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Journal of Control Science and Engineering
Volume 2018, Article ID 3298286, 21 pages
https://doi.org/10.1155/2018/3298286
Research Article

Approximate Prediction-Based Control Method for Nonlinear Oscillatory Systems with Applications to Chaotic Systems

1Universidade Estadual de Santa Cruz (UESC), 45662-900 Ilhéus, BA, Brazil
2Lab. J.-L. Lions UMR CNRS, Inria, UPMC University Paris 06, Sorbonne Universités, 7598 Paris, France
3Fundação Getúlio Vargas, Rio de Janeiro, RJ, Brazil
4Instituto Tecnológico de Aeronáutica (ITA), 12228-900 São José dos Campos, SP, Brazil

Correspondence should be addressed to Thiago P. Chagas; moc.liamg@sagahcht

Received 22 October 2017; Revised 15 December 2017; Accepted 19 December 2017; Published 1 March 2018

Academic Editor: Sundarapandian Vaidyanathan

Copyright © 2018 Thiago P. Chagas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Mesquita, E. L. Rempel, and K. H. Kienitz, “Bifurcation analysis of attitude control systems with switching-constrained actuators,” Nonlinear Dynamics, vol. 51, no. 1-2, pp. 207–216, 2008. View at Publisher · View at Google Scholar · View at Scopus
  2. Y. C. Fung, An Introduction to the Theory of Aeroelasticity, Dover Publications, 2002.
  3. V. P. Zhuravlev and D. M. Klimov, “Theory of the shimmy phenomenon,” Mechanics of Solids, vol. 45, no. 3, pp. 324–330, 2010. View at Publisher · View at Google Scholar · View at Scopus
  4. L. F. R. Turci, E. E. N. Macau, and T. Yoneyama, “Efficient chaotic based satellite power supply subsystem,” Chaos, Solitons & Fractals, vol. 42, no. 1, pp. 396–407, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. D. T. Schmitt and P. C. Ivanov, “Fractal scale-invariant and nonlinear properties of cardiac dynamics remain stable with advanced age: a new mechanistic picture of cardiac control in healthy elderly,” American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, vol. 293, no. 5, pp. R1923–R1937, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. G. F. Fussmann, S. P. Ellner, K. W. Shertzer, and J. Hairston N.G., “Crossing the hopf bifurcation in a live predator-prey system,” Science, vol. 290, no. 5495, pp. 1358–1360, 2000. View at Publisher · View at Google Scholar · View at Scopus
  7. A. C.-L. Chian, E. L. Rempel, and C. Rogers, “Complex economic dynamics: chaotic saddle, crisis and intermittency,” Chaos, Solitons & Fractals, vol. 29, no. 5, pp. 1194–1218, 2006. View at Publisher · View at Google Scholar · View at Scopus
  8. R. L. Devaney, A First Course In Chaotic Dynamical Systems, Avalon Publishing, 1992. View at MathSciNet
  9. M. A. F. Sanjuán and C. Grebogi, Recent Progress in Controlling Chaos, World Scientific Publishing Company, 2010.
  10. P. Cvitanović, “Invariant measurement of strange sets in terms of cycles,” Physical Review Letters, vol. 61, no. 24, pp. 2729–2732, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  11. V. Franceschini, C. Giberti, and Z. M. Zheng, “Characterization of the Lorenz attractor by unstable periodic orbits,” Nonlinearity, vol. 6, no. 2, pp. 251–258, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Physics Letters A, vol. 170, no. 6, pp. 421–428, 1992. View at Publisher · View at Google Scholar · View at Scopus
  13. K. Pyragas, “Delayed feedback control of chaos,” Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, vol. 364, no. 1846, pp. 2309–2334, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. T. Ushio, “Limitation of delayed feedback control in nonlinear discrete-time systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 43, no. 9, pp. 815-816, 1996. View at Publisher · View at Google Scholar · View at Scopus
  15. S. Yamamoto, T. Hino, and T. Ushio, “Dynamic delayed feedback controllers for chaotic discrete-time systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 6, pp. 785–789, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. S. Yamamoto, T. Hino, and T. Ushio, “Delayed feedback control with a minimal-order observer for stabilization of chaotic discrete-time systems,” International Journal of Bifurcation and Chaos, vol. 12, no. 5, pp. 1047–1055, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J. Zhu and Y.-P. Tian, “Necessary and sufficient conditions for stabilizability of discrete-time systems via delayed feedback control,” Physics Letters A, vol. 343, no. 1-3, pp. 95–107, 2005. View at Publisher · View at Google Scholar · View at Scopus
