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Set stability of controlled Chua’s circuit under a non-smooth controller with the absolute value

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  • Control Theory
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Abstract

Chua’s circuit is a typical system with the piecewise-smooth vector field via the absolute value. A new piecewise-linear feedback control scheme with the absolute value is introduced to realize stabilization of sets of the controlled Chua’s circuit. The equilibrium sets and the global asymptotical stability with respect to several invariant sets, which may consist of both admissible and boundary equilibrium points or equilibrium manifolds, are investigated for the controlled Chua’s circuit system. The non-smooth controller has a simple structure and available control properties, and it is easy to be applied in engineering practice. It is shown by numerical simulations that this controller can effectively realize chaos control and stabilization of equilibrium states.

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References

  1. H. Layeghi, M. T. Arjmand, H. Salarieh, and A. Alasty, “Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control,” Chaos, Solitons and Fractals, vol. 37, no. 4, pp. 1125–1135, August 2008.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Dadras, H. R. Momeni, and V. J. Majd, “Sliding mode control for uncertain new chaotic dynamical system,” Chaos, Solitons and Fractals, vol. 41, no. 1, pp. 1857–1862, January 2009.

    Article  MATH  Google Scholar 

  3. J. J. Huang, C. D. Li, and X. He, “Stabilization of a memristor-based chaotic system by intermittent control and fuzzy processing,” International Journal of Control, Automation and Systems, vol. 13, no. 3, pp. 643–637, June 2013.

    Article  Google Scholar 

  4. H. T. Yau and C. L. Chenc, “Chaos control of Lorenz systems using adaptive controller with input saturation,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1567–1574, December 2007.

    Article  Google Scholar 

  5. S. Lee, S. W. Sung, and J. Lee, “In-line mixer pH system for the identification of titration curve,” International Journal of Control, Automation and Systems, vol. 11, no. 3, pp. 442–449, June 2013.

    Article  MathSciNet  Google Scholar 

  6. H. N. Agiza, “Chaos synchronization of Lü dynamical system,” Nonlinear Analysis, vol. 58, no. 1–2, pp. 11–20, July 2004.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Saberi Nik and R. A. Van Gorder, “Competitive modes for the Baier-Sahle hyperchaotic flow in arbitrary dimensions,” Nonlinear Dyn., vol. 74, no. 3, pp. 581–590, July 2013.

    Article  Google Scholar 

  8. S. M. Lee and Ju H. Park, “Delay-dependent criteria for absolute stability of uncertain timedelayed Lur’e dynamical systems,” Journal of the Franklin Institute, vol. 347, no. 1, pp. 146–153, February 2010.

    Article  Google Scholar 

  9. S. EKati, J. Saberi-NadjaL, and H. Saberi Nik, “Optimal and adaptive control for a kind of 3D chaotic and 4D hyper-chaotic systems,” Appl. Math. Modell., vol. 38, no. 2, pp. 759–774, January 2014.

    Article  Google Scholar 

  10. A. Sanayei, “Controlling chaos in forced duffing oscillator based on OGY method and generalized Routh-Hurwitz criterion,” Proc. of 2nd International Conference on Computer and Electrical Engineering, pp. 591–595, 2009.

    Google Scholar 

  11. X. Dong, G. Chen, and L. Chen, “Adaptive control of uncertain Duffing oscillator,” Int. J. Bifurcat Chaos, vol. 7, no. 7, pp. 1651–1658, July 1997.

    Article  MathSciNet  Google Scholar 

  12. C. C. Hwang, J. Y. Hsieh, and R. S. Lin, “A linear continuous feedback control of Chua’s circuit,” Chaos, Solitons and Fractals, vol. 8, no. 9, pp. 1507–1515, 1997.

    Article  Google Scholar 

  13. Y. L. Zou and J. Zhu, “Controlling the chaotic nscroll Chua’s circuit with two low pass filters,” Chaos, Solitons and Fractals, vol. 29, no. 2, pp. 400–406, July 2006.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. T. Sun and Y. P. Zhang, “Impulsive control and synchronization of Chua’s oscillators,” Mathematics and Computers in Simulation, vol. 66, no. 12, pp. 499–508, August 2004.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. H. Li, X. Z. Lin, and Y. P. Tian, “Set stabilization of Chua’s circuit via piece-wise linear feedbacks,” Chaos, Solitons and Fractals, vol. 26, no. 2, pp. 571–579, October 2005.

