Automatic Control and Computer Sciences

, Volume 51, Issue 5, pp 311–320 | Cite as

Research on adaptive sliding synchronization of Rikitake chaotic system with single unknown control coefficient

  • Youan Zhang
  • Heng Li
  • Jingmao Liu
  • Ruwei Zeng
  • Junwei Lei


The control of second order system with uncertain parameters and single unknown control coefficient was investigated to solve the synchronization problem of Rikitake chaotic with reduced number of active inputs. In addition, a kind of adaptive strategy was hybrid with sliding mode method, where the adaptive strategy was used to cope with uncertain parameters produced in the process of sliding mode controller design. At last, detailed numerical simulations with both second order systems and synchronous chaotic system were done to testify the rightness of the proposed method and also multi-time random simulations were done to testify the robustness of the controller. In addition, the main conclusion is that the sliding mode control has very good consistency since the strategy formation is almost the same as the controller for system with known control coefficient, and high gain is necessary for system with single uncertain control coefficient.


synchronization stability backstepping control chaotic system uncertainty adaptive control 


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  1. 1.
    Tan, X., Zhang, J., and Yang, Y., Synchronizing chaotic systems using backstepping design, Chaos Solitons Fractals, 2003, vol. 16, pp. 37–45.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Li, Z.G., Wen, C.Y., Soh, Y.C., and Xie, W.X., The stabilization and synchronization of Chuas oscillators via impulsive control, IEEE Trans. Circuits Syst. I: Fundam. Theor. Appl., 2002, vol. 48, pp. 1351–1355.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Yang, T. and Chua, L.O., Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans. Circuits Syst. I: Fundam. Theor. Appl., 1997, vol. 44, pp. 976–988.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Yan, J.J., Lin, J.S., and Liao, T.L., Synchronization of a modified Chua’s circuit system via adaptive sliding mode control, Chaos Solitons Fractals, 2008, vol. 36, pp. 45–52.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fu, S.H. and Pei, L.J., Chaotic synchronization of Chua’s circuits with nonlinear control, Acta Phys. Sin., 2010, vol. 59, pp. 5985–5989.MATHGoogle Scholar
  6. 6.
    Zhang, T. and Feng, G., Output tracking and synchronization of chaotic Chua’s circuit with disturbances via model predictive regulator, Chaos Solitons Fractals, 2009, vol. 39, pp. 810–820.CrossRefGoogle Scholar
  7. 7.
    Suykens, J.A.K., Curran, P.F., Vandewalle, J., and Chua, L.O., Robust synthesis for master-slave synchronization of Lur’e systems, IEEE Trans. Circuits Syst. I Fundam. Theor. Appl., 1999, vol. 46, pp. 841–850.CrossRefMATHGoogle Scholar
  8. 8.
    Cao, J., Li, H.X., and Ho, D.W.C., Synchronization criteria of Lur’e systems with time-delay feedback control, Chaos Solitons Fractals, 2005, vol. 23, pp. 1285–1298.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Han, Q.L., On designing time-varying delay feedback controllers for master-slave synchronization of Lur’e systems, IEEE Trans. Circuits Syst. I: Reg. Pap., 2007, vol. 54, pp. 1573–1583.CrossRefMATHGoogle Scholar
  10. 10.
    Lu, J.L. and Hill, D.J., Global asymptotical synchronization of chaotic Lur’e systems using sampled data: A linear matrix inequality approach, IEEE Trans. Circuits Syst. II: Express Briefs, 2008, vol. 55, pp. 586–590.Google Scholar
  11. 11.
    Zhang, C.K., He, Y., and Wu, M., Improved global asymptotical synchronization of chaotic Lur’e systems with sampled-data control, IEEE Trans. Circuits Syst. II: Express Briefs, 2009, vol. 56, pp. 320–324.CrossRefGoogle Scholar
  12. 12.
    Zhang, T. and Feng, G., Output tracking of piecewise-linear systems via error feedback regulator with application to synchronization of nonlinear Chua’s circuit, IEEE Trans. Circuits Syst. I, 2007, vol. 54, pp. 1852–1863.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Luo, A.C.J., Singularity and Dynamics on Discontinuous Vector Fields, Amsterdam: Elsevier, 2006.MATHGoogle Scholar
  14. 14.
