Skip to main content
SpringerLink
Log in

Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller

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

Abstract

This letter investigates the stabilization of three-dimensional fractional-order chaotic systems, and proposes a single state adaptive-feedback controller for fractional-order chaos control based on Lyapunov stability theory, fractional order differential inequality, and adaptive control theory. The present controller which only contains a single state variable is simple both in design and implementation. Simulation results for several fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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 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., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. CAS: I 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. Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneado’s systems. Chaos Solitons Fractals 26, 1125–1133 (2005)

    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(3), 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. Tam, L.M., SiTou, W.M.: Parametric study of the fractional order Chen–Lee system. Chaos Solitons Fractals 37, 817–826 (2008)

    Article  Google Scholar 

  11. 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 

  12. Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 358, 1–4 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Balochian, S., Sedigh, A.K., Haeri, M.: Stabilization of fractional order systems using a finite number of state feedback laws. Nonlinear Dyn. (2010). doi:10.1007/s11071-010-9916-y

    Google Scholar 

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

    MATH  Google Scholar 

  15. Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Wang, X.J., Li, J., Chen, G.R.: Chaos in the fractional order unified system and its synchronization. J. Franklin Inst. 345, 392–401 (2008)

    Article  Google Scholar 

  18. Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M., Chen, W.C., Lin, K.T., Kang, Y.: Chaos in the Newton– Leipnik system with fractional order. Chaos Solitons Fractals 36, 98–103 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ivo, P.: A note on the fractional-order Chua’s system. Chaos Solitons Fractals 38, 140–147 (2008)

    Article  Google Scholar 

  20. Xu, Z., Liu, C.X.: Realization of fractional-order Liu chaotic system by a new circuit unit. Chin. Phys. B 17, 4033–4038 (2008)

    Article  Google Scholar 

  21. Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Genesio–Tesi systems. Chin. Phys. B 14, 1517–1521 (2005)

    Article  Google Scholar 

  22. Zhang, R.X., Yang, S.P.: Chaos in the fractional-order conjugate Chen system and its circuit simulation. Acta Phys. Sin. 58, 2957–2962 (2009)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang, 050016, China

    Ruoxun Zhang & Shiping Yang

  2. College of Education, Xingtai University, Xingtai, 054001, China

    Ruoxun Zhang

Authors
  1. Ruoxun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiping Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiping Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, R., Yang, S. Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller. Nonlinear Dyn 68, 45–51 (2012). https://doi.org/10.1007/s11071-011-0202-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0202-4

Keywords

Search

Navigation