# High-Precision Acquisition Method of Position Signal of Permanent Magnet Direct Drive Servo Motor at Low Speed

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Motor Position Feedback Acquisition Problem at Low Speed

_{Ts}in each speed control cycle T

_{s}:

_{T}is the torque current constant; and λ is the filter time constant.

## 3. Traditional Interpolation Fitting Method and its Disadvantage Analysis

#### 3.1. Average Acceleration Interpolation

_{p}is still a small value. It can be assumed that the acceleration of the motor remains unchanged in one speed control cycle, and the motor speed in the next control cycle can be estimated from this acceleration, that is, the average acceleration method.

_{p1}and T

_{p2}, respectively. The average acceleration of the motor in these two intervals can be calculated as ${\omega}_{1}$ and ${\omega}_{2}$, so the average acceleration is ${\alpha}_{1}=({\omega}_{2}-{\omega}_{1})/T$. According to the acceleration, the speed in the next interval can be estimated as:

#### 3.2. Polynomial Fit Interpolation

_{0}< 𝑥

_{1}< ⋯ < 𝑥

_{n}= 𝑏 are given in the interval [a, b]. At the same time there is a function y = f(x) which has function values 𝑓(𝑥

_{0}), 𝑓(𝑥

_{1}), ⋯, 𝑓(𝑥

_{n}) at these points. The function $S(x)$ satisfies that it is a cubic polynomial in each interval and is second-order derivable. Integrating the second derivative of S(x) twice in succession gives:

_{j}and M

_{j}

_{+1}are the function values of the function S(x) at points x

_{j}and x

_{j}

_{+1}, respectively. h

_{j}is the width of the interval [x

_{j}, x

_{j}

_{+1}].

_{j}, 𝑥

_{j}

_{+1}], so the following relationship can be obtained by derivation:

_{j}(j = 0, 1, ..., n − 1) in Equation (8) can be solved. Since two nodes cannot guarantee the accuracy of the algorithm and four nodes will greatly increase the complexity of the algorithm, it is not conducive to the realization of the algorithm. Therefore, this paper selects three nodes, that is, the signal of the position sensor is updated twice. When the position update is detected for the second time, the value of the interpolation function S(x) is calculated, and then the future rotor position of the motor is calculated according to the speed control period T

_{s}.

#### 3.3. Analysis of Disadvantages

## 4. Position Signal Acquisition Strategy Based on Position Observation

#### 4.1. Position Signal Acquisition Strategy Based on Full-Order State Observer

_{L}is counted as a part of the disturbance torque T

_{d}. Write the mechanical motion equation of the motor in the form of the space state equation:

_{a}is the drag coefficient; T

_{e}is the electromagnetic torque; T

_{L}is the load torque.

**A**,

**B**,

**C**and the control variable u are, respectively:

_{1}, k

_{2}, k

_{3}to meet the error convergence speed required by the system. In this paper, the poles of the characteristic equation are configured in the form of multiple roots, and the poles are set at −ω

_{0}, which can ensure the stability of the observer and the convergence of the error. The observer gain matrix can be approximated as:

_{0}is the observer bandwidth. The full-order state observer structure designed according to Equation (12) can be represented by Figure 4:

_{d}, which significantly improves the observation accuracy of the position signal. Further considering the form of disturbance, the full-order observer can theoretically realize static-difference-free observation of fixed disturbances. However, in practical applications, disturbances often appear in more complex forms, such as step, sinusoidal and so on. This requires the observer to have a certain ability to track time-varying disturbances, that is, better dynamic performance. The direct method is to expand the state quantity of the motor system.

#### 4.2. Position Signal Acquisition Strategy Based on Extended State Observer

_{0}. The observer gain matrix can be approximated as:

