Advertisement

Non-Integer Order Control of PMSM Drives with Two Nested Feedback Loops

  • Paolo Lino
  • Guido MaioneEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)

Abstract

In industrial control applications, the plants are often represented by integer-order models and controlled by proportional-integral-derivative controllers. The control scheme for permanent magnet synchronous motors (PMSM) includes two nested loops, each employing a PI controller: the inner loop is dedicated to control the current, the outer loop is devoted to control the angular speed. The optimum modulus and symmetrical optimum criteria are widely accepted techniques to tune the two PI controllers. However, to obtain improvements, one may think to apply non-integer order controllers. To this aim, if one uses a fractional-order PI (FOPI) controller in the inner loop, the consequent inner feedback system of non-integer order becomes a non-integer order plant in the outer loop. Then, a FOPI controller should be more effective to control a real example of non-integer order plant. This paper proposes an appropriate design approach to obtain performance and robustness specifications by FOPI controllers in both loops. The approach provides analytical formulas to determine the parameters of the controllers, which are characterized by stability, minimum-phase and interlacing properties. The simulation of a real PMSM shows the effectiveness of the approach and could help to increase the confidence in non-integer order controllers.

Keywords

Nested loops Fractional order PI controllers Permanent magnet synchronous motors Loop shaping Optimum modulus Symmetrical optimum 

Notes

Acknowledgement

This paper is based upon work from COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology).

