Abstract
This paper introduces a closed-form discretization of fractional-order differential or integral Laplace operators. The proposed method depends on extracting the necessary phase requirements from the phase diagram. The magnitude frequency response follows directly due to the symmetry of the poles and zeros of the finite z-transfer function. Unlike the continued fraction expansion technique, or the infinite impulse response of second-order IIR-type filters, the proposed technique generalizes the Tustin operator to derive a first-, second-, third-, and fourth-order discrete-time operators (DTO) that are stable and of minimum phase. The proposed method depends only on the order of the Laplace operator. The resulted discrete-time operators enjoy flat-phase response over a wide range of discrete-time frequency spectrum. The closed-form DTO enables one to identify the stability regions of fractional-order discrete-time systems or even to design discrete-time fractional-order \(PI^{\lambda }D^{\mu }\) controllers. The effectiveness of this work is demonstrated via several numerical simulations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Al-Alaoui, M.A.: Novel digital integrator and differentiator. IEE Electron. Lett. 29(4), 376ā378 (1993)
Al-Alaoui, M.A.: Novel stable higher order s-to-z transforms. IEEE Trans. Circuit. Syst. I: Fundam. Theory Appl. 48(11), 1326ā1329 (2001)
Al-Alaoui, M.A.: Al-Alaoui operator and the \(\alpha \)-approximation for discretization of analog system. FACTA Universitatis (NIS) 19(1), 143ā146 (2006)
Barbosa, R.S., Machado, J.A.T., Ferreira, I.M.: Pole-zero approximations of digital fractionalorder integrators and diferentiators using modeling techniques. In: 16th IFACWorld Congress. Prague, Czech Republic (2005)
Chen, Y., Moore, K.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuit. Syst.I: Fundam. Theory Appl. 49(3), 363ā367 (2002)
Chen, Y., Vinagre, B.M., Podlubny, I.: Continued fraction expansion to discretize fractionalorder derivatives - an expository review. Nonlinear Dyn. 38(1), 155ā170 (2004)
DorÄĆ”k, L., PetrĆ”Å”, I., TerpĆ”k, J., Zborovjan, M.: Comparison of the method for discrete approximation of the fractional order operator. In: Proceedings of the International Carpathian Control Conference (ICCCā2003), pp. 851ā856. High Tatras, Slovak Republic (2003)
El-Khazali, R.: Biquadratic approximation of fractional-order Laplacian operators. In: 2013 IEEE 56th International Midwest Symposium on Circuits and Systems(MWSCAS), pp. 69ā72. Columbus, OH (2013)
El-Khazali, R.: Discretization of fractional-order differentiators and integrators. In: 19th IFAC World Congress, pp. 2016ā2021. Cape Town, South Africa (2014)
Gupta, M., Yadav, R.: Design of improved fractional-order integrators using indirect discretization method. Int. J. Comput. Appl. 59(14) (2012)
Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies, Elsevier, Amsterdam (2006)
Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 59(3), 386ā426 (2011)
Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704ā719 (1986)
Machado, J.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2ā3), 107ā122 (1997)
Machado, J.T.: Fractional-order derivative approximations in discrete-time control systems. Syst. Anal. Model. Simul. 34, 419ā434 (1999)
Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, vol. 2, pp. 963ā968. Lille, France (1996)
Nie, B., Li, W., Ma, H., Wang, D., Liang, X.: Research of direct discretization method of fractional order differentiator/integrator based on rational function approximation. In: Tarn, T.J., Chen, S.B., Fang, G. (eds.) RoboticWelding, Intelligence and Automation. Lecture Notes in Electrical Engineering, pp. 479ā485. Springer, Berlin, Heidelberg (2011)
Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)
Ortigueira, M.D., Serralheiro, A.J.: Pseudo-fractional ARMA modelling using a double Levinson recursion. IET Control Theory Appl. 1(1), 173ā178 (2007)
Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuit. Syst. I: Fundam. Theory Appl. 47(1), 25ā39 (2000)
Podlubny, I.: The Laplace transform method for linear differential equations of the fractional order. Tech. Rep. UEF-02-9, Slovak Academy of Sciences Institute of Experimental Physics, Kosice, Slovakia (1994)
Podlubny, I.: Fractional differential equations, Volume 198: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution. Mathematics in Science and Engineering. Academic Press, San Diego (1998)
Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives and Some of Their Applications. Nauka i Tekhnika, Minsk (1987)
Siami, M., Tavazoei, M.S., Haeri, M.: Stability preservation analysis in direct discretization of fractional order transfer functions. Signal Process. 91(3), 508ā512 (2011)
Vinagre, B., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Fract. Calculus Appl. Anal. 3(3), 231ā248 (2000)
Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fractionalorder differentiator/integrator. J. Franklin Inst. 340(5), 349ā362 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
El-Khazali, R., Machado, J.A.T. (2019). Closed-Form Discretization of Fractional-Order Differential andĀ Integral Operators. In: Agarwal, P., Baleanu, D., Chen, Y., Momani, S., Machado, J. (eds) Fractional Calculus. ICFDA 2018. Springer Proceedings in Mathematics & Statistics, vol 303. Springer, Singapore. https://doi.org/10.1007/978-981-15-0430-3_1
Download citation
DOI: https://doi.org/10.1007/978-981-15-0430-3_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-0429-7
Online ISBN: 978-981-15-0430-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)