## 4 Simulations and Results Analysis
### Step Steering Maneuver  
Firstly, the nominal performance of the 4 WS stability control system was studied. The step responses of the front wheel angle input were studied to verify the nominal performance of the 4 WS vehicle. Suppose the 4 WS vehicle runs at a velocity 90 km/h, and the road friction coefficient is set as 0.85 (dry road). Figure 15 shows the input signal of the front wheel angle. Figure 16 depicts the sideslip angle response and the yaw rate response among the H\(\infty\) controller, the proportional controller, and the H2/H\(\infty\) controller. As can be seen in Figure 16a, the sideslip angle with the proportional controller had large overshoot, and the sideslip angle with the H\(\infty\) controller presented certain fluctuations. Compared with the H\(\infty\) controller and the proportional controller, the H2/H\(\infty\) controller made the sideslip angle response more stable. As shown in Figure 16b, when the 4 WS vehicle was equipped with the H2/H\(\infty\) and H\(\infty\) controllers, the steady state errors between the yaw rate and the desired yaw rate were almost zero, while the steady state error of the vehicle equipped with the proportional controller was 0.02 rad/s. Compared with the H\(\infty\) controller, the yaw rate of the 4 WS vehicle with the H2/H\(\infty\) controller responses was quicker and more stable. It can be concluded that the H2/H\(\infty\) controller outperformed the H\(\infty\) controller and the proportional controller in terms of yaw rate tracking and path tracking. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(m\) (kg) & 5950 \\ \(l_{z}\) (kg\(\cdot\)m\({}^{2}\)) & 8600 \\ \(l_{f}\) (m) & 1.2 \\ \(l_{r}\) (m) & 1.8 \\ \(C_{f}\) (N/rad) & 55,000 \\ \(C_{r}\) (N/rad) & 45,000 \\ \(K_{r}\) & 1.4 \\ \(K_{m}\) (N/A) & 130 \\ \(K_{1}\) (N/m) & 3000 \\ \(w_{0}\) (rad/s) & 79.5 \\ \(\varepsilon_{0}\) & 0.21 \\ \(K_{\mathrm{s}pl}\) (m\({}^{2}\)/s) & 1.35 \\ \(A_{\mathrm{C}}\) (m\({}^{2}\)) & 1.129 \(\times\) 10\({}^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The parameters of the 4 WS vehicle. Figure 15: The front wheel angle step input. Secondly, in order to demonstrate the disturbances resistance capability of the designed controller, the side wind and yaw rate sensor noise were taken under the step input of the front wheel angle mentioned above. A 15 m/s side wind (_Level_ 7 : 13.9 m/s \(\leq V_{wind}\leq\) 17.1 m/s) was considered to simulate the external interferences, as shown in Figure 17. According to the aerodynamic theory, the lateral interference force and the yaw interference torque can be expressed as follows [37]. \[ \begin{array}{l}f=\frac{1}{2}C_{fx}\rho AV_{a}{}^{2}\\ T_{f}=\frac{1}{2}C_{Mz}\rho ALV_{a}{}^{2}\\ V_{a}{}^{2}=V^{2}+V_{wind}{}^{2}\end{array}  \tag{38}\]  
where \(C_{fx}\) is the lateral force coefficient; \(C_{Mz}\) denotes the yaw torque coefficient; \(\rho\) is the air density; \(V_{a}\) represents the flow rate of the synthetic airflow; \(V_{wind}\) is the wind velocity. The sensitivity to the given disturbances is defined as the average absolute relative error between the yaw rate or sideslip angle without disturbances and that with disturbances, which can be expressed as follows. \[\lambda_{sen}=\sum_{k=1}^{n}\frac{|y_{with}(k)-y_{without}(k)|}{|y_{without}( k)|}/n \tag{39}\]  
where \(y_{without}(k)\) denotes the yaw rate or sideslip angle without disturbances at the \(k\)th moment; \(y_{with}(k)\) is the yaw rate or sideslip angle with disturbances at the kth moment; \(n\) represents the numbers of the yaw rate or sideslip angle in the disturbance periods. Figure 16: The step responses of the front wheel angle: (**a**) sideslip angle response; (**b**) yaw rate response. When the sensitivity \(\lambda_{sen}\) to the given disturbances is greater than its threshold \(\lambda_{max}\), the 4 WS vehicle equipped with the designed controller is vulnerable to disturbances; otherwise, the proposed controller is robust against disturbances.