# Sensorless Control Method of High-Speed Permanent Magnet Synchronous Motor Based on Discrete Current Error

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## Abstract

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## 1. Introduction

## 2. Rotor Position Estimation Method and Angle Error Analysis Based on Sliding Mode Observer

#### 2.1. Principle of Estimating Rotor Position Based on Sliding Mode Algorithm

_{α}and u

_{β}are the motor stator terminal voltage and shaft component, respectively; i

_{α}and i

_{β}are the α and β axis components of the motor current, respectively; e

_{α}and e

_{β}are α and β axis components of the motor back electromotive force; R is the stator resistance, and L is the motor inductance.

_{a}and e

_{b}of the motor can be expressed as:

_{e}is the electromechanical angular velocity, ψ

_{f}is the motor rotor flux linkage, θ is the motor rotor position angle.

_{α}and i

_{β}respectively, z

_{α}and z

_{β}and are respectively the α and β axis current error switch signal, and the expression is:

_{s}is the sliding mode gain, and sgn is the sign function. Since z

_{α}and z

_{β}are high-frequency switching signals containing back-EMF information, they need to be passed through a low-pass filter to get the back-EMF estimates ${\widehat{e}}_{\mathsf{\alpha}}$ and ${\widehat{e}}_{\mathsf{\beta}}$; coupled with the quadrature phase-locked loop, the rotor position $\widehat{\theta}$ and speed ${\widehat{\omega}}_{\mathrm{e}}$ can be calculated. The principle block diagram of the rotor position estimation method based on the sliding mode observer is shown in Figure 1.

#### 2.2. Analysis of Estimation Angle Error in the State of High-Speed and Low Carrier Wave Ratio

#### 2.2.1. The Delay of the Sensorless Algorithm

_{α}and z

_{b}, the low-pass filter is an important part of the sensorless algorithm, but the signal will have a certain phase delay after passing through the filter. Therefore, it is necessary to add angle compensation to the traditional sensorless algorithm to compensate for the angle delay caused by the low-pass filter. The specific compensation Formula is:

_{e}is the cut-off frequency of the low-pass filter.

#### 2.2.2. Digital Control Delay

_{s}is the control period, which is equal to the sampling period and the switching period at the same time. The three-phase current is sampled at T

_{s}(k − 1), and then the duty cycle signal is calculated for a certain period of time. The update can only be performed at T

_{s}. Therefore, from the start of the control algorithm to the beginning of the controller to update the pulse width modulation (PWM) output, there is a Control cycle delay.

#### 2.2.3. Other Delay Issues

## 3. Angle Compensation Method Based on Discrete Current Error

#### 3.1. Analysis of Angle Error Judgment Basis

_{d}= 0 at the base speed, and the voltage feedback method is used for the field weakening control above the base speed. It can be seen that, in theory, if the model in the control system is accurate and the rotor position angle has no error, there should be no static difference between the d-axis current and the d-axis reference value. Therefore, in this paper, the presence or absence of the static error of the d-axis current discretized by the Euler approximation method is used as the basis for judging whether there is an error in the estimated angle in the position control. The control method proposed in this paper is shown in Figure 3. The following is a theoretical analysis.

_{d}, u

_{q}, i

_{d}, and i

_{q}are the stator voltage and current in the rotating coordinate system, and ψ

_{f}is the motor rotor flux.

_{f}sinΔθ is the component of the back electromotive force on the γ-axis, and Δθ is the angle error analyzed above. It can be seen from the above formula that at this time, the γ-axis current obtained by Euler’s approximation method has a component e(k−1)sinΔθ formed by the angle error. It is precisely because of the existence of this component that there is a static current difference between i

_{γ}and the current reference value i

_{dref}, and the expression of the current static difference at time k is:

_{γ}(k) should be equal to 0 when there is no error in the position angle and the motor parameters are accurate. Therefore, it is feasible to use this static current difference value as the basis for evaluating the rotor angle error.

#### 3.2. Principle of Angle Compensation Method Based on Discrete Current Error

_{dref}as the given value of the PI controller and i

_{γ}(k) as the feedback value of the PI controller. Figure 4 shows the principle diagram of position error compensation.

