## 3 Modeling Methodology
### MPC-Based CLC Control Strategy by Considering Preceding Vehicle Switching
#### 3.2.2 Constraints with Preceding Vehicle Switching of CLC Control Strategy
Let us consider vehicle A as an example; if the instantaneous constraint switching is applied, the longitudinal safety constraint between vehicles M and A is only satisfied during the LC-T stage but not during the LC-O stage. Consequently, vehicle A is unable to provide a safe lane-changing gap for vehicle M to change lanes at the beginning of the LC-T stage. On the other hand, the sudden switching of constraints may also cause drastic speed changes, thus affecting comfort. The longitudinal safety constraint between vehicles M and A should change gradually with the lateral movement of vehicle M. Therefore, a suitable preceding vehicle switching method is needed to cope with the above problems. Studies have been conducted to apply switching control methods to connected and automated vehicle platoon control and showed superior performance in cooperative driving of heterogeneous nonlinear vehicle platoons [42]. Inspired by the above study, to ensure both the traffic efficiency and the applicability of the proposed CLC control strategy, two linear piecewise functions are designed, as presented in (11) and (12). These functions are integrated in the longitudinal safety constraints of LC-O and LC-T, as presented in (13)-(14) and (15)-(16), respectively. As a result, the constraints can be updated smoothly with the lateral motion of vehicle M.  
\[LPFA(k)=\left\{ \begin{array}{ll}L_{1}&if0\leq y_{M}(k)\leq d/2\\ L_{1}-L_{2}[y_{M}(k)-d/2]&otherwise\end{array} \right.,\ k=1,2,\ldots,N \tag{11}\]  
\[LPFB(k)=\left\{ \begin{array}{ll}L_{3}y_{M}(k)&if\ 0\leq y_{M}(k)\leq d/2\\ L_{4}&otherwise\end{array} \right.,\ k=1,2,\ldots,N \tag{12}\]  
During the LC-O stage, we have:  
\[S_{CM}(k)\geq S_{safe,CM}(k)LPFA(k),\ k=1,2,\ldots,N \tag{13}\]  
\[S_{BA}(k)\geq S_{safe,BA}(k)LPFA(k),\ k=1,2,\ldots,N \tag{14}\]  
During the LC-T stage, we have:  
\[S_{BM}(k)\geq S_{safe,BM}(k)LPFB(k),\ k=1,2,\ldots,N \tag{15}\]  
\[S_{MA}(k)\geq S_{safe,MA}(k)LPFB(k),\ k=1,2,\ldots,N \tag{16}\]  
where \(y_{M}(k)\) denotes the actual lateral distance of vehicle M from the centerline of the original lane. \(LPFA(k)\) is a function that first maintains a constant value and then decreases with a fast speed to a large negative value with the variation of the lateral position of vehicle M. In \(LPFA(*)\), \(L_{1}\) and \(L_{2}\) denote the control parameters, \(L_{1}\) is a positive value equal to 1 and \(L_{2}\) is a large negative value. \(LPFB(*)\) is a function that first increases and then maintains a constant value. Its value rises gradually from 0 to 1 and then remains 1 with the variation of the lateral position of vehicle M. \(L_{3}\) and \(L_{4}\) denote the control parameters of \(LPFB(*)\), which are positive values and equal to \(2/d\) and 1, respectively. The shape of \(LPFA(*)\) is presented in Figure 4. It changes with the lateral position of vehicle M, i.e., \(y_{M}(k)\). In the LC-O stage, i.e., \(0\leq y_{M}(k)\leq d/2\), the value of the function that \(LPFA(*)\) maintains 1. In the LC-T stage, i.e., \(y_{M}(k)>d/2\), the value of \(LPFA(*)\) decreases with a fast speed to a large negative value. In order to explain the physical meaning of introducing a linear piecewise function, i.e., \(LPFA(*)\), in the longitudinal safety distance constraint, the longitudinal distance constraint between vehicles C and M, presented in (13) is illustrated as an example. By introducing \(LPFA(*)\), the right side of inequality (13), i.e., \(S_{safe,CM}(k)LPFA(k)\), maintains \(S_{safe,CM}(k)\) in the LC-O stage, thus indicating that the longitudinal distance constraint between vehicles C and M should always be held in this stage. This physically shows that vehicle C always has an impact on the longitudinal motion of vehicle M in this stage. In the LC-T stage, the right side of inequality \(S_{safe,CM}(k)LPFA(k)\) decreases rapidly from \(S_{safe,CM}(k)\) to a large negative value.