# Model Predictive Controller Design for Vehicle Motion Control at Handling Limits in Multiple Equilibria on Varying Road Surfaces

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## Abstract

**:**

## 1. Introduction

## 2. Vehicle Modeling and Simulation Setup

#### 2.1. Vehicle Modeling

#### 2.1.1. Two-Wheel Planar Vehicle Dynamics

_{A}is considered zero in the above equations since the speed of drift results in neglectable air drag. Due to small values, rolling and pitching dynamics are also neglected. Since the test vehicle is rear-wheel-driven, and there is no braking during this specific drift maneuver, F

_{xF}is considered zero throughout this paper.

#### 2.1.2. Nonlinear Tire Model

#### 2.2. Model Parameter Identification Measurements

## 3. Equilibrium Analysis

## 4. Controller Design

_{x}states and N

_{u}inputs is considered. The system can be described by ordinary differential equations as presented in Section 2.1 and written in the form of Equation (20).

#### 4.1. MPC Formulation

_{s}). When the cost function is minimized, the first element of the calculated control inputs is applied. Crucial parameters are the prediction horizon N

_{p}and the control horizon N

_{c}. N

_{p}defines the capability of the system in terms of how far it can see into the future. The time horizon N

_{p},

_{t}, which corresponds to the prediction horizon N

_{p}can be written in the following form:

_{c}defines the number of control inputs that can be optimized in one time step. One should carefully set these parameters to see the transient of the system in advance, thereby avoiding unnecessarily high computational needs. The MPC uses full state feedback that minimizes the cost function below.

_{xR}at the rear wheel was between 0 N and 7000 N.

#### 4.2. Successive Linearization

## 5. Simulation Results

#### 5.1. Steady-State Drift in a Single Equilibrium

#### 5.2. Steady-State Drift in Multiple Equilibria

#### 5.3. Drifting on Varying Road Surfaces

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Name | Sign | Dimension |
---|---|---|

mass of the vehicle | m | kg |

vehicle moment of inertia around z-axis | I_{z} | kg·m^{2} |

lateral force | F_{y} | N |

lateral force at front wheel | F_{yF} | N |

lateral force at rear wheel | F_{yR} | N |

normal load on the front wheels | F_{zF} | N |

normal load on the rear wheels | F_{zR} | N |

drive force at the rear wheel | F_{xR} | N |

drive force at the front wheel | F_{xF} | N |

air drag | F_{A} | N |

applied torque at the center of gravity | M_{z} | Nm |

yaw-rate | r | rad/s |

longitudinal velocity | V_{x} | m/s |

lateral velocity | V_{y} | m/s |

distance between center of gravity and front axle | a | m |

distance between center of gravity and rear axle | b | m |

derating factor | $\xi $ | - |

cornering stiffness of the front tire | C_{αF} | N/rad |

cornering stiffness of the rear tire | C_{αR} | N/rad |

air drag coefficient | C_{A} | - |

frontal cross-section surface of the vehicle | A | m^{2} |

air density | ρ | kg/m^{3} |

front tire sideslip angle | α_{F} | rad |

rear tire sideslip angle | α_{R} | rad |

sideslip boundary angle of the front tire | ${\alpha}_{sl\_F}$ | rad |

sideslip boundary angle of the front rear | ${\alpha}_{sl\_R}$ | rad |

friction coefficient | µ | - |

vehicle sideslip angle at center of gravity | β | rad |

steering angle | δ | rad |

state vector | x | - |

input vector | u | - |

continuous time state matrices | A_{c}, B_{c}, C_{c}, D_{c} | - |

discrete time state matrices | A_{d}, B_{d}, C_{d}, D_{d} | - |

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**Figure 11.**Vehicle states during the test case showing the handling of rapid change of road surface grip.

Case No. | δ (°) | F_{xR} (N) | V_{y} (m/s) | r (rad/s) | V_{x} (m/s) |
---|---|---|---|---|---|

1 | −20.05 | 4753 | −5.21 | 0.776 | 10 |

2 | −28.65 | 5500 | −6.99 | 0.713 | 10 |

3 | −22.92 | 5254 | −6.36 | 0.735 | 10 |

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**MDPI and ACS Style**

Czibere, S.; Domina, Á.; Bárdos, Á.; Szalay, Z.
Model Predictive Controller Design for Vehicle Motion Control at Handling Limits in Multiple Equilibria on Varying Road Surfaces. *Energies* **2021**, *14*, 6667.
https://doi.org/10.3390/en14206667

**AMA Style**

Czibere S, Domina Á, Bárdos Á, Szalay Z.
Model Predictive Controller Design for Vehicle Motion Control at Handling Limits in Multiple Equilibria on Varying Road Surfaces. *Energies*. 2021; 14(20):6667.
https://doi.org/10.3390/en14206667

**Chicago/Turabian Style**

Czibere, Szilárd, Ádám Domina, Ádám Bárdos, and Zsolt Szalay.
2021. "Model Predictive Controller Design for Vehicle Motion Control at Handling Limits in Multiple Equilibria on Varying Road Surfaces" *Energies* 14, no. 20: 6667.
https://doi.org/10.3390/en14206667