## 4 Design of Disturbance Estimator
### Formulation of DE-FSMC
For the bump-road profile, \(C_{xy}\) between the chassis vertical acceleration signal and the excitation signal coming from the road surface is shown in Figure 7. The maximum values (\(C_{xy}^{\max}\)) and the corresponding frequency (\(f_{C_{xy}^{\max}}\)) related to each method are given in Table 4. The obtained results indicate that the minimum value of \(C_{xy}^{\max}\) is 0.1269 at the frequency of 4.28 Hz in the proposed method. Besides, \(f_{C_{xy}^{\max}}\) deriving from the surveyed controllers is located in a narrow range of low frequencies. Figure 8 presents the control force of each controller. It is clearly seen that the proposed controller can provide the better control performance with the similar magnitudes of control force as those of the FSMCD and AFSM. Figure 5: Acceleration of the chassis mass \(m_{s}\). \begin{table}
\begin{tabular}{c c c} \hline
**Controller** & \(A_{\alpha}\) **(m/s\({}^{2}\))** & \(\tau\) **(s)** \\ \hline Proposed & 0.0190 & 1.869 \\ FSMCD & 0.0256 & 1.321 \\ AFSM & 0.0374 & 0.905 \\ Passive & 0.2394 & 0.264 \\ \hline \end{tabular}
\end{table} 
Table 3: Bump road: the maximum acc. and time delay of acc. response. AFSM: adaptive fuzzy sliding mode; FSMCD: fuzzy sliding mode controller with disturbance observer. Figure 6: Ratio of cross-correlation between the chassis vertical acceleration signal and the excitation signal coming from the bump-road profile. Figure 7: Spectral coherence \(C_{xy}\) between the excitation signal coming from the bump-typed road surface and acceleration signal of the chassis-mass. As a second investigation, the sine-wave road profile is considered as follows [40]. \[ \begin{array}{l}z(t)=Z_{0}\sin(\omega\,t)+\text{random}(0;\,0.001)\quad(\text{m} )\\ \omega=\frac{2\pi}{D}V\end{array}  \tag{54}\]  
In this equation, \(z(t)\) is the vertical displacement of the road surface; \(Z_{0}\) and \(D\) respectively are the amplitude and cycle of the sine-wave; \(V\) denotes the velocity of vehicle along the road; and \(\text{random}(0;\,0.001)\) denotes the random value belonging to \((-0.001,\,+0.001)\). Related to this road profile, random(0; 0.001) indicates the road disturbance surface. By choosing \(Z_{0}=0.07\,\text{m}\), \(D=10\,\text{m}\), and \(V=30\,\text{km}/\text{h}\), the road profile is achieved, as shown in Figure 9. In order to produce this road profile, an amplitude of 0.07 (m) with an excitation frequency of 0.8333 Hz is used, which is equivalent to the angular frequency of 5.236 rad/s. Figures 10-12 and Tables 5 and 6 present the measured results under the sinusoidal road profile. Figure 10 and Table 5 clearly show that the acceleration of the chassis mass is greatly reduced by activating the proposed controller. The maximum acceleration amplitudes \(A_{a}\) of the DE-FSMC, AFSM, and the passive system is identified by 0.3642, 0.5742, 0.7928, and 3.6587 m/s\({}^{2}\), respectively. Table 5 shows the dynamic response delay \(\tau\) between the chassis vertical acceleration signal and the excitation signal coming from the sine-road surface analyzed via CCF. The maximum value of \(\tau\) is determined by 2.545 (s) from the proposed DE-FSMC, while the minimum of \(\tau\) is determined by 0.793 (s) from the passive suspension. Figure 11 and Table 6 provide the ratio of spectral coherence \(C_{xy}\), showing its maximum value \(C_{xy}^{\text{max}}\), and the corresponding frequency \(f_{\text{C}_{xy}^{\text{max}}}\) related to each method. The minimum value of \(C_{xy}^{\text{max}}\) is 0.4672 at the frequency of 24.03 Hz in the proposed control method. The required control force corresponding to each controller is shown in Figure 12. It is clearly seen that the vibration control performance of the proposed controller is excellent with the lower input power. \begin{table}
\begin{tabular}{c c c} \hline
**Controller** & \(A_{a}\) **(m/s\({}^{2}\))** & \(\tau\) **(s)** \\ \hline Proposed & 0.1269 & 4.28 \\ FSMCD & 0.1985 & 5.05 \\ AFSM & 0.2175 & 5.45 \\ Passive & 0.9208 & 47.08 \\ \hline \end{tabular}
\end{table}