# American Institute of Mathematical Sciences

March  2021, 17(2): 827-839. doi: 10.3934/jimo.2019136

## An adaptive dynamic programming method for torque ripple minimization of PMSM

 1 School of Automation and engineering, University of Electronic Science and Technology of China, China 2 The Shenzhen Energy Storage Power Generation Co., Ltd. of China Southern Power Grid, China

* Corresponding author: Qunying Liu

Received  May 2019 Revised  July 2019 Published  March 2021 Early access  November 2019

The imperfect sinusoidal flux distribution, cogging torque, and current measurement errors can cause periodic torque ripple in the permanent magnet synchronous motor (PMSM). These ripples are reflected in the periodic oscillation of the motor speed and torque, causing vibration at low speeds and noise at high speeds. As a high-precision tracking application, ripple degrades the application performance of PMSM. In this paper, an adaptive dynamic programming (ADP) scheme is proposed to reduce the periodic torque ripples. An optimal controller is designed by iterative control algorithm using robust adaptive dynamic programming theory and strategic iterative technique. ADP is combined with the existing Proportional-Integral (PI) current controller and generates compensated reference current iteratively from cycle to cycle so as to minimize the mean square torque error. As a result, an optimization problem is constructed and an optimal controller is obtained. The simulation results show that the robust adaptive dynamic programming achieves lower torque ripple and shorter dynamic adjustment time during steady-state operation, thus meeting the requirements of steady speed state and the dynamic performance of the regulation system.