  18. B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, and E. Schöll, “Refuting the odd-number limitation of time-delayed feedback control,” Physical Review Letters, vol. 98, no. 11, Article ID 114101, 2007. View at Publisher · View at Google Scholar · View at Scopus
  19. W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hövel, and E. Schöll, “Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 76, no. 2, Article ID 026210, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. B. Fiedler, V. Flunkert, P. Hövel, and E. Schöll, “Beyond the odd number limitation of time-delayed feedback control of periodic orbits,” The European Physical Journal Special Topics, vol. 191, no. 1, pp. 53–70, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. T. Ushio and S. Yamamoto, “Prediction-based control of chaos,” Physics Letters A, vol. 264, no. 1, pp. 30–35, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. T. P. Chagas, P.-A. Bliman, and K. H. Kienitz, “New feedback laws for stabilization of unstable periodic orbits,” IFAC Proceedings Volumes, vol. 43, no. 14, pp. 1005–1010, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. M. P. F. de Córdoba and E. Liz, “Prediction-based control of chaos and a dynamic Parrondo’s paradox,” Physics Letters A, vol. 377, no. 10-11, pp. 778–782, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. E. Braverman, C. Kelly, and A. Rodkina, “Stabilisation of difference equations with noisy prediction-based control,” Physica D: Nonlinear Phenomena, vol. 326, pp. 21–31, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. A. Boukabou, A. Chebbah, and N. Mansouri, “Predictive control of continuous chaotic systems,” International Journal of Bifurcation and Chaos, vol. 18, no. 2, pp. 587–592, 2008. View at Publisher · View at Google Scholar · View at Scopus
  26. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. M. A. Khelifa and A. Boukabou, “Design of an intelligent prediction-based neural network controller for multi-scroll chaotic systems,” Applied Intelligence, vol. 45, no. 3, pp. 793–807, 2016. View at Publisher · View at Google Scholar · View at Scopus
  28. T. P. Chagas, P.-A. Bliman, and K. H. Kienitz, “A new method for stabilizing unstable periodic orbits of continuous-time systems. Application to control of chaos,” in Proceedings of the 51st IEEE Conference on Decision and Control (CDC '12), pp. 2146–2151, Maui, Hawaii, USA, December 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. T. P. Chagas, B. A. Toledo, E. L. Rempel, A. C. Chian, and J. A. Valdivia, “Optimal feedback control of the forced van der Pol system,” Chaos, Solitons & Fractals, vol. 45, no. 9-10, pp. 1147–1156, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  30. B. A. Finlayson, The Method of Weighted Residuals and Variational Principles, with Application in Fluid Mechanics, Heat And Mass Transfer, vol. 87 of Mathematics in Science and Engineering, Academic Press, 1972. View at MathSciNet
  31. J. Viladsen and M. L. Michelsen, Solution of Differential Equation Models by Polynomial Approximation, Prentice Hall, 1978.
  32. E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems, Springer-Verlag, 2008.
  33. T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  34. F. Asenjo, B. A. Toledo, V. Muoz, J. Rogan, and J. A. Valdivia, “Optimal control in a noisy system,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 18, no. 3, Article ID 033106, 2008. View at Publisher · View at Google Scholar · View at Scopus
  35. S. P. Han, “A globally convergent method for nonlinear programming,” Journal of Optimization Theory and Applications, vol. 22, no. 3, pp. 297–309, 1977. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Nonlinear Programming 3: Proceedings of The Special Interest Group on Mathematical Programming Symposium, vol. 3, Academic Press, 1978.
  37. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization, Academic Press, 1981. View at MathSciNet
  38. M. J. Powell, A Fast Algorithm for Nonlinearly Constrained Optimization Calculations, vol. 630 of Lecture Notes in Mathematics, Springer, 1978.
  39. E. F. Camacho and C. B. Alba, Model Predictive Control, Advanced Textbooks in Control and Signal Processing, Springer-Verlag, 2nd edition, 2007.