# Permanent Magnet Synchronous Motor Speed Control Based on Improved Active Disturbance Rejection Control

^{*}

## Abstract

**:**

## 1. Introduction

**∙**) functions in the ESO and NLSEF are nonlinear. There is an inflection point close to the origin of the fal(

**∙**) function, leading to the chattering problem of ADRC and adversely affecting the accuracy and robustness of the system. To improve the control performance of ADRC, an improved nonlinear newfal(

**∙**) function, based on the Sigmoid function, is proposed and a first-order improved ADRC is constructed. Firstly, the mathematical model of a PMSM is built. Secondly, the improved first-order TD, ESO, and NLSEF are designed. The equations of each part of the improved first-order ADRC are provided, and an improved first-order ADRC is constructed. The Lyapunov stability of the first-order TD, the second-order ESO, and the system are analysed. Finally, the improved ADRC (I-ADRC) is used in the speed control loop of the PMSM in simulation and experiment. The results show that I-ADRC has stronger disturbance rejection ability and stability than the PI and ADRC controller.

## 2. Mathematical Model of PMSM

- (1)
- The saturation of the iron in the stator of the motor is ignored;
- (2)
- The effects of the eddy current and hysteresis are ignored;
- (3)
- The three phase windings of the stator are symmetrical.

_{d}, u

_{q}, i

_{d}, i

_{q}, L

_{d}, L

_{q}, ψ

_{d}, and ψ

_{q}are the voltage, current, inductance, and flux linkage in the d-q reference frame, ${R}_{s}$ is the phase resistance, and ψ

_{f}is the flux linkage of the permanent magnet. T

_{e}and T

_{L}are electromagnetic torque and load torque, respectively. B is the damping coefficient of the rotor and load, p

_{n}is the pole number, J is the rotor moment of inertia, and ω is the rotational speed of the motor.

_{e}and i

_{q}, ω can be obtained from Equation (1). First-order ADRC in this study is used to better control i

_{q}and improve the performance of the speed loop.

_{d}= L

_{q}= L

_{s}is fulfilled. When i

_{d}is set to 0 in field-oriented control, the torque equation in (1) changes to

## 3. I-ADRC Design of PMSM

_{q}. Figure 1 shows the first-order ADRC of a speed control loop.

#### 3.1. Improved Nonlinear Function Design

**∙**) function, an obvious inflection point exists and the smoothness is poor. The inflection point will reduce the disturbance rejection ability and robustness of ADRC [34].

**∙**) function, a new function is proposed as

**∙**) and newfal(

**∙**) functions are symmetric about the origin. Figure 2 shows the characteristic curves of the fal(

**∙**) and newfal(

**∙**) functions with different parameters. For the convenience of observation, two groups of the curves of fal(

**∙**) and newfal(

**∙**) are shifted to the left and right from the origin by 0.2 respectively.

**∙**) can be observed, whereas the transition of the newfal(

**∙**) curve is smooth. In this study, a first-order ADRC is designed based on the newfal(

**∙**).

#### 3.2. Improved TD Design and Stability Analysis

**∙**) function is used to improve the disturbance rejection ability of a quadrotor aircraft. The TD constructed using the newfal(

**∙**) function is listed below:

**∙**), it has the same sign as e. When e > 0 or e < 0, ${\dot{\mathrm{V}}}_{1}$ < 0 is satisfied. When e = 0, ${\dot{\mathrm{V}}}_{1}$ = 0, according to the Lyapunov stability principle, ${\mathrm{V}}_{1}$ > 0, the reciprocal of ${\mathrm{V}}_{1}$ satisfies ${\mathrm{V}}_{1}$ < 0, and the stability point is asymptotically stable. Therefore, the first order TD is asymptotically stable.

