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Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives—an Expository Review

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Abstract

This paper attempts to present an expository review of continued fraction expansion (CFE) based discretization schemes for fractional order differentiators defined in continuous time domain. The schemes reviewed are limited to infinite impulse response (IIR) type generating functions of first and second orders, although high-order IIR type generating functions are possible. For the first-order IIR case, the widely used Tustin operator and Al-Alaoui operator are considered. For the second order IIR case, the generating function is obtained by the stable inversion of the weighted sum of Simpson integration formula and the trapezoidal integration formula, which includes many previous discretization schemes as special cases. Numerical examples and sample codes are included for illustrations.

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References

  1. Al-Alaoui, M. A. ‘Novel digital integrator and differentiator’, Electronics Letters29(4), 1993, 376–378.

    Google Scholar 

  2. Al-Alaoui, M. A. ‘A class of second-order integrators and low-pass differentiators’, IEEE Transactions on Circuit and Systems I: Fundamental Theory and Applications42(4), 1995, 220–223.

    Google Scholar 

  3. Al-Alaoui, M. A. ‘Filling the gap between the bilinear and the backward difference transforms: An interactive design approach’, International Journal of Electrical Engineering Education34(4), 1997, 331–337.

    Google Scholar 

  4. Axtell, M. and Bise, E. M. ‘Fractional calculus applications in control systems’, in Proceedings of the IEEE 1990 National Aerospace and Electronics Conference, New York, 1990, pp. 563–566.

  5. Chen, Y. Q. and Vinagre, B. M. ‘A new IIR-type digital fractional order differentiator’, Signal Processing83(11), 2003, 2359–2365.

    Google Scholar 

  6. Chen, Y. Q. and Moore, K. L. ‘Discretization schemes for fractional order differentiators and integrators’, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications49(3), 2002, 363–367.

    Google Scholar 

  7. Chen, Y. Q., Vinagre, B. M., and Podlubny, I. ‘A new discretization method for fractional order differentiators via continued fraction expansion’, in Proceedings of the First Symposium on Fractional Derivatives and Their Applications at The 19th Biennial Conference on Mechanical Vibration and Noise, the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (ASME DETC2003), DETC2003/VIB-48391, Chicago,Illinois, 2003, pp. 1–8,

  8. Machado, J. A. T. (guest editor), ‘Special issue on fractional calculus and applications’, Nonlinear Dynamics29, 2002, 1--385.

    Google Scholar 

  9. Lubich, C. H., ‘Discretized fractional calculus’, SIAM Journal on Mathematical Analysis17(3), 1986, 704–719.

    Google Scholar 

  10. Lurie, Boris J. ‘Three-parameter tunable tilt-integral-derivative (TID) controller’, US Patent US5371670, 1994.

  11. Machado, J. A. T. ‘Analysis and design of fractional-order digital control systems’, Journal of Systems Analysis, Modelling and Simulation27, 1997, 107–122.

    Google Scholar 

  12. Manabe, S., ‘The non-integer integral and its application to control systems’, JIEE (Japanese Institute of Electrical Engineers) Journal80(860), 1960, 589–597.

    Google Scholar 

  13. Manabe, S., ‘The non-integer integral and its application to control systems’, ETJ of Japan6(3/4), 1961, 83–87.

    Google Scholar 

  14. Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

    Google Scholar 

  15. Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.

    Google Scholar 

  16. Ortigueira, M. D. and Machado, J. A. T. (guest editors), ‘Special issue on fractional signal processing and applications’ Signal Processing83(11), 2003, 2285–2480.

    Google Scholar 

  17. Oustaloup, A., ‘Fractional order sinusoidal oscilators: Optimization and their use in highly linear FM modulators’, IEEE Transactions on Circuits and Systems28(10), 1981, 1007–1009.

    Google Scholar 

  18. Oustaloup, A., La dérivation non entière, HERMES, Paris, 1995.

    Google Scholar 

  19. Oustaloup, A., Levron, F., Nanot, F., and Mathieu, B. ‘Frequency band complex non integer differentiator: Characterization and synthesis’, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications47(1), 2000, 25–40.

