## 3 Simulations for Control Performances
### Results and Discussions  
The purpose of the simulation is to prove the robustness of the proposed controller. To achieve this goal, both the displacement and velocity at the seat position subjected to road excitation are investigated and compared with two existing controllers; Compared Controller 1 and Compared Controller 2. Figures 4-6 present vibration control performance of the proposed controller, Compared Controller 1 and Compared Controller 2, respectively. In the results, (a) and (b) represent control responses for a few minutes motion, while (c) and (c) represent control performance for one minute to observe more detailed profiles of the displacement and velocity components at the seat position. It is clearly seen from the results that the proposed controller provides the best vibration control performance in terms of the magnitudes of the displacement and velocity. Specifically, the maximum peak of the displacement is identified by 0.00441 (m) by activating the proposed controller, while it is extremely high in the Compared Controllers 1 and 2. It is also understood from the signal profiles of the enlarged view (d) that both the proposed controller and Compared Controller 1 follow the model of adaptive fuzzy PID control showing the switching motion. This behavior can be expected from both the prescribed sliding surface and the choice of robustness function, given by Equations (15) and (28), respectively. On the other hand, it is clearly seen from the results of Compared Controller 2 that vibration control performance is not good exhibiting unstable motion. In fact, the constraints functions in the input control of the Compared Controller 2 are not sufficient to capture the control energy to overcome the imposed disturbance. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Parameters of sliding surface \([k_{1},k_{2}]\) & \([1,\,0.02]\) \\ Constant of PID \(K_{P}\) & \(10\) \\ Constant of PID \(K_{I}\) & \(150\) \\ Constant of PID \(K_{D}\) & \(50\) \\ Maximum damping force & \(1000\) N \\ Riccati’s parameter \(\Gamma\) & \(0.01\) \\ Riccati’s matrix \(\mathbf{Q}\) & \([-2\,0;\,0\,-2]\) \\ Observer matrix \(\mathbf{Q}_{0}\) & \([1\,0;\,0\,1]\) \\ Observer matrix \(\mathbf{W}_{0}\) & \([30;10]\) \\ Constants of adaptation laws \((\mu_{1}-\mu_{6})\) & \(700\) \\ Initial state for proposed control & \(\left[\begin{array}{cc}0.035&2.5\\ 0.035&2.5\end{array}\right]\) \\ Initial state for observer & \(\left[\begin{array}{cc}0.035&2.5\\ 0.035&2.5\end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the seat suspension for Compared Controller 1 [23]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Maximum damping force & \(1000\) N \\ Observer matrix \(\mathbf{Q}_{0}\) & \([1\,0;\,0\,1]\) \\ Observer matrix \(\mathbf{W}_{0}\) & \([30;10]\) \\ Robustness parameter \(\gamma\) & \(0.00001\) \\ Robustness parameter \(v\) & \(1\) \\ Robustness parameter \(\varepsilon\) & \(1\) \\ Robustness parameter \(\delta\) & \(0.1\) \\ Robustness vector \(y_{0}\) & \(\left[\begin{array}{cc}-67170&-3.975.10^{3}\end{array}\right]\) \\ Robustness matrix \(\mathbf{L}\) & \(\left[\begin{array}{cc}5.10^{-8}&6.10^{-10};\,\,5.10^{-5}&1.10^{-6}\end{array}\right]\) \\ Robustness matrix \(\mathbf{G}_{d}\) & \(\left[\begin{array}{cc}1&1;\,\,1&1\end{array}\right]\) \\ Initial state for control & \(\left[\begin{array}{cc}0.035&2.5\end{array}\right]\) \\ Initial state for observer & \(\left[\begin{array}{cc}0.035&2.5\end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the seat suspension for Compared Controller 2 [24]. Figure 4: Displacement and velocity of the seat system using the proposed controller: (**a,b**) general view, (**c,d**) enlarged view. Figure 5: _Cont_. Figure 5: Displacement and velocity of the seat system of the Compared Controller 1: (**a,b**) general view, (**c,d**) enlarged view.