## 4 Simulation Results
### Nominal Road Friction Coefficient of 0.7  
The simulation results with a medium road friction coefficient of 0.7, which is 22% less than the nominal value, are shown in Figures 9-13. Assuming the AES is not aware of the overestimation of the road friction coefficient, the nominal value of 0.9 is still used to design the evasive path and the nominal cornering stiffnesses are still used in the prediction model.  
Path tracking responses are shown in Figure 9. The AES is assumed to be triggered at 3 s. SISO MPC cannot complete the lane change maneuver and loses the directional stability. Conversely, MIMO MPC can track the desired evasive path and complete the lane change maneuver successfully. Similar to the analysis of Figure 8 in Section 4.1, we found that MIMO MPC has a lane change response faster than that of SISO MPC at the beginning of lane change, as shown in Figure 10. Meanwhile, the peak heading angle error of MIMO MPC is smaller than that of SISO MPC at about 4 s due to minimizing \(y_{e}\) and \(\psi_{e}\) simultaneously via both steering and DYM. From the yaw rate response, we can observe that the vehicle with SISO MPC starts to spin continuously after 4 s.  
As can be seen from Figure 11, SISO MPC requires steering angles much larger than those of MIMO MPC after 4 s. Although SISO MPC tries to correct the steering angle as fast as possible, the steering angle response is not fast enough due to the angular rate limitation and actuator dynamics. Figure 12 shows that the side-slip angle of SISO MPC increases rapidly with large yaw rate after passing \(\beta=0\). It also confirms the directional instability of SISO MPC. On the other hand, the \(\beta-r\) phase plane plot of MIMO MPC converges to the origin after completing the lane change maneuver.  
Figures 13 and 14 show the responses of tire slip angle and lateral tire forces at four wheels. The dashed lines indicate the thresholds of the tire slip angles to saturate lateral  
Figure 8: \(\beta-r\) phase plane plots for \(\mu=0.9\).  
Figure 7: Actuator responses for \(\mu=0.9\).  
tire forces. The front lateral tire forces of SISO MPC are found to be saturated at 4.1 s. Even though the steering angle of SISO MPC in Figure 11 is still increasing after 4 s, the corresponding lateral tire forces are not increased due to saturation as shown in Figure 14. Since the lateral tire force is no longer increased with the increment of the tire slip angle, the prediction model used in MPC is not accurate anymore. Thus, SISO MPC cannot effectively control the vehicle to track the desired evasive path and prevent directional instability. Although the rear tire slip angles of MIMO MPC are saturated around 4.5 s, the front tire slip angles are not saturated. Thus, the front tires can still provide more lateral tire forces via increasing the steering angle input if necessary. Meanwhile, MIMO MPC has additional DYM which can improve the responses of \(r\) and prevent saturating the reference command of the steering angle. Figure 15 shows the responses of tire workload ratios. The CA distributes the longitudinal tire forces as shown in Figure 16 to the wheels with lower tire workload ratios in order to generate the required DYM without saturating the tire forces; thus, demonstrating better, more robust control performance and handling enhancement against model uncertainty due to reduced road friction coefficient.  
Figure 10: Responses of vehicle dynamics for \(\mu=0.7\).  
Figure 9: Path tracking responses for \(\mu=0.7\).  
Figure 11: Actuator responses for \(\mu=0.7\).  
Figure 12: \(\beta-r\) phase plane comparisons for \(\mu=0.7\).  
Figure 13: Responses of tire slip angles for \(\mu=0.7\).  
Figure 16: Responses of longitudinal tire forces for \(\mu=0.7\).  
Figure 14: Responses of lateral tire forces for \(\mu=0.7\).  
Figure 15: Responses of tire workload ratios for \(\mu=0.7\).