Nonlinear Dynamics

, Volume 38, Issue 1–4, pp 305–321 | Cite as

Tuning of PID Controllers Based on Bode’s Ideal Transfer Function

  • Ramiro S. Barbosa
  • J. A. Tenreiro Machado
  • Isabel M. Ferreira


This paper presents a new strategy for tuning PID controllers based on a fractional reference model. The model is represented as an ideal closed-loop system whose open-loop is given by the Bode’s ideal transfer function. The PID controller parameters are determined by the minimization of the integral square error (ISE) between the time responses of the desired fractional reference model and of the system with the PID controller. The resulting closed-loop system (with the PID controller) has the desirable feature of being robust to gain variations with step responses exhibiting an iso-damping property. Several examples are presented that demonstrate the effectiveness and validity of the proposed methodology.

Key words:

Bode’s ideal transfer function fractional calculus fractional-order systems ISE optimization PID tuning 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Oldham, K. B. and Spanier, J. The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.Google Scholar
  2. 2.
    Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Google Scholar
  3. 3.
    Podlubny, I. Fractional Differential Equations, Academic Press, San Diego, California, 1999.Google Scholar
  4. 4.
    Hilfer, R. Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.Google Scholar
  5. 5.
    Oustaloup, A. La Dérivation Non Entière, Hermès, Paris, 1995.Google Scholar
  6. 6.
    Machado, J. A. T. ‘Analysis and design of fractional-order digital control systems’, SAMS Journal Systems Analysis, Modelling, Simulation 27, 1997, 107–122.Google Scholar
  7. 7.
    Machado, J. A. T. ‘Discrete-time fractional-order controllers’, FCAA Fractional Calculus and Applied Analysis 4(1), 2001, 47–66.Google Scholar
  8. 8.
    Vinagre, B. M., Podlubny, I., Hernández, A., and Feliu, V. ‘Some approximations of fractional order operators used in control theory and applications’, FCAA Fractional Calculus and Applied Analysis 3(3), 2000, 231–248.Google Scholar
  9. 9.
    Manabe, S. ‘The non-integer integral and its application to control systems’, ETJ of Japan 6(3/4), 1961, 83–87.Google Scholar
  10. 10.
    Aström, K. and Hägglund, T. PID Controllers: Theory, Design, and Tuning, Instrument Society of America, North Carolina, 1995.Google Scholar
  11. 11.
    Martins de Carvalho, J. L. Sistemas de Controle Automático, LTC-Livros Técnicos e Científicos Editora S. A., Rio de Janeiro, 2000.Google Scholar
  12. 12.
    Podlubny, I. ‘Fractional-order systems and PIλDμ-controllers’, IEEE Transactions on Automatic Control 44(1), 1999, 208–213.Google Scholar
  13. 13.
    Bode, H. W. Network Analysis and Feedback Amplifier Design, Van Nostrand, New York, 1945.Google Scholar
  14. 14.
    Gorenflo, R. and Mainardi, F. ‘Fractional oscillations and Mittag-Leffler functions’, Pre-print A-14/96, Free University of Berlin, Berlin, 1996, pp. 1–22.Google Scholar
  15. 15.
    Barbosa, R. S., Machado, J. A. T., and Ferreira, I. M. ‘A fractional calculus perspective of PID tuning’, in the Proceedings of the ASME International 19th Biennial Conference on Mechanical Vibration and Noise (VIB’03), Chicago, Illinois, September 2–6, 2003, CD-ROM.Google Scholar
  16. 16.
    Chen, Y. Q., Hu, C. H., and Moore, K. L. ‘Relay feedback tuning of robust PID controllers with iso-damping property’, in the Proceedings of the 42nd IEEE conference on Decision and Control, Maui, Hawaii, December 9–12, 2003, pp. 2180–2185.Google Scholar
  17. 17.
    Zhuang, M. and Atherton, D. P. ‘Automatic tuning of optimum PID controllers’, IEE Proceedings-Part D: Control Theory and Applications 140(3), 1993, 216–224.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Ramiro S. Barbosa
    • 1
  • J. A. Tenreiro Machado
    • 1
  • Isabel M. Ferreira
    • 2
  1. 1.Department of Electrotechnical Engineering, Institute of Engineering of PortoRua Dr. António Bernardino de AlmeidaPortoPortugal
  2. 2.Department of Electrotechnical Engineering, Faculty of Engineering of PortoRua Dr. Roberto FriasPortoPortugal

Personalised recommendations