The processes with fractional order delay and PI controller design using particle swarm optimization
In this study, the stability analysis of systems with fractional order delay is presented. Besides, PI controller design using particle swarm optimization (PSO) technique for such systems is also presented. The PSO algorithm is used to obtain the controller parameters within the stability region. As it is known that it is not possible to investigate the stability of systems with fractional order delay using analytical methods such as the Routh-Hurwitz criterion. Furthermore, stability analysis of such systems is quite difficult. In this study, for stability testing of such systems, an approximation method previously introduced in the literature by the corresponding author is used. In addition, the unit step responses have been examined to evaluate the systems' performances. It should be noted that examining unit step responses of systems having fractional-order delay is not possible due to the absence of analytical methods. One of the aims of this study is to overcome this deficiency by using the proposed approximation method. Besides, a solution to the question of which controller parameter values should be selected in the stability region, which provides the calculation of all stabilizing PI controllers, is proposed using the PSO algorithm.
[1] Gu, K., Kharitonov, V.L., & Chen, J. (2003). Stability of Time-Delay Systems. Birkh¨auser Boston, MA.
[2] Eriksson, L., Oksanen, T., & Mikkola, K.(2009). PID controller tuning rules for inte- grating processes with varying time-delays. Journal of the Franklin Institute, 346(5), 470–487.
[3] Han, Q.-L. (2005). Absolute stability of time- delay systems with sector-bounded nonlinear- ity. Automatica, 41(12), 2171–2176.
[4] Ozturk, N., & Uraz, A. (1984). An analytic stability test for a certain class of distributed parameter systems with a distributed lag. IEEE Transactions on Automatic Control, 29(4), 368–370.
[5] Ozturk, N., & Uraz, A. (1985). An anal- ysis stability test for a certain class of distributed parameter systems with delays. IEEE Transactions on Circuits and Systems, 32(4), 393–396.
[6] Ozturk, N. (1990). An application of two di- mensional stability criterion to a special class of distributed parameter systems. Proceedings of IECON ’90: 16th Annual Conference of IEEE Industrial Electronics Society. , 368-371.
[7] Ozturk, N. (1995). Stability independent of distributed lag for a special class of dis- tributed parameter systems. Proceedings of 34th IEEE Conference on Decision and Con- trol. 3245-3246.
[8] Chen, C.F., & Chiu, R.F. (1973). Evaluation of irrational and transcendental transfer func- tions via the fast Fourier transformt. Interna- tional Journal of Electronics, 35(2), 267–276.
[9] Bourquin, J.J., & Trick, T.N. (1969). Stabil- ity of a class of lumped-distributed systems. Journal of the Franklin Institute, 287(5), 363–378.
[10] Toumani, R. (1973). On the stability of lumped-distributed networks. IEEE Transac- tions on Circuit Theory, 20(5), 606–608.
[11] Juchem, J., Chevalier, A., Dekemele, K., & Loccufier, M. (2021). First order plus frac- tional diffusive delay modeling: intercon- nected discrete systems. Fractional Calculus and Applied Analysis, 24(5), 1535-1558.
[12] Ozyetkin, M.M. (2022). An approximation method and PID controller tuning for systems having integer order and non-integer order de- lay. Alexandria Engineering Journal, 61(12), 11365-11375.
[13] Ozyetkin, M.M. (2018). A simple tuning method of fractional order PIλ-PDμ con- trollers for time delay systems. ISA Transac- tions, 74, 77–87.
[14] Onat, C. (2013). A new concept on PI de- sign for time delay systems: weighted geomet- rical center. International Journal of Inno- vative Computing, Information and Control, 9(4), 1539-1556.
[15] Ozyetkin, M.M., & Astekin, D. (2022). Pade approximation for time delay systems and a new design method for the fractional order PI controller. Journal of the Faculty of Engi- neering and Architecture of Gazi University, 38 (2) , 639-652.
[16] Ozyetkin, M.M., Onat, C., & Tan, N.(2012). Zaman Gecikmeli Sistemler i¸cin PIλ Denet¸ci Tasarımı . Otomatik Kontrol T¨urk Milli Komitesi (TOK-2012), 428-433.
[17] Ozyetkin, M.M., & Birdane, H. (2022). Par¸cacık s¨ur¨u optimizasyonu tabanlı PI denetleyici parametrelerinin elde Edilmesi ve sistem tasarımı . In: C. O(¨)zalp, ed. M¨uhendis-lik Alanında Teori ve Ara¸stırmalar. Ser¨uven Yayınevi, Izmir, TR, 249-278.
