American Institute of Mathematical Sciences

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

An adaptive dynamic programming method for torque ripple minimization of PMSM

Tengfei Yan 1, , Qunying Liu 1,, , Bowen Dou 2, , Qing Li 2, and  Bowen Li 1,

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.

Keywords: Permanent magnet synchronous motor, torque ripple minimization, robust adaptive dynamic programming, error compensation, vector control.
Mathematics Subject Classification: Primary: 93C40, 93A30; Secondary: 93C73.
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 & 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.  Google Scholar

[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.  Google Scholar

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J. Chu, Suppressing speed ripples of permanent magnetic synchronous motor based on a method, Trans. of China Electrotechnical Society, 24 (2009), 43-49.   Google Scholar

[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.  Google Scholar

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J. Fiala and F. H. Guenther, Handbook of intelligent control: Neural, fuzzy, and adaptive approaches, Neural Networks, 7 (1994), 851-852.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[9]

F. L. Lewis and D. Liu, Reinforcement learning and approximate dynamic programming for feedback control, John Wiley & Sons, 2013. doi: 10.1002/9781118453988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. M. Pardalos, Approximate dynamic programming: solving the curses of dimensionality, Optimization Methods and Software, 24 (2009), 155. doi: 10.1080/10556780802583108.  Google Scholar

[16]

M. Pinsky, Introduction to Fourier Analysis and Wavelets, 102, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.  Google Scholar

[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.  Google Scholar

[18]

J. Si, et al., Handbook of Learning and Approximate Dynamic Programming, John Wiley & Sons, 2004. doi: 10.1002/9780470544785.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Chu, Suppressing speed ripples of permanent magnetic synchronous motor based on a method, Trans. of China Electrotechnical Society, 24 (2009), 43-49.   Google Scholar

[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.  Google Scholar

[6]

J. Fiala and F. H. Guenther, Handbook of intelligent control: Neural, fuzzy, and adaptive approaches, Neural Networks, 7 (1994), 851-852.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

F. L. Lewis and D. Liu, Reinforcement learning and approximate dynamic programming for feedback control, John Wiley & Sons, 2013. doi: 10.1002/9781118453988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. M. Pardalos, Approximate dynamic programming: solving the curses of dimensionality, Optimization Methods and Software, 24 (2009), 155. doi: 10.1080/10556780802583108.  Google Scholar

[16]

M. Pinsky, Introduction to Fourier Analysis and Wavelets, 102, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.  Google Scholar

[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.  Google Scholar

[18]

J. Si, et al., Handbook of Learning and Approximate Dynamic Programming, John Wiley & Sons, 2004. doi: 10.1002/9780470544785.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Configuration of an ADP-based control system
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Figure 2.  The controller block diagram of PMSM
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Figure 3.  The controller block diagram of PMSM
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Figure 4.  Speed response with ADP controller at 500r/min
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Figure 5.  Speed response Fourier analysis with ADP controller at 500r/min
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Figure 6.  Speed response with ADP controller at 50r/min
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Figure 7.  Speed response Fourier analysis with ADP controller at 50r/min
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Figure 8.   
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Figure 9.   
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Figure 10.  Torque response with ADP and PI controller at 1.6Nm
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Figure 11.  Torque response with ADP and PI controller at 9.0Nm
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Table 1.  parameters of PMSM
CharacteristicSymbolValue
Stator phase resistanceR2.875Ω
d and q-axes${L_d} = {L_q}$8.5mH
Number of pole pairs${p_n}$4
viscous dampingB0.008 N. m. s
Torque constant${K_t}$1.05 N. m
Rotational inertiaJ0.003kg.m2
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CharacteristicSymbolValue
Stator phase resistanceR2.875Ω
d and q-axes${L_d} = {L_q}$8.5mH
Number of pole pairs${p_n}$4
viscous dampingB0.008 N. m. s
Torque constant${K_t}$1.05 N. m
Rotational inertiaJ0.003kg.m2
Table 2.  response at different reference speed
Speed at 500r/minSpeed range(r/min)Fluctuation error
ADP controller499.8802-500.12530.2451
PI controller495.6341-502.87797.2438
Speed at 50r/minSpeed range(r/min)Fluctuation error
ADP controller49.7712-50.29420.5230
PI controller46.3281-53.55267.2279
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Speed at 500r/minSpeed range(r/min)Fluctuation error
ADP controller499.8802-500.12530.2451
PI controller495.6341-502.87797.2438
Speed at 50r/minSpeed range(r/min)Fluctuation error
ADP controller49.7712-50.29420.5230
PI controller46.3281-53.55267.2279
Table 3.  response at different load torque
Load torque at 1.6NmTorque range(Nm)Fluctuation error
ADP controller1.5603-1.64280.0825
PI controller0.3482-3.05702.7088
Load torque at 9.0NmTorque range(Nm)Fluctuation error
ADP controller8.9603-9.15620.1959
PI controller6.9523-12.13285.1805
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Load torque at 1.6NmTorque range(Nm)Fluctuation error
ADP controller1.5603-1.64280.0825
PI controller0.3482-3.05702.7088
Load torque at 9.0NmTorque range(Nm)Fluctuation error
ADP controller8.9603-9.15620.1959
PI controller6.9523-12.13285.1805
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