Tuning the implementable structures of fractional-order PID controllers for control of FOPDT processes

Document Type : Article


Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran


This study presents a set of rules for optimal tuning a class of integer-order controllers, known as implementable fractional-order PID controllers, to be applied in control of first-order-plus-dead-time (FOPDT) processes. To this aim, the approach of so-called “tuning based on the implementable form of the controller” is applied instead of the common approach of “tuning based on the ideal form of the controller”. Consequently, no contradiction is found between the behavior of the tuned controller and that of the implemented controller. Also, algebraic relations between the values of cost functions, which are defined based on integral square error (ISE) and integral square time error (ISTE) performance indices, and free parameters of the implementable controller are established. Tuning implementable fractional-order PID controllers via the proposed rules guarantees that the values of performance indices are reduced in comparison with the case of using optimal PID controllers. In addition to numerical results, experimental results are also provided to demonstrate the effectiveness of the proposed tuning rules in practical applications.


[1] Dzieliński, A., Sarwas, G. and Sierociuk, D. “Comparison and validation of integer and fractional order ultracapacitor models”, Advances in Difference Equations, 2011(1), pp. 11 (2011).
[2] Radwan, A.G., Soliman, A.M. and Elwakil, A.S. “Design equations for fractional‐order sinusoidal oscillators: Four practical circuit examples”, International Journal of Circuit Theory and Applications, 36(4), pp. 473-492 (2008).
[3] Gabano, J.D., Poinot, T. and Kanoun, H. “Identification of a thermal system using continuous linear parameter-varying fractional modelling”, IET Control Theory and Applications, 5(7), pp. 889-899 (2011).
[4] Xu, J. and Li, J. “Stochastic dynamic response and reliability assessment of controlled structures with fractional derivative model of viscoelastic dampers”, Mechanical Systems and Signal Processing, 72, pp. 865-896 (2016).
[5] Kumar, S. “A new fractional analytical approach for treatment of a system of physical models using Laplace transform”, Scientia Iranica, 21(5), pp. 1693-1699 (2014).
[6] Ionescu, C.M. and De Keyser, R.. “Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease”, IEEE Transactions on Biomedical Engineering, 56(4), pp. 978-987 (2009).
[7] Taghavian, H. and Tavazoei, M.S. “Analytic solution of a system of linear distributed order differential equations in the Reimann-Liouville sense”, Scientia Iranica, DOI: 10.24200/SCI.2018.20335, (2018).
[8] Podlubny, I. “Fractional-order systems and PI D   controllers”, IEEE Transactions on Automatic Control,
44(1), pp. 208-214 (1999).
[9] Luo, Y. and Chen, Y. “Fractional order [proportional derivative] controller for a class of fractional order systems”, Automatica, 45(10), pp. 2446-2450 (2009).
[10] Padula, F. and Visioli, A. “Tuning rules for optimal PID and fractional-order PID controllers”, Journal of Process Control, 21(1), pp. 69-81 (2011).
[11] Fergani, N. and Charef, A. “Process step response based fractional PI D   controller parameters tuning for
desired closed loop response”, International Journal of Systems Science, 47(3), pp. 521-532 (2016).
[12] Rahimian, M.A. and Tavazoei, M.S. “Improving integral square error performance with implementable fractional‐order PI controllers”, Optimal Control Applications and Methods, 35(3), pp. 303-323 (2014).
[13] Monje, C.A., Vinagre, B.M., Feliu, V., et al. “Tuning and auto-tuning of fractional order controllers for industry  applications”, Control Engineering Practice, 16(7), pp. 798-812 (2008).
[14] Li, H., Luo, Y. and Chen, Y. “A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments”, IEEE Transactions on Control Systems Technology, 18(2), pp. 516-520 (2010).
[15] Roy, P. and Roy, B.K. “Fractional order PI control applied to level control in coupled two tank MIMO system with experimental validation”, Control Engineering Practice, 48, pp. 119-135 (2016).
[16] Khubalkar, S., Chopade, A., Junghare, A., et al. “Design and realization of stand-alone digital fractional order PID controller for buck converter fed DC motor”, Circuits, Systems and Signal Processing, 35(6), pp. 2189-2211 (2016).