Citation: Tengfei Yan, Qunying Liu, Bowen Dou, Qing Li, Bowen Li. An adaptive dynamic programming method for torque ripple minimization of PMSM. Journal of Industrial and Management Optimization, 2021, 17 (2) : 827-839. doi: 10.3934/jimo.2019136
##### References:
 [1] T. Banks, Matrix Theory, Nuclear Phys. B Proc. Suppl., 67 (1997), 180-224.  doi: 10.1016/S0920-5632(98)00130-3. [2] H. J. Brascamp and E. H. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Advances in Math., 20 (1976), 151-173.  doi: 10.1016/0001-8708(76)90184-5. [3] Y. Cho, et al., Torque-ripple minimization and fast dynamic scheme for torque predictive control of permanent-magnet synchronous motors, IEEE Transactions on Power Electronics, 30 (2015), 2182–2190. doi: 10.1109/TPEL.2014.2326192. [4] J. Chu, Suppressing speed ripples of permanent magnetic synchronous motor based on a method, Trans. of China Electrotechnical Society, 24 (2009), 43-49. [5] S. U. Chung, et al., Fractional slot concentrated winding PMSM with consequent pole rotor for a low-speed direct drive: Reduction of rare earth permanent magnet, IEEE Trans. on Energy Conversion, 30 (2015), 103–109. doi: 10.1109/TEC.2014.2352365. [6] J. Fiala and F. H. Guenther, Handbook of intelligent control: Neural, fuzzy, and adaptive approaches, Neural Networks, 7 (1994), 851-852. [7] D. C. Hanselman, Minimum torque ripple, maximum efficiency excitation of brushless permanent magnet motors, IEEE Transactions on Industr. Electronics, 41 (1994), 292-300.  doi: 10.1109/41.293899. [8] B. H. Lam, et al., Torque ripple minimization in PM synchronous motors an iterative learning control approach, IEEE Internat. Conference on Power Electronics and Drive Systems, 1999. doi: 10.1109/PEDS.1999.794551. [9] F. L. Lewis and D. Liu, Reinforcement learning and approximate dynamic programming for feedback control, John Wiley & Sons, 2013. doi: 10.1002/9781118453988. [10] D. Ma and H. Lin, Accelerated iterative learning control of speed ripple suppression for a seeker servo motor, Algorithms, 11 (2018). doi: 10.3390/a11100152. [11] G. Madescu, et al., Effects of stator slot magnetic wedges on the induction motor performances, Optimization of Electrical and Electronic Equipment (OPTIM), 13th International Conference on IEEE, 2012. doi: 10.1109/optim.2012.6231861. [12] S. G. Min and B. Sarlioglu, Advantages and characteristic analysis of slotless rotary PM machines in comparison with conventional laminated design using statistical technique, IEEE Trans. on Transpor. Electrification, 4 (2018), 517-524.  doi: 10.1109/TTE.2018.2810230. [13] A. R. I. Mohamed and E. F. El-Saadany, A current control scheme with an adaptive internal model for torque ripple minimization and robust current regulation in PMSM drive systems, IEEE Trans. on Energy Conversion, 23 (2008), 92-100.  doi: 10.1109/TEC.2007.914352. [14] N. Nakao, Suppressing pulsating torques: Torque ripple control for synchronous motors, IEEE Industry Appl. Magazine, 21 (2015), 33-44.  doi: 10.1109/MIAS.2013.2288383. [15] P. M. Pardalos, Approximate dynamic programming: solving the curses of dimensionality, Optimization Methods and Software, 24 (2009), 155. doi: 10.1080/10556780802583108. [16] M. Pinsky, Introduction to Fourier Analysis and Wavelets, 102, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102. [17] W. Qian, S. K. Panda and J. X. Xu, Torque ripple minimization in PM synchronous motors using iterative learning control, IEEE Transactions on Power Electronics, 19 (2004), 272-279.  doi: 10.1109/TPEL.2003.820537. [18] J. Si, et al., Handbook of Learning and Approximate Dynamic Programming, John Wiley & Sons, 2004. doi: 10.1002/9780470544785. [19] R. Song, W. Xiao and Q. Wei, Neuro-control to energy minimization for a class of chaotic systems based on ADP algorithm, in International Conference on Intelligent Science and Big Data Engineering, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2013. doi: 10.1007/978-3-642-42057-3_78. [20] P. J. Werbos, Using ADP to understand and replicate brain intelligence: The next level design?, in Neurodynamics of Cognition and Consciousness, Understanding Complex Systems, Springer, Berlin, Heidelberg, 2007,109–123. doi: 10.1007/978-3-540-73267-9_6. [21] J. Wu, et al., Adaptive dual heuristic programming based on delta-bar-delta learning rule, in International Symposium on Neural Networks, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2011, 11–20. doi: 10.1007/978-3-642-21111-9_2. [22] C. Xia, et al., A novel direct torque control of matrix converter-fed PMSM drives using duty cycle control for torque ripple reduction, IEEE Trans. on Industr. Electronics, 61 (2013), 2700–2713. doi: 10.1109/TIE.2013.2276039. [23] Y. Yan, et al., Torque ripple minimization of PMSM using PI type iterative learning control, 40th Annual Conference of the IEEE Industrial Electronics Society, 2014. doi: 10.1109/IECON.2014.7048612. [24] R. Yuan and Z. Q. Zhu, Reduction of both harmonic current and torque ripple for dual three-phase permanent-magnet synchronous machine using modified switching-table-based direct torque control, IEEE Trans. on Industr. Electronics, 62 (2015), 6671-6683.  doi: 10.1109/TIE.2015.2448511. [25] J. P. Yun, et al., Torque ripples minimization in PMSM using variable step-size normalized iterative learning control, IEEE Conference on Robotics, Automation and Mechatronics, 2006. doi: 10.1109/RAMECH.2006.252747. [26] Z. Zhu, Q. S. Ruangsinchaiwanich and D. Howe, Synthesis of cogging-torque waveform from analysis of a single stator slot, IEEE Trans. on Indust. Appl., 42 (2006), 650-657.  doi: 10.1109/TIA.2006.872930.

show all references

##### References:
 [1] T. Banks, Matrix Theory, Nuclear Phys. B Proc. Suppl., 67 (1997), 180-224.  doi: 10.1016/S0920-5632(98)00130-3. [2] H. J. Brascamp and E. H. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Advances in Math., 20 (1976), 151-173.  doi: 10.1016/0001-8708(76)90184-5. [3] Y. Cho, et al., Torque-ripple minimization and fast dynamic scheme for torque predictive control of permanent-magnet synchronous motors, IEEE Transactions on Power Electronics, 30 (2015), 2182–2190. doi: 10.1109/TPEL.2014.2326192. [4] J. Chu, Suppressing speed ripples of permanent magnetic synchronous motor based on a method, Trans. of China Electrotechnical Society, 24 (2009), 43-49. [5] S. U. Chung, et al., Fractional slot concentrated winding PMSM with consequent pole rotor for a low-speed direct drive: Reduction of rare earth permanent magnet, IEEE Trans. on Energy Conversion, 30 (2015), 103–109. doi: 10.1109/TEC.2014.2352365. [6] J. Fiala and F. H. Guenther, Handbook of intelligent control: Neural, fuzzy, and adaptive approaches, Neural Networks, 7 (1994), 851-852. [7] D. C. Hanselman, Minimum torque ripple, maximum efficiency excitation of brushless permanent magnet motors, IEEE Transactions on Industr. Electronics, 41 (1994), 292-300.  doi: 10.1109/41.293899. [8] B. H. Lam, et al., Torque ripple minimization in PM synchronous motors an iterative learning control approach, IEEE Internat. Conference on Power Electronics and Drive Systems, 1999. doi: 10.1109/PEDS.1999.794551. [9] F. L. Lewis and D. Liu, Reinforcement learning and approximate dynamic programming for feedback control, John Wiley & Sons, 2013. doi: 10.1002/9781118453988. [10] D. Ma and H. Lin, Accelerated iterative learning control of speed ripple suppression for a seeker servo motor, Algorithms, 11 (2018). doi: 10.3390/a11100152. [11] G. Madescu, et al., Effects of stator slot magnetic wedges on the induction motor performances, Optimization of Electrical and Electronic Equipment (OPTIM), 13th International Conference on IEEE, 2012. doi: 10.1109/optim.2012.6231861. [12] S. G. Min and B. Sarlioglu, Advantages and characteristic analysis of slotless rotary PM machines in comparison with conventional laminated design using statistical technique, IEEE Trans. on Transpor. Electrification, 4 (2018), 517-524.  doi: 10.1109/TTE.2018.2810230. [13] A. R. I. Mohamed and E. F. El-Saadany, A current control scheme with an adaptive internal model for torque ripple minimization and robust current regulation in PMSM drive systems, IEEE Trans. on Energy Conversion, 23 (2008), 92-100.  doi: 10.1109/TEC.2007.914352. [14] N. Nakao, Suppressing pulsating torques: Torque ripple control for synchronous motors, IEEE Industry Appl. Magazine, 21 (2015), 33-44.  doi: 10.1109/MIAS.2013.2288383. [15] P. M. Pardalos, Approximate dynamic programming: solving the curses of dimensionality, Optimization Methods and Software, 24 (2009), 155. doi: 10.1080/10556780802583108. [16] M. Pinsky, Introduction to Fourier Analysis and Wavelets, 102, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102. [17] W. Qian, S. K. Panda and J. X. Xu, Torque ripple minimization in PM synchronous motors using iterative learning control, IEEE Transactions on Power Electronics, 19 (2004), 272-279.  