Modelling and Quadratic Dynamic Matrix Control of a DC Motor System with Non-linearities Consideration

Abstract

DC motor’s models that have often been used to test and validate control algorithms with, even the most robust ones, have usually been simplistic: limiting the generalization and applicability of the results obtained. This article presents a modeling approach for a DC motor system consisting of a Buck converter and driving a load at the end of the shaft, in which the nonlinearities of each of its sub-parts are taken into account in the model of the overall system. The model was simulated and its step responses were compared to those of the linear model and conventional nonlinear models with a significant difference (P-value<0.001). This model was then used to test the Fast Nonlinear Quadratic Dynamic Matrix Control (FNLQDMC). The letter has demonstrated good performance in simulation in terms of setpoint tracking, constraint handling, and computational load savings during online optimization problem solving.

Introduction

I. INTRODUCTION

DC motors are widely used in industry due to its simple and miniaturized structure, wide and precise speed range, high torque for low input voltages, ease of reversing rotation direction while acting reversibly and returning energy to the line during braking and speed reduction times, even for large motors subjected to high loads [1]-[15]. In high-precision operations, the proven control performance of the DC motor is sought after [1], [4], [16]-[18]. However, its mechatronic nature and coupling to other devices with which it is required to operate often give rise to several sources of nonlinearities, making its modeling and hence its controllability complex, especially when they are not taken into consideration in the design of the model of the overall system on which it depends. These nonlinearities are mostly due to friction torque, converter, and the armature reaction [2], [3]. Therefore, the study of the nonlinear behavior of a DC motor constitutes a useful effort, first for obtaining a more representative model, and also for the design and validation of high-performance controllers. Indeed, DC Motors remains a good testing ground for advanced control algorithms due to its simple theory and extensibility to other disciplines [5], [7]-[9], [13], [14], [19]-[21]. In the modeling of DC Motors, the traditional approach is to limit oneself to electromechanical models (motor connected to a load) [22]. These models are generally linear. And in the contrary case, the usual tendency is to limit oneself only to the nonlinearities introduced by the friction torque of the load. To this end, several models have been proposed and used to accurately approximate the nonlinear behavior of the rotating mechanical load of the DC Motor in order to improve the performance of its control system [1], [10], [12], [15], [23]-[25]. However, Ibbini and Zakaria in [26] demonstrated that DC Motors are generally considered as linear system by neglecting the effect of the induced magnetic reaction or by assuming that the compensating windings completely eliminate such an effect. Thus, they proposed a variable model of the mechanical constant of the motor to take into account the nonlinearities of the induced reaction in the DC Motor model. This model will be used in [9] for the validation of an adaptive control algorithm based on neural networks. However, in the model of [26], the nonlinearities of the friction torque are neglected.

Lyshesvki in [26] later demonstrated that the static converter must be taken into account in the formulation of the DC Motor model and that its nonlinear dynamics cannot be neglected in practice. Because in industry, the DC Motor does not work alone, it is often coupled to a speed controller that serves as its control in the torque-speed plane. Thus, a new modeling approach for the DC motor while adding the converter to the classical electromechanical system has emerged [17], [27], one of the major advantages of this model being the possibility of using the converter duty cycle to control the entire system directly. However, in the model of [17], the nonlinearities of the friction torque and those of the armature reaction are neglected.

This paper deals with the modeling of a DC Motor as a mechatronic system consisting of an electronic part, an electrical part, and a mechanical part, in which the nonlinearities of each of these sub-parts are explicitly taken into account in the formulation of the model of the overall system. A permanent magnet DC motor driving a mechanical load at the end of the shaft and coupled to a Buck converter is considered. The Fast Nonlinear Quadratic Dynamic Matrix Control (FNLQDMC) algorithm of [28] will be tested by this model. This paper is structured as follows: after the introduction in section 1, the system is modeled in section 2, section 3 briefly describes the FNLQDMC algorithm, results are presented and analyzed in section 4, and section 5 presents the conclusion.

