# A Novel Adaptive PID Controller with Application to Vibration Control of a Semi-Active Vehicle
## 3 Application to Vibration Control
### Simulation Results and Discussions  
In this study, vibration control performance and characteristics of the proposed hybrid controller are firstly investigated through computer simulations prior to implementing the controller on the real system. In the computer simulations, the MR seat suspension system is excited by three different road profiles as shown in Figure 3: random bump, regular bump, and random step wave bump. These signals are collected from the practical conditions of the roads and the equation as shown in [32]. The parameters of the seat suspension and MR damper are listed in Tables 1 and 2. The applied current (42) is calculated using the dynamic parameters shown in Table 3. The maximal damping force of the manufactured MR damper is 1000 N (\(\pm 5\%\)), and the maximal applied current is 2 A. As mentioned above, the fuzzy model for variables of the system is established based on the online model. These models include displacement and acceleration values of the MR seat suspension system. It is remarked that these values are extracted from the experiment results obtained at applied current 2 A. The outputs of these fuzzy models are shown in Table 4. It is noted that it includes six clusters, and then the output of fuzzy rules is also six results. The sigma value for Gaussian function of the fuzzy model is chosen as 0.4 [27]. The constant values of the sliding surface \([k_{1},\ k_{2}]\) are chosen as \([0.3,1]\). The initial value \(\Gamma\) of the Riccati-like equation is chosen by 200. Besides, the matrix \(\mathbf{Q}\) of the Riccati-like equation is chosen as \(\mathbf{Q}=[-2\ 0;\ 0\ -2]\)[33], and the value \(\beta\) is 0.02. The constants \(\eta_{1},\eta_{2},\eta_{3},\eta_{4},\eta_{5},\eta_{6}\) of adaptation laws are chosen as 700 for all. The values of \(\varepsilon_{f},\ \varepsilon_{g},\ \varepsilon_{\Gamma},\ \varepsilon_{KP},\ \varepsilon_{KI},\ \varepsilon_{KD}\) of the expanded adaptation laws are chosen by 10 for all. The values of \(\delta_{1},\ \delta_{2},\ \delta_{3},\ \delta_{4},\ \delta_{5},\ \delta_{6}\) are chosen by 0.05 for all. It is noteworthy that the constants of sliding surface and the adaptation laws are found by trial-and-error method. In order to investigate inherent characteristics of the proposed hybrid controller, three different excitation signals are applied to MR seat suspension system as shown in Figure 3. From the values of these signals, the proposed control scheme calculates and finds the optimal values. The displacements of the seat are presented in Figure 4. It is clearly observed that the displacements after applying the proposed controller are always less than the initial values. It is clearly also seen that that the proposed controller agrees well to the response of the observer which directly indicates assured stability of the proposed control system. From Figure 4(a1,a2), the maximal value of the displacements before controlling is approximately 0.08 m, and then after controlling, this value is less than 0.003 m. It is also identified from Figure 4(b1,b2), the maximal values of the displacement before and after controlling are 0.06 and 0.002 m, respectively. Similarly, from the results shown in Figure 4(c1,c2), the maximal values are identified by 0.06 and 0.005 m before and after controlling, respectively. From these results, it is understood that the vibrations caused from random road profiles are substantially reduced. This is one of excellent benefits of the proposed hybrid controller designed to take account of system uncertainties and external disturbances. It is remarked here that the initial constants are required for any simulation of control system. These constants are chosen by the designer. There are two ways for applying these constants. The first way is to use the initial constant as the dominant vibration, and then evaluate the control responses. The second way is to use both the initial constants and the vibration together. This approach can clearly show full view of responses of the control system.