theor. Phys. 35, 311–322 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44, 554–566 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lazopoulos, K.A.: Non-local continuum mechanics and fractional calculus. Mech. Res. Commun. 33, 753–757 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Wu, X.J., Li, J., Chen, G.R.: Chaos in the fractional order unified system and its synchronization. J. Franklin Inst. 345, 392–401 (2008) zbMATHCrossRefGoogle Scholar
  17. 17.
    Ge, Z.M., Ou, C.Y.: Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals 35, 705–717 (2008) CrossRefGoogle Scholar
  18. 18.
    Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Phys. A, Stat. Mech. Appl. 387, 57–70 (2008) CrossRefGoogle Scholar
  19. 19.
    Matouk, A.E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol–Duffing circuit. Commun. Nonlinear Sci. Numer. Simul. 16, 975–986 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Delavari, H., Ghaderi, R., Ranjbar, A., Momani, S.: Fuzzy fractional order sliding mode controller for nonlinear systems. Commun. Nonlinear Sci. Numer. Simul. 15, 963–978 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Wu, X.J., Lu, H.T., Shen, S.L.: Synchronization of a new fractional-order hyperchaotic system. Phys. Lett. A 373, 2329–2337 (2009) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang, J.R., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262–272 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Zhang, R.X., Yang, S.P.: Designing synchronization schemes for a fractional-order hyperchaotic system. Acta Phys. Sin. 57, 6837–6843 (2008) zbMATHGoogle Scholar
  24. 24.
    Mophou, G.M.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61, 68–78 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Wang, X.Y., He, Y.J., Wang, M.J.: Chaos control of a fractional order modified coupled dynamos system. Nonlinear Anal. 71, 6126–6134 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Zhou, P., Zhu, W.: Function projective synchronization for fractional-order chaotic systems. Nonlinear Anal., Real World Appl. 12, 811–816 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Zheng, Y.G., Nian, Y.B., Wang, D.J.: Controlling fractional order chaotic systems based on Takagi–Sugeno fuzzy model and adaptive adjustment mechanism. Phys. Lett. A 375, 125–129 (2010) CrossRefGoogle Scholar
  28. 28.
    Dadras, S., Momeni, H.R.: Control of a fractional-order economical system via sliding mode. Phys. A, Stat. Mech. Appl. 389, 2434–2442 (2010) CrossRefGoogle Scholar
  29. 29.
    Deng, W.H.: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal., Real World Appl. 72, 1768–1777 (2009) Google Scholar
  30. 30.
    Asheghan, M.M., Beheshti, M.T.H., Tavazoei, M.S.: Robust synchronization of perturbed Chen’s fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16, 1044–1051 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Peng, G., Jiang, Y.: Two routes to chaos in the fractional Lorenz system with dimension continuously varying. Phys. A, Stat. Mech. Appl. 389, 4140–4148 (2010) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yang, Q.G., Zeng, C.B.: Chaos in fractional conjugate Lorenz system and its scaling attractors. Commun. Nonlinear Sci. Numer. Simul. 15, 4041–4051 (2010) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Di-yi Chen
    • 1
    • 2
  • Yu-xiao Liu
    • 1
  • Xiao-yi Ma
    • 1
    Email author
  • Run-fan Zhang
    • 1
  1. 1.Electrical DepartmentNorthwest A&F UniversityYanglingChina
  2. 2.College of Mechanical and Electric EngineeringNorthwest A&F UniversityYanglingChina

Personalised recommendations