Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 512104, 10 pages
http://dx.doi.org/10.1155/2015/512104
Research Article

A Comparative Analysis of Laguerre-Based Approximators to the Grünwald-Letnikov Fractional-Order Difference

Department of Electrical, Control and Computer Engineering, Opole University of Technology, Ulica Sosnkowskiego 31, 45-272 Opole, Poland

Received 17 August 2014; Revised 30 December 2014; Accepted 31 December 2014

Academic Editor: Baocang Ding

Copyright © 2015 Rafał Stanisławski 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. R. S. Barbosa and J. A. T. Machado, “Implementation of discrete-time fractional- order controllers based on LS approximations,” Acta Polytechnica Hungarica, vol. 3, no. 4, pp. 5–22, 2006. View at Google Scholar · View at Scopus
  2. Y. Chen, B. Vinagre, and I. Podlubny, “A new discretization methodfor 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.
  3. B. M. Vinagre, I. Podlubny, A. Hernandez, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 945–950, 2000. View at Google Scholar
  4. P. J. Kootsookos and R. C. Williamson, “FIR approximation of fractional sample delay systems,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 43, no. 3, pp. 269–271, 1996. View at Publisher · View at Google Scholar · View at Scopus
  5. C. X. Jiang, J. E. Carletta, and T. T. Hartley, “Implementation of fractional-order operators on field programmable gate arrays,” in Advances in Fractional Calculus, pp. 333–346, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. 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
  7. 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 (MMAR '12), pp. 64–69, Miedzyzdroje, Poland, August 2012. View at Publisher · View at Google Scholar · View at Scopus
  8. R. Stanisławski and K. J. Latawiec, “Normalized finite fractional differences: computational and accuracy breakthroughs,” International Journal of Applied Mathematics and Computer Science, vol. 22, no. 4, pp. 907–919, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. M. A. Al-Alaoui, “Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems,” Electrical Engineering, vol. 90, no. 6, pp. 455–467, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. M. A. Al-Alaoui, “Simulation and discretization of fractional-order systems,” in Proceedings of the International Conference on Genetics and Evlutionary Methods, pp. 249–255, Las Vegas, Nev, USA, 2009.
  11. M. A. Al-Alaoui, “Class of digital integrators and differentiators,” IET Signal Processing, vol. 5, no. 2, pp. 251–260, 2011. View at Publisher · View at Google Scholar · View at Scopus
  12. Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 3, pp. 363–367, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Y. Chen, B. M. Vinagre, and I. Podlubny, “Continued fraction expansion approaches to discretizing fractional order derivatives–-an expository review,” Nonlinear Dynamics, vol. 38, no. 1–4, pp. 155–170, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. Gupta and R. Yadav, “New improved fractional order differentiator models based on optimized digital differentiators,” The Scientific World Journal, vol. 2014, Article ID 741395, 11 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  15. B. T. Krishna, “Studies on fractional order differentiators and integrators: a survey,” Signal Processing, vol. 91, no. 3, pp. 386–426, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. R. Stanisławski, “New Laguerre filter approximators to the Grünwald-LETnikov fractional difference,” Mathematical Problems in Engineering, vol. 2012, Article ID 732917, 21 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. R. Stanisławski, Advances in Modeling of Fractional Difference Systems—New Accuracy, Stability and Computational Results, vol. 243 of Studia i Monografie, Opole University of Technology Press, Opole, Poland, 2013.
  18. R. Stanislawski, K. J. Latawiec, and M. Lukaniszyn, “The steady-state error issue in Laguerre-based modeling of the GL fractional difference,” in Proceedings of the IEEE 22nd International Symposium on Industrial Electronics (ISIE '13), pp. 1–6, Taipei, Taiwan, May 2013. View at Publisher · View at Google Scholar · View at Scopus
  19. Y. Ferdi, “Computation of fractional order derivative and integral via power series expansion and signal modeling,” Nonlinear Dynamics, vol. 46, no. 1-2, pp. 1–15, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. G. Maione, “On the Laguerre rational approximation to fractional discrete derivative and integral operators,” IEEE Transactions on Automatic Control, vol. 58, no. 6, pp. 1579–1585, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. Z. Liao, “A new definition of fractional derivatives based on truncated left-handed Grunwald-Letnikov formula with 0<α<1 and mediancorrection,” Abstract and Applied Analysis, vol. 2014, Article ID 914386, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. R. Stanisławski, W. P. 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 (MMAR '11), pp. 142–145, Miedzyzdroje, Poland, August 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. C. Monje, Y. Chen, B. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls, Springer, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  24. P. Ostalczyk, “Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains,” International Journal of Applied Mathematics and Computer Science, vol. 22, no. 3, pp. 533–538, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  25. R. Stanislawski, “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.
  26. M. Buslowicz, “Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix,” Bulletin of the Polish Academy of Sciences: Technical Sciences, vol. 60, no. 4, pp. 809–814, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. M. Buslowicz 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 MathSciNet · View at Scopus
  28. 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 Google Scholar · View at Scopus
  29. 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 MathSciNet · View at Scopus
  30. I. Podlubny, Fractional Differential Equations, Academic Press, Orlando, Fla, USA, 1999. View at MathSciNet
  31. 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 · View at MathSciNet · View at Scopus
  32. P. M. van den Hof, P. S. Heuberger, and J. Bokor, “System identification with generalized orthonormal basis functions,” Automatica, vol. 31, no. 12, pp. 1821–1834, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  33. Y. Q. Chen and K. L. Vinagre, “A new IIR-type digital fractional order differentiator,” Signal Processing, vol. 83, no. 11, pp. 2359–2365, 2003. View at Publisher · View at Google Scholar · View at Scopus
  34. M. Siami, M. Saleh Tavazoei, and M. Haeri, “Stability preservation analysis in direct discretization of fractional order transfer functions,” Signal Processing, vol. 91, no. 3, pp. 508–512, 2011. View at Publisher · View at Google Scholar · View at Scopus
  35. B. M. Vinagre, Y. Q. Chen, and I. Petráš, “Two direct Tustin discretization methods for fractional-order differentiator/integrator,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 340, no. 5, pp. 349–362, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus