Hybrid modeling and predictive control of intelligent vehicle longitudinal velocity considering nonlinear tire dynamics

  • Xiaoqiang SunEmail author
  • Houzhong Zhang
  • Yingfeng Cai
  • Shaohua Wang
  • Long Chen
Original paper


A hybrid model predictive control (HMPC) strategy is proposed in this paper to autonomously regulate intelligent vehicle longitudinal velocity considering nonlinear tire dynamics. Since the tire longitudinal dynamics, which has significant influence on vehicle longitudinal velocity control performance, exhibits highly nonlinear dynamical behaviors, the piecewise affine (PWA) identification is conducted firstly based on experimental data to accurately model the tire longitudinal dynamics. On this basis, due to that the intelligent vehicle needs to be operated in two distinct modes (drive and brake) for autonomous velocity regulation and because of the affine submodel switching behaviors of the PWA-identified tire model, the intelligent vehicle longitudinal dynamics control process considered in this work can be regarded as a hybrid system with both continuous variables and discrete operating modes. Thus, the mixed logical dynamical framework is further used to model the intelligent vehicle longitudinal dynamics, and a HMPC controller, which allows us to optimize the switching sequences of the operation modes (binary control inputs) and the torques acted on the wheels (continuous control inputs), is tuned based on online mixed-integer quadratic programming. Simulation results finally demonstrate the effectiveness of the proposed HMPC controller for intelligent vehicle longitudinal velocity regulation under typical driving conditions.


Intelligent vehicle Longitudinal dynamics Hybrid modeling Model predictive control Nonlinear tire dynamics 

List of symbols


Vehicle acceleration along the forward direction


Windward area


Number of the PWA submodels


Aerodynamic resistance coefficient

\(f_{\mathrm{R} }\)

Rolling resistance coefficient


Vehicle accelerating resistance

\(F_{\mathrm{G} }\)

Vehicle climbing resistance


Vehicle rolling resistance


Vehicle aerodynamic resistance


Tire vertical load


Longitudinal forces generated by the left driving tire

\(F_{\mathrm{xr} }\)

Longitudinal forces generated by the right driving tire


Coefficient matrices of the polyhedral region


Coefficient matrices of the polyhedral region


Acceleration of gravity


Road slope angle

\(\kappa \)

Longitudinal slip coefficient


Number of the data points


Vehicle curb weight


Vehicle loading weight


Rolling resistance torque


PWA model orders


PWA model orders


Positive penalty weighting parameters


Positive penalty weighting parameters


Control horizon


Effective wheel rolling radius


Drive torque acted on the wheel


Brake torque acted on the wheel


Rolling resistance torque


MLD system inputs


Speed of the tire–road interface


Speed of the left tire–road interface


Speed of the right tire–road interface


Initial vehicle velocity


Vehicle actual velocity


PWA model output

\({\vartheta }_{i }\)

Parameter vectors defining each submodel

\({\varphi }(t)\)

Regression vector of the PWA model

\({\chi }_{i}\)

Whole polyhedral region of the affine submodels

\({\varTheta }\)

Moment of inertia of the wheels

\({\rho }_{\mathrm{a}}\)

Air density

\({\varOmega }_{\mathrm{w}}\)

Wheel angular velocity



Mixed logical dynamical


Piecewise affine


Hybrid model predictive control


Intelligent transportation systems


Model predictive control


Mixed-integer quadratic programming


Multi-parametric programming technology



This work was supported by the National Natural Science Foundation of China (Grant Nos. 51705207, U1564201), the China Postdoctoral Science Foundation (Grant No. 2019T120401, 2017M611728), the Postdoctoral Research Foundation of Jiangsu Province (Grant No. 1701112B) and the Six Talent Peaks Project of Jiangsu Province (Grant No. GDZB-163).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Automotive Engineering Research InstituteJiangsu UniversityJiangsuPeople’s Republic of China
  2. 2.School of Automotive and Traffic EngineeringJiangsu UniversityJiangsuPeople’s Republic of China

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