doi: 10.1109/TPEL.2003.820537. [18] J. Si, et al., Handbook of Learning and Approximate Dynamic Programming, John Wiley & Sons, 2004. doi: 10.1002/9780470544785. [19] R. Song, W. Xiao and Q. Wei, Neuro-control to energy minimization for a class of chaotic systems based on ADP algorithm, in International Conference on Intelligent Science and Big Data Engineering, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2013. doi: 10.1007/978-3-642-42057-3_78. [20] P. J. Werbos, Using ADP to understand and replicate brain intelligence: The next level design?, in Neurodynamics of Cognition and Consciousness, Understanding Complex Systems, Springer, Berlin, Heidelberg, 2007,109–123. doi: 10.1007/978-3-540-73267-9_6. [21] J. Wu, et al., Adaptive dual heuristic programming based on delta-bar-delta learning rule, in International Symposium on Neural Networks, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2011, 11–20. doi: 10.1007/978-3-642-21111-9_2. [22] C. Xia, et al., A novel direct torque control of matrix converter-fed PMSM drives using duty cycle control for torque ripple reduction, IEEE Trans. on Industr. Electronics, 61 (2013), 2700–2713. doi: 10.1109/TIE.2013.2276039. [23] Y. Yan, et al., Torque ripple minimization of PMSM using PI type iterative learning control, 40th Annual Conference of the IEEE Industrial Electronics Society, 2014. doi: 10.1109/IECON.2014.7048612. [24] R. Yuan and Z. Q. Zhu, Reduction of both harmonic current and torque ripple for dual three-phase permanent-magnet synchronous machine using modified switching-table-based direct torque control, IEEE Trans. on Industr. Electronics, 62 (2015), 6671-6683.  doi: 10.1109/TIE.2015.2448511. [25] J. P. Yun, et al., Torque ripples minimization in PMSM using variable step-size normalized iterative learning control, IEEE Conference on Robotics, Automation and Mechatronics, 2006. doi: 10.1109/RAMECH.2006.252747. [26] Z. Zhu, Q. S. Ruangsinchaiwanich and D. Howe, Synthesis of cogging-torque waveform from analysis of a single stator slot, IEEE Trans. on Indust. Appl., 42 (2006), 650-657.  doi: 10.1109/TIA.2006.872930.
Configuration of an ADP-based control system
The controller block diagram of PMSM
The controller block diagram of PMSM
Speed response with ADP controller at 500r/min
Speed response Fourier analysis with ADP controller at 500r/min
Speed response with ADP controller at 50r/min
Speed response Fourier analysis with ADP controller at 50r/min
Torque response with ADP and PI controller at 1.6Nm
Torque response with ADP and PI controller at 9.0Nm
parameters of PMSM
 Characteristic Symbol Value Stator phase resistance R 2.875Ω d and q-axes ${L_d} = {L_q}$ 8.5mH Number of pole pairs ${p_n}$ 4 viscous damping B 0.008 N. m. s Torque constant ${K_t}$ 1.05 N. m Rotational inertia J 0.003kg.m2
 Characteristic Symbol Value Stator phase resistance R 2.875Ω d and q-axes ${L_d} = {L_q}$ 8.5mH Number of pole pairs ${p_n}$ 4 viscous damping B 0.008 N. m. s Torque constant ${K_t}$ 1.05 N. m Rotational inertia J 0.003kg.m2
response at different reference speed
 Speed at 500r/min Speed range(r/min) Fluctuation error ADP controller 499.8802-500.1253 0.2451 PI controller 495.6341-502.8779 7.2438 Speed at 50r/min Speed range(r/min) Fluctuation error ADP controller 49.7712-50.2942 0.5230 PI controller 46.3281-53.5526 7.2279
 Speed at 500r/min Speed range(r/min) Fluctuation error ADP controller 499.8802-500.1253 0.2451 PI controller 495.6341-502.8779 7.2438 Speed at 50r/min Speed range(r/min) Fluctuation error ADP controller 49.7712-50.2942 0.5230 PI controller 46.3281-53.5526 7.2279
response at different load torque
 Load torque at 1.6Nm Torque range(Nm) Fluctuation error ADP controller 1.5603-1.6428 0.0825 PI controller 0.3482-3.0570 2.7088 Load torque at 9.0Nm Torque range(Nm) Fluctuation error ADP controller 8.9603-9.1562 0.1959 PI controller 6.9523-12.1328 5.1805
 Load torque at 1.6Nm Torque range(Nm) Fluctuation error ADP controller 1.5603-1.6428 0.0825 PI controller 0.3482-3.0570 2.7088 Load torque at 9.0Nm Torque range(Nm) Fluctuation error ADP controller 8.9603-9.1562 0.1959 PI controller 6.9523-12.1328 5.1805
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