# Study on a Second-Order Adaptive Sliding-Mode Observer Control Algorithm for the Sensorless Permanent Magnet Synchronous Motor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Physical and Mathematical Model of a PMSM

_{1}X

_{2}, Y

_{1}Y

_{2}, and Z

_{1}Z

_{2}are symmetrically installed in a circular space, with X

_{1}, Y

_{1}, and Z

_{1}being the first ends of each winding and X

_{2}, Y

_{2}, and Z

_{2}being the last ends. The first end outflow and the last end inflow are specified as the positive direction of the current. According to the right-hand screw rule, the direction of the magnetic field generated by each winding is specified as the positive direction of the axis of the winding. Then, using these three directions as the reference axes of the spatial coordinate axes, a three-phase stationary coordinate system named $A-B-C$ is established. The counterclockwise direction is specified as the positive direction of the angle and angular velocity. In Figure 1, the two-phase stationary coordinate system $\alpha -\beta $ is used to fix the $\alpha $-axis and keep it stationary on the axis of the stator A-phase winding, while the $\beta $-axis is 90° ahead compared to the $\alpha $-axis. The two-phase rotating coordinate system $d-q$ consists of the orthogonal $d$-axis and $q$-axis. The direction of the $d$-axis is the direction of the flux linkage of the permanent magnet, and the angle between the $d$-axis and the $A$-axis is $\theta $. The coordinate transformation from $A-B-C$ to $\alpha -\beta $ is a Park transformation, and the transformation from $\alpha -\beta $ to $d-q$ is a Clarke transformation. The algorithm proposed in this paper is based on the mathematical model in the $\alpha -\beta $ coordinate system.

## 3. Second-Order Adaptive Sliding-Mode Observer Algorithm

#### 3.1. Design of Second-Order SMO

#### 3.2. Adaptive Estimation of the Back-EMF

#### 3.3. Calculation of Rotor Velocity and Position

## 4. Experiment Verification and Results Analysis

#### 4.1. Analysis of Rotor Velocity and Rotor Position

#### 4.2. Analysis of Rotor Velocity Error and Rotor Position Error

#### 4.3. Analysis of the Estimated Back-EMF and the Three-Phase Current

#### 4.4. Stability Analysis under Mismatch of Stator Resistance

#### 4.5. Comparison of Results of Different Control Methods

## 5. Conclusions

- (1)
- This algorithm effectively weakens the jitter of the traditional SMO algorithm, reduces the velocity error and position error, and improves the velocity and position tracking performance of the system.
- (2)
- The estimated back-EMF noise is smaller, and there are fewer high-order harmonics of the current after adding load, making this algorithm more resistant to sudden load changes and giving it high robustness.
- (3)
- Compared with other PMSM position-sensorless control algorithms, the estimation accuracy of rotor velocity and position of the algorithm proposed in this paper is higher.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Parameters | Values |
---|---|

Stator resistance (R_{s}/Ω) | 3 |

Stator inductance (L_{s}/H) | 0.01 |

Permanent magnet flux linkage (ψ_{f}/(Wb)) | 0.175 |

Moment of inertia (J/(kg·m^{^2})) | 0.001 |

Number of pole pairs | 4 |

Rated power (P_{n}/kW) | 1.2 |

Data Source | Algorithm | Rotor Velocity Error (r/min) | Rotor Position Error (rad) |
---|---|---|---|

Reference [32] | An improved SMO based on tanh(x) | 1 | 0.05 |

Reference [33] | Fuzzy SMO | 2 | - |

Reference [34] | Sensorless control based on PLL | 5 | 0.087 |

Reference [35] | A novel MRAS algorithm | 1.5 | - |

This paper | Traditional SMO | 9.95 | 0.049 |

This paper | Second-order adaptive SMO | 0.94 | 0.022 |

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**MDPI and ACS Style**

Yao, G.; Cheng, Y.; Wang, Z.; Xiao, Y.
Study on a Second-Order Adaptive Sliding-Mode Observer Control Algorithm for the Sensorless Permanent Magnet Synchronous Motor. *Processes* **2023**, *11*, 1636.
https://doi.org/10.3390/pr11061636

**AMA Style**

Yao G, Cheng Y, Wang Z, Xiao Y.
Study on a Second-Order Adaptive Sliding-Mode Observer Control Algorithm for the Sensorless Permanent Magnet Synchronous Motor. *Processes*. 2023; 11(6):1636.
https://doi.org/10.3390/pr11061636

**Chicago/Turabian Style**

Yao, Guozhong, Yuanpeng Cheng, Zhengjiang Wang, and Yuhan Xiao.
2023. "Study on a Second-Order Adaptive Sliding-Mode Observer Control Algorithm for the Sensorless Permanent Magnet Synchronous Motor" *Processes* 11, no. 6: 1636.
https://doi.org/10.3390/pr11061636