# A Novel Data-Driven Modeling and Control Design Method for Autonomous Vehicles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Motivation

## 2. Formulation of the Control-Oriented LPV Model Using Data-Driven Approach

#### 2.1. Acquisition of Data from Simulations

- (1)
- longitudinal velocity (${v}_{x}$)
- (2)
- angular velocity of the wheels (${\omega}_{x,y}$), $x\in \{front,rear\}$, $y\in \{left,right\}$
- (3)
- steering angle ($\delta $)
- (4)
- yaw-rate ($\dot{\psi}$)
- (5)
- accelerations (${a}_{x}$, ${a}_{y}$)
- (6)
- lateral velocity (${v}_{y}$).
- (7)
- side-slips of the wheels (${\alpha}_{x}$), $x\in \{front,rear\}$.

#### 2.2. Categorization of Instances

## 3. Parameters Optimization and Determination of Scheduling Parameters

#### 3.1. Fundamentals of the Applied Machine-Learning-Based Method

#### 3.2. Parameter Selection of the Control-Oriented Model

#### 3.3. Evaluation of the Data-Driven LPV Models

## 4. Path Following LPV Control Design Using the Data-Driven Model

- Minimization of the lateral error: As mentioned, the goal of the control design is to guarantee the trajectory tracking of the vehicle thus the error between the measured and the reference lateral positions must be minimized:$$\begin{array}{c}\hfill {z}_{2}={y}_{ref}-y,\phantom{\rule{28.45274pt}{0ex}}\left|{z}_{2}\right|\to min,\end{array}$$

- Minimization of the yaw-rate error: In order to achieve smooth trajectory tracking, a reference yaw-rate is also prescribed, which also must be tracked by the vehicle:$$\begin{array}{c}\hfill {z}_{1}={\dot{\psi}}_{ref}-\dot{\psi},\phantom{\rule{28.45274pt}{0ex}}\left|{z}_{1}\right|\to min,\end{array}$$

- Minimization of the interventions: Due to the energy consumption, the interventions also must be minimized during the operation of the vehicle:$$\begin{array}{cc}\hfill {z}_{3}& =\delta ,\phantom{\rule{34.14322pt}{0ex}}|{z}_{3}|\to min.\hfill \end{array}$$$$\begin{array}{cc}\hfill {z}_{4}& ={M}_{d},\phantom{\rule{25.6073pt}{0ex}}\left|{z}_{4}\right|\to min.\hfill \end{array}$$

