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Closed-Form Discretization of Fractional-Order Differential and Integral Operators

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Fractional Calculus (ICFDA 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 303))

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.

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References

  1. Al-Alaoui, M.A.: Novel digital integrator and differentiator. IEE Electron. Lett. 29(4), 376–378 (1993)

    Article  Google Scholar 

  2. Al-Alaoui, M.A.: Novel stable higher order s-to-z transforms. IEEE Trans. Circuit. Syst. I: Fundam. Theory Appl. 48(11), 1326–1329 (2001)

    Article  Google Scholar 

  3. Al-Alaoui, M.A.: Al-Alaoui operator and the \(\alpha \)-approximation for discretization of analog system. FACTA Universitatis (NIS) 19(1), 143–146 (2006)

    Google Scholar 

  4. 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)

    Google Scholar 

  5. Chen, Y., Moore, K.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuit. Syst.I: Fundam. Theory Appl. 49(3), 363–367 (2002)

    Article  MathSciNet  Google Scholar 

  6. Chen, Y., Vinagre, B.M., Podlubny, I.: Continued fraction expansion to discretize fractionalorder derivatives - an expository review. Nonlinear Dyn. 38(1), 155–170 (2004)

    Article  Google Scholar 

  7. Dorčák, L., Petráš, I., Terpák, J., Zborovjan, M.: Comparison of the method for discrete approximation of the fractional order operator. In: Proceedings of the International Carpathian Control Conference (ICCC’2003), pp. 851–856. High Tatras, Slovak Republic (2003)

    Google Scholar 

  8. El-Khazali, R.: Biquadratic approximation of fractional-order Laplacian operators. In: 2013 IEEE 56th International Midwest Symposium on Circuits and Systems(MWSCAS), pp. 69–72. Columbus, OH (2013)

    Google Scholar 

  9. El-Khazali, R.: Discretization of fractional-order differentiators and integrators. In: 19th IFAC World Congress, pp. 2016–2021. Cape Town, South Africa (2014)

    Google Scholar 

  10. Gupta, M., Yadav, R.: Design of improved fractional-order integrators using indirect discretization method. Int. J. Comput. Appl. 59(14) (2012)

    Article  Google Scholar 

  11. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies, Elsevier, Amsterdam (2006)

    Book  Google Scholar 

  12. Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 59(3), 386–426 (2011)

    Article  MathSciNet  Google Scholar 

  13. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)

    Article  MathSciNet  Google Scholar 

  14. Machado, J.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997)

    MATH  Google Scholar 

  15. Machado, J.T.: Fractional-order derivative approximations in discrete-time control systems. Syst. Anal. Model. Simul. 34, 419–434 (1999)

    MATH  Google Scholar 

  16. Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, vol. 2, pp. 963–968. Lille, France (1996)

    Google Scholar 

  17. 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. 479–485. Springer, Berlin, Heidelberg (2011)

    Google Scholar 

  18. Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  19. Ortigueira, M.D., Serralheiro, A.J.: Pseudo-fractional ARMA modelling using a double Levinson recursion. IET Control Theory Appl. 1(1), 173–178 (2007)

    Article  Google Scholar 

  20. 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), 25–39 (2000)

    Article  Google Scholar 

  21. 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)

    Google Scholar 

  22. 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)

    Google Scholar 

  23. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives and Some of Their Applications. Nauka i Tekhnika, Minsk (1987)

    MATH  Google Scholar 

  24. Siami, M., Tavazoei, M.S., Haeri, M.: Stability preservation analysis in direct discretization of fractional order transfer functions. Signal Process. 91(3), 508–512 (2011)

    Article  Google Scholar 

  25. 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), 231–248 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fractionalorder differentiator/integrator. J. Franklin Inst. 340(5), 349–362 (2003)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. ECE Department, Khalifa University, Abu Dhabi, United Arab Emirates

    Reyad El-Khazali

  2. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    J. A. Tenreiro Machado

Authors
  1. Reyad El-Khazali
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  2. J. A. Tenreiro Machado
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Corresponding author

Correspondence to Reyad El-Khazali .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Anand International College of Engineering, Jaipur, Rajasthan, India

    Praveen Agarwal

  2. Department of Mathematics and Computer Science, Çankaya University, Ankara, Turkey

    Dumitru Baleanu

  3. School of Engineering (MESA Lab), University of California, Merced, Merced, CA, USA

    YangQuan Chen

  4. Department of Mathematics, The University of Jordan, Amman, Jordan

    Shaher Momani

  5. Institute of Engineering, Polytechnic Institute of Porto, Porto, Portugal

    José António Tenreiro Machado

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El-Khazali, R., Machado, J.A.T. (2019). Closed-Form Discretization of Fractional-Order Differential and 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

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