An Estimation of Accuracy of Oustaloup Approximation

  • Krzysztof OprzędkiewiczEmail author
  • Wojciech Mitkowski
  • Edyta Gawin
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 440)


In the paper a new accuracy estimation method for Oustaloup approximation is presented. Oustaloup approximation is a fundamental tool to describe fractional-order systems with the use of integer-order, proper transfer function. The accuracy of approximation can be estimated via comparison of impulse responses for plant and Oustaloup approximation. The impulse response of the plant was calculated with the use of an accurate analytical formula and it can be interpreted as a standard. Approach presented in the paper can be applied to effective tuning of Oustaloup approximant for given application (for example in FO PID controller). The use of proposed method does not require us to know time response of a modeled controller. The proposed methodology can be easily generalized to another known approximations. Results of simulations show that the good performance of approximation is reached for low order and narrow angular frequency range.


Fractional order transfer function Oustaloup approximation 



This paper was partially supported by the AGH (Poland)—project no and partially supported by the AGH (Poland)—project no


  1. 1.
    Caponetto, R., Dongola, G., Fortuna, L., Petras, I.: Fractional Order Systems. Modeling and Control Applications. World Scientific Series on Nonlinear Science, Series A, vol. 72. World Scientific Publishing (2010)Google Scholar
  2. 2.
    Kaczorek, T.: Selected Problems in Fractional Systems Theory. Springer (2011)Google Scholar
  3. 3.
    Mitkowski, W., Skruch, P.: Fractional-order models of the supercapacitors in the form of RC ladder networks. Bull. Polish Acad. Sci. Tech. Sci. 61(3), 581–587 (2013)Google Scholar
  4. 4.
    Mitkowski, W.: Finite-dimensional approximations of distributed RC networks. Bull. Polish Acad. Sci. Tech. Sci. 62(2), 263–269 (2014)Google Scholar
  5. 5.
    Mitkowski, W., Obrączka, A.: Simple identification of fractional differential equation. Solid State Phenom. 180, 331–338 (2012)CrossRefGoogle Scholar
  6. 6.
    Mitkowski, W., Oprzędkiewicz, K.: Application of fractional order transfer functions to modeling of high-order systems. 7 IFIP Conference, Klagenfurt (2013)Google Scholar
  7. 7.
    Oprzędkiewicz, K.: A Strejc model-based, semi- fractional (SSF) transfer function model. Automatyka/Automatics; AGH UST 2012 vol. 16 no. 2, pp. 145–154 (2012). Direct link to text:
  8. 8.
    Merikh-Bayat, F.: Rules for selecting the parameters of Oustaloup recursive approximation for the simulation of linear feedback systems containing PIλDµ controller. Commun. Nonlinear Sci. Numer. Simul. 17, 1852–1861 (2012)Google Scholar
  9. 9.
    Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. I 47(1), 25–39 (2000)Google Scholar
  10. 10.
    Petras, I.: Realization of fractional order controller based on PLC and its utilization to temperature control. Transfer inovaci nr 14(2009), 34–38 (2009)Google Scholar
  11. 11.
    Mitkowski, W., Oprzędkiewicz, K.: An estimation of accuracy of Charef approximation. Theoretical developments and applications of non-integer order systems. In: Domek, S., Dworak, P. (eds.) 7th Conference on Non-Integer Order Calculus and its Applications (Lecture Notes in Electrical Engineering; ISSN 1876–1100; vol. 357). Szczecin, Poland, pp. 71–80. Springer (2016)Google Scholar
  12. 12.
    Isermann, R., Muenchhof, M.: Identification of Dynamic Systems. An Introduction with Applications. Springer (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Krzysztof Oprzędkiewicz
    • 1
    Email author
  • Wojciech Mitkowski
    • 1
  • Edyta Gawin
    • 2
  1. 1.Faculty of Electrotechnics, Automatics, Informatics and Biomedical Engineering, Department of Automatics and Biomedical EngineeringAGH University of Science and TechnologyKrakówPoland
  2. 2.Polytechnic InstituteHigh Vocational School in TarnówTarnówPoland

Personalised recommendations