# Analytical Investigation of Vertical Force Control in In-Wheel Motors for Enhanced Ride Comfort
## 4 Conclusions  
This study investigated the impact of vertical force control in IWMs on enhancing ride comfort in electric vehicles (EVs) using a dynamic vehicle model and a proportional ride-blending controller. The implementation and analysis of the system's frequency response demonstrated that the proportional ride-blending controller effectively reduces vertical vibrations of the sprung mass, thereby improving ride comfort. This research converted the state-space model into a transfer function to evaluate the system's response. Road profiles were generated according to ISO 8608 standards  
Figure 17: Time response of the front wheel force for the road profile classes C to D. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Type of Road Profile**} & **The Most Appropriate** & **Percentage Decrease in** \\  & **Value of \(K_{p}\)** & **Frequency-Weighted Acceleration** \\  & **(Compared with Open-Loop)** \\ \hline \(k=3\) (Class A to B) & 300 & 7.5756\% \\ \(k=4\) (Class B to C) & 150 & 3.9632\% \\ \(k=5\) (Class C to D) & 75 & 2.0119\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The summary of the appropriate value of \(K_{p}\) for each type of road profile. and converted into Power Spectral Density (PSD) inputs for simulation. The frequency-weighted acceleration (\(a_{w}\)) was calculated based on ISO 2631 standards to assess ride comfort improvements. The findings revealed that increasing the proportional gain (\(K_{p}\)) reduced both the frequency-weighted acceleration and the RMS of the vertical acceleration of the sprung mass. However, due to the torque limitations of the IWMs, the gain could not be increased indefinitely. In summary, this study provides a comprehensive analysis of using vertical force control in IWMs to enhance ride comfort in EVs. The results suggest that advanced suspension technologies, including the proportional ride-blending controller, can effectively address the challenges posed by increased unsprung mass in IWM vehicles, offering a promising approach to improve ride quality and vehicle dynamics. Conceptualization, S.C. and G.P. ; data curation, S.C. and G.P. ; formal analysis, C.B., S.C. and G.P. ; funding acquisition, G.P. ; investigation, S.C.; methodology, C.B. and S.C.; project administration, G.P. ; resources, G.P. ; software, C.B. ; supervision, S.C. and G.P. ; validation, C.B. ; visualization, C.B. and G.P. ; writing--original draft, C.B., S.C. and G.P. ; writing--review and editing, C.B., S.C. and G.P. All authors have read and agreed to the published version of the manuscript. This research project is supported by the Thailand Science Research and Innovation Fund of Chulalongkorn University [Estimation algorithms and control systems for electric vehicle] and the Ratchadaphieskomphot Endowment Fund of Chulalongkorn University, grant number CU_GI_67_004_2100_001. Not applicable. Any inquiry can be directly sent to the corresponding author. This research project is supported by the Second Century Fund (C2F), Chulalongkorn University, Bangkok, Thailand ([PERSON]). The authors declare no conflicts of interest. The state-space representation of the open-loop system according to Equation (13) is shown as follows:  
\[\dot{x}=A_{ol}x+B_{ol}u\]  
\[y=C_{ol}x+D_{ol}u\]  
where  
\[x= \begin{bmatrix}z_{b}&\dot{z}_{b}&\theta&\dot{\theta}\end{bmatrix} ^{T},\]  
\[A_{ol}= \begin{bmatrix}0&1&0&0\\ -\frac{k_{f}+k_{r}}{m}&-\frac{\varepsilon_{f}+c_{r}}{m}&\frac{\varepsilon_{ bf}-ck_{r}}{m}&\frac{b_{fc}-ck_{r}}{m}&\frac{b_{fc}-ck_{r}}{m}\\ 0&0&0&1\\ \frac{b_{fc}-ck_{r}}{l_{z}}&\frac{b_{fc}-cc_{r}}{l_{z}}&-\frac{b^{2}k_{f}+ \varepsilon_{b}k_{r}}{l_{z}}&-\frac{b^{2}\varepsilon_{f}+c^{2}c_{r}}{l_{z}} \end{bmatrix} ,\]  
\[B_{ol}= \begin{bmatrix}0&0&0\\ \frac{k_{f}}{m}&\frac{k_{r}}{m}&\frac{c_{f}}{m}&\frac{c_{r}}{m}\\ 0&0&0&0\\ -\frac{k_{f}}{l_{z}}&\frac{ck_{r}}{l_{z}}&-\frac{b_{fc}l_{z}}{l_{z}}&\frac{ cc_{r}}{l_{z}}\end{bmatrix} ,\]  
\[C_{ol}= \begin{bmatrix}0&1&0&0\end{bmatrix} ,\]  
\[D_{ol}= \begin{bmatrix}0&0&0&0\end{bmatrix}