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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Sliding Mode Control of Fractional Lorenz-Stenflo Hyperchaotic System

Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 445--455 | DOI:10.5890/DNC.2015.11.007

Jian Yuan; Bao Shi

Institute of System Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai, 264001, China

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Abstract

This paper proposes sliding mode control for the 4-D fractional order Lorenz-Stenflo hyperchaotic system. Two methods are utilized: one is based on the frequency distributed model of fractional integral operator; and the other is based on the Mittag-Leffler stability theorem and the Caputo operator property. Both of the two methods involve two steps: firstly, constructing a fractional order sliding surface; secondly, designing a single sliding control law for suppression of the nominal plant. Numerical simulations are carried out to verify the efficiency of the theoretical results.

Acknowledgments

This study was supported by a grant from the Natural Science Foundation of the Province Shandong of China (ZR2014AM006).

References

  1. [1]  Diethelm, K. (2010), The Analysis of Fractional Differential Equations: An Application-oriented Exposition Using Differential Operators of Caputo Type, Springer Verlag.
  2. [2]  Podlubny, I. (1998), Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press.
  3. [3]  Caponetto, R. (2010), Fractional order systems: modeling and control applications,World Scientific Publishing Company.
  4. [4]  Mathieu Moze, J.S. and Oustaloup, A. (2007), LMI Characterization of Fractional Systems Stability, in: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 419??434.
  5. [5]  Petráš, I. (2011), Fractional-Order Nonlinear Systems, Springer verlag: Berlin Heidelberg.
  6. [6]  Hegazi, A.S., Ahmed, E., and Matouk, A.E.(2013), On chaos control and synchronization of the commensurate fractional order Liu system, J. Sci. Commun., 18, 1193–1202.
  7. [7]  Li, C.and Chen,G. (2004), Chaos in the fractional order Chen system and its control, Chaos. Soliton. Fract., 22, 549- 554.
  8. [8]  Li, C.P., Deng, C.P., and Xu,D. (2006), Chaos synchronization of the Chua system with a fractional order, Physica. A, 360, 171-185.
  9. [9]  Kuntanapreeda, S. (2012), Robust synchronization of fractional-order unified chaotic systems via linear control, Comput. Math. Appl., 63, 183-190.
  10. [10]  Yan,J.and Li,C. (2007), On chaos synchronization of fractional differential equations, Chaos. Soliton. Fract., 32, 725- 735.
  11. [11]  Bhalekar, S. and Daftardar-Gejji, V. (2010), Synchronization of different fractional order chaotic systems using active control, J. Sci. Commun., 15, 3536-3546.
  12. [12]  Asheghan, M.M., Hamidi Beheshti, M.T., and Tavazoei, M.S.(2011), Robust synchronization of perturbed Chen’s fractional-order chaotic systems, J. Sci. Commun., 16, 1044-1051.
  13. [13]  Pan, L., Zhou, W., Zhou, L., and Sun, K. (2011), Chaos synchronization between two different fractional-order hyperchaotic systems, J. Sci. Commun., 16, 2628-2640.
  14. [14]  Odibat, Z. (2012), A note on phase synchronization in coupled chaotic fractional order systems, Nonlinear. Anal. Real., 13, 779-789.
  15. [15]  Mahmoudian, M., Ghaderi,R., Ranjbar,A., Sadati, J., Hosseinnia, S.H. and Momani, S. (2010), Synchronization of Fractional-Order Chaotic System via Adaptive PID Controller, in: New Trends in Nanotechnology and Fractional Calculus Applications, Springer, pp. 445-452.
  16. [16]  L.X. Liu, J., and Zhao, J.C.( 2011), Prediction-control based feedback control of a fractional order unified chaotic system, 2011 Chinese Control and Decision Conference (CCDC), 2093-2097.
  17. [17]  Radwan, A.G., Moaddy, K., Salama, K.N., Momani, S.and Hashim, I. (2013), Control and switching synchronization of fractional order chaotic systems using active control technique, J. Adv. Res..
  18. [18]  Yin, C., Zhong, S.M. and Chen, W.F. (2012), Design of sliding mode controller for a class of fractional-order chaotic systems, J. Sci. Commun., 17, 356-366.
  19. [19]  Aghababa, M.P. (2012), Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller, J. Sci. Commun., 17, 2670-2681.
  20. [20]  Dadras, S. and Momeni, H.R. (2010), Control of a fractional-order economical system via sliding mode, Physica. A, 389, 2434-2442.
  21. [21]  Chen, D.Y., Liu, Y.X., Ma, X.Y. and Zhang, R.F. (2012), Control of a class of fractional-order chaotic systems via sliding mode, Nonlinear Dynam., 67, 893-901.
  22. [22]  Faieghi, M.R., Delavari, H. and Baleanu, D. (2012), Control of an uncertain fractional-order Liu system via fuzzy fractional-order sliding mode control, J. Vib. Control, 18, 1366-1374.
  23. [23]  Faieghi, M.R., Delavari, H., and Baleanu, D. (2013), A note on stability of sliding mode dynamics in suppression of fractional-order chaotic systems, Comput. Math. Appl., 66(5), 832-837.
  24. [24]  Yin, C., Dadras, S.,and Zhong, S.M. (2012), Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional-order chaotic systems with fully unknown parameters, J. Franklin. I., 349, 3078-3101.
  25. [25]  Yin, C., Dadras, S., Zhong, S.M. and Chen, Y. (2013), Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach, Appl. Math. Model., 37, 2469-2483.
  26. [26]  Trigeassou, J.C., Maamri, N., Sabatier, J. and Oustaloup, A. (2011), A Lyapunov approach to the stability of fractional differential equations, Signal Processing, 91, 437-445.
  27. [27]  Yuan, J., Shi, B.and Ji, W.Q. (2013), Adaptive sliding mode control of a novel class of fractional chaotic systems, Advances in Mathematical Physics, 2013.
  28. [28]  Li, Y., Chen, Y. and Podlubny, I. (2010),Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Math. Appl., 59, 1810-1821.
  29. [29]  Aguila-Camacho, N., Duarte-Mermoud, M.A. and Gallegos, J.A. (2014), Lyapunov functions for fractional order systems, J. Sci. Commun., 19, 2951-2957.
  30. [30]  Achar, B. N. N., Lorenzo, C. F.and Hartley, T. T., (2007),The Caputo fractional derivative: Initialization issues relative to fractional differential equation, in: J. Sabatier, O. P. Agrawal, J. T. Machado(Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, pp. 27-42.
  31. [31]  Utkin, V.I. (1992), Sliding modes in control and optimization, Springer, pp.286.

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