    Article  MathSciNet  Google Scholar 

  16. Q. Wu, S. Onyshko, N. Sepehri, and A. B. Thornton-Trump, “On construction smooth Lyapunov functions for non-smooth systems,” Int. J. control, vol. 69, no. 3, pp. 443–457, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Sheitz and B. Paden, “Lyapunov stability theory of non-smooth systems,” IEEE Trans. on Automatic Control, vol. 39, no. 9, pp. 1910–1914, September 1994.

    Article  Google Scholar 

  18. Q. Wu and N. Sepehri, “On Lyapunov stability analysis of non-smooth systems with applications to control engineering,” International Journal of Non-Linear Mechanics, vol. 36, no. 7, pp. 1153–1161, October 2001.

    Article  MATH  MathSciNet  Google Scholar 

  19. B. Fiedler, S. Liebscher, and J. C. Alexander, “Generic Hopf bifurcation from lines of equilibria without parameters: I, theory,” J. Diff. Equs., vol. 167, no. 1, pp. 16–35, October 2000.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Fiedler, S. Liebscher, and J. C. Alexander, “Generic Hopf bifurcation from lines of equilibria without parameters: III, binary oscillations,” International Journal of Bifurcation and Chaos, vol. 10, no. 7, pp. 1613–1621, July 2000.

    Article  MathSciNet  Google Scholar 

  21. C. Robinson, Dynamical Systems, Stability, Symbolic Dynamics and Chaos, 2nd ed., CRC Press, New York, 1998.

    Google Scholar 

  22. S. H. Li and Y. P. Tian, “Set stabilization of a modified Chua’s circuit,” International Journal of Bifurcation and Chaos, vol. 15, no. 2, pp. 567–604, February 2005.

    Article  MATH  MathSciNet  Google Scholar 

  23. M. di Bernardoa, A. Nordmark, and G. Olivar, “Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems,” Physica D, vol. 237, no. 1, pp. 119–136, January 2008.

    Article  MathSciNet  Google Scholar 

  24. Y. Lin, E. D. Sontag, and Y. Wang, “Recent results on Lyapunov-theoretic techniques for nonlinear stability,” Proc. of American Control Conference, pp. 1771–1775, 1994.

    Google Scholar 

  25. K. Sandra and C. S. Anibal, “Homoclinic bifurcations in Chua’s circuit,” Physica A, vol. 262, no. 1–2, pp. 144–152, January 1999.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, China

    Shi-Hui Fu

  2. Department of Dynamics and Control, Beihang University, Bei**g, 100191, China

    Qi-Shao Lu

Authors
  1. Shi-Hui Fu
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  2. Qi-Shao Lu
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Corresponding author

Correspondence to Shi-Hui Fu.

Additional information

Recommended by Editor Ju Hyun Park.

This journal was supported by China Natural Sciences Foundation (Grant No. U1204106 and 11372282) and National Key Projects (Grant Nos. 2012CB821200 and 2012CB821202).

Shi-Hui Fu is received her Ph.D. degree in Engineering from Beihang University in 2006. Her research interests include nonlinear dynamics, control, and bifurcation.

Qi-Shao Lu is a Professor of the Department of Dynamics and Control of School of Aeronautical Science and Engineering, Beihang University (BUAA) in Bei**g, China. He is the Vice Chief-Editor of Acta Mechanics Sinica, as well as members of some academic institutes. His research interests include nonlinear dynamics and control in nonlinear vibration systems, elastic structures, rotor systems and multi-body systems, neuro dynamics, nonlinear differential equations, and so on. Prof. Lu has visited many universities and institutes in USA, UK, Canada, Russia and Hong Kong. He won the National Natural Science Award of China (2nd class) in 2003 and other prizes from Chinese government and universities. He has been the Chair of the Second International Conference on Dynamics, Vibration and Control (ICDVC-2006) and the Co-chair of the First International Conference on Cognitive Neuro dynamics (ICCN-2009), and so on.

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Fu, SH., Lu, QS. Set stability of controlled Chua’s circuit under a non-smooth controller with the absolute value. Int. J. Control Autom. Syst. 12, 507–517 (2014). https://doi.org/10.1007/s12555-013-0372-6

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  • DOI: https://doi.org/10.1007/s12555-013-0372-6

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