_{p}is the proportional coefficient, and k

_{i}is the integral coefficient. If the error value is not zero and the value is large, it indicates that there is an angle error at this time, and the angle compensation algorithm needs to be executed, and then the cycle is repeated until the d-axis discrete error is zero. Compensating the calculated Δθ behind the estimated position angle can compensate for all the errors analyzed in Section 2.2.

#### 3.3. Variable Proportional Integral Coefficient PI Regulator

_{γ}, Δi

_{γ}is positively correlated with the back-EMF. It can be seen that, for actual working conditions, the relationship between the position angle error and the current static difference is time-varying. Therefore, it is necessary to use a variable proportional integral coefficient PI regulator to ensure that the angle compensation value can be obtained quickly and accurately at different speeds.

_{p}and k

_{i}are the proportional and integral coefficients of the regulator, respectively, and e(t) is the deviation signal. Corresponding to this article, u(t) is the position angle error value Δθ, and e(t) is the current static difference value Δi

_{γ}.

_{p}and k

_{i}parameters have different correction effects on the deviation. Increasing the value of k

_{p}can improve the response speed and reduce the steady-state deviation, but if k

_{p}is too large, the overshoot will increase; the k

_{i}parameter is mainly used to eliminate the static error, k

_{i}the larger the value, the smaller the static difference, but if k

_{i}is too large, the transition process of the system becomes longer. From this, the following adjustment mechanism can be determined, that is, k

_{p}is positively correlated with the current static difference Δi

_{γ}. When the error Δi

_{γ}is large, the larger k

_{p}value should be selected, and when the error is small, the smaller k

_{p}value should be selected, so choose The corresponding positive correlation function corrects k

_{p}; while k

_{i}is negatively correlated with the static current difference value. When the error is large, choose a smaller value of k

_{i}, and when the error is small, choose a larger value of k

_{i}, so you need to choose the corresponding negative value. The correlation function modifies k

_{i}. The positive correlation function and negative correlation function used in this article are shown in Equation (14). Figure 5 shows the graphs of these two functions.

_{p}and k

_{i}values are corrected by the selected function, and the rebuilt k

_{p}and k

_{i}function expressions are:

_{po}and k

_{io}are the initial PI parameters, and k

_{1}and k

_{2}are correction coefficients. Only the k

_{1}and k

_{2}parameters need to be adjusted to achieve fast and accurate variable proportional integral control.

_{1}, and Figure 7 is the histogram of overshoot, stabilization time, and static difference under different k

_{2}when k

_{1}is determined.

_{1}values is analyzed. From the perspective of static difference, with the increase of k

_{1}, the static difference is gradually smaller, but the reduction range is also smaller. When k

_{1}is 7, 8, and 9, the static difference is almost the same, but the overshoot will increase after exceeding 7, so the k

_{1}value is determined to be 7. However, there is still a static error at this time. After determining k

_{1}, the appropriate k

_{2}value is selected from Figure 7 to continue to eliminate the static error. The static error gradually decreases with the increase of k

_{2}, and there is almost no static error at 500. The tracking value can be stabilized near the given value, so the k

_{2}value is determined to be 500.

#### 3.4. Parameter Robustness Analysis of the Proposed Method

_{s}ω

_{e}):

_{δ}is relatively small. When there is a 20% inductance parameter mismatch, that is, the value of 1-m is ±0.2, and the absolute value of the first term in Equation (20) is less than 2. Taking the motor used in this article as an example, the inductance value is 3 mH, the rotor flux linkage value is 0.15 Wb, the number of pole pairs of the motor is 2, and Δθ is 1.5 T

_{s}ω

_{e}when only the digital control delay is calculated. When the speed is equal to 9000 r/min, the angle error value is 0.353 rad at this time. At this time, the value of the second term of Equation (20) is at least 17.65, which is much larger than the first term, so the first term of Equation (20) can be ignored. The proposed method is less affected by the misalignment of motor parameters and works well at 0.8 times inductance mismatch and 1.2 times inductance mismatch with strong parameter robustness.

## 4. Experimental Verification and Result Analysis

#### 4.1. Experimental Platform Parameters

_{PWM}time of each switching cycle. Therefore, after adopting the double sampling and double update mode, the sampling delay is shortened to 0.5 T

_{PWM}, the PWM update delay is shortened to 0.25 T

_{PWM}, and the total control delay of the system is 0.75 T

_{PWM}. This control mode can effectively reduce the impact of digital control delay, especially in high-speed and low carrier wave ratio conditions can effectively improve the quality of the three-phase current waveform.