[17] Sayyaf, N. and Tavazoei, M.S. “Robust Fractional-Order Compensation in the Presence of Uncertainty in a Pole/Zero of the Plant”, IEEE Transactions on Control Systems Technology, 26(3), pp. 797- 812 (2018).
[18] Sayyaf, N. and Tavazoei, M.S. “Desirably Adjusting Gain Margin, Phase Margin and Corresponding Crossover  Frequencies Based on Frequency Data”, IEEE Transactions on Industrial Informatics, 13(5), pp. 2311-2321 (2017).
[19] Badri, V. and Tavazoei, M.S. “Some Analytical Results on Tuning Fractional-Order [Proportional-Integral] Controllers for Fractional-Order Systems”, IEEE Transactions on Control Systems Technology, 24(3), pp. 1059-1066 (2016).
[20] Karaboga, D. and Akay, B. “Proportional-integral-derivative controller design by using artificial bee colony, harmony search and the bees algorithms”, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 224(7), pp. 869-883 (2010).
[21] Kesarkar, A.A. and Selvaganesan, N. “Tuning of optimal fractional-order PID controller using an artificial bee colony algorithm”, Systems Science and Control Engineering, 3(1), pp. 99-105 (2015).
[22] Padula, F. and Visioli, A. “Optimal tuning rules for proportional-integral-derivative and fractional-order proportional-integral-derivative controllers for integral and unstable processes”, IET Control Theory and Applications, 6(6), pp. 776-786 (2012).
[23] Chang, L.Y. and Chen, H.C. “Tuning of fractional PID controllers using adaptive genetic algorithm for active magnetic bearing system”, WSEAS Transactions on Systems, 8(1), pp. 158-167 (2009).
[24] Cao, J.Y. and Cao, B.G. “Design of fractional order controllers based on particle swarm optimization”, 1ST IEEE Conference on Industrial Electronics and Applications, pp. 1-6 (2006).
[25] Ateş, A. and Yeroglu, C. “Optimal fractional order PID design via Tabu Search based algorithm”, ISA Transactions, 60, pp. 109-118 (2016).
[26] Das, S., Pan, I. and Das, S. “Multi-objective LQR with optimum weight selection to design FOPID controllers for delayed fractional order processes”, ISA Transactions, 58, pp. 35-49 (2015).
[27] Tavazoei, M.S. and Haeri, M. “Rational approximations in the simulation and implementation of fractional-order dynamics: A descriptor system approach”, Automatica, 46(1), pp. 94-100 (2010).
[28] Rahimian, M.A. and Tavazoei, M.S. “Optimal tuning for fractional-order controllers: an integer-order approximating filter approach”, ASME Journal of Dynamic Systems, Measurement and Control, 135(2), (2013).
[29] Kinney, T.B. “Tuning process controllers”, Chemical Engineering, 90(19), pp. 67-72 (1983).
[30] Podlubny, I., Petraš, I., Vinagre, B.M., et al. “Analogue realizations of fractional-order controllers”, Nonlinear Dynamics, 29(1), pp. 281-296 (2002). [31] Machado, J.A. “Delay approximation of fractional integrals”, Asian Journal of Control, 15(3), pp. 713-722 (2013).
[32] Ogata, K. “Modern Control Engineering”, Prentice Hall of India, New Delhi (1982).
[33] Sundaravadivu, K., Arun, B. and Saravanan, K. “Design of fractional order PID controller for liquid level control of spherical tank”, IEEE International Conference on Control System, Computing and Engineering (ICCSCE), pp. 291-295 (2011). [34] Walton, K. and Marshall, J.E. “Closed form solution for time delay systems' cost functionals”, International Journal of Control, 39(5), pp. 1063-1071 (1984).
[35] Kealy, T. and O'Dwyer, A. “Analytical ISE calculation and optimum control system design”, In Proceedings of the Irish Signals and Systems Conference, Limerick, Ireland, pp.418-423 (2003).
[36] Shmakov, S.L. “A universal method of solving quartic equations”, International Journal of Pure and Applied Mathematics, 71(2), pp. 251-259 (2011).
[37] Kreyszig, E. and Norminton, E.J. “Advanced engineering mathematics”, 4th Edn., Wiley, New York (1993).
[38] Walton, K., Ireland, B. and Marshall, J.E. “Evaluation of weighted quadratic functional for time-delay systems”, International Journal of Control, 44(6), pp. 1491-1498 (1986).
[39] Nocedal, J., Wright, S.J. “Numerical optimization”, Springer-Verlag, New York.
[40] Courant, R. “Differential and integral calculus”, 2th Edn., John Wiley and Sons.
[41] Zhuang, M. and Atherton, D.P. “Automatic tuning of optimum PID controllers”, In IEE Proceedings D (Control Theory and Applications), 140(3), pp. 216-224 (1993).
[42] Nash, J.C. “Compact numerical methods for computers: linear algebra and function minimization”, CRC press.
[43] Padula, F. and Visioli, A. “Advances in Robust Fractional Control”, Springer (2015).