    Jason, J.G., Kathryn, W.J., et al., A simplified adaptive robust back-stepping approach using sliding modes and a z-swapping identifier, Proceedings of the American Control Conference, Denver, 2003.Google Scholar
  15. 15.
    Zhou, Y., Wu, Y., and Hu, Y., Robust backstepping sliding mode control of a class of uncertain MIMO nonlinear systems, 2007 IEEE International Conference on Control and Automation, Guangzhou, 2007.Google Scholar
  16. 16.
    Lee, T. and Kim, Y., Nonlinear adaptive flight control using back-stepping and neural networks controller, J. Guid. Control Dyn., 2001, vol. 24, no. 4, pp. 675–682.CrossRefGoogle Scholar
  17. 17.
    Zhu Kai, Qi Naiming, and Qin Changmao, Adaptive sliding mode controller design for BTT missile based on back-stepping control, J. Astronaut., 2010, vol. 31, no. 3, pp. 769–773.Google Scholar
  18. 18.
    Chen, Y., Dong, C.Y., Wang, Q., et al., Reaction-jet and aerodynamics compound control missile autopilot design based on adaptive fuzzy sliding mode control via backstepping, Acta Aeronaut. Astronaut. Sin., 2007, vol. 28, pp. 1141–1145.Google Scholar
  19. 19.
    Jay, F., Manu, S., and Marios, P., Backstepping based flight control with adaptive function approximation, J. Guid. Control Dyn., 2007, vol. 30, no. 2, pp. 322–336.CrossRefGoogle Scholar
  20. 20.
    Tsung-Ying Chiang, Jui-Sheng Lin, Teh-Lu Liao, and Jun-Juh Yan, Antisynchronization of uncertain unified chaotic systems with dead-zone nonlinearity, Nonlinear Anal., 2008, vol. 68, pp. 2629–2637.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    May, R.M., Simple mathematical models with very complicated dynamics, Nature, 1976, vol. 261, pp. 459–462.CrossRefMATHGoogle Scholar
  22. 22.
    Feigenbaum, M.J., Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 1978, vol. 19, pp. 25–52.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Pecora, L.M. and Carroll, T.L., Synchronization in chaotic systems, Phys. Rev. Lett., 1990, vol. 64, pp. 821–824.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ge, S.S., Wang, C., and Lee, T.H., Adaptive backstepping control of a class of chaotic systems, Int. J. Bifurcation Chaos, 2000, vol. 10, no. 5, pp. 1140–1156.MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ge, S.S. and Wang, C., Adaptive control of uncertain Chua’s circuits, IEEE Trans. Circuits Syst., 2000, vol. 47, no. 9, pp. 1397–1402.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Fradkov, A.L. and Markov, A.Yu., Adaptive synchronization of chaotic systems based on speed gradient method and pacification, IEEE Trans. Circuits Syst., 1997, vol. 44, no. 10, pp. 905–912.CrossRefGoogle Scholar
  27. 27.
    Dong, X. and Chen, L., Adaptive control of the uncertain Duffing oscillator, Int. J. Bifurcation Chaos, 1997, vol. 7, no. 7, pp. 1651–1658.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Tao Yang, Chun-Mei Yang, and Lin-Bao Yang, A detailed study of adaptive control of chaotic systems with unknown parameters, Dyn. Control, 1998, vol. 8, pp. 255–267.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Yassen, M.T., Chaos control of chaotic dynamical systems using backstepping design, Chaos Solitons Fractals, 2006, vol. 27, pp. 537–548.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Fengxiang Chen, Lin Chen, and Weidong Zhang, Stabilization of parameters perturbation chaotic system via adaptive backstepping technique, Appl. Math. Comput., 2008, vol. 200, pp. 101–109.MathSciNetMATHGoogle Scholar
  31. 31.
    Yassen, M.T., Adaptive chaos control and synchronization for uncertain new chaotic dynamical system, Phys. Lett. A, 2006, vol. 350, pp. 36–43.CrossRefMATHGoogle Scholar
  32. 32.
    Jianping Yan and Changpin Li, On synchronization of three chaotic systems, Chaos Solitons Fractals, 2005, vol. 23, pp. 1683–1688.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  • Youan Zhang
    • 1
  • Heng Li
    • 2
  • Jingmao Liu
    • 3
  • Ruwei Zeng
    • 2
  • Junwei Lei
    • 4
  1. 1.Department of Electrical and Electronic EngineeringYantai Nanshan UniversityYantaiChina
  2. 2.Receiving and Training Center of New EquipmentsNaval Aeronautical and Astronautical UniversityYantaiChina
  3. 3.Shandong Nanshan International Flight Co., LTDYantai, ShandongChina
  4. 4.Department of Control EngineeringNaval Aeronautical and Astronautical UniversityYantai, ShandongChina

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