# A Novel Adaptive PID Controller with Application to Vibration Control of a Semi-Active Vehicle
## 3 Application to Vibration Control
### Simulation Results and Discussions
Hence, the controller can be evaluated exactly compared with the first way. Therefore, the second way is chosen in this work. On the other hand, the vibrations at the position of the driver are evaluated and presented in Figure 5. The vibration control performances at the driver position are similar to the results shown in Figure 4. This point is clearly expressed in Equation (40). It is noted here that the sliding surfaces subjected to two different road profiles go to zero during control action. This is possible since when the sliding mode converges zero, the control \(u\) given in Equation (17) is not zero because of the control \(u_{1}\) given in Equation (14). The control \(u_{1}\) belongs to the fuzzy neural networks model which determines the fuzzified values for the control force. This energy from the control \(u\) and the semi-active control of MR damper acts on the system. Thus, the control input always exists although the sliding surface converges to zero. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Data**} & \multicolumn{3}{c}{**Centroid Vector**} \\ \cline{2-3}  & \multicolumn{2}{c}{**Displacement (\(\times 10^{-2}\) m)**} & **Acceleration (m/s\({}^{2}\))** \\ \hline Random bump & \([-5.428-0.178-0.098\) 0.182 0.253 5.598] & \([-6.433-0.651-0.128\) 0.698 1.140 7.185] \\ Regular bump & \([-5.428-0.178-0.098\) 0.182 0.253 5.598] & \([-6.433-0.651-0.128\) 0.698 1.140 7.185] \\ Random step wave & \([-8.926-0.018-0.007\) 0.004 0.026 6.017] & \([-11.250-1.355-0.520\) 0.085 0.906 7.661] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Centroid vectors of fuzzy rules using Gaussian model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Linear spring stiffness \(k_{0}\) & 0 N/m \\ Viscous damping coefficient \(c_{a}\) & 990 Ns/m \\ Viscous damping coefficient influenced by \(v\): \(c_{b}\) & 3095 Ns/m V \\ Stiffness of \(\varphi\): \(\alpha_{a}\) & 545 N/m \\ Stiffness of \(\varphi\) influenced by \(v\): \(\alpha_{b}\) & 620 N/mV \\ Positive parameter of hysteresis loop \(\phi\) & 4 \\ Positive parameter of hysteresis loop \(\kappa\) & 48 \\ Positive parameter of hysteresis loop \(\lambda\) & 48 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters to calculate the applied current of MR damper. **Remark 4**. : _As mentioned above, the practical values shown in Figure 3 are used in experiment. However, they must be calibrated to fix the system for preventing a breakdown or dangerous accident. Hence, the results of experiment are different a little from the simulation results. This does not alter the objective of the proposed control system. It is noted that the data sample time for simulation is 30 s. In the experiment, the sample time is chosen to meet the operation of the seat's suspension system (capacity of the hydraulic system). In this experiment, the sample time is 3 s for regular bump excitation, and 5 s for random step wave excitation. Because of the difference, it is hard to plot both the simulation and the experiment results in one figure. However, the effectiveness of the proposed controller is similar in both the simulation and experiment._  
Figure 4: Simulation results of the displacement \(x_{\text{s}}\) of the seat: (**a1,a2**) random-bump road; (**b1,b2**) regular-bump road; (**c1,c2**) random step wave road. **Remark 5**. : _In this study, three road profiles are used in order to simulate the input signals. The road profiles shown in Figure 5a,c are collected from the real roads. The road profile shown in Figure 5b is generated from the Equation (40) given in [32]. These road profiles are used to investigate the vibrations transferred from wheels of the vehicles installed with MR dampers. Hence, these profiles can be adopted for experimental testing to emulate the practical road conditions as shown in Figure 6. It is noted that the design of vehicle suspension systems has been improved over the past decades, but this does not contribute vibration control of the seat position.