An Estimation of Accuracy of Oustaloup Approximation
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.
KeywordsFractional order transfer function Oustaloup approximation
This paper was partially supported by the AGH (Poland)—project no 184.108.40.2065 and partially supported by the AGH (Poland)—project no 220.127.116.117.
- 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.Kaczorek, T.: Selected Problems in Fractional Systems Theory. Springer (2011)Google Scholar
- 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.Mitkowski, W.: Finite-dimensional approximations of distributed RC networks. Bull. Polish Acad. Sci. Tech. Sci. 62(2), 263–269 (2014)Google Scholar
- 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.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: http://journals.bg.agh.edu.pl/AUTOMAT/2012.16.2/automat.2012.16.2.145.pdf
- 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.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.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.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.Isermann, R., Muenchhof, M.: Identification of Dynamic Systems. An Introduction with Applications. Springer (2011)Google Scholar