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Chen et al. Complex Eng Syst 2023;3:8 I http://dx.doi.org/10.20517/ces.2022.50 Page 13 of 15

Figure 4. Torque and adhesion rate of each tire. ( = 0.9, = 100 km/h).

to PID control, and with both methods, velocity fluctuations occurred close to the reference line. However,
without control, the velocity dropped considerably. As shown inFigure 5B, the vehicle without control lost
stability and deviated from the designated trajectory. The hierarchical time-varying MPC control reduced
the maximum lateral displacement by approximately 0.2 m compared to that achieved with PID control. As
depicted in Figure 5C and D, under hierarchical time-varying MPC control, the yaw rate and sideslip angle
tracked the reference curves very well. The performance of PID control was slightly inferior in comparison,
while the case without control performed the worst and the vehicle diverged from the set trajectory. With
both PID control and hierarchical time-varying MPC control, the yaw rate control effect was stronger than the
sideslip angle control effect because the sideslip angle is more difficult to control than the yaw rate. However,
with hierarchical time-varying MPC control, the sideslip angle was less than 2.5 , which is within the safety
◦
limit.

The torque and adhesion rate of each tire are shown in Figure 6. According to Figure 6A, the torques acting
on the left front and rear tires are similar but not equal, and the torques acting on the right front and rear tires
are similar but not equal. However, the adhesion rates of the two left tires are almost equal, and the adhesion
rates of the two right tires are almost equal, as shown in Figure 6B. This finding indicates that the proposed
algorithm can secure vehicle safety and ensure that the adhesion rates of the two tires on the same side of the
vehicle are as close to each other as possible. Unlike on the high-adhesion road, the torques and adhesion rates
of each of the tires are lower, which is consistent with the actual situation.

6. CONCLUSIONS
In this study, 3DOF reference vehicle model and a 7DOF nonlinear vehicle model were developed. A novel hi-
erarchical time-varying MPC control strategy was proposed for 4WID EVs by considering vehicle stability and
adhesion efficiency. A time-varying MPC controller was designed to reduce system error in the linearization
process.

In the co-simulation, two typical conditions were adopted to demonstrate the performance of the proposed
method. The DLC maneuver was performed on high- and low-adhesion roads to verify the effectiveness of
the proposed control strategy. The results indicated that the proposed hierarchical time-varying MPC control
strategy was able to enhance vehicle handling stability effectively. Furthermore, the lower torque allocation
algorithm was able to improve the adhesion efficiency of each tire.