# A Novel Adaptive PID Controller with Application to Vibration Control of a Semi-Active Vehicle
## 4 Experimental Results and Discussions
The maximum acceleration at the driver position is found from Figure 8(d1,d2) and the values are 1.67057 and 2.22271 m/s\({}^{2}\) for the proposed control and the control of [5], respectively. Figure 7: Experiment results under regular bump excitation: (**a1**) displacement; (**a2**) enlarged view of the displacement; (**b1**) velocity; (**b2**) enlarged view of the velocity; (**c1**) acceleration at the seat; (**c2**) enlarged view of acceleration at the seat; (**d1**) acceleration of the driver; (**d2**) enlarged view of the acceleration of the driver. Figure 8: Experiment results under random-step-wave excitation: (**a1**) displacement; (**a2**) enlarged view of the displacement; (**b1**) velocity; (**b2**) enlarged view of the velocity; (**c1**) acceleration at the seat; (**c2**) enlarged view of the acceleration at the seat; (**d1**) acceleration of the driver; (**d2**) enlarged view of the acceleration of the driver. From these results obtained via experimental implementation, it is clear that the proposed controller provides the better vibration control performances than the comparative controller under two different road profiles. The comparative control results are also good performances, but the performance degree is less than the proposed controller. This indicates that the proposed controller has a higher robustness under uncertain operating conditions of the control system. The proposed controller proves that the combination of two or more control logics can bring better control responses since the inherent advantages of each controller are retained to guarantee robust stability against any disturbance and the uncertainty of the system. It is noted here that according to the standard ISO 2631-1, the acceleration at the driver position should be less than 2.5 m/s\({}^{2}\)[17]. From the results presented in Figures 7 and 8, the accelerations of the proposed controller shown in Figure 7(d1,d2) and Figure 8(d1,d2), are always less than the standard value of ISO 2631-1. The applied current for regular bump excitation and random-step-wave excitation is shown in Figure 9. The maximum applied currents of regular bump excitation are 1.487 A and 1.494 A for the proposed control and the control in [5], respectively. The maximum applied currents of random-step-wave excitation are 1.4908 and 1.492 A for the proposed control and the control in [5], respectively. In order to evaluate vibration control performance, two standard factors are used in this study: ToD (Transmissibility of Displacement) and ToA (Transmissibility of Acceleration) values. The ToD values mean the ratio of the input weighted displacement to the seat frame's weighted displacement. The floor and seat frame displacement signals are analyzed to find the vibration transmissibility response of the seat suspension. Similarly, the ToA value means the ratio of the input weighted acceleration to the seat frame's weighted acceleration. The floor and seat frame acceleration signals are analyzed to find the vibration transmissibility response of the seat suspension. The ToD and ToA values are expressed as  
\[ToD=\frac{d_{scat}}{d_{floor}} \tag{43}\]  
Figure 9: Applied control current: (**a1,a2**) regular bump excitation; (**b1,b2**) random-step-wave excitation. \[ToA=\frac{a_{seat}}{a_{floor}} \tag{44}\]  
where, \(d_{seat}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}d^{2}_{seat,i}}\), \(d_{floor}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}d^{2}_{floor,i}}\), \(a_{seat}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}a^{2}_{seat,i}}\), \(a_{floor}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}a^{2}_{floor,i}}\). It is noted that the ToD and ToA values are calculated based on the root mean square equation as given in \(d_{seat}\), \(d_{floor}\), \(a_{seat}\) and \(a_{floor}\). It has been identified that the ToD values of the regular bump excitation are 0.262262, 0.289031, and 0.274320 for the proposed control, the comparative control in [5], and the control in [17], respectively.