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Gáspár, P.; Szabó, Z.; Bokor, J.; Németh, B. Robust Control Design for Active Driver Assistance Systems; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Pham, T.P.; Sename, O.; Dugard, L. Real-time Damper Force Estimation of Vehicle Electrorheological Suspension: A NonLinear Parameter Varying Approach. Part of special issue: 3rd IFAC Workshop on Linear Parameter Varying Systems LPVS 2019: Eindhoven, Netherlands, 4–6 November 2019. IFAC-PapersOnLine
**2019**, 52, 94–99. [Google Scholar] [CrossRef] - Rosolia, U.; Borrelli, F. Learning How to Autonomously Race a Car: A Predictive Control Approach. IEEE Trans. Control. Syst. Technol.
**2020**, 28, 2713–2719. [Google Scholar] [CrossRef] [Green Version] - Nemeth, B.; Gaspar, P.; Peni, T. Nonlinear analysis of vehicle control actuations based on controlled invariant sets. Int. J. Appl. Math. Comput. Sci.
**2016**, 26, 31–43. [Google Scholar] [CrossRef] [Green Version] - Ribeiro, A.M.; Fioravanti, A.R.; Moutinho, A.; de Paiva, E.C. Nonlinear state-feedback design for vehicle lateral control using sum-of-squares programming. Veh. Syst. Dyn.
**2020**, 1–27. [Google Scholar] [CrossRef] - Yang, G.; Zhao, Y. Motion Stability Analysis of Vehicle with Four Wheel Steering System Considering Tire Nonlinearity. In Proceedings of the 2010 3rd International Congress on Image and Signal Processing (CISP2010), Yantai, China, 16–18 October 2010; pp. 3433–3437. [Google Scholar]
- Sadri, S.; Wu, Q. Lateral Stability Analysis of On-road Vehicles Using Lyapunov’s Direct Method. In Proceedings of the 2012 Intelligent Vehicles Symposium, Alcala de Henares, Spain, 3–7 June 2012; pp. 821–826. [Google Scholar]
- Huijun, G.; Weichao, S.; Shen, Y.; Okyay, K. Stability Control for Lateral Vehicle Motion with Uncertain Parameters and External Nonlinearities. In Proceedings of the 32nd Chinese Control Conference, Xi’an, China, 26–28 July 2013. [Google Scholar]
- Corno, M.; Panzani, G.; Roselli, F.; Giorelli, M.; Azzolini, D.; Savaresi, S.M. An LPV Approach to Autonomous Vehicle Path Tracking in the Presence of Steering Actuation Nonlinearities. IEEE Trans. Control. Syst. Technol.
**2020**, 24, 956–970. [Google Scholar] [CrossRef] - Gaspar, P.; Szabo, Z.; Bokor, J. A grey-box identification of an LPV vehicle model with side slip angle estimation. In Proceedings of the 2007 American Control Conference, Zurich, Switzerland, 4–7 September 2007. [Google Scholar]
- Rodonyi, G.; Bokor, J. Uncertainty Identification for a Nominal LPV Vehicle Model Based on Experimental Data. In Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain, 15 December 2005; pp. 2682–2687. [Google Scholar]
- Kuutti, S.; Bowden, R.; Jin, Y.; Barber, P.; Fallah, S. A Survey of Deep Learning Applications to Autonomous Vehicle Control. IEEE Trans. Intell. Transp. Syst.
**2020**, 1–22. [Google Scholar] [CrossRef] - Hubschneider, C.; Bauer, A.; Doll, J.; Weber, M.; Klemm, S.; Kuhnt, F.; Zollner, J.M. Integrating end-to-end learned steering into probabilistic autonomous driving. In Proceedings of the 2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), Yokohama, Japan, 16–19 October 2017; pp. 1–7. [Google Scholar]
- Rausch, V.; Hansen, A.; Solowjow, E.; Liu, C.; Kreuzer, E.; Hedrick, J.K. Learning a deep neural net policy for end-to-end control of autonomous vehicles. In Proceedings of the 2017 American Control Conference (ACC), Seattle, WA, USA, 24–26 May 2017; pp. 4914–4919. [Google Scholar]
- Pomerleau, D. Knowledge-Based Training of Artificial Neural Networks for Autonomous Robot Driving. Robot. Learn.
**1993**, 233, 13–43. [Google Scholar] - Cavanini, L.; Ferracuti, F.; Longhi, S.; Monteriu, A. LS-SVM for LPV-ARX Identification: Efficient Online Update by Low-Rank Matrix Approximation. In Proceedings of the 2020 International Conference on Unmanned Aircraft Systems (ICUAS), Athens, Greece, 1–4 September 2020; pp. 1590–1595. [Google Scholar] [CrossRef]
- Romano, R.A.; dos Santos, P.L.; Pait, F.; Perdicoulis, T.; Ramos, J.A. Machine learning barycenter approach to identifying LPV state-space models. In Proceedings of the 2016 American Control Conference (ACC), Boston, MA, USA, 6–8 July 2016; pp. 6351–6356. [Google Scholar] [CrossRef] [Green Version]
- Bao, Y.; Velni, J.M. Data-Driven Linear Parameter-Varying Model Identification Using Transfer Learning. IEEE Control. Syst. Lett.
**2021**, 5, 1579–1584. [Google Scholar] [CrossRef] - Abdufattokhov, S.; Muhiddinov, B. Stochastic Approach for System Identification using Machine Learning. In Proceedings of the 2019 Dynamics of Systems, Mechanisms and Machines (Dynamics), Omsk, Russia, 5–7 November 2019; pp. 1–4. [Google Scholar] [CrossRef]
- Biagetti, G.; Crippa, P.; Falaschetti, L.; Turchetti, C. Machine learning regression based on particle bernstein polynomials for nonlinear system identification. In Proceedings of the 2017 IEEE 27th International Workshop on Machine Learning for Signal Processing (MLSP), Tokyo, Japan, 25–28 September 2017; pp. 1–6. [Google Scholar] [CrossRef]
- Rosolia, U.; Borrelli, F. Learning Model Predictive Control for Iterative Tasks. A Data-Driven Control Framework. IEEE Trans. Autom. Control.
**2018**, 63, 1883–1896. [Google Scholar] [CrossRef] [Green Version] - Fliess, M.; Join, C. Model-free control. Int. J. Control.
**2013**, 86, 2228–2252. [Google Scholar] [CrossRef] [Green Version] - Formentin, S.; De Filippi, P.; Corno, M.; Tanelli, M.; Savaresi, S.M. Data-Driven Design of Braking Control Systems. IEEE Trans. Control. Syst. Technol.
**2013**, 21, 186–193. [Google Scholar] [CrossRef] - Palmieri, G.; Baric, M.; Glielmo, L.; Borrelli, F. Robust vehicle lateral stabilisation via set-based methods for uncertain piecewise affine systems. Veh. Syst. Dyn.
**2012**, 50, 861–882. [Google Scholar] [CrossRef] - Fenyes, D.; Nemeth, B.; Gaspar, P. Analysis of autonomous vehicle dynamics based on the big data approach. In Proceedings of the European Control Conference, Limassol, Cyprus, 12–15 June 2018; pp. 219–224. [Google Scholar]
- Rajamani, R. Vehicle Dynamics and Control; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Wang, Y.; Witten, I.H. Pace Regression; (Working Paper 99/12); University of Waikato, Department of Computer Science: Hamilton, New Zealand, 1999. [Google Scholar]
- Shibata, R. An optimal selection of regression variables. Biometrika
**1981**, 68, 45–54. [Google Scholar] [CrossRef] - Gill, P.E.; Murray, W.; Wright, M. Practical Optimization; Academic Press: London, UK, 1981. [Google Scholar]
- Coleman, T.F.; Li, Y. A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on some of the Variables. SIAM J. Optim.
**1996**, 6, 1040–1058. [Google Scholar] [CrossRef]

