Algebraic Methods in Autotuning Design: Implementation and Simulations

  • Roman ProkopEmail author
  • Jiří Korbel
  • Radek Matušů
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 348)


Autotuners represent a combination of a relay feedback identification test and some control design method. In this contribution, models with up to three parameters are estimated by means of a single asymmetrical relay experiment. Then a stable low order transfer function with a time delay term is identified by a relay experiment. Autotuning principles then combine asymmetrical relay feedback tests with a control synthesis. Two algebraic control syntheses then are presented in this paper. The first one is based on the ring of proper and stable rational functions R PS. The second one utilizes a special ring R MS, a set of RQ-meromorphic functions. In both cases, controller parameters are derived through a general solution of a linear Diophantine equation in the appropriate ring. A final controller can be tuned by a scalar real parameter m>0. The presented philosophy covers a generalization of PID controllers and the Smith-like control structure. This contribution deals with the implementation of proposed autotuners and presents some illustrative examples. A Matlab toolbox for automatic design and simulation was developed and various simulations performed and analyzed.


Algebraic control design Diophantine equation Relay experiment Autotuning Pole-placement problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Åström, K.J., Hägglund, T.: Automatic tuning of simple regulators with specifica-tion on phase and amplitude margins. Automatica 20, 645–651 (1984)CrossRefzbMATHGoogle Scholar
  2. 2.
    Åström, K.J., Hägglund, T.: PID Controllers: Theory, Design and Tuning. Instrumental Society of America, Research Triangle Park (1995)Google Scholar
  3. 3.
    Garpinger, O., Hägglund, T., Åström, K.J.: Criteria and trade-offs in PID design. Preprints of IFAC Conferencieon Advancesin Control, Brescia, Italy (2012)Google Scholar
  4. 4.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Applied Math. Sciences, vol. 99. Springer, New York (1993)zbMATHGoogle Scholar
  5. 5.
    Hang, C.C., Åström, K.J., Wang, Q.C.: Relay feedback auto-tuning of process con-trollers – a tutorial review. Journal of Process Control 12(6), 143–162 (2002)CrossRefGoogle Scholar
  6. 6.
    Kaya, I., Atherton, D.P.: Parameter estimation from relay autotuning with asymmetric limit cycle data. Journal of Process Control 11(4), 429–439 (2001)CrossRefGoogle Scholar
  7. 7.
    Leva, A., Bascetta, L., Schiavo, F.: Model based PI/PID autotuning with fast relay identification. Ind. Eng.Chem. Res. 45, 4052–4062 (2006)CrossRefGoogle Scholar
  8. 8.
    Luyben, W.L.: Getting more identification from relay-feedback tests. Ind. Eng.Chem. Res. 40, 4391–4402 (2001)CrossRefGoogle Scholar
  9. 9.
    Majhi, S.: Relay based identification of processes with time delay. Journal of Process Control 17, 93–101 (2007)CrossRefGoogle Scholar
  10. 10.
    Majhi, S., Atherton, D.P.: Autotuning and controller design for unstable time delay processes. Preprints of UKACC Conf. an Control, pp. 769–774 (1998)Google Scholar
  11. 11.
    Morari, M., Zafiriou, E.: Robust Process Control. Prentice Hall, New Jersey (1989)Google Scholar
  12. 12.
    Morilla, F., Gonzáles, A., Duro, N.: Auto-tuning PID controllers in terms of relative damping. Preprints of IFAC Workshop PID 2000, pp. 161–166 (2000)Google Scholar
  13. 13.
    O’Dwyer, A.: Handbook of PI and PID controller tuning rules. Imperial College Press, London (2003)CrossRefGoogle Scholar
  14. 14.
    Panda, R.C., Yu, C.C.: Analytical expressions for relay feedback responses. Journal of Process Control 13, 489–501 (2003)CrossRefGoogle Scholar
  15. 15.
    Pecharromán, R.R., Pagola, F.L.: Control design for PID controllers auto-tuning based on improved identification. Preprints of IFAC Workshop PID 2000, pp. 89–94 (2000)Google Scholar
  16. 16.
    Prokop, R., Korbel, J.: PI autotuners based on biased relay identification. Preprints of the 15th IFAC World Congress, Prague (2005)Google Scholar
  17. 17.
    Prokop, R., Korbel, J., Prokopová, Z.: Relay feedback autotuning – A polynomial approach. Preprints of 24th European Conf. on Modelling and Simulation, Kuala Lumpur (2010)Google Scholar
  18. 18.
    Prokop, R., Korbel, J., Líška, O.: A novel principle for relay-based autotuning. International Journal of Mathematical Models and Methods in Applied Science 5(7), 1180–1188 (2011)Google Scholar
  19. 19.
    Prokop, R., Korbel, J., Pekař, L.: Algebraic methods in autotuning design, Part I: Theory and design. This Journal (2014)Google Scholar
  20. 20.
    Thyagarajan, T., Yu, C.C.: Improved autotuning using shape factor from relay feedback. Preprints of IFAC World Congress (2002)Google Scholar
  21. 21.
    Vítečková, M., Víteček, A.: Plant identification by relay methods. In: Dudas, L. (ed.) Engineering the Future, pp. 242–256. Sciyo, Rijeka (2010)Google Scholar
  22. 22.
    Yu, C.C.: Autotuning of PID Controllers. Springer, London (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of applied informaticsTomas Bata University in ZlínZlínCzech Republic

Personalised recommendations