## 2 Car-Following Dynamics Model
### Model Predictive Control Framework  
To incorporate control increments and enable reference tracking, the system is augmented by defining the extended state vector:\[\kappa_{k}=\begin{bmatrix}x_{k}\\ u_{k-1}\end{bmatrix}. \tag{11}\]  
The tracking error at each time step is then provided by the following:  
\[e_{k}=y_{\text{ref},k}-y_{k}=y_{\text{ref},k}-Cx_{k}. \tag{12}\]  
To track both relative distance and velocity, the output matrix \(C\) is defined as follows:  
\[C=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\end{bmatrix}. \tag{13}\]  
The objective of the MPC is to minimize the cumulative tracking error and control effort over a prediction horizon \(N_{p}\). The cost function is expressed as follows:  
\[J=\sum_{k=0}^{N_{p}-1}\left(e_{k}^{\top}Qe_{k}+\Delta u_{k}^{\top}R\Delta u_{ k}\right)+\rho e^{2}. \tag{14}\]  
Here, \(Q\succeq 0\) and \(R\succ 0\) are the weighting matrices for state errors and control increments, respectively. The slack variable \(\varepsilon\) allows constraint relaxation, with \(\rho>0\) being its associated penalty.  
The optimization is subject to the following constraints:  
\[u_{\text{min}} \leq u_{k}\leq u_{\text{max}}, \tag{15}\] \[\Delta u_{\text{min}} \leq\Delta u_{k}\leq\Delta u_{\text{max}},\] (16) \[y_{\text{min}} \leq y_{k}\leq y_{\text{max}}. \tag{17}\]  
These constraints ensure safety, comfort, and actuator feasibility during control execution.  
The constrained optimization problem is cast into the standard quadratic programming (QP) form:  
\[\min_{\Delta U}\quad\frac{1}{2}\Delta U^{\top}H\Delta U+f^{\top}\Delta U, \tag{18}\]  
where \(\Delta U\) is the stacked vector of control increments over the prediction horizon. The matrices \(H\) and \(f\) are derived from the system dynamics and the cost function. The QP is solved using MATLAB's quadprog solver, and only the first element of \(\Delta U\) is applied in each sampling period, following the receding horizon principle.  
The weighting matrices \(Q\) and \(R\) define the relative importance of each control objective. \(Q\in\mathbb{R}^{3\times 3}\) is diagonal and positive semi-definite:  
\[Q=\begin{bmatrix}\alpha_{1}&0&0\\ 0&\alpha_{2}&0\\ 0&0&\alpha_{3}\end{bmatrix},\quad R=\beta. \tag{19}\]  
Here, \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) correspond to the weights for relative distance \(\Delta d\), relative velocity \(\Delta v\), and acceleration \(a_{f}\), respectively. Larger values indicate higher priority in minimizing the associated state error.  
The weight configuration of the fixed-MPC controller in this study is devised with reference to the multi-objective control structure presented in [22], which outlines a representative MPC framework for adaptive cruise control in electric vehicles. Drawing inspiration from its design philosophy, we assign weighting coefficients of 100, 1.0, and 1.0 to the spacing error, velocity error, and acceleration terms, respectively, thereby prioritizing safety while ensuring smooth and responsive control behavior. Additionally, a control increment penalty of 1000 is applied to discourage excessive actuation. This configuration is adopted as a baseline for comparative analysis, offering a balanced representation of conventional fixed-weight control strategies.  
\[\alpha_{1}=100,\quad\alpha_{2}=1.0,\quad\alpha_{3}=1.0,\quad\beta=1000. \tag{20}\]  
This parameter set provides a balanced trade-off among tracking accuracy, smoothness, and constraint satisfaction and has demonstrated stable performance under standard conditions such as the WLTC cycle.