Frequency response based curve fitting approximation of fractional-order PID controllers

Bingi, Kishore ; Ibrahim, Rosdiazli ; Karsiti, Mohd Noh ; Hassam, Sabo Miya ; Harindran, Vivekananda Rajah

International Journal of Applied Mathematics and Computer Science, Tome 29 (2019) no. 2, pp. 311-326.

Résumé

Fractional-order PID (FOPID) controllers have been used extensively in many control applications to achieve robust control performance. To implement these controllers, curve fitting approximation techniques are widely employed to obtain integer-order approximation of FOPID. The most popular and widely used approximation techniques include the Oustaloup, Matsuda and Cheraff approaches. However, these methods are unable to achieve the best approximation due to the limitation in the desired frequency range. Thus, this paper proposes a simple curve fitting based integer-order approximation method for a fractional-order integrator/differentiator using frequency response. The advantage of this technique is that it is simple and can fit the entire desired frequency range. Simulation results in the frequency domain show that the proposed approach produces better parameter approximation for the desired frequency range compared with the Oustaloup, refined Oustaloup and Matsuda techniques. Furthermore, time domain and stability analyses also validate the frequency domain results.

Détail

Keywords: curve fitting, fractional order PID controller, frequency response, integer order approximation, Oustaloup approximation, Matsuda approximation
Mots-clés : dopasowywanie krzywej, ułamkowy regulator PID, odpowiedź częstotliwościowa

Bibliographie

[1] Atherton, D.P., Tan, N. and Yüce, A. (2014). Methods for computing the time response of fractional-order systems, IET Control Theory Applications 9(6): 817–830.

[2] Balas, G., Chiang, R., Packard, A. and Safonov, M. (2007). Robust Control Toolbox 3: User’ Guide, MathWorks, Natick, MA.

[3] Bingi, K., Ibrahim, R., Karsiti, M.N., Hassan, S.M. And Harindran, V.R. (2018a). A comparative study of 2DOF PID and 2DOF fractional order PID controllers on a class of unstable systems, Archives of Control Sciences 28(4): 635–682.

[4] Bingi, K., Ibrahim, R., Karsiti, M.N., Hassan, S.M. And Harindran, V.R. (2018b). Real-time control of pressure plant using 2DOF fractional-order PID controller, Arabian Journal for Science and Engineering 44(3): 2091–2102.

[5] Caponetto, R. (2010). Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore.

[6] Das, S. (2011). Functional Fractional Calculus, Springer, Berlin/Heidelberg.

[7] de Oliveira Valério, D.P.M. (2005). Fractional Robust System Control, PhD thesis, Universidade Técnica de Lisboa, Lisboa.

[8] Deniz, F.N., Alagoz, B.B., Tan, N. and Atherton, D.P. (2016). An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators, ISA Transactions 62: 154–163.

[9] Djouambi, A., Charef, A. and Besançon, A.V. (2007). Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function, International Journal of Applied Mathematics and Computer Science 17(4): 455–462, DOI: 10.2478/v10006-007-0037-9.

[10] Du, B., Wei, Y., Liang, S. and Wang, Y. (2017). Rational approximation of fractional order systems by vector fitting method, International Journal of Control, Automation and Systems 15(1): 186–195.

[11] Joice Nirmala, R. and Balachandran, K. (2017). The controllability of nonlinear implicit fractional delay dynamical systems, International Journal of Applied Mathematics and Computer Science 27(3): 501–513, DOI: 10.1515/amcs-2017-0035.

[12] Kaczorek, T. (2018). Decentralized stabilization of fractional positive descriptor continuous-time linear systems, International Journal of Applied Mathematics and Computer Science 28(1): 135–140, DOI: 10.2478/amcs-2018-0010.

[13] Khanra, M., Pal, J. and Biswas, K. (2011). Rational approximation and analog realization of fractional order differentiator, 2011 International Conference on Process Automation, Control and Computing (PACC), Coimbatore, India, pp. 1–6.

[14] Khanra, M., Pal, J. and Biswas, K. (2013). Rational approximation and analog realization of fractional order transfer function with multiple fractional powered terms, Asian Journal of Control 15(3): 723–735.

[15] Kishore, B., Ibrahim, R., Karsiti, M.N. and Hassan, S.M. (2017). Fractional-order filter design for set-point weighted PID controlled unstable systems, International Journal of Mechanical Mechatronics Engineering 17(5): 173–179.

[16] Kishore, B., Ibrahim, R., Karsiti, M.N. and Hassan, S.M. (2018). Fractional order set-point weighted PID controller for pH neutralization process using accelerated PSO algorithm, Arabian Journal for Science and Engineering 43(6): 2687–2701.

[17] Krajewski, W. and Viaro, U. (2011). On the rational approximation of fractional order systems, 16th International Conference on Methods and Models in Automation and Robotics (MMAR), Międzyzdroje, Poland, pp. 132–136.

