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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)

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.

Keywords

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

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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

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