## 2 The Design of the Control System
### Model Predictive Control Considering Driving Style and Slope  
Model predictive control is a feedback control algorithm that solves a multi-objective optimization problem in the predictive time domain in real time online based on a dynamic mathematical model of the system, and then calculates the most appropriate control action and achieves control of the system through continuous iterative operations. The algorithm has the advantages of generalization, immunity to disturbances, robustness, and excellent dynamic performance, without the need for an exact model. Therefore, model predictive control can be a good solution to the problems of model mismatch, time lag, and non-linearity caused by the complex environment of the vehicle. The basic principle of model predictive control is to predict the future output of the system based on the existing model, the current state of the system, and the future control quantities, and to achieve the control purpose by solving the constrained optimization problem on a rolling basis, with three basic properties: predictive model, feedback correction, and rolling optimization. Due to the time lag in the braking system, the engine system, and the acquisition and processing of sensor signals, there is a delay between the actual acceleration and the desired acceleration during the longitudinal motion of the car. In order to track the desired acceleration more accurately, the tracking of the car's acceleration can be considered  
Figure 7: Design surface diagrams of fuzzy rules. as a first-order inertial link and incorporated into the control system, whose transfer characteristics can be expressed as Equation (9). \[G(s)=\frac{K}{\tau\cdot s+1} \tag{9}\]  
where, \(K\) is the link gain. In model predictive control based on driving style, the influence of driving style constraints on the state and control quantities of the system needs to be considered. In this study, the driving style constraints are mainly manifested as the effects of acceleration and acceleration rate of change, with different constraints for the acceleration and deceleration conditions, respectively. For an aggressive driving style, drivers will use greater throttle opening, faster acceleration, and a higher rate of change in acceleration when accelerating, higher deceleration, and a wider opening of the brake pedal when decelerating, while normal and calm drivers will be more cautious, so the constraints on acceleration and the rate of change in acceleration are shown in Equations (10)-(12). \[u_{min}\leq u(k+i)\leq u_{max},i=0,\ldots,\ N_{c}-1 \tag{10}\]  
\[\alpha\Delta u_{min,ac}\leq\Delta u_{ac}(k+i)\leq\alpha\Delta u_{max,ac},i=0, \ldots,N_{c}-1 \tag{11}\]  
\[\alpha\Delta u_{min,de}\leq\Delta u_{de}(k+i)\leq\alpha\Delta u_{max,de},i=0, \ldots,N_{c}-1 \tag{12}\]  
In the above equations, \(U\) is the acceleration constraint and the lower two rows are the acceleration change constraint on acceleration and the acceleration constraint on deceleration, respectively. For the effect of driving style, \(\alpha=1.2\) for aggressive drivers, \(\alpha=1\) for moderate drivers and \(\alpha=0.8\) for calm drivers, as below. Equations (13) and (14) were used to make decisions about their rate of change in acceleration. \[\Delta u_{min}\leq\Delta u(k+i)\leq\Delta u_{max},i=0,\ldots,\ N_{c}-1 \tag{13}\]  
\[\Delta u_{max}=\left\{ \begin{matrix}\alpha\Delta u_{max,ac}&\text{if }a_{des}(k-1)\geq 0 \\ min(\alpha\Delta u_{max,de},\alpha\Delta u_{max,ac}-a_{des}(k-1))&\text{if }a_{des}(k-1)<0 \end{matrix} \right. \tag{14}\]  
In this section, according to the aforementioned adaptive model predictive control with driving style recognition and fuzzy rule construction, an adaptive model predictive control model considering driving style and slope was constructed using Carsim 2019.1 and MATLAB/Simulink 2018a. The model built in this paper was validated by setting different road conditions of slope and real driving conditions.