Skip to main content

Advertisement

SpringerLink
Log in

LMI-based stabilization of a class of fractional-order chaotic systems

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

Abstract

Based on the theory of stabilization of fractional-order LTI interval systems, a simple controller for stabilization of a class of fractional-order chaotic systems is proposed in this paper. We consider the structure of the chaotic systems as fractional-order LTI interval systems due to the limited amplitude of chaotic trajectories. We introduce a simple feedback controller for the interval system and then, based on a recently established theorem for stabilization of interval systems, we reach to a linear matrix inequality (LMI) problem. Solving the LMI yields an appropriate decoupling feedback control law which suffices to bring the chaotic trajectories to the origin. Several illustrative examples are given which show the effectiveness of the method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  2. Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I, Regul. Pap. 42, 485–490 (1995)

    Article  Google Scholar 

  3. Matouk, A.E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol–Duffing circuit. Commun. Nonlinear Sc.i Numer. Simulations 16, 975–986 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bhalekar, S., Gejji, V.D.: Synchronization of different fractional order chaotic systems using active control. Commun. Nonlinear Sc.i Numer. Simulations 15, 3536–3546 (2010)

    Article  MATH  Google Scholar 

  5. Faieghi, M.R., Delavari, H.: Chaos in fractional-order Genesio–Tesi system and its synchronization. Commun. Nonlinear Sc.i Numer. Simulations 17, 731–741 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Faieghi, M.R., Delavari, H., Baleanu, D.: Control of an uncertain fractional-order Liu system via fuzzy fractional-order. J. Vib. Control (2011). doi:10.1177/1077546311422243

    Google Scholar 

  7. Zhou, P., Ding, R.: Chaotic synchronization between different fractional-order chaotic systems. J. Franklin Inst. 348, 2839–2848 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, D., Liu, Y., Ma, X., Zhang, R.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 67(1), 893–901 (2011)

    Article  MathSciNet  Google Scholar 

  9. Lin, T.C., Kou, C.H.: H∞ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach. ISA Trans. 50, 548–556 (2011)

    Article  Google Scholar 

  10. Lu, J., Chen, Y.: Robust stability and stabilization of fractional-order interval systems with the fractional order α: the 0 < α<1 case. IEEE Trans. Autom. Control 55, 152–158 (2010)

    Article  Google Scholar 

  11. Kuntanapreeda, S.: Robust synchronization of fractional-order unified chaotic systems via linear control. Comput. Math. Appl. 63, 183–190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Tavazoei, M.S., Haeri, M.: Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Process. 1, 171–181 (2007)

    Article  Google Scholar 

  13. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garrappa, R.: On linear stability of predictor–corrector algorithms for fractional differential equations. Int. J. Comput. Math. 87, 2281–2290 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Miyaneh Branch, Islamic Azad University, Miyaneh, Iran

    Mohammadreza Faieghi

  2. Department of Mechanical and Aerospace Engineering, Faculty of Engineering, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand

    Suwat Kuntanapreeda

  3. Department of Electrical Engineering, Hamedan University of Technology, Hamedan, 65155, Iran

    Hadi Delavari

  4. Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  5. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

Authors
  1. Mohammadreza Faieghi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suwat Kuntanapreeda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hadi Delavari
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hadi Delavari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Faieghi, M., Kuntanapreeda, S., Delavari, H. et al. LMI-based stabilization of a class of fractional-order chaotic systems. Nonlinear Dyn 72, 301–309 (2013). https://doi.org/10.1007/s11071-012-0714-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0714-6

Keywords

Search

Navigation