II. MODELING OF DC MOTOR SYSTEM

Unlike most of the DC motor systems available in the literature, whose studies are often limited to the individual DC motor without taking into account the effects of its interaction with other systems with which it is required to operate, this paper describes a system in which the DC motor is only a sub-part of a system consisting of a Buck converter and a mechanical load at its shaft as illustrated in Fig. 1.

Conclusion

In this work, a modeling approach for the DC motor has been presented. A permanent magnet DC motor system, consisting of an electronic part, an electrical part, and a mechanical part, has been considered. This approach is based on the decomposition and consideration of the nonlinearities of each of these functional subparts of the motor in formulating the complete model of the system. At the end of the process, the obtained model was compared to the simplistic models of the same system with the simplifying assumptions of the conventional models. The simulation results obtained show a significant difference between the models and a dominance of the nonlinearities of the armature reaction on the complete system model. Used to execute the FNLQDMC algorithm, the latter presented good simulation performance in terms of setpoint tracking, constraint handling, and calculation load economy during online optimization problem solving. These results encourage the use of the proposed complete nonlinear model of the DC motor system for the implementation and validation of high-performance control algorithms. However, it should be noted that the results presented in this work are based on simulations, and further experimental validation is necessary to confirm the effectiveness of the proposed approach in practical applications. Furthermore, the modeling approach presented in this work can be extended to other types of motors and systems with nonlinearities, offering a promising research direction for control engineers and researchers in the field of nonlinear control systems.