#### 3.3. Improved Second Order ESO Design and Stability Analysis

_{1}= z

_{1}− y and ε

_{2}= z

_{2}− x. Subtract Equation (5) from Equation (13), the error state equation of the system can be obtained as

_{21}= ε

_{1}and ε

_{22}= $\dot{\epsilon}$

_{1}, then Equation (14) can be transformed into

_{02}> 0, it can be derived

#### 3.4. Stability Analysis of I-ADRC

## 4. Simulation and Experimental Results Analysis

_{q}and id by the Clarke and Park transformation. Both the torque and speed of the PMSM can be controlled by controlling the torque current i

_{q}. When i

_{d}= 0, the flux is completely supplied by the permanent magnet and all the current of the motor is used to generate the electromagnetic torque.

#### 4.1. Analysis of Simulation Results

**∙**) function. The target speed was set to 1000 rpm, and a 5 N·m load was suddenly applied at 0.05 s and unloaded at 0.1 s. The control performance of the three algorithms was compared based on the simulation results.

#### 4.1.1. Speed Simulation Results Analysis

#### 4.1.2. Analysis of Three Phase Current Simulation Results

#### 4.1.3. Analysis of Torque Simulation Results

#### 4.2. Analysis of Experimental Results

#### 4.2.1. No Load Speed Regulation Result Analysis

#### 4.2.2. Analysis of Load Speed Test Results

#### 4.2.3. Analysis of 3 Phase Current Test Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Comparison of characteristic curves of functions fal(

**∙**) and newfal(

**∙**). In (

**a**), α takes different values and in (

**b**), δ takes different values.

**Figure 4.**I-ADRC simulation structure diagram: (

**a**) overall simulation diagram; (

**b**) I-ADRC module; (

**c**) TD module; (

**d**) ESO module; (

**e**) NLSEF module and (

**f**) ESO module.

**Figure 7.**Torque simulation curves: (

**a**) complete torque simulation curve; (

**b**) partial enlargement A and (

**c**) partial enlargement B.

**Figure 8.**LINKS RS PMSM console: (

**a**) servo control platform; (

**b**) load system and (

**c**) controller and driver.

**Figure 11.**Three phase current test curve: (

**a**) complete current curve; (

**b**) local current curve. The test results of speed and current are consistent with simulation results.

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

Rated power (W) | 100 |

Pole pairs | 4 |

Rated voltage (V) | 36 |

Rated speed (rpm) | 3000 |

Instantaneous | 0.954 |

Rated current (A) | 4.6 |

Back EMF coefficient (mV/rpm) | 5.35 |

Line resistance (Ω) (25 °C) | 0.75 |

Line inductance (mH) | 2 |

Weight (kg) | 0.8 |

Rated torque (N·m) | 0.318 |

Encoder | 1250 |

I-ADRC Component | Symbol | Value |
---|---|---|

First order TD | ${\alpha}_{0}$ | 1.25 |

${\delta}_{0}$ | 0.01 | |

${a}_{0}$ | 500 | |

$K$ | 450 | |

Second order ESO | ${\beta}_{01}$ | 30 |

${\beta}_{02}$ | 5 | |

${\alpha}_{1}$ | 0.25 | |

${\delta}_{1}$ | 0.01 | |

${a}_{1}$ | 90 | |

$b$ | 110 | |

NLSEF | ${\alpha}_{2}$ | 1.1 |

${\delta}_{2}$ | 0.01 | |

${a}_{2}$ | 60 | |

${\beta}_{1}$ | 220 |

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**MDPI and ACS Style**

Shi, Z.; Zhang, P.; Lin, J.; Ding, H.
Permanent Magnet Synchronous Motor Speed Control Based on Improved Active Disturbance Rejection Control. *Actuators* **2021**, *10*, 147.
https://doi.org/10.3390/act10070147

**AMA Style**

Shi Z, Zhang P, Lin J, Ding H.
Permanent Magnet Synchronous Motor Speed Control Based on Improved Active Disturbance Rejection Control. *Actuators*. 2021; 10(7):147.
https://doi.org/10.3390/act10070147

**Chicago/Turabian Style**

Shi, Zhaoyao, Pan Zhang, Jiachun Lin, and Hongyu Ding.
2021. "Permanent Magnet Synchronous Motor Speed Control Based on Improved Active Disturbance Rejection Control" *Actuators* 10, no. 7: 147.
https://doi.org/10.3390/act10070147