#### 4.2. Experimental Analysis

_{d}, and i

_{q}waveforms of the motor from 1500 r/min to 6700 r/min.

## 5. Conclusions

- (1)
- The proposed method uses the static difference of the d-axis current discretized by Euler’s approximation method as the basis for judging whether there is an angle estimation error, that is when there is an angle estimation error, there is a static difference between the actual value of the d-axis current and the reference value, and otherwise, there is no error.
- (2)
- Since the static difference value of the d-axis current changes with the speed change, a PI controller with variable proportional and integral coefficient is required, that is, the input value of the PI controller is the given value of the d-axis current, and the feedback value is the discretized d-axis current, The output value is the position angle error compensation amount. By using different proportional integral coefficients at different speeds, it is ensured that the position angle can be accurately and quickly estimated using the proposed sensorless control method at different speeds, and the estimation error is unchanged.
- (3)
- Through theoretical analysis and experimental verification, it can be seen that the proposed method can accurately compensate the estimated rotor position in the case of high-speed and low carrier wave ratio, improve the control accuracy, and have strong parameter robustness. In the case of the model, mismatch caused inductance parameter changes, the angle error can still be accurately compensated to ensure the motor performance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Block diagram of sensorless control for high-speed permanent magnet synchronous motor based on discrete current error.

**Figure 9.**The experimental results with the traditional method: (

**a**) Speed and dq axis current when the speed is increased from 1500 r/min to 6700 r/min; (

**b**) Estimated position angle, actual position angle and error under 1500 r/min; (

**c**) Estimated position angle, actual position angle and error at 6700 r/min; (

**d**) Phase A current at 6700 r/min.

**Figure 10.**The experimental results with the proposed method: (

**a**) Speed and dq axis current when the speed is increased from 1500 r/min to 9000 r/min; (

**b**) Estimated position angle, actual position angle and error under 6700 r/min; (

**c**) Estimated position angle, actual position angle and error at 9000 r/min; (

**d**) Phase A current at 9000 r/min.

**Figure 11.**The speed-up experimental results with the proposed method when the program parameter is equal to 0.8 times the nominal inductance: (

**a**) Rotating speed and dq axis current at 1500 r/min to 9000 r/min; (

**b**) Estimated position angle, actual position angle and error at 9000 r/min; (

**c**) A-phase current at 9000 r/min.

**Figure 12.**The speed-up experimental results with the proposed method when the program parameter is equal to 1.2 times the nominal inductance: (

**a**) Rotating speed and dq axis current at 1500 r/min to 9000 r/min; (

**b**) Estimated position angle, actual position angle and error at 9000 r/min; (

**c**) A-phase current at 9000 r/min.

P_{N}/kW | T_{N}/N·m | ω_{N}/r/min | J/kg·m^{2} | L_{d}, L_{q} /mH | n_{p} | R_{s}/Ω | ψ_{f}/Wb |
---|---|---|---|---|---|---|---|

3.7 | 11.8 | 3000 | 0.0012 | 3 | 2 | 0.38 | 0.15 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Du, D.; Yu, Q.; Zhang, H.; Li, C.; Guo, L.; Gu, X.; Li, X.
Sensorless Control Method of High-Speed Permanent Magnet Synchronous Motor Based on Discrete Current Error. *World Electr. Veh. J.* **2023**, *14*, 69.
https://doi.org/10.3390/wevj14030069

**AMA Style**

Wang Z, Du D, Yu Q, Zhang H, Li C, Guo L, Gu X, Li X.
Sensorless Control Method of High-Speed Permanent Magnet Synchronous Motor Based on Discrete Current Error. *World Electric Vehicle Journal*. 2023; 14(3):69.
https://doi.org/10.3390/wevj14030069

**Chicago/Turabian Style**

Wang, Zhiqiang, Dezheng Du, Qi Yu, Haifeng Zhang, Chen Li, Liyan Guo, Xin Gu, and Xinmin Li.
2023. "Sensorless Control Method of High-Speed Permanent Magnet Synchronous Motor Based on Discrete Current Error" *World Electric Vehicle Journal* 14, no. 3: 69.
https://doi.org/10.3390/wevj14030069