## 3 Modeling Methodology
### MPC-Based CLC Control Strategy by Considering Preceding Vehicle Switching
#### 3.2.3 Objective Functions with Preceding Vehicle Switching of CLC Control Strategy  
(1) Longitudinal distance tracking control with preceding vehicle switching  
The longitudinal distance tracking error is described as the difference between the actual longitudinal distances between vehicles M, A, and their preceding vehicles and the expected longitudinal distance between these two vehicles. The expected longitudinal distance \(d_{exp,M}(k)\) and \(d_{exp,A}(k)\) are set according to the constant time headway rule. \[d_{exp,M}(k)=v_{M}(k)*t_{hd}+d_{0},\;k=1,\;2,\;\ldots,\;N \tag{23}\]  
Figure 5: The trajectory curve of linear piecewise function \(LPFB\). \[d_{exp,A}(k)=v_{A}(k)*t_{hd}+d_{0},\;k=1,\;2,\;\ldots,\;N \tag{24}\]  
In the LC-O stage, the longitudinal distance tracking errors between vehicles C and M, and B and A are computed using (25) and (26) as follows:  
\[ex_{M1}(k)=S_{CM}(k)-d_{exp,M}(k),\;k=1,\;2,\;\ldots,\;N \tag{25}\]  
\[ex_{A1}(k)=S_{BA}(k)-d_{exp,A}(k),\;k=1,\;2,\;\ldots,\;N \tag{26}\]  
where \(t_{hd}\) denotes the expected time headway, which is a predetermined value. In the LC-T stage, the longitudinal distance tracking errors between vehicles B and M, and M and A are calculated by using (27) and (28) as follows:  
\[ex_{M2}(k)=S_{BM}(k)-d_{exp,M}(k),\;k=1,\;2,\;\ldots,\;N \tag{27}\]  
\[ex_{A2}(k)=S_{MA}(k)-d_{exp,A}(k),\;k=1,\;2,\;\ldots,\;N \tag{28}\]  
The preceding vehicle switching problem during the lane-changing process should be considered not only in the constraints but also in the optimization objectives. Based on the ideas of the previous longitudinal distance constraints construction between vehicles C and M, and B and A, the linear piecewise function \(LPFB(*)\) is also applied to soften the change in the longitudinal distance tracking errors from the LC-O stage to LC-T stage when the preceding vehicle switches. It is evident from Figure 4 that this function can map the proportion of the switching of preceding vehicles to a range of \([0,1]\), and take half of the total lateral distance \(d/2\) as the watershed because of the two stages during the whole lane change. When the lateral distance is less than \(d/2\), i.e., in the LC-O stage, the proportion coefficient gradually grows from 0 to 1 with a slope of \(2/d\). On the contrary, when it is greater than \(d/2\), i.e., in the LC-T stage, the proportion coefficient remains unchanged at 1. Based on the aforementioned analysis, the longitudinal distance tracking errors of vehicles M and A are rewritten as (29) and (30). The linear piecewise function \(LPFB(*)\) is used to control the weight values of the two preceding vehicles that should be switched and tracked. \[\widetilde{ex}_{M}(k)=LPFB(k)ex_{M2}(k)+[1-LPFB(k)]ex_{M1}(k),\;k=1,2,\ldots,N \tag{29}\]  
\[\widetilde{ex}_{A}(k)=LPFB(k)ex_{A2}(k)+[1-LPFB(k)]ex_{A1}(k),\;k=1,2,\ldots,N \tag{30}\]  
where \(\widetilde{ex}_{M}(k)\) and \(\widetilde{ex}_{A}(k)\) represent the improved longitudinal distance tracking errors of vehicles M and its original and new preceding vehicles (i.e., vehicles C and B), and vehicles A and its original and new preceding vehicles (i.e., vehicles B and M), respectively. (2) Longitudinal speed tracking control with preceding vehicle switching  
In addition, vehicles M and A should track their expected longitudinal distances while tracking the longitudinal speeds of their preceding vehicles during the process of lane change. Therefore, longitudinal speed-tracking errors should also be considered. In the LC-O stage, the longitudinal speed tracking errors between vehicles C and M, and B and A are expressed as (31) and (32). \[ev_{M1}(k)=v_{M}(k)-v_{C}(k),k=1,2,\ldots,N \tag{31}\]  
Figure 5:0  
In the LC-T stage, the longitudinal speed tracking errors between vehicles B and M, and M and A are expressed by using (33) and (34).