# Adaptive Cruise Control for Cut-In Scenarios Based on Model Predictive Control Algorithm
## 4 Host Vehicle Trajectory Tracking Control
### Inter-Vehicle Longitudinal Kinematics Model  
In this paper, constant time headway (CTH) is adopted for the MPC-ACC. In the CTH strategy, the time headway is a fixed value, and the inter-vehicle distance is proportional to the host vehicle's speed. In order to ensure safety, the minimum safe distance is considered in the strategy, as shown in the following equation:  
\[\Delta x_{des}=t_{h}v_{H}+x_{0} \tag{31}\]  
where \(\Delta x_{des}\) is the desired inter-vehicle distance, \(t_{h}\) is the time headway and \(x_{0}\) is minimum safe distance added to ensure safety, which is constant. When the host vehicle follows the preceding vehicle in a stable state, the trajectory of the preceding vehicle needs to be used as the reference trajectory of the MPC-ACC. At the same time, to consider the safety of following, the reference trajectory of the host vehicle is obtained:  
\[x_{ref}(k)=x_{H}(k)-\Delta x_{des} \tag{32}\]  
where \(x_{ref}(k)\) is the longitudinal reference trajectory of the host vehicle and \(x_{H}(k)\) is the host vehicle's coordinates. According to the inter-vehicle longitudinal kinematics, as shown in Figure 19, the longitudinal kinematics model of the vehicles can be obtained, as shown using the following equations  
\[\Delta x(k+1)=\Delta x(k)+v_{rel}(k)T_{s}+\frac{1}{2}(a_{P}(k)-a_{H}(k))T_{s}^ {2} \tag{33}\]  
\[v_{rel}(k+1)=v_{rel}(k)+[a_{P}(k)-a_{H}(k)]T_{s} \tag{34}\]  
\[v_{H}(k+1)=v_{H}(k)+a_{H}(k)T_{s} \tag{35}\]  
where \(a_{H}(k)\), \(a_{P}(k)\) is acceleration of the host vehicle and the preceding vehicle, respectively, \(v_{rel}(k)\) is the relative velocity, which satisfies the equation \(v_{rel}(k)=v_{H}(k)-v_{P}(k)\), where is \(v_{H}\) the host vehicle velocity and \(v_{P}\) is the preceding vehicle's relative distance, which satisfies \(\Delta x(k)=x_{P}(k)-x_{H}(k)\), \(x_{P}(k)\) is the preceding vehicle's coordinates, and \(T_{s}\) is the sampling time. In ACC, a hierarchical control is usually adopted. Because of the foundation of lower-level control, the vehicle cannot usually immediately reach a desired acceleration output using the upper-level control. Therefore, the following equations is used to reflect the delay characteristics of the acceleration control response [27]. \[a_{H}(k+1)=(1-\frac{T_{s}}{\tau})a_{H}(k)+\frac{T_{s}}{\tau}u(k) \tag{36}\]  
\[Jerk_{P}(k+1)=-\frac{1}{\tau}a_{P}(k)+\frac{1}{\tau}u(k) \tag{37}\]  
where \(u(k)\) is the desired acceleration output by the upper control system and \(Jerk_{H}\) is the jerk of the host vehicle. Select relative distance, relative vehicle velocity, host vehicle velocity, host vehicle acceleration, and the host vehicle jerk as the state quantities of the longitudinal kinematics model:  
\[x(k)=[\Delta x(k),v_{rd}(k),v_{H}(k),a_{H}(k),Jerk_{H}(k)]^{T} \tag{38}\]  
Select the relative distance and the relative velocity as the output vector of the longitudinal kinematics model:  
\[y_{c}(k)=[\Delta x(k),v_{rel}(k)]^{T} \tag{39}\]  
The acceleration of the preceding vehicle is regarded as a disturbance of the control system, so the state equation of the control system can be obtained:  
\[x(k+1)=\mathbf{A}\mathbf{x}(k)+\mathbf{B}u(k)+\mathbf{G}w(k) \tag{40a}\]  
\[x_{ref}(k)=x_{H}(k)-\Delta x_{des} \tag{32}\]0  
where,  
\[\mathbf{A}=\left[ \begin{array}{cccc}1&T_{s}&0&-T_{s}^{2}/2&0\\ 0&1&0&-T_{s}&0\\ 0&0&1&0&0\\ 0&0&0&1-T_{s}/\tau&0\\ 0&0&0&-1/\tau&0\end{array} \right]\mathbf{,}\mathbf{B}=\left[ \begin{array}{c}0\\ 0\\ 0\\ T_{s}/\tau\\ 1/\tau\end{array} \right]\]  
\[\mathbf{G}=\left[ \begin{array}{c}T_{s}^{2}/2\\ T_{s}\\ T_{s}\\ 0\\ 0\end{array} \right]\mathbf{,}\mathbf{C}=\left[ \begin{array}{c}1&0\\ 0&1\\ 0&0\\ 0&0\\ 0&0\end{array} \right]^{T}\mathbf{,}\mathbf{Z}=\left[ \begin{array}{c}0\\ 0\end{array} \right]\]  
Equations (40a) and (40b) is the longitudinal kinematics control model of the ACC system.