# Nonlinear MPC-Based Acceleration Slip Regulation for Distributed Electric Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. System Model Establishment

#### 2.1. Vehicle Longitudinal Movement

#### 2.2. Four Wheels’ Rotation Movement

#### 2.3. Tire Longitudinal Force Calculation Model

#### 2.4. Longitudinal Slip Calculation Model

## 3. Control Strategy

#### 3.1. PID-Based Speed Controller Design

#### 3.2. NMPC-Based Slip Controller Design

#### 3.2.1. Model Discretization

#### 3.2.2. Description of Optimization Problem

#### 3.3. Design of Intervention and Exit Mechanisms

## 4. Simulation and Analysis

#### 4.1. Low Adhesion Road

^{2}. After 2.2 s, the NMPC is turned off, and acceleration rapidly drops close to 0, because the reference speed has been tracked at this time, and thus, continuing to accelerate becomes unnecessary. The simulation results of ASR-PID are similar to those of ASR-MPC. The difference is that the first arrival reference speed of ASR-PID is slightly slower than that of ASR-MPC, because the latter has better slip rate control. In contrast with WASR, acceleration is maintained at 1.65 m/s

^{2}most of the time, and the slip rate is close to 1. After reaching the reference speed, acceleration cannot be reduced in time, and the simulation reports an error at 4.6 s. This working condition is extremely dangerous and should be avoided. In the torque diagram, MPC and PID are the slip rate control output torque and driver output torque, respectively, in ASR control. ASR is the final output torque of the front and rear axle motors. Under the action of the intervention and exit mechanisms, the front and rear axle motors can always select a more suitable torque as the output.

_{min}.

#### 4.2. High Adhesion Road

_{min}, and the front wheel slip rate can effectively track the optimal slip rate. However, the rear wheel slip rate cannot track the optimal slip rate, because the load is transferred backward during the acceleration process, and the rear axle wheels require a larger motor torque to generate the same slip rate. The slip rate cannot continue increasing due to the limitation of the motor torque. The control effect of ASR-PID is basically consistent with that of ASR-MPC. In WASR control, the wheel slip rate is large at 0.2–3 s. The handling stability of the vehicle is considerably reduced at this time. This condition is more dangerous, and the rear wheel slip rate is always kept at a small value due to load transfer. When speed is greater than v

_{min}, the slip rate can track the optimal slip rate if flag = 1, and the slip rate remains below the optimal slip rate if flag = 0. Therefore, the designed algorithm can also play an important role in a road surface with a high adhesion coefficient. The expected results can be achieved regardless of the starting and acceleration.

#### 4.3. Docking Road

#### 4.4. Split Road

#### 4.5. Robustness Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Road Surface | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{\lambda}}_{\mathit{o}\mathit{p}\mathit{t}}$ | ${\mathit{\mu}}_{\mathbf{max}}$ |
---|---|---|---|---|---|

Snow | 0.1964 | 94.129 | 0.0646 | 0.06 | 0.19 |

Wet asphalt | 0.8570 | 33.822 | 0.3470 | 0.13 | 0.80 |

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

Vehicle Mass | $m$ | 1710 kg |

Height of the vehicle c.g. | ${h}_{g}$ | 0.552 m |

Distance from c.g. to front axle | $a$ | 1.216 m |

Distance from c.g. to Rear axle | $b$ | 1.613 m |

Rolling radius of the tyre | $R$ | 0.32 m |

Front area | $A$ | 2.3157 m^{2} |

Front motor peak power | ${P}_{f\mathrm{max}}$ | 130 kW |

Front motor peak torque | ${T}_{f\mathrm{max}}$ | 225 Nm |

Rear motor peak power | ${P}_{r\mathrm{max}}$ | 60 kW |

Rear motor peak torque | ${T}_{r\mathrm{max}}$ | 170 Nm |

Road Surface | Condition | Algorithm | A | B | C | D | E |
---|---|---|---|---|---|---|---|

Low adhesion road | Starting | ASR-MPC | 2.2 s | 2.2 s | N | 0 | 0 |

ASR-PID | 2.3 s | 2.3 s | N | 0 | 0 | ||

WASR | 2.6 s | In | Y | 0 | 0 | ||

Acceleration | ASR-MPC | 2.2 s | 2.2 s | N | 0 | 0 | |

ASR-PID | 2.4 s | 2.4 s | N | 0 | 0 | ||

WASR | 2.6 s | In | Y | 0 | 0 | ||

High adhesion road | Starting | ASR-MPC | 2.3 s | 2.3 s | N | 0 | 0 |

ASR-PID | 2.35 s | 2.35 s | N | 0 | 0 | ||

WASR | 2.6 s | 3.2 s | Y | 0 | 0 | ||

Acceleration | ASR-MPC | 2.3 s | 2.3 s | N | 0 | 0 | |

ASR-PID | 2.3 s | 2.3 s | N | 0 | 0 | ||

WASR | 2.6 s | 3.2 s | Y | 0 | 0 | ||

Docking road | Starting | ASR-MPC | 2.9 s | 2.9 s | N | 0 | 0 |

ASR-PID | 2.9 s | 2.9 s | N | 0 | 0 | ||

WASR | 3.1 s | 4.2 s | Y | 0 | 0 | ||

Acceleration | ASR-MPC | 2.8 s | 2.8 s | N | 0 | 0 | |

ASR-PID | 2.8 s | 2.8 s | N | 0 | 0 | ||

WASR | 2.8 s | 3.6 s | Y | 0 | 0 | ||

Split road | Starting | ASR-MPC | 2.9 s | 2.9 s | N | 24.8° | 0.08 m |

ASR-PID | 2.9 s | 2.9 s | N | 26.1° | 0.08 m | ||

WASR | 2.7 s | In | Y | 360° | 7.3 m | ||

Acceleration | ASR-MPC | 2.8 s | 2.8 s | N | 25.2° | 0.07 m | |

ASR-PID | 2.8 s | 2.8 s | N | 26.2° | 0.07 m | ||

WASR | 2.6 s | In | Y | 360° | 5.6 m |

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**MDPI and ACS Style**

Shi, W.; Jiang, Y.; Shen, Z.; Yu, Z.; Chu, H.; Liu, D.
Nonlinear MPC-Based Acceleration Slip Regulation for Distributed Electric Vehicles. *World Electr. Veh. J.* **2022**, *13*, 200.
https://doi.org/10.3390/wevj13110200

**AMA Style**

Shi W, Jiang Y, Shen Z, Yu Z, Chu H, Liu D.
Nonlinear MPC-Based Acceleration Slip Regulation for Distributed Electric Vehicles. *World Electric Vehicle Journal*. 2022; 13(11):200.
https://doi.org/10.3390/wevj13110200

**Chicago/Turabian Style**

Shi, Wentong, Yuyao Jiang, Zuying Shen, Zhongjing Yu, Hongqing Chu, and Dengcheng Liu.
2022. "Nonlinear MPC-Based Acceleration Slip Regulation for Distributed Electric Vehicles" *World Electric Vehicle Journal* 13, no. 11: 200.
https://doi.org/10.3390/wevj13110200