Using fractional-order integrator to control chaos in single-input chaotic systems


This paper deals with a fractional calculus based control strategy for chaos suppression in the 3D chaotic systems. It is assumed that the structure of the controlled chaotic system has only one control input. In the proposed strategy, the controller has three tuneable parameters and the control input is constructed as fractional-order integration of a linear combination of linearized model states. The tuning procedure is based on the stability theorems in the incommensurate fractional-order systems. To evaluate the performance of the proposed controller, the design method is applied to suppress chaotic oscillations in a 3D chaotic oscillator and in the Chen chaotic system.

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  1. 1.

    Andrievsky, B.R., Fradkov, A.L.: Control of chaos: Methods and applications. Part I: Methods. Autom. Remote Control 64(5), 673–713 (2003)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Chen, G., Yu, X.: Chaos Control: Theory and Applications. Springer, Berlin (2003)

    MATH  Google Scholar 

  3. 3.

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

    MATH  Google Scholar 

  4. 4.

    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2001)

    Google Scholar 

  5. 5.

    Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Lune, B.J.: Three-parameter tuneable tilt-integral derivative (TID) controller. US Patent US5371 670 (1994)

  7. 7.

    Lazarevic, M.P.: Finite time stability analysis of PDα fractional control of robotic time-delay systems. Mech. Res. Commun. 33, 269–279 (2006)

    Article  MathSciNet  Google Scholar 

  8. 8.

    Maione, G., Lino, P.: New tuning rules for fractional PIα controllers. Nonlinear Dyn. 49, 251–257 (2007)

    Article  Google Scholar 

  9. 9.

    Oustaloup, A., Sabatier, J., Lanusse, P.: From fractal robustness to CRONE control. Fractional Calc. Appl. Anal. 2(1), 1–30 (1999)

    MATH  MathSciNet  Google Scholar 

  10. 10.

    Oustaloup, A., Moreau, X., Nouillant, M.: The CRONE suspension. Control Eng. Pract. 4(8), 1101–1108 (1996)

    Article  Google Scholar 

  11. 11.

    Oustaloup, A.: La Commande CRONE. Hermes, Paris (1991)

    MATH  Google Scholar 

  12. 12.

    Raynaud, H.F., Inoh, A.Z.: State-space representation for fractional-order controllers. Automatica 36, 1017–1021 (2000)

    MATH  Article  Google Scholar 

  13. 13.

    Monje, C.A., Feliu, V.: The fractional-order lead compensator. In: IEEE International Conference on Computational Cybernetics. Vienna, Austria, August 30–September 1 (2004)

  14. 14.

    Feliu-Batlle, V., Rivas Perez, R., Sanchez Rodriguez, L.: Fractional robust control of main irrigation canals with variable dynamic parameters. Control Eng. Pract. 15, 673–686 (2007)

    Article  Google Scholar 

  15. 15.

    Suárez, J.I., Vinagre, B.M., Calderón, A.J., Monje, C.A., Chen, Y.Q.: Using fractional calculus for lateral and longitudinal control of autonomous vehicles. In: Lecture Notes in Computer Science, vol. 2809. Springer, Berlin (2004)

    Google Scholar 

  16. 16.

    Calderon, A.J., Vinagre, B.M., Feliu, V.: Fractional-order control strategies for power electronic buck converters. Signal Process. 86, 2803–2819 (2006)

    Article  Google Scholar 

  17. 17.

    Silva, M.F., Machado, J.A.T., Lopes, A.M.: Fractional-order control of a hexapod robot. Nonlinear Dyn. 38, 417–433 (2004)

    MATH  Article  Google Scholar 

  18. 18.

    Moreau, X., Ramus-Serment, C., Oustaloup, A.: Fractional differentiation in passive vibration control. Nonlinear Dyn. 29, 343–362 (2002)

    MATH  Article  Google Scholar 

  19. 19.

    Sabatier, J., Poullain, S., Latteux, P., Thomas, J.L., Oustaloup, A.: Robust speed control of a low damped electromechanical system based on CRONE control: application to a four mass experimental test bench. Nonlinear Dyn. 38, 383–400 (2004)

    MATH  Article  Google Scholar 

  20. 20.

    Manabe, S.: A suggestion of fractional-order controller for flexible spacecraft attitude control. Nonlinear Dyn. 29, 251–268 (2002)

    MATH  Article  Google Scholar 

  21. 21.

    Tavazoei, M.S., Haeri, M.: Chaos control via a simple fractional-order controller. Phys. Lett. A 372, 798–807 (2008)

    Article  Google Scholar 

  22. 22.

    Tavazoei, M.S., Haeri, M., Jafari, S.: Fractional controller to stabilize fixed points of uncertain chaotic systems: theoretical and experimental study. J. Syst. Control Eng. (2008). doi:10.1243/09596518JSCE481

    MATH  Google Scholar 

  23. 23.

    Deng, W., Li, C., Lü, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)

    Article  Google Scholar 

  24. 24.

    Kailath, T.: Linear Systems. Prentice-Hall, New York (1980)

    MATH  Google Scholar 

  25. 25.

    Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  26. 26.

    Huijberts, H.: Linear controllers for the stabilization of unknown steady states of chaotic systems. IEEE Trans. Circuits Syst. I 53(10), 2246–2254 (2006)

    Article  MathSciNet  Google Scholar 

  27. 27.

    Ahlfors, L.V.: Complex Analysis, 2nd edn. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

  28. 28.

    Dorf, R.C., Bishop, R.H.: Modern Control Systems, 7th edn. Addison Wesley, Menlo Park (1995)

    Google Scholar 

  29. 29.

    Dorcak, L., Petras, I., Terpak, J., Zborovjan, M.: Comparison of the methods for discrete approximation of the fractional-order operator. Acta Montanistica Slovaca 8(4), 236–239 (2003)

    Google Scholar 

  30. 30.

    Elwakil, A., Kennedy, M.: Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices. IEEE Trans. Circuits Syst. I 48(3), 289–307 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  31. 31.

    Chen, G., Ueta, T.: Yet another attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  32. 32.

    Tavazoei, M.S., Haeri, M., Attari, M., Bolouki, S., Siami, M.: More details on analysis of fractional-order van der Pol oscillator. Submitted to J. Vib. Control (2007)

  33. 33.

    Tavazoei, M.S., Haeri, M., Jafari, S., Bolouki, S., Siami, M.: Some applications of fractional calculus in suppression of chaotic oscillations. Submitted to IEEE Trans. Ind. Electron. (2008)

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Correspondence to Mohammad Haeri.

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Tavazoei, M.S., Haeri, M., Bolouki, S. et al. Using fractional-order integrator to control chaos in single-input chaotic systems. Nonlinear Dyn 55, 179–190 (2009).

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  • Fractional-order integrator
  • Chaos
  • Control
  • Stabilization