[18] Lazarevic, M., Rapaic, M. & Sekara, T.(2014). Introduction to fractional calculus with brief historical background . In: V. Mladenov, & N. Mastorakis, eds. Advanced Topics on Applications of Fractional Calcu- lus on Control Problems, System Stability and Modeling. WSEAS Press, 3-16.
[19] Yusuf, A., Qureshi, S., Mustapha, U.T., Musa, S.S., & Sulaiman, T.A. (2022). Frac- tional modeling for improving scholastic per- formance of students with optimal control. In- ternational Journal of Applied and Computa- tional Mathematics, 8(1).
[20] Muresan, C.I., & Ionescu, C.M. (2020). Gen- eralization of the FOPDT model for identifi- cation and control purposes. Processes, 8(6), 682.
[21] Ucar, E., Ucar, S., Evirgen, F., & Ozdemir, N. (2021). A Fractional SAIDR model in the frame of Atangana-Baleanu derivative. Frac- tal and Fractional, 5, 32.
[22] Evirgen, F. (2023). Transmission of Nipah virus dynamics under Caputo fractional de- rivative. Journal of Computational and Ap- plied Mathematics, 418, 114654.
[23] Lorenzo, C.F., & Hartley, T.T. (2002). Vari- able order and distributed order fractional op- erators. Nonlinear Dynamics, 29, 57–98.
[24] Podlubny, I. (1999). Fractional-order systems and PIλ Dμ controllers. IEEE Transactions on Automatic Control, 44(1), 208–214.
[25] O(¨)zyetkin, M.M., Yeroˇglu, C., Tan, N., &Taˇgluk M.E. (2010). Design of PI and PID controllers for fractional order time delay systems. IFAC Proceedings Volumes, 43(2), 355–360.
[26] Yuce, A., & Tan, N., (2021). On the approxi- mate inverse Laplace transform of the transfer function with a single fractional order. Trans- actions of The Institute of Measurement and Control, 43(6), 1376-1384.
[27] Yuce, A., & Tan, N. (2019). Inverse laplace transforms of the fractional order transfer functions. Proceedings of 11th International Conference on Electrical and Electronics En- gineering (ELECO), 775-779.
[28] Kennedy, J., & Eberhart, R. (1995). Par- ticle swarm optimization. Proceedings of In- ternational Conference on Neural Networks (ICNN), 4, 1942-1948.
[29] Campo, A.B. (2012). PID control design. In:V. Katsikis, ed. In MATLAB - A Fundamen- tal Tool for Scientific Computing and Engi- neering Applications, IntechOpen.
[30] Vastrakar, N.K., & Padhy, P.K. (2013). Sim- plified PSO PI-PD controller for unstable pro- cesses. Proceedings of 4th International Con- ference on Intelligent Systems, Modelling and Simulation, 350-354.
[31] Zennir, Y., Mechhoud, E.A., Seboui, A., & Bendib, R. (2017). Multi-controller ap- proach with PSO-PIλ Dμ controllers for a robotic wrist. Proceedings of 5th Interna- tional Conference on Electrical Engineering- Boumerdes(ICEE-B), 1-7.
[32] Liu, J., Wang, H., & Zhang, Y. (2015). New result on PID controller design of LTI systems via dominant eigenvalue assignment. Auto- matica, 62, 93–97.
[33] Tan, N. (2005). Computation of stabilizing PI and PID controllers for processes with time delay. ISA Transactions, 44(2), 213–223.
[34] Hamamci, S.E., & Tan, N. (2006). Design of PI controllers for achieving time and fre- quency domain specifications simultaneously. ISA Transactions, 45(4), 529–543.
[35] Hohenbichler, N. (2009). All stabilizing PID controllers for time delay systems. Automat- ica, 45(11), 2678–2684.
[36] Hwang, C., & Cheng, Y.C. (2006). A numeri- cal algorithm for stability testing of fractional delay systems. Automatica. 42(5), 825-831.
[37] Ozyetkin, M.M. (2022). PD controller de- sign and stability analysis for systems hav- ing fractional order delay. Journal of Scien- tific Reports-A, 050, 254-269.
[38] Monje, C.A., Chen, Y., Vinagre, B.M., Xue,D., & Feliu, V. (2010). Fractional-order Sys- tems and Controls Fundamentals and Appli- cations. Springer, London.