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

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Abstract

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.

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References

  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. Liu, Y.J., Yang, Q.G.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Anal., Real World Appl. 11, 2563–2572 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Harb, A.M., Abdel-Jabbar, N.: Controlling Hopf bifurcation and chaos in a small power system. Chaos Solitons Fractals 18, 1055–1063 (2003)

    Article  MATH  Google Scholar 

  4. Ditto, W.L.: Applications of chaos in biology and medicine. Chaos Chang. Nat. Sci. Med. 376, 175–202 (1996)

    Google Scholar 

  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)

    Article  Google Scholar 

  6. Lamba, P., Hudson, J.L.: Experiments on bifurcations to chaos in a forced chemical reactor. Chem. Eng. Sci. 42, 1–8 (1987)

    Article  Google Scholar 

  7. Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4, 75–89 (1977)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jumarie, G.: Fractional master equation: non-standard analysis and Liouville–Riemann derivative. Chaos Solitons Fractals 12, 2577–2587 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Elwakil, S.A., Zahran, M.A.: Fractional integral representation of master equation. Chaos Solitons Fractals 10, 1545–1558 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. El-Misiery, A.E.M., Ahmed, E.: On a fractional model for earthquakes. Appl. Math. Comput. 178, 207–211 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  13. El-Sayed, A.M.A.: Fractional-order diffusion-wave equation. Int. J. Theor. Phys. 35, 311–322 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44, 554–566 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lazopoulos, K.A.: Non-local continuum mechanics and fractional calculus. Mech. Res. Commun. 33, 753–757 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  22. Wang, J.R., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262–272 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, R.X., Yang, S.P.: Designing synchronization schemes for a fractional-order hyperchaotic system. Acta Phys. Sin. 57, 6837–6843 (2008)

    MATH  Google Scholar 

  24. Mophou, G.M.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61, 68–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhou, P., Zhu, W.: Function projective synchronization for fractional-order chaotic systems. Nonlinear Anal., Real World Appl. 12, 811–816 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Electrical Department, Northwest A&F University, Shaanxi, Yangling, 712100, China

    Di-yi Chen, Yu-xiao Liu, Xiao-yi Ma & Run-fan Zhang

  2. College of Mechanical and Electric Engineering, Northwest A&F University, Shaanxi, Yangling, 712100, China

    Di-yi Chen

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  1. Di-yi Chen
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  2. Yu-xiao Liu
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  3. Xiao-yi Ma
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  4. Run-fan Zhang
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Correspondence to Xiao-yi Ma.

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Chen, Dy., Liu, Yx., Ma, Xy. et al. Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn 67, 893–901 (2012). https://doi.org/10.1007/s11071-011-0002-x

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