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Two direct Tustin discretization methods for fractional-order differentiator/integrator. (English) Zbl 1051.93031
The aim of this paper is to obtain discrete equivalents to the fractional integrodifferential operators in the Laplace domains $$s^{\pm r}$$ with $$(0< r< 1)$$.
Two direct discretization methods based on the Tustin transformation are presented. For the first method a recursion scheme is derived. In the second method, continued fraction expansion is used. The methods are illustrated and compared by an application example. A fractional order controller with $$D(s)= s^{0.5}$$ is used for a double integrator plant. Robustness properties of the closed-loop systems with unity negative feedback are illustrated.

##### MSC:
 93B40 Computational methods in systems theory (MSC2010) 93C55 Discrete-time control/observation systems 93B17 Transformations
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##### References:
 [1] Hilfer, R., Applications of fractional calculus in physics, (2000), World Scientific Singapore · Zbl 0998.26002 [2] Manabe, S., The non-integer integral and its application to control systems, JIEE (Japan. inst. electr. eng.) J., 80, 860, 589-597, (1960) [3] M. Axtell, E.M. Bise, Fractional calculus applications in control systems, in: Proceedings of the IEEE 1990 National Aerospace and Electronics Conference, New York, USA, 1990, pp. 563-566. [4] L. Dorčák, Numerical models for simulation the fractional-order control systems, in: UEF-04-94, The Academy of Science, Institute of Experimental Physics, Kosice, Slovak Republic, 1994, pp. 1-12. [5] D. Matignon, Stability result on fractional differential equations with applications to control processing, in: IMACS-SMC Proceedings, Lille, France, 1996, pp. 963-968. [6] Oustaloup, A., La Dérivation non entière, (1995), HERMES Paris [7] Petráš, I., The fractional-order controllersmethods for their synthesis and applications, J. electr. eng., 50, 9-10, 284-288, (1999) [8] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [9] B.M. Vinagre, YangQuan Chen. Lecture Note on Fractional Calculus Applications in Automatic Control and Robotics, The 41st IEEE CDC2002 Tutorial Workshop ♯ 2, 2002, Dec., Las Vegas, Nevada [10] Vinagre, B.; Podlubny, I.; Hernández, A.; Feliu, V., Some approximations of fractional order operators used in control theory and applications, Fractional calculus appl. anal., 3, 3, 231-248, (2000) · Zbl 1111.93302 [11] Chen, Y.Q.; Moore, K.L., Discretization schemes for fractional order differentiators and integrators, IEEE trans. circuits systems-I fund. theory appl., 49, 3, 363-367, (2002) · Zbl 1368.65035 [12] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004 [13] Lubich, C.H., Discretized fractional calculus, SIAM J. math. anal., 17, 3, 704-719, (1986) · Zbl 0624.65015 [14] J.F. Claerbout, Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting, Blackwell Scientific Publications, Oxford 1976, [15] Press, W.H.; Vetterling, W.T.; Teukolsky, S.A.; Flannery, B.P., Numerical recipes in C, (1992), Cambridge University Press Cambridge · Zbl 0778.65003 [16] Khinchin, A.I.; Khinchin, A.Ya.; Eagle, H., Continued fractions, (1997), Dover (Inc Scripta Technica, Translator) New York [17] V.-G. Rao, D.-S. Bernstein, Naive control of the double integrator, IEEE Control System Mag. (2001) 21, 86-97. [18] V. Feliu, K.-S. Rattan, H.-B. Brown, Adaptive control of a single-link flexible manipulator, IEEE Control System Mag. (1990) 10, 29-33. [19] Debnath, L., Integral transforms and their applications, (1995), CRC Press Boca Raton, FL · Zbl 0920.44001
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