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Nonlinear Dynamics

, Volume 54, Issue 3, pp 263–282 | Cite as

Fractional control of heat diffusion systems

  • Isabel S. JesusEmail author
  • J. A. Tenreiro Machado
Review Paper

Abstract

The concept of differentiation and integration to non-integer order has its origins in the seventeen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated and compared. Simulations are presented assessing the performance of the proposed fractional-order algorithms.

Keywords

PID tuning Fractional calculus Fractional-order systems ISE and ITSE optimization 

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References

  1. 1.
    Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974) Google Scholar
  2. 2.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) zbMATHGoogle Scholar
  3. 3.
    Battaglia, J.L., Cois, O., Puigsegur, L., Oustaloup, A.: Solving an inverse heat conduction problem using a non-integer identified model. Int. J. Heat Mass Transf. 44, 2671–2680 (2001) zbMATHCrossRefGoogle Scholar
  4. 4.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, Partial Differential Equations. Wiley–Interscience II, New York (1962) Google Scholar
  5. 5.
    Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Automat. Contr. 44(1), 208–213 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Vinagre, B.M., Chen, Y.Q., Petráš, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. Franklin Inst. 340(5), 349–362 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion to discretize fractional order derivatives-an expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Barbosa, R.S., Tenreiro Machado, J.A., Silva, M.F.: Time domain design of fractional differintegrators using least-squares. Signal Process. 86(10), 2567–2581 (2006) CrossRefzbMATHGoogle Scholar
  9. 9.
    Zhuang, M., Atherton, D.P.: Automatic tuning of optimum PID controllers. IEE Proc., Control Theory Appl. 140(3), 216–224 (1993) zbMATHCrossRefGoogle Scholar
  10. 10.
    Petrás, I., Vinagre, B.M.: Practical application of digital fractional-order controller to temperature control. Acta Montan. Slovaca 7(2), 131–137 (2002) Google Scholar
  11. 11.
    Barbosa, R.S., Tenreiro Machado, J.A., Ferreira, I.M.: Tuning of PID controllers based on Bode’s ideal transfer function. Nonlinear Dyn. 38(1/4), 305–321 (2004) zbMATHCrossRefGoogle Scholar
  12. 12.
    Chen, Y.Q., Moore, K.L.: Relay feedback tuning of robust PID controllers with iso-damping property. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 35(1), 23–31 (2005) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Tenreiro Machado, J., Jesus, I., Boaventura Cunha, J., Tar, J.K.: Fractional Dynamics and Control of Distributed Parameter Systems. In: Intelligent Systems at the Service of Mankind, Vol. 2, pp. 295–305. Ubooks (2006) Google Scholar
  14. 14.
    Crank, J.: The Mathematics of Diffusion. Oxford University Press, London (1956) zbMATHGoogle Scholar
  15. 15.
    Gerald, C.F., Wheatley, P.O.: Applied Numerical Analysis. Addison Wesley, Reading (1999) Google Scholar
  16. 16.
    Farlow, S.J.: Partial Differential Equations for Scientists and Engineers. Wiley, New York (1993) zbMATHGoogle Scholar
  17. 17.
    Chen, Y.Q.: Ubiquitous fractional order controls? In: The Second IFAC Symposium on Fractional Derivatives and Applications—IFAC–FDA06, July, Portugal (2006) Google Scholar
  18. 18.
    Jesus, I.S., Barbosa, R.S., Tenreiro Machado, J.A., Boaventura Cunha, J.: Strategies for the control of heat diffusion systems based on fractional calculus. In: IEEE-ICCC 2006—IEEE International Conference on Computational Cybernetics, August 3–8, Estonia (2006) Google Scholar
  19. 19.
    Smith, O.J.M.: Closed control of loops with dead time. Chem. Eng. Process. 53, 217–219 (1957) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Electrotechnical EngineeringInstitute of Engineering of PortoPortoPortugal

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