# Adaptive Cruise Control Based on Model Predictive Control with Constraints Softening
## 5 Case Studies
The velocity of the host vehicle with MO-ACC ranges between 21.98 m/s and 29.67 m/s and its acceleration ranges between -4.56 m/s\({}^{2}\) and 2.05 m/s\({}^{2}\) in response to the preceding vehicle; while the velocity of the host vehicle with ACC-Soften ranges between 21.07 m/s to 29.48 m/s and its acceleration ranges between -3.35m/s\({}^{2}\) and 2.57m/s\({}^{2}\) in response to the preceding vehicle. Compared with ACC-Soften, the variation range in terms of acceleration with MO-ACC shows that the host vehicle accelerates and decelerates more frequently during the first 40 s.  
Figure 3e indicates that the maximum jerk for MO-ACC exceeds 3.70 m/s\({}^{3}\). From Equations (3)-(5), the jerk can be determined by the actual acceleration and the desired acceleration. The controller is a process of optimizing and the rolling process. At the time interval around 45 s, the simulated actual acceleration for ACC-soften closely approximates a constant value, as can be seen in the Figure 3d. While the simulated desired acceleration for ACC-soften is not a constant value, as can be seen in the Figure 3f. According to Equation (5), jerk is determined by the actual acceleration and the desired acceleration and the jerk is not zero around 45 s. Researches indicate that the maximum absolute value of jerk suitable for passengers is 2.0 m/s\({}^{3}\). Otherwise, the passengers will experience discomfort [25]. The average jerk values of ACC-Soften and MO-ACC are \(-\)0.2455 m/s\({}^{3}\) and \(-\)0.2648 m/s\({}^{3}\), respectively, which indicates that the ride comfort of ACC-Soften is 7.2% better compared with that of MO-ACC. During the emergent brake stage of the preceding vehicle for the last 10 seconds, both ACC-Soften and MO-ACC use larger braking acceleration to avoid collisions; whilst at the same time, comfort and economy are also guaranteed to a certain extent. The simulation results show that the driving comfort and the economy are drastically improved under the premise of ensuring safety and car following, at the same time, collision can be avoided with maximum braking acceleration when the preceding vehicle is in emergency braking. The acceleration of the vehicle is closely related to the economic efficiency [26; 27]. According to Figure 3d and Table 1, the average acceleration of ACC-Soften is \(-\)0.4849 m/s\({}^{2}\), with a standard deviation of 2.024 m/s\({}^{2}\) and range of 8.853 m/s\({}^{2}\). The average acceleration of MO-ACC is \(-\)0.5072 m/s\({}^{2}\), with a standard deviation of 2.056 m/s\({}^{2}\) and range of 7.576 m/s\({}^{2}\). Compared with MO-ACC, ACC-Soften has a lower acceleration standard deviation, without severe acceleration fluctuations. The economic efficiency of ACC-Soften is increased by 4.4% compared with that of MO-ACC. Figure 4 indicates that the average deviation between the actual inter-distance and the target inter-distance for MO-ACC is 2.698 m with a standard deviation of 3.757 m. The average deviation between the actual inter-distance and the target inter-distance for ACC-Soften is 1.116 m with a standard deviation of 2.536 m. This result shows that ACC-Soften can track the desired inter-distance better than MO-ACC. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Strategy** & **Average (m/s\({}^{2}\))** & **Standard Deviation (m/s\({}^{2}\))** & **Range (m/s\({}^{2}\))** \\ \hline MO-ACC & \(-\)0.5072 & 2.056 & 7.576 \\ ACC-Soften & \(-\)0.4849 & 2.024 & 8.853 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics of the acceleration of the host vehicle.