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Page 12 of 15 Chen et al. Complex Eng Syst 2023;3:8 I http://dx.doi.org/10.20517/ces.2022.50

Figure 3. State comparison of different control modes during DLC maneuver. ( = 0.9, = 100 km/h).

5.1. DLC maneuver on high-adhesion road
The adhesion coefficient on the high-adhesion road was set to 0.9, and the reference vehicle velocity was set
to 100 km/h. Figure 3. depicts a comparison of the typical state parameters by using different control meth-
ods, which are without active control, hierarchical time-varying MPC control, and PID control. According
to Figure 3A, the velocity fluctuation due to the proposed hierarchical time-varying MPC control was smaller
than that due to PID control. As shown in Figure 3B, compared to the without active control and PID control
methods, the hierarchical time-varying MPC control method decreased the maximum lateral displacement
by approximately 0.25 m and 0.11 m, respectively. The yaw rate and sideslip angle of the vehicle under MPC
control were able to follow the ideal curve furthest, which effectively enhanced vehicle handling stability and
safety, as depicted in Figure 3C and D.

The adhesion rate can be expressed as follows:

= / (33)
where = or , = or , and denotes front left, denotes front right, denotes rear left, and denotes
rear right.

The torque and adhesion rate of each tire are depicted in Figure 4. According to Figure 4A, the torque acting
on each tire changed gently. Moreover, the torques acting on the left front and rear tires were similar but not
equal. Likewise, the torques acting on the right front and rear tires were similar but not equal. However, the
adhesion rates of the left two tires were almost equal, and the adhesion rates of the two right tires were almost
equal, as depicted in Figure 4B. This indicates that the proposed algorithm can enhance vehicle safety and
ensure that the adhesion rates of the two tires on the same side are as similar as possible.

5.2. DLC maneuver on low-adhesion road
Generally, a low-adhesion road can reflect the control effect more remarkably. The adhesion coefficient on a
low-adhesionroadsandthereferencevehiclevelocityweresetto0.3and70km/hinthisstudy. Asillustratedin
Figure 5A, the velocity fluctuation due to the hierarchical time-varying MPC control was smaller than that due