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The Scientific World Journal
Volume 2014, Article ID 521625, 8 pages
http://dx.doi.org/10.1155/2014/521625
Research Article

On Fractional Model Reference Adaptive Control

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

Received 19 August 2013; Accepted 30 October 2013; Published 16 January 2014

Academic Editors: A. Atangana, S. C. O. Noutchie, and A. Secer

Copyright © 2014 Bao Shi 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. I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, 2011.
  2. R. Caponetto, Fractional Order Systems: Modeling and Control Applications, vol. 72, World Scientific, 2010.
  3. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198, Academic Press, 1998.
  4. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at Publisher · View at Google Scholar · View at Scopus
  5. C. P. Li and F. R. Zhang, “A survey on the stability of fractional differential equations,” European Physical Journal, vol. 193, no. 1, pp. 27–47, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. Y. Q. Chen, I. Petráš, and D. Xue, “Fractional order control—a tutorial,” in Proceedings of the American Control Conference (ACC '09), pp. 1397–1411, St. Louis, Mo, USA, June 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. B. J. Lurie, Three-Parameter Tunable Tilt-Integral-Derivative (TID) Controller, Google Patents, 1994.
  8. A. Oustaloup, X. Moreau, and M. Nouillant, “The crone suspension,” Control Engineering Practice, vol. 4, no. 8, pp. 1101–1108, 1996. View at Publisher · View at Google Scholar · View at Scopus
  9. I. Podlubny, “Fractional-order systems and PIλDμ-controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999. View at Publisher · View at Google Scholar · View at Scopus
  10. H.-F. Raynaud and A. Zergaïnoh, “State-space representation for fractional order controllers,” Automatica, vol. 36, no. 7, pp. 1017–1021, 2000. View at Publisher · View at Google Scholar · View at Scopus
  11. D. Xue and Y. Q. Chen, “A comparative introduction of four fractional order controllers,” in Proceedings of the 4th World Congress on Intelligent Control and Automation, pp. 3228–3235, Shanghai, China, June 2002. View at Scopus
  12. J. Yuan, B. Shi, W. Ji et al., “Sliding mode control of the fractional order unified chaotic system,” Abstract and Applied Analysis, vol. 2013, Article ID 397504, 13 pages, 2013. View at Publisher · View at Google Scholar
  13. C. Yin, S.-M. Zhong, and W.-F. Chen, “Design of sliding mode controller for a class of fractional-order chaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 356–366, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. A. Razminia and D. Baleanu, “Complete synchronization of commensurate fractional order chaotic systems using sliding mode control,” Mechatronics, vol. 23, no. 7, pp. 873–879, 2013. View at Google Scholar
  15. S. Bao, J. Yuan, and D. Chao, “Pseudo-state sliding mode control of fractional SISO nonlinear systems,” Advances in Mathematical Physics, vol. 2013, Article ID 918383, 7 pages, 2013. View at Publisher · View at Google Scholar
  16. M. P. Aghababa, “Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2670–2681, 2012. View at Publisher · View at Google Scholar · View at Scopus
  17. D. M. Senejohnny and H. Delavari, “Active sliding observer scheme based fractional chaos synchronization,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4373–4383, 2012. View at Publisher · View at Google Scholar · View at Scopus
  18. M. P. Aghababa, “Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique,” Nonlinear Dynamics, vol. 69, no. 1-2, pp. 247–261, 2011. View at Google Scholar
  19. S. Dadras and H. R. Momeni, “Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty,” Journal of Science Communication, vol. 17, no. 1, pp. 367–377, 2012. View at Google Scholar
  20. A. Si-Ammour, S. Djennoune, and M. Bettayeb, “A sliding mode control for linear fractional systems with input and state delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2310–2318, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. M. R. Faieghi, H. Delavari, and D. Baleanu, “A note on stability of sliding mode dynamics in suppression of fractional-order chaotic systems,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 832–837, 2013. View at Google Scholar
  22. M. Pourmahmood, S. Khanmohammadi, and G. Alizadeh, “Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 7, pp. 2853–2868, 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. R. Zhang and S. Yang, “Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach,” Nonlinear Dynamics, vol. 71, no. 1-2, pp. 269–278, 2012. View at Google Scholar
  24. Y. Jian, S. Bao, Z. Xiaoyun et al., “Sliding mode control of the fractional order unified chaotic system,” Abstract and Applied Analysis, vol. 2013, Article ID 397504, 13 pages, 2013. View at Publisher · View at Google Scholar
  25. B. M. Vinagre, I. Petráš, I. Podlubny, and Y. Q. Chen, “Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 269–279, 2002. View at Publisher · View at Google Scholar · View at Scopus
  26. S. Ladaci and A. Charef, “On fractional adaptive control,” Nonlinear Dynamics, vol. 43, no. 4, pp. 365–378, 2006. View at Publisher · View at Google Scholar · View at Scopus
  27. Z. M. Odibat, “Adaptive feedback control and synchronization of non-identical chaotic fractional order systems,” Nonlinear Dynamics, vol. 60, no. 4, pp. 479–487, 2010. View at Publisher · View at Google Scholar · View at Scopus
  28. C. Li and Y. Tong, “Adaptive control and synchronization of a fractionalorder chaotic system,” Pramana, vol. 80, no. 4, pp. 583–592, 2013. View at Google Scholar
  29. L. Chen, S. Wei, Y. Chai, and R. Wu, “Adaptive projective synchronization between two different fractional-order chaotic systems with fully unknown parameters,” Mathematical Problems in Engineering, vol. 2012, Article ID 916140, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  30. O. P. Agrawal, “A general formulation and solution scheme for fractional optimal control problems,” Nonlinear Dynamics, vol. 38, no. 1–4, pp. 323–337, 2004. View at Publisher · View at Google Scholar · View at Scopus
  31. Z. D. Jelicic and N. Petrovacki, “Optimality conditions and a solution scheme for fractional optimal control problems,” Structural and Multidisciplinary Optimization, vol. 38, no. 6, pp. 571–581, 2009. View at Publisher · View at Google Scholar · View at Scopus
  32. S. Djennoune and M. Bettayeb, “Optimal synergetic control for fractional-order systems,” Automatica, vol. 49, no. 7, pp. 2243–2249, 2013. View at Google Scholar
  33. J. C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, “A Lyapunov approach to the stability of fractional differential equations,” Signal Processing, vol. 91, no. 3, pp. 437–445, 2011. View at Publisher · View at Google Scholar · View at Scopus
  34. J. Yuan, S. Bao, and J. Wenqiang, “Adaptive sliding mode control of a novel class of fractional chaotic systems,” Advances in Mathematical Physics, vol. 2013, Article ID 576709, 13 pages, 2013. View at Publisher · View at Google Scholar
  35. J. J. E. Slotine and W. Li, Applied Nonlinear Control, vol. 1, Prentice Hall, Upper Saddle River, NJ, USA, 1991.
  36. K. J. Åström and B. Wittenmark, Adaptive Control, Courier Dover, 2008.
  37. J. Sabatier, O. P. Agrawal, and J. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007.