Odd-Harmonic High Order Repetitive Control

  • Germán A. RamosEmail author
  • Ramon Costa-Castelló
  • Josep M. Olm
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 446)


HORC is mainly used to improve the repetitive control performance robustness under disturbance/reference signals with varying or uncertain frequency. Unlike standard repetitive control, the HORC involves a weighted sum of several signal periods. With a proper selection of the associated weights, this high order function offers a characteristic frequency response in which the high gain peaks located at harmonic frequencies are extended to a wider region around the harmonics. Furthermore, the use of an odd-harmonic internal model will make the system more appropriate for applications where signals have only odd-harmonic components, as in power electronics systems. This Chapter presents an Odd-harmonic High Order Repetitive Controller suitable for applications involving odd-harmonic type signals with varying/uncertain frequency. The open loop stability of internal models used in HORC and the one presented here is analysed. Additionally, as a consequence of this analysis, an anti-windup scheme for repetitive control is proposed.


Unit Circle Nyquist Plot Internal Model Harmonic Frequency Iterative Learn Control 
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  1. 1.
    Chang, W.S., Suh, I.H., Kim, T.W.: Analysis and design of two types of digital repetitive control systems. Automatica 31(5), 741–746 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    da Silva Jr., J.G., Tarbouriech, S.: Antiwindup design with guaranteed regions of stability: an LMI-based approach. IEEE Transactions on Automatic Control 50(1), 106–111 (2005)CrossRefGoogle Scholar
  3. 3.
    di Bruno, F.F.: Note sur une nouvelle formule du calcul diffrentielle. The Quarterly Journal of Pure and Applied Mathematics 1, 359–360 (1857)Google Scholar
  4. 4.
    Flores, J., Gomes da Silva, J., Pereira, L., Sbarbaro, D.: Robust repetitive control with saturating actuators: a LMI approach. In: Proceedings of the American Control Conference (ACC), pp. 4259–4264 (July 2010)Google Scholar
  5. 5.
    Galeani, S., Tarbouriech, S., Turner, M., Zaccarian, L.: A tutorial on modern anti-windup design. European Journal of Control 15, 418–440 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Galeani, S., Teel, A.R., Zaccarian, L.: Constructive nonlinear anti-windup design for exponentially unstable linear plants. Systems & Control Letters 56(5), 357–365 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Grimm, G., Teel, A.R., Zaccarian, L.: The L2 anti-windup problem for discrete-time linear systems: Definition and solutions. Systems & Control Letters 57(4), 356–364 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Griñó, R., Costa-Castelló, R.: Digital repetitive plug-in controller for odd-harmonic periodic references and disturbances. Automatica 41(1), 153–157 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Herrmann, G., Turner, M., Postlethwaite, I.: Linear matrix inequalities in control. In: Turner, M., Bates, D. (eds.) Mathematical Methods for Robust and Nonlinear Control. LNCIS, vol. 367, pp. 123–142. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Hippe, P.: Windup in Control: Its Effects and Their Prevention. Advances in Industrial Control. Springer (2010)Google Scholar
  11. 11.
    Inoue, T.: Practical repetitive control system design. In: Proceedings of the 29th IEEE Conference on Decision and Control, pp. 1673–1678 (1990)Google Scholar
  12. 12.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002)zbMATHGoogle Scholar
  13. 13.
    Kuc̆era, V.: Deadbeat response is L2 optimal. In: Proceedings of the 3rd International Symposium on Communications, Control and Signal Processing, ISCCSP 2008, pp. 154–157 (March 2008)Google Scholar
  14. 14.
    Kuc̆era, V.: Analysis and Design of Discrete Linear Control Systems. Prentice Hall (1991)Google Scholar
  15. 15.
    Pipeleers, G., Demeulenaere, B., Sewers, S.: Robust high order repetitive control: Optimal performance trade offs. Automatica 44, 2628–2634 (2008)zbMATHCrossRefGoogle Scholar
  16. 16.
    Ryu, Y.S., Longman, R.: Use of anti-reset windup in integral control based learning and repetitive control. In: Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics. Humans, Information and Technology, vol. 3, pp. 2617–2622 (October 1994)Google Scholar
  17. 17.
    Sbarbaro, D., Tomizuka, M., de la Barra, B.L.: Repetitive control system under actuator saturation and windup prevention. Journal of Dynamic Systems, Measurement, and Control 131(4), 044505 (2009)CrossRefGoogle Scholar
  18. 18.
    Seron, M., Goodwin, G.C., Braslavsky, J.: Fundamental limitations in filtering and control. Springer, London (1997)zbMATHCrossRefGoogle Scholar
  19. 19.
    Steinbuch, M.: Repetitive control for systems with uncertain period-time. Automatica 38(12), 2103–2109 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Steinbuch, M., Weiland, S., Singh, T.: Design of noise and period-time robust high order repetitive control, with application to optical storage. Automatica 43, 2086–2095 (2007)zbMATHCrossRefGoogle Scholar
  21. 21.
    Sugimoto, K., Inoue, A., Masuda, S.: A direct computation of state deadbeat feedback gains. IEEE Transactions on Automatic Control 38(8), 1283–1284 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Tarbouriech, S., Turner, M.: Anti-windup design: an overview of some recent advances and open problems. Control Theory Applications, IET 3(1), 1–19 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yeol, J.W., Longman, R.W., Ryu, Y.S.: On the settling time in repetitive control systems. In: Proceedings of 17th International Federation of Automatic Control (IFAC) World Congress (July 2008)Google Scholar
  24. 24.
    Zaccarian, L., Teel, A.R.: A common framework for anti-windup, bumpless transfer and reliable designs. Automatica 38(10), 1735–1744 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Zheng, A., Kothare, M.V., Morari, M.: Anti-windup design for internal model control. International Journal of Control 60, 1015–1024 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Germán A. Ramos
    • 1
    Email author
  • Ramon Costa-Castelló
    • 2
  • Josep M. Olm
    • 3
  1. 1.Department of Electrical and Electronic EngineeringUniversidad Nacional de Colombia BogotáColombia
  2. 2.Escola Tècnica Superior d’Enginyeria Industrial de Barcelona (ETSEIB)Universitat Politècnica de Catalunya (UPC) BarcelonaSpain
  3. 3.Department of Applied Mathematics IV Universitat Politècnica de CatalunyaCastelldefelsSpain

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