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Discrete Transfer Function Models for Non Integer Order Inertial System

  • Krzysztof OprzędkiewiczEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 743)

Abstract

In the paper new, discrete, transfer function models of non integer order inertial plant are proposed. These models can be employed to digital modeling of high order dynamic systems, for example heat transfer systems. Models under consideration use Charef approximation and generating functions expressed by schemes given by Euler, Tustin and Al-Aloui. The practical stability and accuracy for all presented models is analysed also. Results are by simulations depicted.

Keywords

Fractional order systems Fractional order transfer function Charef approximation Generating function Practical stability 

Notes

Acknowledgements

This paper was sponsored partially by AGH UST grant no. 11.11.120.815.

References

  1. 1.
    Al-Alaoui, M.A.: Al-Alaoui operator and the \(\alpha \) -approximation for discretization of analog systems. Facta Univ. (Nis)Ser. Electron. Energ. 19(1), 143–146 (2006)CrossRefGoogle Scholar
  2. 2.
    Caponetto, R., Dongola, G., Fortuna, I., Petras, I.: Fractional Order Systems Modeling and Control Applications. World Scientific Series on Nonlinear Science, vol. 72. World Scientific Publishing, New Jersey (2010)Google Scholar
  3. 3.
    Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractional system as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Y.Q.: Oustaloup Recursive Approximation for Fractional Order Differentiators, MathWorks Inc., Matlab Central File Exchange (2003)Google Scholar
  5. 5.
    Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008)zbMATHGoogle Scholar
  6. 6.
    Das, S., Pan, I.: Fractional Order Signal Processing. SpringerBriefs in Applied Sciences and Technology. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-23117-9-2 CrossRefzbMATHGoogle Scholar
  7. 7.
    Dlugosz, M., Skruch, P.: The application of fractional-order models for thermal process modelling inside buildings. J. Build. Phys. 39, 1–13 (2015)Google Scholar
  8. 8.
    Djouambi, A., Charef, A., BesançOn, A.: Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function. Int. J. Appl. Math. Comput. Sci. 17(4), 455–462 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dzielinski, A., Sierociuk, D., Sarwas, G.: Some applications of fractional order calculus. Bull. Pol. Acad. Sci. Techn. Sci. 58(4), 583–592 (2010)zbMATHGoogle Scholar
  10. 10.
    Garrappa, R., Maione, G.: Fractional Prabhakar derivative and applications in anomalous dielectrics: a numerical approach. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds.) Theory and Applications of Non-integer Order Systems. LNEE, vol. 407, pp. 429–439. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-45474-0_38 CrossRefGoogle Scholar
  11. 11.
    Kaczorek, T.: Practical stability of positive fractional discrete-time systems. Bull. Pol. Acad. Sci. Tech. Sci. 56(4), 313–317 (2008)Google Scholar
  12. 12.
    Kaczorek, T.: Selected Problems in Fractional Systems Theory. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok (2014)zbMATHGoogle Scholar
  14. 14.
    Maione, G.: High-speed digital realizations of fractional operators in the delta domain. IEEE Trans. Autom. Control 56(3), 697–702 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mitkowski, W., Skruch, P.: Fractional-order models of the supercapacitors in the form of RC ladder networks. Bull. Pol. Acad. Sci. Tech. Sci. 61(3), 581–587 (2013)Google Scholar
  16. 16.
    Obraczka, A., Mitkowski, W.: The comparison of parameter identification methods for fractional partial differential equation. Solid State Phenom. 210, 265–270 (2014)CrossRefGoogle Scholar
  17. 17.
    Oprzedkiewicz, K., Mitkowski, W., Gawin, E.: Parameter identification for non integer order, state space models of heat plant. In: MMAR 2016: 21th International Conference on Methods and Models in Automation and Robotics, 29 August–01 September 2016, pp. 184–188. Miedzyzdroje, Poland (2016). ISBN:978-1-5090-1866-6, 978-837518-791-5Google Scholar
  18. 18.
    Oprzędkiewicz, K., Kołacz, T.: A non integer order model of frequency speed control in AC motor. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds.) Challenges in Automation, Robotics and Measurement Techniques. AISC, vol. 440, pp. 287–298. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29357-8_26 CrossRefGoogle Scholar
  19. 19.
    Ostalczyk, P.: Discrete Fractional Calculus. Applications in control and image processing. Series in Computer Vision, vol. 4. World Scientific Publishing, Singapore (2016)zbMATHGoogle Scholar
  20. 20.
    Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circ. Syst. I Fundam. Theor. Appl. I 47(1), 25–39 (2000)Google Scholar
  21. 21.
    Stanislawski, R., Latawiec, K.J., Lukaniszyn, M.: A comparative analysis of Laguerre-based approximators to the Grunwald-Letnikov fractional-order difference. Math. Probl. Eng. 2015, 10 (2015). Article ID 512104MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.AGH University of Science and TechnologyKrakowPoland

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