## 4 Frequency Response of Vehicle Handling Dynamics  
This section shows a theoretical analysis of the effect of the DYC input on the vehicle lateral motion compared to the case without control for the loaded vehicle. As the baseline reference, the frequency response of the vehicle lateral motion for the vehicle without loading will also be shown to confirm the performance of the controller to maintain the handling dynamics of the vehicle. By substituting the equation of the DYC control law into the equations of lateral and yaw motions and then using Laplace transformation with zero initial condition, the transfer function from the front steering angle to the yaw rate, the body side slip angle, and the lateral acceleration, in the case of the vehicle equipped with the proposed DYC system, as indicated in the previous section, can be calculated as the following equations.  
\[\frac{r(s)}{\delta_{f}(s)}=\frac{A_{2}s^{2}+A_{1}s+A_{0}}{(T_{FF}s+1)\{s^{2}+( -a_{11}-a_{22}-hk_{r})s+a_{11}(a_{22}+hk_{r})-a_{21}a_{12}\}}, \tag{33}\]  
\[\frac{\beta\left(s\right)}{\delta_{f}(s)}=\frac{B_{2}s^{2}+B_{1}s+B_{0}}{(T_{ FF}s+1)\{s^{2}+(-a_{11}-a_{22}-hk_{r})s+a_{11}(a_{22}+hk_{r})-a_{21}a_{12}\}}, \tag{34}\]  
\[\frac{a_{y}\left(s\right)}{\delta_{f}(s)}=V\left(s\cdot\frac{\beta\left(s \right)}{\delta_{f}(s)}+\frac{r\left(s\right)}{\delta_{f}(s)}\right) \tag{35}\]  
where each coefficient can be described as follows  
\[A_{0}=a_{21}b_{11}-a_{11}b_{21},\] \[A_{0}=a_{21}b_{11}-a_{11}b_{21},\] \[A_{1}=b_{21}+(a_{21}b_{11}-a_{11}b_{21})T_{FF}-a_{11}hK_{FF},\] \[A_{2}=b_{21}T_{FF}+hK_{FF},\] \[B_{0}=a_{12}b_{21}-(a_{22}+hk_{r})b_{11},\] \[B_{1}=b_{11}+\{a_{12}b_{21}-(a_{22}+hk_{r})b_{11}\}T_{FF}+a_{12} hK_{FF},\] \[B_{2}=b_{11}T_{FF}.\]  
Here, if we set the values of feed-forward and feedback compensators \(K_{FF}\), \(T_{FF}\), and \(k_{r}\) to zero, then the transfer function of the case without control can be achieved. By substituting the key vehicle dynamics parameters in the case of unloaded vehicle and the loaded vehicle, the transfer function of lateral vehicle dynamics can be obtained. As an example of theoretical analysis on the controller performance, the constant vehicle velocity is set to 80 km/h and the loading condition in the rear is 80 kg. Under this condition, the frequency responses of the yaw rate, the body side slip angle, and the lateral acceleration, with respect to the front steering angle of three cases, (1. Unloaded/without DYC, 2. Loaded/without DYC, and 3. Loaded/with DYC) are shown in Figures 7-9, respectively.  
Figure 8: Frequency response of side slip angle with respect to front steering angle.  
Figure 7: Frequency response of yaw rate with respect to front steering angle.  
As can be noticed from the frequency response plots shown in Figures 7-9, in the case of the loaded vehicle without direct yaw moment control, the phase delay of the vehicle response is larger than the other cases. This corresponds to bad responsiveness of the vehicle motion with respect to the steering maneuver. Next, in the low frequency region, the steady-state gain of the vehicle response in the case of the loaded vehicle without direct yaw moment control is larger when compared with the case of the unloaded vehicle. This means that the sensitivity of vehicle motion to the steering angle is getting higher, which may lead to unstable vehicle motion in high speed region. In contrast, by applying direct yaw moment control to the loaded vehicle, the phase delay problem can be improved, and the excessive vehicle response gain to the steering angle can also be effectively compensated by the proposed DYC. In addition, it is verified that the proposed DYC can maintain the vehicle handling dynamics to be insensitive to the loading condition as it still makes the response of the loaded vehicle close to that of the unloaded vehicle.