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Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?

Author

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  • J. Alberto Conejero

    (Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain)

  • Jonathan Franceschi

    (Department of Mathematics “F. Casorati”, Università degli Studi di Pavia, 27100 Pavia, Italy)

  • Enric Picó-Marco

    (Departamento de Ingeniería de Sistemas y Automática, Universitat Politècnica de València, 46022 Valencia, Spain)

Abstract

The concept of a fractional derivative is not at all intuitive, starting with not having a clear geometrical interpretation. Many different definitions have appeared, to the point that the need for order has arisen in the field. The diversity of potential applications is even more overwhelming. When modeling a problem, one must think carefully about what the introduction of fractional derivatives in the model can provide that was not already adequately covered by classical models with integer derivatives. In this work, we present some examples from control theory where we insist on the importance of the non-local character of fractional operators and their suitability for modeling non-local phenomena either in space (action at a distance) or time (memory effects). In contrast, when we encounter completely different nonlinear phenomena, the introduction of fractional derivatives does not provide better results or further insight. Of course, both phenomena can coexist and interact, as in the case of hysteresis, and then we would be dealing with fractional nonlinear models.

Suggested Citation

  • J. Alberto Conejero & Jonathan Franceschi & Enric Picó-Marco, 2022. "Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?," Mathematics, MDPI, vol. 10(15), pages 1-18, August.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2719-:d:877848
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    References listed on IDEAS

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    1. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    2. Duarte Valério & Manuel D. Ortigueira & António M. Lopes, 2022. "How Many Fractional Derivatives Are There?," Mathematics, MDPI, vol. 10(5), pages 1-18, February.
    3. Adeel Ahmad Jamil & Wen Fu Tu & Syed Wajhat Ali & Yacine Terriche & Josep M. Guerrero, 2022. "Fractional-Order PID Controllers for Temperature Control: A Review," Energies, MDPI, vol. 15(10), pages 1-28, May.
    4. Vasily E. Tarasov & Svetlana S. Tarasova, 2020. "Fractional Derivatives and Integrals: What Are They Needed For?," Mathematics, MDPI, vol. 8(2), pages 1-22, January.
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