Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review

Nonlinear Dynamics

Volumn 38, Issue 1-4, 2004, Pages 155-170

  Chen, Yangquan ^a   Vinagre, Blas M ^b   Podlubny, Igor ^c  

^a Utah State University   (United States)

^b UNIVERSITY OF EXTREMADURA   (Spain)

^c TECHNICAL UNIVERSITY OF KO ICE   (Slovakia)

Author keywords

Al Alaoui operator; Discretization; Fractional differentiator; Fractional order differentiator; Fractional order dynamic systems; Tustin operator

Indexed keywords

FUNCTIONS; IMPULSE RESPONSE; LAPLACE TRANSFORMS; MATHEMATICAL OPERATORS; ROBUSTNESS (CONTROL SYSTEMS); TIME DOMAIN ANALYSIS;

AL-ALAOUI OPERATOR; DISCRETIZATION; FRACTIONAL DIFFERENTIATOR; FRACTIONAL-ORDER DIFFERENTIATOR; FRACTIONAL-ORDER DYNAMIC SYSTEMS; TUSTIN OPERATOR;

FRACTALS;

EID: 15544372160     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-004-3752-x     Document Type: Conference Paper

References (35)

1. Al-Alaoui, M. A. 'Novel digital integrator and differentiator', Electronics Letters 29(4), 1993, 376-378.
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 Applications 42(4), 1995, 220-223.
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 Education 34(4), 1997, 331-337.
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 Processing 83(11), 2003, 2359-2365.
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 Applications 49(3), 2002, 363-367.
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 Dynamics 29, 2002, 1 - 385.
9. Lubich, C. H., 'Discretized fractional calculus', SIAM Journal on Mathematical Analysis 17(3), 1986, 704-719.
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 Simulation 27, 1997, 107-122.
12. Manabe, S., 'The non-integer integral and its application to control systems', JIEE (Japanese Institute of Electrical Engineers) Journal 80(860), 1960, 589-597.
13. Manabe, S., 'The non-integer integral and its application to control systems', ETJ of Japan 6(3/4), 1961, 83-87.
14. Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
15. Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.
16. Ortigueira, M. D. and Machado, J. A. T. (guest editors), 'Special issue on fractional signal processing and applications' Signal Processing 83(11), 2003, 2285-2480.
17. Oustaloup, A., 'Fractional order sinusoidal oscilators: Optimization and their use in highly linear FM modulators', IEEE Transactions on Circuits and Systems 28(10), 1981, 1007-1009.
18. Oustaloup, A., La dérivation non entière, HERMES, Paris, 1995.
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 Applications 47(1), 2000, 25-40.
20. Oustaloup, A. and Mathieu, B., Lacommande CRONE: du scalaire au multivariable, HERMES, Paris, 1999.
21. Oustaloup, A., Mathieu, B., and Lanusse, P., 'The CRONE control of resonant plants: Application to a flexible transmission', European Journal of Control 1(2), 1995, pp. 113-121.
22. Oustaloup, A., Moreau, X., and Nouillant, M., 'TheCRONE suspension', Control Engineering Practice 4(8), 1996, 1101-1108.
23. Oustaloup, A., Sabatier, J., and Lanusse, P., 'From fractal robustness to CRONE control', Fractionnal Calculus and Applied Analysis 2(1), 1999, 1-30.
24. Petráš, I., 'The fractional-order controllers: Methods for their synthesis and application', Journal of Electrical Engineering 50(9-10), 1999, 284-288.
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. μ-Controllers', IEEE Transactions Automatic Control 44(1), 1999, 208-214.
27. Raynaud, H.-F. and Zergaïnoh, A., 'State-space representation for fractional order controllers', Automatica 36, 2000, 1017-1021.
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.
29. Tseng, C.-C., 'Design of fractional order digital FIR differentiator', IEEE Signal Processing Letters 8(3), 2001, 77-79.
30. Tseng, C.-C., Pei, S.-C., and Hsia, S.-C., 'Computation of fractional derivatives using fourier transform and digital FIR differentiator', Signal Processing 80, 2000, 151-159.
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 Analysis 3(3), 2000, 231-248.
34. Vinagre, B. M., Chen, Y. Q., and Petras, I. 'Two direct tustin discretization methods for fractional-order differentiator/integrator', The Journal of Franklin Institute 340(5) 2003, 349-362.
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.