Abstract and Applied Analysis

Stability Analysis and Control of a New Smooth Chua's System

Guopeng Zhou, Jinhua Huang, Xiaoxin Liao, and Shijie Cheng

Full-text: Open access

Abstract

This paper is concerned with the stability analysis and control of a new smooth Chua's system. Firstly, the chaotic characteristic of the system is confirmed with the aid of the Lyapunov exponents. Secondly, it is proved that the system has globally exponential attractive set and positive invariant set. For the three unstable equilibrium points of the system, a linear controller is designed to globally exponentially stabilize the equilibrium points. Then, a linear controller and an adaptive controller are, respectively, proposed so that two similar types of smooth Chua's systems are globally synchronized, and the estimation errors of the uncertain parameters converge to zero as t tends to infinity. Finally, the numerical simulations are also presented.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 620286, 10 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393447695

Digital Object Identifier
doi:10.1155/2013/620286

Mathematical Reviews number (MathSciNet)
MR3049413

Zentralblatt MATH identifier
1271.93114

Citation

Zhou, Guopeng; Huang, Jinhua; Liao, Xiaoxin; Cheng, Shijie. Stability Analysis and Control of a New Smooth Chua's System. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 620286, 10 pages. doi:10.1155/2013/620286. https://projecteuclid.org/euclid.aaa/1393447695


Export citation

References

  • L. O. Chua, “The genesis of Chua's circuit,” Archiv fur Elektronik und Ubertragungstechnik, vol. 46, no. 4, pp. 250–257, 1992.
  • L. P. Shil'nikov, “Chua's circuit: rigorous results and future problems,” International Journal of Bifurcation and Chaos, vol. 4, no. 3, pp. 489–519, 1994.
  • L. O. Chua, C. W. Wu, A. Huang, and G.-Q. Zhong, “A universal circuit for studying and generating chaos I: routes to chaos,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 10, pp. 732–744, 1993.
  • L. O. Chua, “A zoo of strange attractor from the canonical Chua's circuits,” in Proceedings of the 35th Midwest Symposium on Circuits and Systems, vol. 2, pp. 916–926, 1992.
  • Y. F. Wang and J. G. Jiang, “The chaotic phenomena analysis of asymmetric nonlinear Chua's circuit,” Systems Engineering and Electronics, vol. 29, no. 12, pp. 2029–2031, 2007.
  • K. S. Tang, K. F. Man, G. Q. Zhong, and G. Chen, “Generating chaos via $x\vert x\vert $,” IEEE Transactions on Circuits and Systems I, vol. 48, no. 5, pp. 636–641, 2001.
  • X. Liao, P. Yu, S. Xie, and Y. Fu, “Study on the global property of the smooth Chua's system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 10, pp. 2815–2841, 2006.
  • H. M. Deng, T. Li, Q. H. Wang et al., “Shaped Chua's chaotic system and its synchronization problem,” Systems Engineering and Electronics, vol. 31, no. 3, pp. 638–641, 2009.
  • G. A. Leonov, A. I. Bunin, and N. Koksch, “Attractor localization of the Lorenz system,” Zen and the Art of Motorcycle Maintenance, vol. 67, no. 12, pp. 649–656, 1987.
  • F. Zhou, Z. Y. Wang, G. P. Zhou, and F. X. Zhen, “Synchronization of two unsmooth Chua's circuits,” Mathematica Applicata, vol. 25, no. 2, pp. 382–388, 2012.
  • X. X. Liao, H. G. Luo, Y. L. Fu et al., “Positive invariant set and the globally exponentially attractive set of Lorenz system group,” Science in China-E, vol. 37, no. 6, pp. 757–769, 2007.
  • X. X. Liao, “New results for globally attractive set and positive invariant set of Lorenz system and application of chaos control and synchronization,” Science in China-E, vol. 34, no. 12, pp. 1404–1419, 2004.
  • J. G. Jian, X. L. Deng, and J. F. Wang, “New results of globally exponentially attractive set and synchronization controlling of the Qi chaotic system,” Advances in N Eural Networks-ISNN, pp. 643–650, 2010.
  • F. Q. Wang and C. X. Liu, “A new criterion for chaos and hyperchaos synchronization using linear feedback control,” Physics Letters A, vol. 360, no. 2, pp. 274–278, 2006.
  • H. R. Koofigar, F. Sheikholeslam, and S. Hosseinnia, “Robust adaptive synchronization for a general class of uncertain chaotic systems with application to Chuas circuit,” Chaos, vol. 21, no. 4, Article ID 043134, 2011.
  • H. G. Zhang, W. Huang, Z. L. Wang et al., “Adaptive synchronization between two different chaotic systems with unknown parameters,” Physics Letters A, vol. 350, no. 5-6, pp. 363–366, 2006.
  • F. Zhou, Z. Y. Wang, and G. P. Zhou, “Adaptive chronization of some modified smooth Chua's circuit,” Journal of Nanjing University of Information Science and Techology, vol. 4, no. 2, pp. 186–189, 2012.
  • X. X. Liao, Theory Methods and Application of Stability, Huazhong University of Science and Technology Press, 2nd edition, 2010.
  • C. L. Phillips and J. Parr, Feedback Control Systems, Prentice Hall, 5th edition, 2010.
  • M. Kristic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, NY, USA, 1995.
  • L. Liu, Z. Chen, and J. Huang, “Parameter convergence and minimal internal model with an adaptive output regulation problem,” Automatica, vol. 45, no. 5, pp. 1306–1311, 2009.
  • S. Tong and Y. Li, “Observer-based fuzzy adaptive control for strict-feedback nonlinear systems,” Fuzzy Sets and Systems, vol. 160, no. 12, pp. 1749–1764, 2009.
  • G. Zhou and C. Wang, “Deterministic learning from control of nonlinear systems with disturbances,” Progress in Natural Science, vol. 19, no. 8, pp. 1011–1019, 2009.