## 6 Conclusions  
In this work, a new adaptive fuzzy sliding mode controller associated with the disturbance estimator (DE-FSMC) was designed, and its effectiveness was validated by adopting the vibration control of a semi-active vehicle suspension system installed with MRD. Firstly, an adaptive control law was formulated based on Lyapunov stability, so that closed-loop stability and finite-time convergence of tracking error could be maintained. Then, a set of solutions for estimating compensative quantity  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Controller** & \(\mathbf{C_{xy}^{\max}}\) & \(\mathbf{f_{C_{xy}^{\max}}}\) **(Hz)** \\ \hline
**Proposed** & 0.4672 & 24.03 \\
**AFSM** & 0.5195 & 70.54 \\
**Passive** & 0.8140 & 89.92 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Sine road: the spectral coherence at corresponding frequency.  
Figure 11: Ratio of cross-correlation between the chassis vertical acceleration signal and the excitation signal.  
Figure 12: Control force corresponding to each controller.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Controller** & \(\mathbf{A_{s}}\) **(m/s\({}^{2}\))** & \(\mathbf{\tau}\) **(s)** \\ \hline
**Proposed** & 0.3642 & 2.545 \\
**AFSM** & 0.5742 & 1.359 \\
**Passive** & 3.6587 & 0.793 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Sine road: the acceleration and time delay.  
for the impact of UAD on a control system and for avoiding the singular cases of the fuzzy-based function approximation were provided to increase convergence ability and reduce the calculating cost. Subsequently, the effectiveness of the proposed DE-FSMC was verified by undertaking an experimental implementation of the controllers. Vibration control evaluation were investigated under two road profile types; the bump- and sine-typed road with disturbance surfaces, and the following results were achieved.  
(1) Vibration control performances under two different road files have verified that the proposed DE-FSMC shows a better performance than the comparative FSMCD and AFSM controllers. Specifically, the maximum acceleration amplitude at the bump road profile is identified by 0.0190, 0.0256, 0.0374 and 0.2394 m/s\({}^{2}\) by utilizing the DE-FSMC, FSMCD, AFSM and the passive case, respectively.  
(2) It has demonstrated from the cross-correlation function (CCF) that the proposed controller is less sensitive to the excitation signals coming from the road surface status compared with other controllers. Specifically, the minimum value of \(C_{xy}^{\max}\) (the ratio of spectral coherence) is 0.4672 at the frequency of 24.03 Hz in the proposed control method, while it is 0.5195 in the AFSM under the sinusoidal road profile.  
(3) It has been shown that the proposed controller requires the smallest input power in order to achieve excellent vibration control performances. This directly indicates that the updated adaptive control gains determined from the DE-FSMC functionally work well for the semi-active vehicle suspension system featuring MRD dampers.  
This work was supported by the Incheon National University Research Grant in 2016.  
[PERSON] designed a disturbance estimator and integrated with the main controller and wrote the manuscript, and [PERSON] did experimental tests and analyzed the results in terms of vibration control performance. [PERSON] designed a new control algorithm of the fuzzy sliding mode controller and wrote the manuscript.  
The authors declare that there is no conflict of interest.