# Parallel Hybrid Electric Vehicle Modelling and Model Predictive Control
## 1 Introduction  
Controllers for HEVs powertrains and speeds can be included model-free or model-based. Model-free controllers are mostly used with heuristic, fuzzy, neuro, AI, or human virtual and augmented reality. The use of model-free methods will be presented in the next part of this study. Model-based controllers can be used with a conventional adaptive PID, \(H_{\text{2}}\), \(H_{\infty}\), or sliding mode. However, all conventional control methods cannot include the real-time dynamic constraints of the vehicle physical limits, the surrounding obstacles, and the environment (road and weather) conditions. Therefore, a MPC with horizon state and open loop control prediction subject to dynamic constraints are mainly used to control as real-time the HEV speeds and torques. Due to the limit size of this paper, we have reviewed some of the most recent research of MPC applications for HEVs. In this paper, vehicle dynamic formulas and calculations are referred to the reference [1]. A recent modelling and control of the dual clutch transmission for HEVs are presented in [2], in which a new controller was designed for synchronizing the dual clutch transmission (DCT) with higher performances and lower fuel consumption. Another MPC for an autonomous driving vehicle is developed in [3], in which the MPC was used to drive the HEV to exactly track the given feasible trajectories. Moreover, a controller for a hybrid dual clutch transmission powertrain for a HEV is introduced in [4], in which the ICE and EM were driven by a DCT powertrain. A MPC for a HEV with linear parameters and a varying model is presented in [5], in which the MPC controller was designed to improve the fuel economy of the power-split HEV. A MPC for a HEV is necessary not only to control the torque and speed, but also to control the gas emission and improve the fuel economy. The authors of [6] developed aMPC with a multi-objective function for HEVs for fuel economy and exhaust emission and collision detection as well as to optimize the vehicle speed and engine torque. A new MPC design for HEVs with adaptive cruise of autonomous electric vehicles was presented in [7]. A hybrid MPC to optimize the HEV mode selection was introduced in [8], in which the vehicle thermal management was controlled by this MPC subject to decision-making algorithms. Fuel economy and lower emission were also controlled by a MPC with outer approximation and semi-convex cut generation, as presented in [9]. Due to the recent world commitment to limit the increase in global warming and stop using fossil fuels, plug-in and the pure electric vehicles are expanding. MPC algorithms are also being developed to control the plug-in hybrid vehicle (PHEV), as shown in [10]. In this study, a non-linear MPC is designed to control the torque-split and to optimize fuel management. The authors of [11] also present a data-based scenario MPC framework to optimize power consumption. Non-linear model predictive control (NMPC) has been widely used due to the rapid increase in computer capacity and speeds. Computers now can calculate a real-time solution from complex non-linear functions. Therefore, the authors of [12] provide a MPC for the non-linear energy management of the power split HEV. Energy efficiency management for HEVs is now also extended in communication among vehicles, as shown in [13], in which a MPC framework is proposed to generate the optimal torque and velocity by connecting the information from vehicle to vehicle. The authors of [14] reviewed the latest model-based controllers in the market to assess the improvement of energy management for HEVs, and, in this study, the MPC was used to calculate optimal energy, torque, and speed. Since a MPC is a model-based algorithm, difficulties will arise when there are existing mismatches between the model and the plant or the plant uncertainties. These mismatches and uncertainties may lead to the instability of the controller. Robust model predictive control (RMPC) algorithms are, therefore, developed to deal to these uncertainties.