## 5. Simulation and Experiment

#### 5.1. Simulation

#### 5.2. Experimental Verification

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Khatri, F.I.; Robinson, B.S.; Semprucci, M.D.; Boroson, D.M. Lunar Laser Communication Demonstration operations architecture. Acta Astronaut.
**2015**, 111, 77–83. [Google Scholar] [CrossRef] - Wang, J.; Wang, G.; Bai, R.; Li, B.; Zhou, Y. Ground simulation method for arbitrary distance optical transmission of a free- space laser communication system based on an optical fiber nanoprobe. J. Opt. Commun. Netw.
**2017**, 9, 1136–1144. [Google Scholar] [CrossRef] - Biswas, A.; Srinivasan, M.; Rogalin, R.; Piazzolla, S.; Liu, J.; Schratz, B.; Wong, A.; Alerstam, E.; Wright, M.; Roberts, W.T.; et al. Status of NASA’s deep space optical communication technology demonstration. In Proceedings of the 2017 IEEE International Conference on Space Optical Systems and Applications (ICSOS), Naha, Japan, 14–16 November 2017. [Google Scholar] [CrossRef]
- SLiu, S.B.; Giusti, A.; Althoff, M. Velocity Estimation of Robot Manipulators: An Experimental Comparison. IEEE Open J. Control. Syst.
**2023**, 2, 1–11. [Google Scholar] [CrossRef] - Guo, C.; Gao, X.; Zhang, Q.; Zhu, Y. Fault Tolerance Method of Low-Resolution Hall Sensor in Permanent Magnet Synchronous Machine. IEEE Access
**2022**, 10, 119162–119169. [Google Scholar] [CrossRef] - Wu, Z.; Zuo, S.; Huang, Z.; Hu, X.; Chen, S.; Liu, C.; Zhuang, H. Effect of Hall Errors on Electromagnetic Vibration and Noise of Integer-Slot Inset Permanent Magnet Synchronous Motors. IEEE Trans. Transp. Electrif.
**2023**, 9, 522–533. [Google Scholar] [CrossRef] - Oh, S.; Park, J.; Jung, H.; Lee, K.; Lim, H. A Method for Improving Initial Driving Vibration of Electric Scooter with Low Resolution Position Sensors. In Proceedings of the 2022 25th International Conference on Electrical Machines and Systems (ICEMS), Chiang Mai, Thailand, 29 November–2 December 2022. [Google Scholar] [CrossRef]
- Zhang, J.; Jiang, Y.; Li, X.; Huo, M.; Luo, H.; Yin, S. An adaptive remaining useful life prediction approach for single battery with unlabeled small sample data and parameter uncertainty. Reliab. Eng. Syst. Saf.
**2022**, 222, 108357. [Google Scholar] [CrossRef] - Zhang, J.; Jiang, Y.; Li, X.; Luo, H.; Yin, S.; Kaynak, O. Remaining Useful Life Prediction of Lithium-Ion Battery with Adaptive Noise Estimation and Capacity Regeneration Detection. IEEE/ASME Trans. Mechatron.
**2023**, 28, 632–643. [Google Scholar] [CrossRef] - Cavus, B.; Aktas, M. MPC-Based Flux Weakening Control for Induction Motor Drive with DTC for Electric Vehicles. IEEE Trans. Power Electron.
**2023**, 38, 4430–4439. [Google Scholar] [CrossRef] - Zhang, J.; Zhang, K.; An, Y.; Luo, H.; Yin, S. An Integrated Multitasking Intelligent Bearing Fault Diagnosis Scheme Based on Representation Learning Under Imbalanced Sample Condition. IEEE Trans. Neural Netw. Learn. Syst.
**2023**, 1–12. [Google Scholar] [CrossRef] - Zhang, J.; Li, X.; Tian, J.; Jiang, Y.; Luo, H.; Yin, S. A variational local weighted deep sub-domain adaptation network for remaining useful life prediction facing cross-domain condition. Reliab. Eng. Syst. Saf.
**2023**, 231, 108986. [Google Scholar] [CrossRef] - Sun, X.; Tang, X.; Tian, X.; Wu, J.; Zhu, J. Position Sensorless Control of Switched Reluctance Motor Drives Based on a New Sliding Mode Observer Using Fourier Flux Linkage Model. IEEE Trans. Energy Convers.
**2022**, 37, 978–988. [Google Scholar] [CrossRef] - Liu, J.M.; Zhu, Z.Q. Novel sensorless control strategy with injection of high-frequency pulsating carrier signal into stationary reference frame. IEEE Trans. Ind. Appl.
**2014**, 50, 2574–2583. [Google Scholar] [CrossRef] - Bernardes, T.; Montagner, V.F.; Grundling, H.A.; Pinheiro, H. Discrete-time sliding mode observer for sensorless vector control of permanent magnet synchronous machine. IEEE Trans. Ind. Electron.
**2014**, 61, 1679–1691. [Google Scholar] [CrossRef] - Bounasla, N.; Barkat, S.; Benyoussef, E.; Tounsi, K. Sensorless sliding mode control of a five-phase PMSM using extended Kalman filter. In Proceedings of the 2016 8th International Conference on Modelling, Identification and Control (ICMIC), Algiers, Nigeria, 15–17 November 2016. [Google Scholar] [CrossRef]
- Wang, G.; Liu, R.; Zhao, N.; Ding, D.; Xu, D. Enhanced linear ADRC strategy for HF pulse voltage signal injection-based sensorless IPMSM drives. IEEE Trans. Power Electron.
**2018**, 34, 514–525. [Google Scholar] [CrossRef] - Fu, D.; Zhao, X.; Zhu, J. A novel robust super-twisting nonsingular terminal sliding mode controller for permanent magnet linear synchronous motors. IEEE Trans. Power Electron.
**2022**, 37, 2936–2945. [Google Scholar] [CrossRef] - Ismail, S.; Shabri, A.; Samsudin, R. A hybrid model of Self-organizing Maps (SOM) and Least Square Support Vector Machine (LSSVM) for time-series forecasting. Expert Syst. Appl.
**2011**, 38, 10574–10578. [Google Scholar] [CrossRef] - Shen, W.; Liu, S.; Liu, M. Adaptive sliding mode control of hydraulic systems with the event trigger and finite-time disturbance observer. Inf. Sci.
**2021**, 569, 55–69. [Google Scholar] [CrossRef] - De Angelo, C.; Bossio, G.; Solsona, J.; Garcia, G.; Valla, M. Mechanical sensorless speed control of permanent-magnet AC motors driving an unknown load. IEEE Trans. Ind. Electron.
**2006**, 53, 406–414. [Google Scholar] [CrossRef] - Ying, F.; Li, Z. A wide-speed mode observer for sensorless direct torque control of a new self-decelerating permanent magnet in-wheel motor. Trans. China Electrotech. Soc.
**2014**, 29, 141–148. [Google Scholar] - Solsona, J.; Valla, M. Disturbance and nonlinear Luenberger observers for estimating mechanical variables in permanent magnet synchronous motors under mechanical parameters uncertainties. IEEE Trans. Ind. Electron.
**2003**, 50, 717–725. [Google Scholar] [CrossRef] - Fei, W.; Luk, P.C.-K. Torque ripple reduction of a direct-drive permanent-magnet synchronous machine by material-efficient axial pole pairing. IEEE Trans. Ind. Electron.
**2012**, 59, 2601–2611. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**Interpolation method for estimating position simulation results. (

