## 3 Application to Vibration Control
### Computer Simulations  
In this study, the proposed controller was evaluated from both computer simulation and experimental realization. To emulate the road profiles in practical environment, two road excitations were adopted: random bump road and random step wave road. These excitations have different properties, and were collected from the real roads. The random bump road has small differences of magnitude of vibration, and it was used for checking the stability under small difference of amplitude and/or small disturbances. For the random step wave excitation, the large difference of amplitude shows severe disturbance which exists inside the performance. Hence, these excitations were the best choice for evaluating the control performance related to the road profiles. It is remarked again that the parameters of the seat suspension are listed in [9]. The maximal force of the MR damper was 1000 N (\(\pm 5\%\)), and the maximal current was 2 A. The parameters of the fuzzy model are presented in [9] with six clusters, and the sigma value for Gaussian function was chosen as 0.4. The constant values of the sliding surface \([k_{1},k_{2}]\) for two excitations were chosen as \([0.02,1]\). The initial value \(\Gamma_{n}\) of the Riccati-like equation was chosen as 0.01. The value \(\Gamma_{n}\) of as 0.01 was initial one for the adaptation law in Equation (28). The adaptation law is the derivative function, hence \(\Gamma_{n}\) must be chosen as a positive value. The value of 0.01 was finally chosen based on the trial-and-error method by observing control performance. Besides, the matrix \(\mathbf{Q}_{n}\) of the Riccati-like equation was chosen as \(\mathbf{Q}_{n}=[-2\ 0;0\ -2]\), and the value \(\beta\) was 20,000. The values of \(\eta_{1},\eta_{2},\eta_{3},\eta_{4},\eta_{5},\eta_{6}\) of adaptation laws were chosen as 700 for all. The values of \(\varepsilon_{nf}\), \(\varepsilon_{ng}\), \(\varepsilon_{nT}\), \(\varepsilon_{nKP}\), \(\varepsilon_{nKI}\), and \(\varepsilon_{nKD}\) were chosen as 10 for all. The values of \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), \(\delta_{4}\), \(\delta_{5}\), and \(\delta_{6}\) were chosen as 0.05 for all. The values of \(K_{P},K_{I},K_{D}\) were chosen as (10, 150, 50), and (10, 150, 10) for random bump road and random step wave road, respectively. In this simulation, the initial states used were [0.01 2.5] for the dynamic states, [0.06 0] for the observer states, and 3.5 m/s\({}^{2}\) for the initial acceleration. In this simulation, the fourth-order Runge-Kutta method was applied for solving the differential equation. Simulation results are shown in Figures 2-5. In Figure 2, the displacement of the seat before and after controlling is presented. The controlled displacement of the seat is less than the initial vibration. The initial amplitude of vibration before and after using the control is in range of \(-0.06\) to \(0.08\) m, as shown in Figure 2a, and \(-0.002\) to \(0.004\) m, as shown in Figure 2c, respectively. Subsequently, the velocity shown in Figure 2b,d of the seat is also decreased with the stability. The power spectral density of these vibrations and the driver position is shown in Figure 3. It is clearly observed that the proposed control is very efficient for controlling the random bump road excitation. This result directly indicates that the vibration with small difference of amplitude and trivial disturbance can be controlled well by using the proposed controller. The simulated results under the random step wave road are shown in Figures 4 and 5. In Figure 4, the displacement of the seat is much lower than the initial vibration. The initial amplitude of vibration before and after using the control is in range of \(-1.5\) to \(1.5\) m, as shown in Figure 4a, and \(-0.004\) to \(0.004\) m, as shown in Figure 4c, respectively. The velocity shown in Figure 4b,c of the proposed control is also stable without large vibration. These results can be seen in Figure 5 with the power spectral density.