## 7 Proportional--Integral--Derivative (PID) Controller Enhanced with Deep Learning (DL)
### System Vibration Control Under Impulse and Random Disturbances
#### 7.4.2 Control System Affected by Random Disturbance  
In the scenario where a continuous random disturbance affects the cantilever beam, the system's response is carefully analyzed. The reference input is a unit step with an amplitude zero, and Figure 23 illustrates the random disturbance applied to the system. To observe the system's reaction to this disturbance, Figure 24 shows the resulting output signal before any control is applied.  
Figure 23: Random Disturbance of the System.  
Figure 22: Voltage Response of PZT Sensor in the Cantilever with DL-PID Controller and without.  
The trained deep neural network is employed to tune the PID controller by processing the uncontrolled system output under random disturbance conditions. The network predicts the optimal PID parameters necessary to reduce and mitigate the beam's vibrations. By dynamically adjusting the PID gains, the network ensures that the control response is both precise and effective.  
The PID parameters tuned by the network are shown in Table 5, further demonstrating its capability to infer appropriate PID parameters across various test conditions. It shows the adaptability of the trained network in tailoring the PID controller to meet the specific requirements of the system.  
As seen in Figure 24, the uncontrolled voltage response of the beam is monitored. Based on these data, the deep learning network tunes the PID controller, resulting in the control signal shown in Figure 25. This signal is specifically designed to counteract the detected random disturbances by adjusting the PID parameters, enabling the controller to stabilize the beam's vibrations efficiently.  
Figure 26 demonstrates the system's behavior after applying the deep learning-tuned PID controller. The figures highlight how the optimized PID gains result in superior vibration control. This approach underlines the advantage of integrating deep learning with traditional PID control, offering more adaptive and precise vibration mitigation.  
Figure 27 illustrates the difference in system performance before and after applying the tuned PID controller using deep learning. The beam's vibrations are noticeably reduced. With the DL-PID controller, the Integral Time Absolute Error (ITAE) is reduced to 3.2347, indicating the system's effectiveness in minimizing both the amplitude and duration of the error. Moreover, the mean squared error (MSE) of 0.7631 further highlights the predictive precision of the deep learning approach in optimizing the PID controller, resulting in more accurate and efficient vibration control.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(K_{p}\) & \(K_{i}\) & \(K_{d}\) \\ \hline
661.4 & \(-\)2049.3 & 182.6 \\ \hline \hline \end{tabular}
\end{table}
Table 5: PID Controller Parameters Tuned Using Deep Learning of the System Affected by Random Disturbance.  
Figure 24: Voltage Response of PZT Sensor before Vibration Control (under Random Disturbance).  
Figure 27: Voltage Response of PZT Sensor in the Cantilever with DL-PID Controller and without.  
Figure 26: Voltage Response of PZT Sensor after Vibration Control with DL-PID Controller (under Impulse Disturbance).  
Figure 25: System Control Signal.