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

  • Reyad El-KhazaliEmail author
  • J. A. Tenreiro Machado
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 303)


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.


Fractional calculus Transfer function Discrete-time operator Discrete-time integro-differential operators Frequency response 


  1. 1.
    Al-Alaoui, M.A.: Novel digital integrator and differentiator. IEE Electron. Lett. 29(4), 376–378 (1993)CrossRefGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 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. 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. 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)MathSciNetCrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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. 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. 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. 10.
    Gupta, M., Yadav, R.: Design of improved fractional-order integrators using indirect discretization method. Int. J. Comput. Appl. 59(14) (2012)CrossRefGoogle Scholar
  11. 11.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies, Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  12. 12.
    Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 59(3), 386–426 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Machado, J.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997)zbMATHGoogle Scholar
  15. 15.
    Machado, J.T.: Fractional-order derivative approximations in discrete-time control systems. Syst. Anal. Model. Simul. 34, 419–434 (1999)zbMATHGoogle Scholar
  16. 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. 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. 18.
    Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)zbMATHGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 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. 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. 23.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives and Some of Their Applications. Nauka i Tekhnika, Minsk (1987)zbMATHGoogle Scholar
  24. 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)CrossRefGoogle Scholar
  25. 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)MathSciNetzbMATHGoogle Scholar
  26. 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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.ECE DepartmentKhalifa UniversityAbu DhabiUnited Arab Emirates
  2. 2.Department of Electrical Engineering, Institute of EngineeringPolytechnic of PortoPortoPortugal

Personalised recommendations