### THE USAGE OF LAMBERT W FUNCTION FOR IDENTIFICATION AND SPEED CONTROL OF A DC MOTOR

**DOI Number**

**First page**

**Last page**

#### Abstract

The paper proposes a new method of identifying the linear model of a DC motor. The parameter estimation is based on the closed-loop step response of the DC motor under a proportional controller. For the application of the method, a deliberate delay of the measured speed was introduced. The paper considers the speed regulation of the direct current motor with negligible inductance by applying 1-DOF and 2-DOF, proportional integral retarded controllers. The proportional and integral gain of the PI retarded controllers was received by using a pole placement method on the identified model. The Lambert W function was applied for the identification and in designing the controller with the purpose of finding the rightmost poles of the closed-loop as well as the boundary conditions for selecting the gain of the PI controller. The robustness of the calculated controllers was considered under the effect of an disturbance, uncertainty in each of the DC motor parameters as well as perturbations in time delay.

#### Keywords

#### Full Text:

PDF#### References

R. Gerov, Z. Jovanović, “Lambert W function application to a direct current motor speed regulation with a delay”, In Proceedings of the 13th International Conference on Applied Electromagnetics - PES 2017, Niš, Serbia, 2017, pp. O6-3.

A. Hughes, “Electric Motors and Drives: Fundamentals, Types and Applications”, 3nd, Elsevier, 2006.

H. A. Toliyat (Ed.), G. B. Kliman (Ed.), “Handbook of Electric Motors”, 2nd, Boca Raton: CRC Press, 2004.

T. Umeno, Y. Hori, “Robust speed control of dc servomotors using modern two degrees-of-freedom controller design”, IEEE Transactions on Industrial Electronics, vol. 38, no. 5, 1991, pp. 363–368.

V. L´echapp´e, O. Salas, J. Le´on, F. Plestan, E. Moulay, A. Glumineau, “Predictive control of disturbed systems with input delay: experimental validation on a DC motor”, In Proceedings of the IFAC, 2015, vol. 48, no. 12, pp. 292–297.

M. Ruderman, J. Krettek, F. Hoffmann, T. Bertram, “Optimal State Space Control of DC Motor”, In Proceedings of the IFAC, 2008, vol. 41, no. 2, pp. 5796–5801.

A. Ramirez, R. Garrido, S. Mondie, “Velocity control of servo system using an integral retarded algorithm”, Elsevier, ISA Transactions, vol. 58, pp. 357–366, 2015.

K. Matsuo, T. Miura and T. Taniguchi, “Speed control of a dc motor system through delay time variant network,” In Proceedings of the SICE-ICASE International Joint Conference, 2006, pp. 399–404.

W. Lord, J. H. Hwang, “DC Servomotors: Modeling and parameter determination", IEEE Transactions on Industry Aplications”, vol. 1A-13, no. 3, pp. 234–243. 1977.

R. Garrido, R. Miranda, “Closed loop identification of a DC servomechanism”, In Proceedings ogf the 2006 IEEE International Power Electronics Congress, 2006, pp.1–5.

K. Radojka, A. Sanja, S. Danilo, “Recursive Least Squares Method in Parameters Identification of DC Motors Models”, Facta Universitatis, Series: Electronics and Energetics, vol. 18, no. 3, pp. 467–478, 2005.

C. S. Chen, S. C. Lin, S. M. Wang, Y. M. Hong, “Adaptive control based on reduced-order parameter identification and disturbance observer for linear motor”, In Proceedings of the ASME 2007 Conference on International Manufacturing Science and Engineering, Atlanta, Georgia, USA, 2007, pp. 751–758.

A. A. Al-Qassar, M. Z. Othman, “Experimental Determination of Electrical and Mechanical DC Motor using Genetic Elman Neural Network”, Journal of Engineering Science and Technology, vol. 3, no. 2, pp. 190–169, 2008.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D.E. Knuth, “On the Lambert W Function,”, Advances in Computational Mathematics, vol. 5, 1996, pp. 329–359.

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

S. Yi, P. W. Nelson, A. G. Ulsoy, “Time delay systems: Analysis and control using the Lambert W function”, World Scientific, 2010b.

S. Yi, P. W. Nelson, A. G. Ulsoy, “DC Motor Control Using the Lambert W Function Approach”, In Proceedings of the IFAC, 2012, vol. 45, no. 14, pp. 49–54.

S. Yi, P. W. Nelson, A. G. Ulsoy, “The Lambert W Function Approach to Time Delay Systems and the LambertW_DDE Toolbox,” In Proceedings of the IFAC, 2012, vol. 45, no. 14, pp. 114–119.

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.

R. M. Corless, H. Ding, N. J. Higham, D. J. Jeffrey, “The solution of S*exp(S)=A is not always the Lambert W function of A”, In Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation ISSAC 2007, Waterloo, Ontario, Canada, 2007, pp. 116–121.

R. R. Alla, J. S. Lather, G. L. Pahuja: “Comparison of PI controller performance for first order systems with time delay”, Journal of Engineering Science and Technology, vol. 12, no. 4, pp. 1081–1091, 2017.

R. Gerov, Z. Jovanović, “Primena proporcionalno-integralnog kontrolera za sisteme prvog reda sa kašnjenjem korišćenjem Lambert W funkcije za podešavanje polova”, Zbornik 61. Konferencije za elektroniku, telekomunikacije, računarstvo, automatiku i nuklearnu tehniku, ETRAN 2017, Kladovo, Srbija, 2017, pp. AU1.8.1-5.

S. M. Özer, S. Yıldız, A. Iftar, “Optimization-based PV/PI design for a DC-motor system with delayed feedback,” In Proceedings of the 26th Mediterranean Conference on Control and Automation, Zadar, Crotia, 2018, pp. 39–45.

S. Ayasun, "Stability analysis of time-delayed DC motor speed control system", Turkish Journal of Electrical Engineering & Computer Sciences, vol. 21, pp. 381–393, 2013.

G. J. Silva, A. Datta, S. P. Bhattacharyya: "PI stabilization of first-order systems with time delay", Automatica, vol. 37, pp. 2025–2031, 2001.

R. Sipahi, S. I. Niculescu, C. T. Abdallah, W. Michiels, K. Gu, "Stability and Stabilization of Systems with Time Delay. Limitations and Opportunities", 2010, https://digitalrepository.unm.edu/ece_rpts/35.

W. Michiels, S.I. Niculescu, L. Moreau, "Using delays and time-varying gains to improve the static output feedback stabilizability of linear systems: a comparison", IMA Journal of Mathematical Control and Information, vol. 21, no. 4, 2004, pp. 393–418, 2004.

Inteco, Modular Servo System-User’s Manual, [Online]. Available: www.inteco.com.pl, 2008.

### Refbacks

- There are currently no refbacks.

ISSN: 0353-3670 (Print)

ISSN: 2217-5997 (Online)

COBISS.SR-ID 12826626