About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 732917, 21 pages
http://dx.doi.org/10.1155/2012/732917
Research Article

New Laguerre Filter Approximators to the Grünwald-Letnikov Fractional Difference

Institute of Control and Computer Engineering, Opole University of Technology, ul. Proszkowska 76, 45-758 Opole, Poland

Received 9 September 2012; Accepted 16 November 2012

Academic Editor: Alex Elias-Zuniga

Copyright © 2012 Rafał Stanisławski. 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. H. Delavari, A. N. Ranjbar, R. Ghaderi, and S. Momani, “Fractional order control of a coupled tank,” Nonlinear Dynamics, vol. 61, no. 3, pp. 383–397, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. I. Petráš and B. Vinagre, “Practical application of digital fractionalorder controller to temperature control,” Acta Montanistica Slovaca, vol. 7, no. 2, pp. 131–137, 2002.
  3. D. Riu, N. Retiére, and M. Ivanes, “Turbine generator modeling by non-integer order systems,” in Proceedings of the International Conference on Electric Machines and Drives, pp. 185–187, Cambridge, Mass, USA, 2001.
  4. V. Zaborowsky and R. Meylaov, “Informational network traffic model based on fractional calculus,” in Proceedings of International Conference Info-tech and Info-net (ICII '01), vol. 1, pp. 58–63, Beijing, China, 2001.
  5. S. Hu, Z. Liao, and W. Chen, “Reducing noises and artifacts simultaneously of low-dosed X-ray computed tomography using bilateral filter weighted by Gaussian filtered sinogram,” Mathematical Problems in Engineering, vol. 2012, Article ID 138581, 14 pages, 2012. View at Publisher · View at Google Scholar
  6. C. Junyi and C. Binggang, “Fractional-order control of pneumatic position servosystems,” Mathematical Problems in Engineering, vol. 2011, Article ID 287565, 14 pages, 2011. View at Publisher · View at Google Scholar
  7. T. Kaczorek, “Practical stability of positive fractional discrete-time linear systems,” Bulletin of the Polish Academy of Sciences, Technical Sciences, vol. 56, no. 4, pp. 313–317, 2008. View at Scopus
  8. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  9. D. Sierociuk and A. Dzieliński, “Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation,” International Journal of Applied Mathematics and Computer Science, vol. 16, no. 1, pp. 129–140, 2006.
  10. Ch. Lubich, “Discretized fractional calculus,” SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704–719, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. D. Ortigueira, “Introduction to fractional linear systems. Part 2: discrete-time case,” IEE Proceedings, vol. 147, no. 1, pp. 71–78, 2000. View at Publisher · View at Google Scholar · View at Scopus
  12. P. Ostalczyk, “The non-integer difference of the discrete-time function and its application to the control system synthesis,” International Journal of Systems Science, vol. 31, no. 12, pp. 1551–1561, 2000. View at Publisher · View at Google Scholar · View at Scopus
  13. I. Petráš, L. Dorčák, and I. Koštial, “The modelling and analysis of fractional-order control systems in discrete domain,” in Proceedings of the International Conference on Computational Creativity (ICCC '00), pp. 257–260, High Tatras, Slovakia, 2000.
  14. A. Dzieliński and D. Sierociuk, “Stability of discrete fractional order state-space systems,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1543–1556, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. R. Barbosa and J. Machado, “Implementation of discrete-time fractional-order controllers based on LS approximations,” Acta Polytechnica Hungarica, vol. 3, pp. 5–22, 2006.
  16. Y. Chen, B. Vinagre, and I. Podlubny, “A new discretization method for fractional order differentiators via continued fraction expansion,” in Proceedings of ASME Design Engineering Technical Conferences (DETC '03), vol. 340, pp. 349–362, Chicago, Ill, USA, 2003.
  17. B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 231–248, 2000. View at Zentralblatt MATH
  18. C. C. Tseng, S. C. Pei, and S. C. Hsia, “Computation of fractional derivatives using Fourier transform and digital FIR differentiator,” Signal Processing, vol. 80, no. 1, pp. 151–159, 2000. View at Publisher · View at Google Scholar · View at Scopus
  19. C. Monje, Y. Chen, B. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls, Springer, London, UK, 2010.
  20. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  21. R. Stanisławski, “Identification of open-loop stable linear systems using fractional orthonormal basis functions,” in Proceedings of the 14th International Conference on Methods and Models in Automation and Robotics, pp. 935–985, Miedzyzdroje, Poland, 2009.
  22. R. Stanisławski and K. J. Latawiec, “Normalized finite fractional differences: the computational and accuracy breakthroughs,” International Journal of Applied Mathematics and Computer Science, vol. 22, no. 4, 2012.
  23. R. Stanisławski and K. J. Latawiec, “Modeling of open-loop stable linear systems using a combination of a finite fractional derivative and orthonormal basis functions,” in Proceedings of the 15th International Conference on Methods and Models in Automation and Robotics (MMAR '10), pp. 411–414, Miedzyzdroje, Poland, August 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. G. Maione, “A digital, noninteger order differentiator using Laguerre orthogonal sequences,” International Journal of Intelligent Control and Systems, vol. 11, pp. 77–81, 2006.
  25. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
  26. M. Busłowicz and T. Kaczorek, “Simple conditions for practical stability of positive fractional discrete-time linear systems,” International Journal of Applied Mathematics and Computer Science, vol. 19, no. 2, pp. 263–269, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. R. Stanisławski, W. Hunek, and K. J. Latawiec, “Finite approximations of a discrete-time fractional derivative,” in Proceedings of the 16th International Conference on Methods and Models in Automation and Robotics, pp. 142–145, Miedzyzdroje, Poland, 2011.
  28. P. S. C. Heuberger, P. M. J. Van den Hof, and B. Wahlberg, Modelling and Identification with Rational Orthogonal Basis Functions, Springer, London, UK, 2005. View at Publisher · View at Google Scholar
  29. P. M. J. Van den Hof, P. S. C. Heuberger, and J. Bokor, “System identification with generalized orthonormal basis functions,” Automatica, vol. 31, no. 12, pp. 1821–1834, 1995, Special issue on trends in system identification (Copenhagen, 1994). View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. C. Boukis, D. P. Mandic, A. G. Constantinides, and L. C. Polymenakos, “A novel algorithm for the adaptation of the pole of Laguerre filters,” IEEE Signal Processing Letters, vol. 13, no. 7, pp. 429–432, 2006. View at Publisher · View at Google Scholar · View at Scopus
  31. S. T. Oliveira, “Optimal pole conditions for Laguerre and twoparameter Kautz models: a survey of known results,” in Proceedings of the 12th IFAC Symposium on System Identification (SYSID '00), pp. 457–462, Santa Barbara, Calif, USA, 2000.
  32. K. J. Latawiec, R. Stanisławski, W. P. Hunek, and M. Łukaniszyn, “Adaptive finite fractional difference with a time-varying forgetting factor,” in Proceedings of the 17th International Conference on Methods and Models in Automation and Robotics, pp. 64–69, Miedzyzdroje, Poland, 2012.