Nonlinear Dynamics

, Volume 56, Issue 4, pp 401–407 | Cite as

Approximating fractional derivatives in the perspective of system control

  • J. A. Tenreiro Machado
  • Alexandra Galhano
Original Paper


The theory of fractional calculus goes back to the beginning of the theory of differential calculus, but its application received attention only recently. In the area of automatic control some work was developed, but the proposed algorithms are still in a research stage. This paper discusses a novel method, with two degrees of freedom, for the design of fractional discrete-time derivatives. The performance of several approximations of fractional derivatives is investigated in the perspective of nonlinear system control.


Fractional calculus Control Discretization schemes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, San Diego (1974) Google Scholar
  2. 2.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993) MATHGoogle Scholar
  3. 3.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  4. 4.
    Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21(5), 741–748 (1983) MATHCrossRefGoogle Scholar
  5. 5.
    Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi 133, 425–430 (1986) CrossRefGoogle Scholar
  6. 6.
    Le Méhauté, A.: Fractal Geometries: Theory and Applications. Penton, Cleveland (1991) MATHGoogle Scholar
  7. 7.
    Oustaloup, A.: La commande CRONE: commande robuste d’ordre non entier. Hermes, Paris (1991) MATHGoogle Scholar
  8. 8.
    Anastasio, T.J.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern. 72(1), 69–79 (1994) CrossRefGoogle Scholar
  9. 9.
    Oustaloup, A.: La dérivation non entier: théorie, synthèse et applications. Hermes, Paris (1995) MATHGoogle Scholar
  10. 10.
    Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7(9), 1461–1477 (1996) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Machado, J.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997) MATHGoogle Scholar
  12. 12.
    Podlubny, I.: Fractional-order systems and PI λ D μ-controllers. IEEE Trans. Automat. Contr. 44(1), 208–213 (1999) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  14. 14.
    Al-Alaoui, M.A.: Novel digital integrator and differentiator. Electron. Lett. 29(4), 376–378 (1993) CrossRefGoogle Scholar
  15. 15.
    Al-Alaoui, M.A.: Filling the gap between the bilinear and the backward-difference transforms: an interactive design approach. Int. J. Electr. Eng. Educ. 34(4), 331–337 (1997) Google Scholar
  16. 16.
    Smith, J.M.: Mathematical Modeling and Digital Simulation for Engineers and Scientists, 2nd edn. Wiley, New York (1987) Google Scholar
  17. 17.
    Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst.—I. Fundam. Theory Appl. 49(3), 363–367 (2002) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Tseng, C.C.: Design of fractional order digital fir differentiators. IEEE Signal Process. Lett. 8(3), 77–79 (2001) CrossRefGoogle Scholar
  19. 19.
    Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. J. Franklin Inst. 340(5), 349–362 (2003) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Chen, Y.Q., Vinagre, B.M.: A new IIR-type digital fractional-order differentiator. Signal Process. 83(11), 2359–2365 (2003) MATHCrossRefGoogle Scholar
  21. 21.
    Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Barbosa, R.S., Machado, J.T., Ferreira, I.M.: Least-squares design of digital fractional-order operators. In: Proceedings of the First IFAC Workshop on Fractional Differentiation and Its Applications, Bordeaux, France, pp. 434–439 (2004) Google Scholar
  23. 23.
    Barbosa, R.S., Machado, J.T., Silva, M.F.: Time domain design of fractional differintegrators using least squares approximations. Signal Process. 86(10), 2567–2581 (2006) CrossRefGoogle Scholar
  24. 24.
    Machado, J.T., Galhano, A.: A new method for approximating fractional derivatives: application in non-linear control. In: ENOC 2008—6th EUROMECH Conference, Saint Petersburg, Russia (2008) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institute of Engineering of PortoDepartment of Electrical EngineeringPortoPortugal

Personalised recommendations