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Fractional Calculus: Fundamentals and Applications

  • J. A. Tenreiro MachadoEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 198)

Abstract

This paper presents the fundamental aspects of the theory of Fractional Calculus. Several approximation methods for the calculation of fractional-order derivatives are discussed. The application of Fractional Calculus in automatic control systems and their main properties are also analyzed.

References

  1. 1.
    D. Baleanu, J.T. Machado, A. Luo, Fractional Dynamics and Control (Springer, New York, 2011)Google Scholar
  2. 2.
    D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods (World Scientific Publishing Company, Amsterdam, 2012)CrossRefzbMATHGoogle Scholar
  3. 3.
    G.E. Carlson, C.A. Halijak, Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process. IEEE Trans. Circuit Theory 10, 210–213 (1964)CrossRefGoogle Scholar
  4. 4.
    R. Hilfer, Applications of fractional calculus in physics (World Scientific, Singapore, 2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol. 204 (Elsevier, Amsterdam, 2006)Google Scholar
  6. 6.
    J.T. Machado, Analysis and design of fractional-order digital control systems. Systems Anal. Model. Simul. 27(2–3), 107–122 (1997)zbMATHGoogle Scholar
  7. 7.
    J.T. Machado, Fractional-order derivative approximations in discrete-time control systems. Syst. Anal. Model. Simul. 34, 419–434 (1999)zbMATHGoogle Scholar
  8. 8.
    J.T. Machado, Discrete-time fractional-order controllers. Fract. Calculus Appl. Anal. 4(1), 47–66 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    J.T. Machado, A.M. Galhano, Approximating fractional derivatives in the perspective of system control. Nonlinear Dyn. 56(4), 401–407 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)zbMATHGoogle Scholar
  12. 12.
    C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional-Order Systems and Con- trols (Springer, London, 2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)zbMATHGoogle Scholar
  14. 14.
    Oustaloup, A.: La Commande CRONE: Commande Robuste d’Ordre Non Entier, Hermes, Paris (1991)Google Scholar
  15. 15.
    I. Petráš: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, Heidelberg, 2011)Google Scholar
  16. 16.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)zbMATHGoogle Scholar
  17. 17.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives (Gordon and Breach Science Publishers, Yverdon, 1993)zbMATHGoogle Scholar
  18. 18.
    V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles (Fields and Media, Springer, 2010)CrossRefzbMATHGoogle Scholar
  19. 19.
    D. Valerio, J.S. da Costa, An Introduction to Fractional Control (IET, Stevenage, 2012)CrossRefzbMATHGoogle Scholar
  20. 20.
    S. Westerlund, Dead Matter Has Memory (Causal Consulting, Kalmar, 2002)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Electrical Engineering, Institute of EngineeringPolytechnic of PortoPortoPortugal

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