## 4 Design of Disturbance Estimator
### Formulation of DE-FSMC  
The controller DE-FSMC is a combination of the \(B\)-controller, which directly indicates the FSMC and the DE. With reference to the proposed theory, as well as the structure and operating principle of the suspension system as abovementioned, these two parts are established. The state space model is written as follows. \[\mathbf{x}(t)=\left[x_{1},\;x_{2}\right]^{T}=\left[z_{s},\;\dot{z}_{s}\right]^ {T} \tag{47}\]  
Figure 4: Database extracted from the measured results. Figure 3: Experimental apparatus for a quarter vehicle semi-active suspension system with a MRD. LVDT: linear variable differential transformer. From Figure 2, the spring and damping forces are denoted by functions of the displacement and velocity parameters as follows. \[f_{s}=k_{s}(x_{1}-z_{r}),f_{d}=c_{s}(x_{2}-\dot{z}_{r}) \tag{48}\]  
The dynamic response of the sprung mass is then written as below. \[\left\{ \begin{array}{l}m_{s}(t)\ddot{z}_{s}+k_{s}(x_{1}-z_{r})+c_{s}(x_{2}- \dot{z}_{r})=-u(t)+d(t)\\ y=z_{r}\end{array} \right. \tag{49}\]  
Then, from (51), Equation (1) is re-written for the vehicle suspension system as follows. \[\left\{ \begin{array}{l}{z_{s}}^{(2)}=f_{0}(\mathbf{x},t)+g_{01}(\mathbf{x}, t)u(t)+g_{02}(t)D(t)\\ y=x_{1}\end{array} \right. \tag{50}\]  
\[f_{0}(\mathbf{x},t)=-\  
sliding mode controller proposed in [12] are considered. In this work, in order to show quantitative results, the maximum magnitude of chassis acceleration \(A_{a}\) is defined and used.  
\[A_{a}=\underset{i=1...m}{\max}\Big{|}\tilde{z}_{s}^{i}\Big{|} \tag{53}\]  
where, \(P\) is the number of samples and \(\tilde{z}_{s}\) is the vertical acceleration of the chassis. Two road profiles used for surveys are the bump and sinusoidal type with disturbance surfaces. The MRD is excited to make vibrations with an amplitude of 0.0861 (m) and an angular frequency of 1.9134 (rad/s). It is noted that the speed of the car crossing over the bump profile is decided by 2.34 km/h. The vertical acceleration on the bump road profile is shown in Figure 5. It is identified that the maximum acceleration magnitude \(A_{a}\) defined in Equation (80) related to the DE-FSMC is 0.0190 m/s\({}^{2}\), which is smaller than that of the fuzzy sliding mode controller with disturbance observer (FSMCD) and AFSM, whose values are 0.0256 and 0.0374 m/s\({}^{2}\), respectively. It is noted that the fuzzy sliding mode controller with disturbance observer (FSMCD) used in this work as a comparative controller is the modified one developed in [1], in which the disturbance observer is also used like the proposed one. The AFSM used in this work is the modified one proposed in [30] to adapt to the semi-active suspension system. The numerical results of acceleration identified from the figure are given in Table 3. Figure 6 and Table 3 reflect the dynamic response delay \(\tau\) between the chassis vertical acceleration signal and the excitation signal coming from the bump-road surface analyzed via cross-correlation function (CCF). The maximum value of \(\tau\) is identified as 1.869 (s) for the proposed DE-FSMC, while the minimum of \(\tau\) is calculated by 0.264 (s) for the passive suspension. This indicates that when controlled by the proposed DE-FSMC, the vibration of the chassis is less sensitive to the road surface status compared with the passive AFSM and FSMCD. This is an advantage for controlling vibration by avoiding the resonant status. This aspect is more clearly seen via the ratio of spectral coherence \(C_{xy}\) provided in figfig:applsci-07-01053-f007 and Table 4. The spectral coherence can identify the frequency-domain correlation between two databases. The values of \(C_{xy}\) tending towards zero indicate that the corresponding frequency components are uncorrelated, and conversely for the tending towards 1 [38; 39].