    Google Scholar 

  20. Oustaloup, A. and Mathieu, B., Lacommande CRONE: du scalaire au multivariable, HERMES, Paris, 1999.

    Google Scholar 

  21. Oustaloup, A., Mathieu, B., and Lanusse, P., ‘The CRONE control of resonant plants: Application to a flexible transmission’, European Journal of Control1(2), 1995, pp. 113–121.

    Google Scholar 

  22. Oustaloup, A., Moreau, X., and Nouillant, M., ‘TheCRONE suspension’, Control Engineering Practice4(8), 1996, 1101–1108.

    Google Scholar 

  23. Oustaloup, A., Sabatier, J., and Lanusse, P., ‘From fractal robustness to CRONE control’, Fractionnal Calculus and Applied Analysis2(1), 1999, 1–30.

    Google Scholar 

  24. Petráš, I., ‘The fractional-order controllers: Methods for their synthesis and application’, Journal of Electrical Engineering50(9–10), 1999, 284–288.

    Google Scholar 

  25. Podlubny, I., Fractional-order systems and fractional-order controllers’, Technical Report UEF-03-94, Slovak Academy of Sciences. Institute of Experimental Physics, Department of Control Engineering. Faculty of Mining, University of Technology, Kosice, Slovak Republic, November 1994.

  26. Podlubny, I., ‘Fractional-order systems and PIλDμ-Controllers’, IEEE Transactions Automatic Control44(1), 1999, 208–214.

    Google Scholar 

  27. Raynaud, H.-F. and Zergaïnoh, A., ‘State-space representation for fractional order controllers’, Automatica36, 2000, 1017–1021.

    Google Scholar 

  28. Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives and Some of Their Applications. Nauka i Technika, Minsk, Russia, 1987.

    Google Scholar 

  29. Tseng, C.-C., ‘Design of fractional order digital FIR differentiator’, IEEE Signal Processing Letters8(3), 2001, 77–79.

    Google Scholar 

  30. Tseng, C.-C., Pei, S.-C., and Hsia, S.-C., ‘Computation of fractional derivatives using fourier transform and digital FIR differentiator’, Signal Processing80, 2000, 151–159.

    Google Scholar 

  31. Vinagre, B. M., Petras, I., Merchan, P., and Dorcak, L., ‘Two digital realisation of fractional controllers: Application to temperature control of a solid’, in Proceedings of the European Control Conference (ECC2001), Porto, Portugal, September 2001, pp. 1764–1767.

  32. Vinagre, B. M., Podlubny, I., Hernandez, A., and Feliu, V.‘On realization of fractional-order controllers’, in Proceedings of the Conference Internationale Francophoned’Automatique, Lille, France, July 2000.

  33. Vinagre, B. M., Podlubny, I., Hernandez, A., and Feliu, V.‘Some approximations of fractional order operators used in control theory and applications’, Fractional Calculus and Applied Analysis3(3), 2000, 231–248.

    Google Scholar 

  34. Vinagre, B. M., Chen, Y. Q., and Petras, I. ‘Two direct tustin discretization methods for fractional-order differentiator/integrator’, The Journal of Franklin Institute340(5) 2003, 349–362.

    Google Scholar 

  35. Vinagre, B. M. and Chen, Y. Q., ‘Lecture notes on fractional calculus applications in automatic control and robotics’in The 41st IEEE CDC2002 Tutorial Workshop No. 2, B. M. Vinagre and Y. Q. Chen (eds.), retrieved from http://mechatronics. ece.usu.edu/foc/cdc02_tw2_ln.pdf, Las Vegas, Nevada, 2002, pp. 1–310.

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Authors and Affiliations

  1. Center for Self-Organizing and Intelligent Systems (CSOIS), UMC 4160, College of Engineering, Utah State University, Logan, UT, 84322-4160, U.S.A.

    Yangquan Chen

  2. Department of Electronic and Electromechanical Engineering, Industrial Engineering School, University of Extremadura, 06071, Badajoz, Spain

    Blas M. Vinagre

  3. Department of Informatics and Process Control, Technical University of Košice, 04200, Košice, Slovak Republic

    Igor Podlubny

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  1. Yangquan Chen
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  2. Blas M. Vinagre
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  3. Igor Podlubny
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Chen, Y., Vinagre, B. & Podlubny, I. Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives—an Expository Review. Nonlinear Dyn 38, 155–170 (2004). https://doi.org/10.1007/s11071-004-3752-x

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