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

Reyad El-Khazali, J. A.Tenreiro Machado

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


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 controllers. The effectiveness of this work is demonstrated via several numerical simulations.

Original languageBritish English
Title of host publicationFractional Calculus - ICFDA 2018
EditorsPraveen Agarwal, Praveen Agarwal, Praveen Agarwal, Dumitru Baleanu, YangQuan Chen, Shaher Momani, José António Tenreiro Machado
Number of pages17
ISBN (Print)9789811504297
StatePublished - 2019
EventInternational Conference on Fractional Differentiation and its Applications, ICFDA 2018 - Amman, Jordan
Duration: 16 Jul 201818 Jul 2018

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


ConferenceInternational Conference on Fractional Differentiation and its Applications, ICFDA 2018


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


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