References

[1] Cong, S., Li, G., & Feng, X. (2010, July). Parameters identification of nonlinear DC motor model using compound evolution algorithms. In Proceedings of the World Congress on Engineering (Vol. 1). [2] Virgala, I., & Peter Frankovsky, M. K. (2013). Friction effect analysis of a DC motor. Am. J. Mech. Eng, 1(1), 1-5. [3] Miková, ?., Virgala, I., & Kelemen, M. (2015). Speed Control of a DC Motor Using PD and PWM Controllers. In Solid State Phenomena (Vol. 220, pp. 244-250). Trans Tech Publications. [4] Sundareswaran, K., & Sreedevi, V. T. (2008, November). DC motor speed controller design through a colony of honey bees. In TENCON 2008-2008 IEEE Region 10 Conference (pp. 1-6). IEEE. [5] Tripathi, R. P., Singh, A. K., & Gangwar, P. (2023). Fractional order adaptive Kalman filter for sensorless speed control of DC motor. International Journal of Electronics, 110(2), 373-390. [6] Goyat, S., & Tushir, M. (2016). Controlling of DC Motor Characteristics and Non Linear Behavior using Modeling and Simulation. International Journal of Electronics, Electrical and Computational System (IJEECS), ISSN 2348-117X, Volume 5, Issue 8 [7] Tufenkci, S., Alagoz, B. B., Kavuran, G., Yeroglu, C., Herencsar, N., & Mahata, S. (2023). A theoretical demonstration for reinforcement learning of PI control dynamics for optimal speed control of DC motors by using Twin Delay Deep Deterministic Policy Gradient Algorithm. Expert Systems with Applications, 213, 119192. [8] Yang, X., Deng, W., & Yao, J. (2022). Neural network based output feedback control for DC motors with asymptotic stability. Mechanical Systems and Signal Processing, 164, 108288. [9] Horng, J. H. (1999). Neural adaptive tracking control of a DC motor. Information sciences, 118(1-4), 1-13. [10] Mahajan, N. P., & Deshpande, S. B. (2013). Study of nonlinear behavior of DC motor using modeling and simulation. International Journal of Scientific and Research Publications, 3(3), 576-581. [11] Dursun, E. H., & Durdu, A. (2016). Speed control of a DC motor with variable load using sliding mode control. International Journal of Computer and Electrical Engineering, 8(3), 219. [12] Ruderman, M., Krettek, J., Hoffmann, F., & Bertram, T. (2008). Optimal state space control of DC motor. IFAC Proceedings Volumes, 41(2), 5796-5801. [13] Aribowo, W., Supari, B. S., & Suprianto, B. (2022). Optimization of PID parameters for controlling DC motor based on the aquila optimizer algorithm. International Journal of Power Electronics and Drive Systems (IJPEDS), 13(1), 808-2814. [14] Izci, D., Ekinci, S., Zeynelgil, H. L., & Hedley, J. (2022). Performance evaluation of a novel improved slime mould algorithm for direct current motor and automatic voltage regulator systems. Transactions of the Institute of Measurement and Control, 44(2), 435-456. [15] Goyat, S. and Tushir, M. (2016). Analysis dc motor non linear behavior using modelingand simulation. Int. J. Adv. Res. Electr. Electron. In strum. Eng., 5(1) :263–268. [16] Kara, T., & Eker, I. (2004). Nonlinear modeling and identification of a DC motor for bidirectional operation with real time experiments. Energy Conversion and Management, 45(7-8), 1087-1106. [17] Lyshevski, S. E. (1999). Nonlinear control of mechatronic systems with permanent-magnet DC motors. Mechatronics, 9(5), 539-552. [18] Canudas, C., Astrom, K., & Braun, K. (1987). Adaptive friction compensation in DC-motor drives. IEEE Journal on Robotics and Automation, 3(6), 681-685. [19] Holkar, K. and Waghmare, L. (2010). Discrete model predictive control for dc drive using orthonormal basis function.UKACC International Conference on CONTROL 2010, 2010 p. 435 – 440 [20] Sahoo, S., Subudhi, B., and Panda, G. (2015). Optimal speed control of dc motor using linear quadratic regulator and model predictive control. In 2015 International Conference on Energy, Power and Environment: Towards Sustainable Growth (ICEPE), pages 1–5. IEEE. [21] Suman, S. K. and Giri, V. K. (2016). Speed control of dc motor using optimization techniques based pid controller. In 2016 IEEE International Conference on Engineering and Technology (ICETECH), pages 581–587. IEEE. [22] G?owacz, Z., & G?owacz, W. (2007). Mathematical model of DC motor for analysis of commutation processes. Electrical Power Quality and Utilisation. Journal, 13(2), 65-68. [23] Liu, Z., & Jiang, L. (2011, August). PWM speed control system of DC motor based on AT89S51. In Electronic and Mechanical Engineering and Information Technology (EMEIT), 2011 International Conference on (Vol. 3, pp. 1301-1303). IEEE. [24] Rengifo Rodas, C. F., Castro Casas, N., & Bravo Montenegro, D. A. (2017). A performance comparison of nonlinear and linear control for a DC series motor. Ciencia en Desarrollo, 8(1), 41-50. [25] Jang, J. O., & Jeon, G. J. (2000). A parallel neuro-controller for DC motors containing nonlinear friction. Neurocomputing, 30(1-4), 233-248. [26] Ibbini, M. S., & Zakaria, W. S. (1996). Nonlinear control of DC machines. Electric machines and power systems, 24(1), 21-35. [27] Bitjoka, L., Ndje, M., Boum, A. T., & Song-Manguelle, J. (2017). Implementation of quadratic dynamic matrix control on arduino due ARM cortex-M3 microcontroller board. Journal of Engineering Technology (ISSN: 0747-9964), 6(2), 682-695. [28] Zenere, A. and Zorzi, M. (2018). On the coupling of model predictive control and robust kalman ?ltering. IET Control Theory & Applications, 12(13) :1873–1881. [29] Ndje, M., Bitjoka, L., Boum, A. T., Mbogne, D. J. F., Bu?oniu, L., Kamgang, J. C., & Djogdom, G. V. T. (2021). Fast constrained nonlinear model predictive control for implementation on microcontrollers. IFAC-PapersOnLine, 54(4), 19-24. [30] Ndje, M., Yome, J. N., Boum, A. T., Bitjoka, L., & Kamgang, J. C. (2018). Dynamic matrix control and tuning parameters analysis for a DC motor system control. Engineering, Technology & Applied Science Research, 8(5), 3416-3420.

Copyright

Copyright © 2023 Martial Ndje, Felix J Ngouem, Golam Guidkaya, Sylvain Zango Nkeutia, David Jaures Fotsa-Mbogne, Gilde Vanel Tchane Djogdom, Laurent Bitjoka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.