**a**) is the sensor output position feedback. (

**b**) is the average acceleration interpolation method. (

**c**) is the cubic spline interpolation method. (

**d**) is the partial enlargement of (

**b**,

**c**).

**Figure 7.**Simulation results of observer based position signal interpolation strategy. (

**a**) is the full-order observer compensation. (

**b**) is the extended state observer compensation.

Parameter | Numerical Value | Parameter | Numerical Value |
---|---|---|---|

Rated voltage | 24 V | Rotational inertia | 5.58 kg·mm^{2} |

Rated current | 0.35 A | Damping coefficient | 5.12 × 10^{−6} |

Rated speed | 2650 r/min | DC Bus Voltage | 31 V |

Stator resistance | 2 Ω | Speed control frequency | 2 kHz |

Stator inductance | 1.23 mH | Switching frequency | 20 kHz |

Polar logarithm | 4 | Rated torque | 60 mNm |

Linkage | 0.013 Wb | Peak cogging torque | 42 mNm |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, D.; Dong, Z.; Bu, F.; Gu, Z.; Guo, Z.
High-Precision Acquisition Method of Position Signal of Permanent Magnet Direct Drive Servo Motor at Low Speed. *Energies* **2023**, *16*, 4491.
https://doi.org/10.3390/en16114491

**AMA Style**

Zhang D, Dong Z, Bu F, Gu Z, Guo Z.
High-Precision Acquisition Method of Position Signal of Permanent Magnet Direct Drive Servo Motor at Low Speed. *Energies*. 2023; 16(11):4491.
https://doi.org/10.3390/en16114491

**Chicago/Turabian Style**

Zhang, Deli, Zhaopeng Dong, Feifei Bu, Zijie Gu, and Zitao Guo.
2023. "High-Precision Acquisition Method of Position Signal of Permanent Magnet Direct Drive Servo Motor at Low Speed" *Energies* 16, no. 11: 4491.
https://doi.org/10.3390/en16114491