Accuracy Estimation of Digital Fractional Order PID Controller
In the paper an accuracy estimation method for digital fractional order PID controller is discussed. Integral and derivative parts of a controller are approximated with the use of Power Series Expansion (PSE) and Continuous Fraction Expansion (CFE) methods. These approximations are fundamental tools to modeling fractional-order elements with the use of integer-order, discrete, proper transfer function in the form of FIR or IIR filter. The accuracy of each approximation is a function of its order and other parameters. It can be estimated via comparison of step responses: analytical and approximated in sample moments. The step response expressed by accurate analytical formula can be interpreted as a standard. Approach presented in the paper can be applied during implementation of FO PID at each digital platform (microcontroller, PLC). Results of simulations show, that the CFE approximation allows us to build a FO PID controller so accurate, as constructed with the use of PSE, but much more simple to implementation, because its order is significantly lower.
KeywordsDigital fractional order PID controller PSE approximation CFE approximation
This paper was supported by the AGH (Poland) – project no 18.104.22.1685.
- 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, Singapore (2010)Google Scholar
- 2.Dorcak, L., Petras, I., Terpak, J., Zbrojovan, M.: Comparison of the methods for discrete approximation of the fractional-order operator. In: Proceedings of the Carpathian Control Conference, pp. 851–856 (2003)Google Scholar
- 4.Petras, I.: Realization of fractional order controller based on PLC and its utilization to temperature control. Transfer Inovaci 14, 34–38 (2009)Google Scholar
- 6.Al Aloui, M.A.: Dicretization methods of fractional parallel PID controllers. In: Proceedings of 6th IEEE International Conference on Electronics, Circuits and Systems, pp. 327–330 (2009)Google Scholar
- 8.Mitkowski, W., Oprzędkiewicz, K.: An estimation of accuracy of Charef approximation. In: Domek, S., Dworak, P. (eds.) Theoretical Developments and Applications of Non-integer Order Systems : 7th Conference on Non-integer Order Calculus and Its Applications. Lecture Notes in Electrical Engineering, vol. 357, pp. 71–80. Springer (2016)Google Scholar
- 9.Oprzędkiewicz, K., Mitkowski, W., Gawin, E.: An estimation of accuracy of Oustaloup approximation. In: Szewczyk, R. et al. (eds.) Challenges in Automation, Robotics and Measurement Techniques. Advances in Intelligent Systems and Computing, vol. 440, pp. 299–307. Springer, Heidelberg (2016)Google Scholar