References

  1. 1.
    Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Nonlinear noninteger order circuits and systems - an introduction. In: Chua, L. (ed.) World Scientific Series on Nonlinear Science. Series A, vol. 38. World Scientific, Singapore (2000)zbMATHGoogle Scholar
  2. 2.
    Åström, K.J., Hägglund, T.: PID Controllers: Theory, Design, and Tuning, 2nd edn. Instrument Society of America, Research Triangle Park (1995)Google Scholar
  3. 3.
    Caponetto, R., Dongola, G.: A numerical approach for computing stability region of FO-PID controller. J. Franklin Inst. 350(4), 871–889 (2013).  https://doi.org/10.1016/j.jfranklin.2013.01.017MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caponetto, R., Dongola, G., Fortuna, L., Petráš, I.: Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore (2010)CrossRefGoogle Scholar
  5. 5.
    Caponetto, R., Dongola, G., Pappalardo, F., Tomasello, V.: Auto-tuning and fractional order controller implementation on hardware in the loop system. J. Optim. Theory Appl. 156(1), 141–152 (2013).  https://doi.org/10.1007/s10957-012-0235-yMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, Y.Q.: Ubiquitous fractional order controls? In: Proceedings of the 2nd IFAC Symposium on Fractional Derivatives and Its Applications, Porto, Portugal, vol. 2, pp. 168–173, 19–21 July 2006Google Scholar
  7. 7.
    Chen, Y.Q., Petras, I., Xue, D.: Fractional order control – a tutorial. In: 2009 American Control Conference, Hyatt Regency Riverfront, St. Louis, MO, USA, 10–12 June 2009Google Scholar
  8. 8.
    Kalman, R.E.: When is a linear control system optimal? Trans. ASME J. Basic Eng. 86(Series D), 51–60 (1964)CrossRefGoogle Scholar
  9. 9.
    Kessler, C.: Das symmetrische optimum. Regelungstechnik 6, 395–400, 432–436 (1958)Google Scholar
  10. 10.
    Jalloul, A., Trigeassou, J.-C., Jelassi, K., Melchior, P.: Fractional order modeling of rotor skin effect in induction machines. Nonlinear Dyn. 73(1), 801–813 (2013)CrossRefGoogle Scholar
  11. 11.
    Lino, P., Maione, G.: Loop-shaping and easy tuning of fractional-order proportional integral controllers for position servo systems. Asian J. Control. 15(3), 796–805 (2013).  https://doi.org/10.1002/asjc.556MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lino, P., Maione, G., Padula, F., Stasi, S., Visioli, A.: Synthesis of fractional-order PI controllers and fractional-order filters for industrial electrical drives. IEEE/CAA J. Autom. Sinica 4(1), 58–69 (2017).  https://doi.org/10.1109/JAS.2017.7510325MathSciNetCrossRefGoogle Scholar
  13. 13.
    Luo, Y., Chen, Y.Q.: Fractional Order Motion Controls. Wiley, Chichester (2013)Google Scholar
  14. 14.
    Lurie, B.J., Enright, P.J.: Classical Feedback Control: With MATLAB. Control Engineering Series. Munro, N. (ed.) Marcel Dekker, Inc., New York, Basel (2000)Google Scholar
  15. 15.
    Lutz, H., Wendt, W.: Taschenbuch der Regelungstechnik. 4. Korregierte Auflage, Verlag Harri Deutsch, Frankfurt am Main (2002)Google Scholar
  16. 16.
    Maione, G., Lino, P.: New tuning rules for fractional \({PI}^\alpha \) controllers. Nonlinear Dyn. 49(1–2), 251–257 (2007).  https://doi.org/10.1007/s11071-006-9125-xCrossRefzbMATHGoogle Scholar
  17. 17.
    Maciejowski, J.M.: Multivariable Feedback Design. Addison-Wesley, Wokingham (1989)zbMATHGoogle Scholar
  18. 18.
    Maione, G.: Continued fractions approximation of the impulse response of fractional order dynamic systems. IET Control Theory Appl. 2(7), 564–572 (2008).  https://doi.org/10.1049/iet-cta:20070205MathSciNetCrossRefGoogle Scholar
  19. 19.
    Maione, G.: Conditions for a class of rational approximants of fractional differentiators/integrators to enjoy the interlacing property. In: Bittanti, S., Cenedese, A., Zampieri, S. (eds.) Proceedings of the 18th IFAC World Congress (IFAC WC 2011), Università Cattolica del Sacro Cuore, IFAC Proceedings, Milan, Italy, 28 August–2 September 2011, vol. 18, Part 1, pp. 13984–13989 (2011).  https://doi.org/10.3182/20110828-6-IT-1002.01035CrossRefGoogle Scholar
  20. 20.
    Mansouri, R., Bettayeb, M., Djennoune, S.: Approximation of high order integer systems by fractional order reduced-parameters models. Math. Comput. Model. 51, 53–62 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Monje, C.A., Calderon, A.J., Vinagre, B.M., Feliu, V.: The fractional order lead compensator. In: Proceedings of the 2nd IEEE International Conference on Computational Cybernetics (ICCC 2004), Vienna, Austria, 30 August–1 September 2004, pp. 347–352 (2004)Google Scholar
  22. 22.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)CrossRefGoogle Scholar
  23. 23.
    Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.Q.: Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng. Pract. 16, 798–812 (2008)CrossRefGoogle Scholar
  24. 24.
    Oldenbourg, R.C., Sartorius, H.: A uniform approach to the optimum adjustments of control loops. In: Oldenburger, R. (ed.) Frequency Response. The Macmillan Co., New York (1956)Google Scholar
  25. 25.
    Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)zbMATHGoogle Scholar
  26. 26.
    Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers. Springer, Dordrecht (2011)CrossRefGoogle Scholar
  27. 27.
    Oustaloup, A.: La Commande CRONE. Command Robuste d’Ordre Non Entiér. Editions Hermés, Paris (1991)Google Scholar
  28. 28.
    Podlubny, I.: Fractional-order systems and \({PI}^\lambda {D}^\mu \) controllers. IEEE Trans. Autom. Control. 44(1), 208–214 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, 367–386 (2002)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Retiere, N.M., Ivanès, M.S.: Modeling of electrical machines by implicit derivative half order systems. IEEE Power Eng. Rev. 18(9), 62–64 (1998)Google Scholar
  31. 31.
    Retiere, N.M., Ivanès, M.S.: An introduction to electrical machines modeling by non integer order systems. Application to double-cage induction machine. IEEE Trans. Energy Convers. 14(4), 1026–1032 (1999)CrossRefGoogle Scholar
  32. 32.
    Shah, P., Agashe, S.: Review of fractional PID controller. Mechatronics 38, 29–41 (2016).  https://doi.org/10.1016/j.mechatronics.2016.06.005CrossRefGoogle Scholar
  33. 33.
    Valério, D., Sá da Costa, J.: Tuning of fractional PID controllers with Ziegler-Nichols-type rules. Signal Process. 86, 2771–2784 (2010)CrossRefGoogle Scholar
  34. 34.
    Voda, A.A., Landau, I.D.: A method for auto-calibration of PID controllers. Automatica 31(1), 41–53 (1995)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Yu, W., Luo, Y., Pi, Y., Chen, Y.Q.: Fractional-order modeling of a permanent magnet synchronous motor velocity servo system: method and experimental study. In: Proceedings of the 2014 International Conference on Fractional Differentiation and Its Applications, Catania, Italy, 23–25 June 2014Google Scholar
  36. 36.
    Zhao, C., Xue, D., Chen, Y.Q.: A fractional order PID tuning algorithm for a class of fractional order plants. In: Proceedings of the IEEE International Conference on Mechatronics and Automation, Niagara Falls, Canada, 29 July–1 August 2005, vol. 1, pp. 216–221 (2005)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Electrical and Information EngineeringPolytechnic University of BariBariItaly

Personalised recommendations