Nonlinear Dynamics

, Volume 62, Issue 1–2, pp 447–452 | Cite as

Optimal tuning of fractional controllers using genetic algorithms

  • J. A. Tenreiro Machado
Original Paper


This study addresses the optimization of fractional algorithms for the discrete-time control of linear and non-linear systems. The paper starts by analyzing the fundamentals of fractional control systems and genetic algorithms. In a second phase the paper evaluates the problem in an optimization perspective. The results demonstrate the feasibility of the evolutionary strategy and the adaptability to distinct types of systems.


Fractional derivatives Fractional calculus Optimization Genetic algorithms 


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.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  5. 5.
    Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983) MATHCrossRefGoogle Scholar
  6. 6.
    Oustaloup, A.: La Commande CRONE: Commande Robuste d’Ordre non Entier. Hermes, Paris (1991) MATHGoogle Scholar
  7. 7.
    Anastasio, T.J.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern. 72(1), 69–79 (1994) CrossRefGoogle Scholar
  8. 8.
    Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tenreiro Machado, J.A.: Analysis and design of fractional-order digital control systems. J. Syst. Anal. Model. Simul. 27, 107–122 (1997) MATHGoogle Scholar
  10. 10.
    Nigmatullin, R.: The statistics of the fractional moments: Is there any chance to “read quantitatively” any randomness? Signal Process. 86(10), 2529–2547 (2006) MATHCrossRefGoogle Scholar
  11. 11.
    Tarasov, V.E., Zaslavsky, G.M.: Fractional dynamics of systems with long-range interaction. Commun. Nonlinear Sci. Numer. Simul. 11(8), 885–898 (2006) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Berlin (2007). ISBN:978-1-4020-6041-0 MATHGoogle Scholar
  13. 13.
    Tenreiro Machado, J.A.: Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 14(9–10), 3492–3497 (2009) CrossRefGoogle Scholar
  14. 14.
    Baleanu, D.: About fractional quantization and fractional variational principles. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2520–2523 (2009) CrossRefGoogle Scholar
  15. 15.
    Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Autom. Control 44(1), 208–213 (1999) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Tenreiro Machado, J.A.: Discrete-time fractional-order controllers. J. Fract. Calc. Appl. Anal. 4, 47–66 (2001) MATHMathSciNetGoogle 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.
    Barbosa, R.S., Tenreiro Machado, J.A., Silva, M.: Time domain design of fractional differintegrators using least squares approximations. Signal Process. 86(10), 2567–2581 (2006) MATHCrossRefGoogle Scholar
  22. 22.
    Al-Alaoui, M.A.: Novel digital integrator and differentiator. Electron. Lett. 29(4), 376–378 (1993) CrossRefGoogle Scholar
  23. 23.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975) Google Scholar
  24. 24.
    Goldenberg, D.E.: Genetic Algorithms in Search Optimization, and Machine Learning. Addison-Wesley, Reading (1989) Google Scholar
  25. 25.
    Tenreiro Machado, J.A.: Calculation of fractional derivatives of noisy data with genetic algorithms. Nonlinear Dyn. 57(1–2), 253–260 (2009) MATHCrossRefGoogle Scholar
  26. 26.
    Tenreiro Machado, J.A., Galhano, A., Oliveira, A.M., Tar, J.K.: Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms. Commun. Nonlinear Sci. Numer. Simul. 15, 482–490 (2010) CrossRefGoogle Scholar
  27. 27.
    Maiti, D., Acharya, A., Chakraborty, M., Konar, A., Janarthanan, R.: Tuning PID and PIλDδ controllers using the integral time absolute error criterion. In: IEEE Forth International Conference on Information and Automation for Sustainability, December 12–14, 2008, Colombo, Sri Lanka Google Scholar
  28. 28.
    Cao, J.-Y., Cao, B.-G.: Design of fractional order controller based on particle swarm optimization. Int. J. Control Autom. Syst. 4(6), 775–781 (2006) Google Scholar
  29. 29.
    Valério, D., Sá da Costa, J.: Tuning of fractional controllers minimising H2 and H norms. Acta Polytech. Hung. 3(4), 55–70 (2006) Google Scholar
  30. 30.
    Biswas, A., Das, S., Abraham, A., Dasgupta, S.: Design of fractional-order PIλDμ controllers with an improved differential evolution. Eng. Appl. Artif. Intell. 22, 343–350 (2009) CrossRefMathSciNetGoogle Scholar
  31. 31.
    Barbosa, R.S., Tenreiro Machado, J.A.: Describing function analysis of systems with impacts and backlash. Nonlinear Dyn. 29(1–4), 235–250 (2002) MATHCrossRefGoogle Scholar
  32. 32.
    Duarte, F., Tenreiro Machado, J.A.: Describing function of two masses with backlash. Nonlinear Dyn. 56(4), 409–413 (2009) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dept. of Electrical EngineeringInstitute of Engineering of PortoPortoPortugal

Personalised recommendations