Parameter | Notion | Value | Unit |
---|---|---|---|

Mass of the car | m | 1690 | kg |

Yaw-inertia | J | 4192 | kg/m${}^{2}$ |

Location of front axis from COG | ${l}_{1}$ | 1.11 | m |

Cornering stiffness of front wheels | ${C}_{1}$ | 155,160 | N/rad |

Location of rear axis from COG | ${l}_{2}$ | 1.66 | m |

Cornering stiffness of rear wheels | ${C}_{2}$ | 114,659 | N/rad |

Front drag area of the car | A | 1.8 | m${}^{2}$ |

Height of COG | h | 0.56 | m |

Type of front suspensions | - | Independent | - |

Mass of front suspensions | ${m}_{s,f}$ | 85 | kg |

Type of rear suspensions | - | Independent | - |

Mass of rear suspensions | ${m}_{s,r}$ | 85 | kg |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fényes, D.; Németh, B.; Gáspár, P.
A Novel Data-Driven Modeling and Control Design Method for Autonomous Vehicles. *Energies* **2021**, *14*, 517.
https://doi.org/10.3390/en14020517

**AMA Style**

Fényes D, Németh B, Gáspár P.
A Novel Data-Driven Modeling and Control Design Method for Autonomous Vehicles. *Energies*. 2021; 14(2):517.
https://doi.org/10.3390/en14020517

**Chicago/Turabian Style**

Fényes, Dániel, Balázs Németh, and Péter Gáspár.
2021. "A Novel Data-Driven Modeling and Control Design Method for Autonomous Vehicles" *Energies* 14, no. 2: 517.
https://doi.org/10.3390/en14020517