Radmila Gerov, Zoran Jovanović

DOI Number
First page
Last page


The paper explores the Proportional-derivative controller for a double integrator plus dead time processes, which is a challenging control problem, that is designed based on the existing Proportional-integrative controller for integrator plus dead time processes. The PD controller is extended with an integral action and an ideal PID controller is received. The parameters of both controllers are received by using the pole placement technique, whereby the transcendent characteristics equation of the closed loop system is solved by using the Lambert W function. The paper also examines the influence of the desired poles of the system with a closed feedback as well as the influence of the disturbance and the change of the DIPTD processes parameters onto the received control system performances. The results received by simulation, and the quantitative indicators, show that the proposed control system has better performances in comparison to the control systems obtained by other methods in literature.


PID controller, PD controller, double integrator plus dead time processes, pole placement, time delay, Lambert W function

Full Text:



K. Aström, T. Hägglund, "The future of PID control," Control Engineering Practice, vol. 9, pp. 1163–1175, 2001.

J. G. Ziegler, N. B. Nichols, "Optimum settings for automatic controllers", Transactions ASME, vol. 64, pp. 759-768, 1942.

A. O’Dwyer, Handbook of PI and PID controller tuning rules, Imperial College Press, 3nd edition, London, U.K. 2009.

S. Skogestad, “Simple analytic rules for model reduction and PID controller tuning”, Journal of Process Control, vol. 13, no. 4, pp. 291-309, 2003. [Online]. Available: http://folk.ntnu.no/skoge/publications/2003/tuningPID/finalpaper.pdf

J. E. Normey-Rico, E. F. Camacho, Control of Dead-Time Processes, Springer-Verlag, London, 2007.

A. Visioli, Q. C. Zhong, Control of Integral Processes with Dead Time, Advances in Industrial Control series, Springer-Verlag, London, 2011.

D. D. Ruscio, C. Dalen, "Tuning PD and PID Controllers for Double Integrating Plus Time Delay Systems," Modeling, Identification and Control, vol. 38, no. 2, pp. 95-110, 2017. [Online]. Available: http://www.mic-journal.no/PDF/2017/MIC-2017-2-4.pdf

S. B. Stojanović, D. Lj. Debeljković, D. S. Antić, "Finite-Time stability and stabilization of linear time-delay systems," Facta Universitatis, Series: Mechanics, Automatic Control and Robotics, vol. 11, no. 1, pp. 25-36, 2012. [Online]. Available: http://casopisi.junis.ni.ac.rs/files/journals/2/olderissues/acar201201/acar20120103.pdf

O. J. Smith, "Closed control of loops with dead time, Chemical Engineering Progress," vol. 53, pp. 217–219, 1957.

M. S. Matijević, M. R. Stojić, "The robust controller design for processes with dead times," Facta Universitatis, Series: Mechanics, Automatic Control and Robotics, vol. 5, no 17, pp. 131 – 144, 2006. [Online]. Available:http://casopisi.junis.ni.ac.rs/files/journals/2/olderissues/macar200601/macar200601-11nn.pdf

R. Gerov, Z. Jovanović, "Synthesis of PI Controller with a Simple Set-Point Filter for Unstable First-Order Time Delay Processes and Integral plus Time Delay Plant," Elektronika ir Elektrotechnika, vol. 24, no. 2, pp. 3-11, 2018. [Online]. Available: http://dx.doi.org/10.5755/j01.eie.24.2.20629.

R. Gerov, Z. Jovanović, "A Simple Method of Tuning PI Controller for Integrator Dead/Time Processes," 2018 16th IEEE International Symposium on Intelligent Systems and Informatics (SISY), pp. 00051-00056, 2018.

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, D. Knuth, "On the Lambert W function," Advances in Computational Mathematics, vol. 5, pp. 329– 359, 1996.

R. C. Gomez, W. Michiels, "Some special cases in the stability analysis of multi-dimensional time-delay systems using the matrix Lambert W function," Automatica, vol. 53, pp. 339-345, 2015.

S. Yi, P. W. Nelson, A. G. Ulsoy, "Analysis and Control of Time Delayed Systems via the Lambert W Function," IFAC Proceedings Volumes, vol. 41, no. 2, pp. 13414–13419, 2008. [Online]. Available: https://doi.org/10.3182/20080706-5-KR-1001.02272.

S. Yi, P. W. Nelson, A. G. Ulsoy, "Eigenvalue assignment via the Lambert W function for control of time-delay systems," Journal of Vibration and Control, vol. 16, no. 7-8, pp. 961-982, 2010.

S. Yi, P. W. Nelson, A. G. Ulsoy, " Proportional-Integral Control of First-Order Time-Delay Systems via Eigenvalue Assignment," IEEE Transactions on Control Systems Technology, vol. 21, no. 5, pp. 1586-1594, 2013.

S. Yi, P. W. Nelson, A. G. Ulsoy, "Robust control and time-domain specifications for systems of delay differential equations via eigenvalue assignment," Journal of Dynamic Systems, Measurement and Control, vol. 132, no. 3, pp. 031003-1 – 031003-7, 2010.

A. G. Usloy, "Time-Delayed Vibration Control of Two Degree-Of-Freedom Mechanical System for Improved Stability Margins," IFAC-PapersOnLine, vol. 48, no. 12, pp. 001-006, 2015. [Online]. Available: https://doi.org/10.1016/j.ifacol.2015.09.343.

A. G. Usloy, "Time-Delayed Control of SISO Systems for Improved Stability Margins," Journal of Dynamic Systems, Measurement and Control, vol. 137, no. 4, pp. 041014-1 – 041014-12, 2015.

S. Yi, S. Duan, P. W. Nelson, A. G. Ulsoy, "The Lambert W Function Approach to Time Delay systems and the LambertWDDE toolbox," IFAC Proceedings Volumes, vol. 45, no. 14, pp. 114–119, 2012. [Online]. Available: https://doi.org/10.3182/20120622-3-US-4021.00008.

E. Jahanshahi, V. D. Oliveira, C. Grimholt, S. Skogestad, "A comparison between internal model control, optimal pidf and robust controllers for unstable flow in risers," IFAC Proceedings Volumes, vol. 47, no. 3, pp. 5752–5759, 2014. [Online]. Available: https://doi.org/10.3182/20140824-6-ZA-1003.02381

DOI: https://doi.org/10.22190/FUACR1901001G


  • There are currently no refbacks.

Print ISSN: 1820-6417
Online ISSN: 1820-6425