[18] Krajewski, W. and Viaro, U. (2014). A method for the integer-order approximation of fractional-order systems, Journal of the Franklin Institute 351(1): 555–564.

[19] Krishna, B. (2011). Studies on fractional order differentiators and integrators: A survey, Signal Processing 91(3): 386–426.

[20] Li, Z., Liu, L., Dehghan, S., Chen, Y. and Xue, D. (2017). A review and evaluation of numerical tools for fractional calculus and fractional order controls, International Journal of Control 90(6): 1165–1181.

[21] Liang, S., Peng, C., Liao, Z. and Wang, Y. (2014). State space approximation for general fractional order dynamic systems, International Journal of Systems Science 45(10): 2203–2212.

[22] Meng, L. and Xue, D. (2012). A new approximation algorithm of fractional order system models based optimization, Journal of Dynamic Systems, Measurement, and Control 134(4): 044504.

[23] Merrikh-Bayat, F. (2012). Rules for selecting the parameters of Oustaloup recursive approximation for the simulation of linear feedback systems containing PIλDμ controller, Communications in Nonlinear Science and Numerical Simulation 17(4): 1852–1861.

[24] Mitkowski, W. and Oprzedkiewicz, K. (2016). An estimation of accuracy of Charef approximation, in S. Domek and P. Dworak (Eds.), Theoretical Developments and Applications of Non-Integer Order Systems, Springer, Berlin/Heidelberg, pp. 71–80.

[25] Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D. and Feliu-Batlle, V. (2010). Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, Berlin/Heidelberg.

[26] Oprzedkiewicz, K. (2014). Approximation method for a fractional order transfer function with zero and pole, Archives of Control Sciences 24(4): 447–463.

[27] Oustaloup, A., Levron, F., Mathieu, B. and Nanot, F.M. (2000). Frequency-band complex noninteger differentiator: characterization and synthesis, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47(1): 25–39.

[28] Pachauri, N., Singh, V. and Rani, A. (2018). Two degrees-of-freedom fractional-order proportional-integral-derivative-based temperature control of fermentation process, Journal of Dynamic Systems, Measurement, and Control 140(7): 071006.

[29] Petráš, I. (2011a). Fractional derivatives, fractional integrals, and fractional differential equations in Matlab, in A. Assi (Ed.), Engineering Education and Research Using MATLAB, InTech, London, pp. 239–264.

[30] Petráš, I. (2011b). Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, Berlin/Heidelberg.

[31] Poinot, T. and Trigeassou, J.-C. (2003). A method for modelling and simulation of fractional systems, Signal processing 83(11): 2319–2333.

[32] Shah, P. and Agashe, S. (2016). Review of fractional PID controller, Mechatronics 38: 29–41.

[33] Sheng, H., Chen, Y. and Qiu, T. (2011). Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Springer, Berlin/Heidelberg.

[34] Shi, G. (2016). On the nonconvergence of the vector fitting algorithm, IEEE Transactions on Circuits and Systems II: Express Briefs 63(8): 718–722.

[35] Tepljakov, A., Petlenkov, E. and Belikov, J. (2012). Application of Newton’s method to analog and digital realization of fractional-order controllers, International Journal of Microelectronics and Computer Science 2(2): 45–52.

[36] Valério, D., Trujillo, J.J., Rivero, M., Machado, J.T. And Baleanu, D. (2013). Fractional calculus: A survey of useful formulas, The European Physical Journal Special Topics 222(8): 1827–1846.

[37] Vinagre, B., Podlubny, I., Hernandez, A. and Feliu, V. (2000). Some approximations of fractional order operators used in control theory and applications, Fractional Calculus and Applied Analysis 3(3): 231–248.

[38] Wei, Y., Gao, Q., Peng, C. and Wang, Y. (2014a). A rational approximate method to fractional order systems, International Journal of Control, Automation and Systems 12(6): 1180–1186.

[39] Wei, Y., Gao, Q., Peng, C. and Wang, Y. (2014b). A rational approximate method to fractional order systems, International Journal of Control, Automation and Systems 12(6): 1180–1186.

[40] Xue, D. (2017). Fractional-order Control Systems: Fundamentals and Numerical Implementations, Walter de Gruyter GmbH, Berlin.

[41] Xue, D., Chen, Y. and Attherton, D.P. (2007). Linear Feedback Control: Analysis and Design with MATLAB, SIAM, Philadelphia, PA.

[42] Xue, D., Zhao, C. and Chen, Y. (2006). A modified approximation method of fractional order system, Proceedings of the 2006 IEEE International Conference on Mechatronics and Automation, Luoyang, China pp. 1043–1048.

[43] Yüce, A., Deniz, F.N. and Tan, N. (2017). A new integer order approximation table for fractional order derivative operators, IFAC-PapersOnLine 50(1): 9736–9741.