## 4 Experiments
### Results and Disscussion of Experiments  
**Experiment 1**. : _Vehicle Riding on a Rough Concave-Convex Road Profile._  
The rough concave-convex road profile is composed of a bump and a hollow with a sinusoidal disturbance, which is represented Equation (46). \[R_{r}(t)=\left\{ \begin{array}{ll}-1.554{(t-3.5)}^{3}+3.5{(t-3.5)}^{2}+d(t)& fort\in[3.5,5)\\ 1.554{(t-6.5)}^{3}+3.5{(t-6.5)}^{2}+d(t)&fort\in[5,6.5)\\ 1.554{(t-8.5)}^{3}-3.5{(t-8.5)}^{2}+d(t)&fort\in[8.5,10)\\ -1.554{(t-11.5)}^{3}-3.5{(t-11.5)}^{2}+d(t)&fort\in[10,11.5)\\ d(t)&else\end{array} \right. \tag{46}\]  
where the sinusoidal disturbance is \(d(t)=0.175\sin(2\pi t)+0.07\sin(7.5\pi t)\). Figure 10 shows the variation in the displacement of the sprung mass when the quarter car travels on a rough concave-convex road with immediate 3 cm changes. In Figure 10, the blue line shows that the road profiles that are generated by the Road Generator have a \(\pm 0.5\) cm error in the rough concave-convex road profile. The pink dotted line shows the displacement of the sprung mass, which is controlled by the MacPherson struts. Because no additional force is supplied from PM in this case, the maximum displacement of the sprung mass is 2.4 cm, thus, only a 0.6 cm-high vibration is suppressed. Figure 10 shows that the FSB-ASMC, the FSB-ASMC+\(H_{\infty}\) and the GP-FSB-ASMC+\(H_{\infty}\) better suppress vibration than MacPherson struts under the same conditions. The black line in Figure 10 shows that FSB-ASMC suppresses a vibration in the sprung mass of less than 1.2 cm. The green dashed line shows that the FSB-ASMC+\(H_{\infty}\) suppresses a maximum vibration of less than 0.65 cm because it attenuates lumped uncertainties. The GP-FSB-ASMC+\(H_{\infty}\) limits the maximum vibration to less than 0.4 cm, which is \begin{table}
\begin{tabular}{c|c} \hline \hline \multicolumn{2}{c}{**Initial Weights**} & **Control Parameters** \\ \hline \(W_{F}=\left[\begin{array}{cc}0.053,0.003,0.021,0.01,0.038,0.02,0.01,-0.007\\ -0.042,0.017,0.04,-0.01,0.014,0.025,0.01,0.05\end{array}\right]\) & \(c_{1}=3.2\) \\ \(W_{G}=\left[\begin{array}{cc}5.08,\ 0.01,\ 0.017,\ 0.01,\ 0.007,\ 0.013,\ 0.02,\ 0.011\\ 0.018,0.011,\ 0.007,\ 0.021,-0.003,\ 0.008,0.011,\ 0.013\end{array}\right]\) & \(\begin{array}{c}c_{2}=3.3\\ A_{21}=0.1563\\ A_{22}=0.8049\end{array}\) \\ \(W_{G}=\left[\begin{array}{cc}5.08,\ 0.01,\ 0.017,\ 0.01,\ 0.007,\ 0.013,\ 0.02,\ 0.011\\ 0.018,0.011,\ 0.007,\ 0.021,-0.003,\ 0.008,0.011,\ 0.013\end{array}\right]\) & \(\begin{array}{c}\sigma=0.5\\ \eta_{1}=1\\ \eta_{2}=0.01\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the proposed method for a rough concave-convex road profile. Figure 9: Fuzzy partitions and membership functions for the error, the change in error and the fuzzy output. better than either the FSB-ASMC or the FSB-ASMC+\(H_{\infty}\). Table 2 compares the maximum vibration and the root-mean-square error (RMSE) for the vibration, where the RMSE is defined as:  
\[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(value_{vibration}-value_{zero\_level} \right)^{2}} \tag{47}\]  
where \(value_{vibration}\) is the measured vibration, \(value_{zero\_level}\) is the zero displacement of the sprung mass and \(N\) is the total number of the measured vibration data points. Table 3 shows the weights after training for the GP-FSB-ASMC+\(H_{\infty}\). In Figure 10, there is a slight oscillation on the sprung mass because of tire deformation, which is a small variation that depends on the road profile. Using the FLC, the variation in the output of the road generator is shown as a blue line in Figure 10, for which the tracking error is limited to \(\pm 0.5\) cm. The acceleration and the control voltage and the sprung mass for all methods are respectively shown in Figures 11 and 12. It is noted that no additional force is applied to MacPherson struts, thus, Figure 12 shows no control voltage for them. Figure 11: Acceleration of the sprung mass for the rough concave-convex road profile.