## 3 Experimental Results  
### Single-Axis Robotic Arm Controller Experiment and Study  
The robot arm developed in this research has a larger operating range of 0-100 degrees in the fourth axis, and it has no loading effect. In order to explore the experimental results of each controller more clearly, the output response of the fourth axis is used as a comparison. The PID controller is used as the basis of development, the purpose of which is to enable the robot arm to follow the preset reference input to execute the tracking control function, and then to compare the control performance difference between different controllers to verify the possibility of applying the controller to the actual system and evaluating its performance. In the PID controller experiment, the simulated PID control parameters were applied to the real system to observe the real robot arm in closed loop PID control and compare the results between the real system and the simulated system. The actual output response is shown in Figures 9 and 10, and the comparison results are recorded in Table 4.  
Hence, compared to a PID controller that adjusts each parameter one at a time to obtain better tracking results, self-adaptive PID controllers provide a more convenient way to obtain a set of controller parameters in feedback, and the parameters of the self-adaptive PID controller are variable in nature. Theoretically, it is more robust than the fixed parameters. In the adaptive PID controller, the convergence factor \(\eta\) has a significant effect on the convergence of the system response. The following experiment was conducted to adjust the size of the convergence factor of the proportional parameter to observe the effect on the output response of the system under different convergence factors. The parameter settings are shown in Table 6.  
From Figure 13, we can observe that different convergence factors affect the speed of convergence of the error. Theoretically, the larger the convergence factor, the faster the rate of convergence. If it is not limited, it will easily lead to larger steady-state error or dispersion, and conversely, the smaller the convergence factor, the slower the rate of convergence. Since this study limits the control voltage command for the output to the pneumatic valve, excessive convergence factor does not really cause system dispersion when using LMS-PID control. The application of ILC is similar to that of the LMS-PID controller in terms of its internal controller parameters \(W_{p}=1.21\), \(W_{d}=0.042\), and \(W_{i}=0.02\). The learning rate \(L\) has a great influence on the convergence of the system. The following experiments were conducted for three different learning rates (\(L=0.05\); \(L=0.2\); \(L=0.8\)). Comparing the RMSE convergence characteristics of the system with different learning rates, we can observe from Figure 14 that as the learning rate gain increases, the RMSE convergence rate also increases. However, if the learning rate is larger than a certain threshold, the RMSE will not be able to converge the error and the whole control system will be scattered, which is called the overlearning of the system. Therefore, when using iterative learning control, it is important to avoid inputting excessive learning rates, or to limit the number of learning cycles to avoid system scattering. \begin{table}
\begin{tabular}{c c} \hline
**Initial Controller Parameters** & \(\mathbf{W_{p}=0;W_{d}=0;W_{i}=0}\) \\ \hline Convergence factor 1 & \(\eta_{p}=2\times 10^{-6}\); \(\eta_{d}=2\times 10^{-6}\); \(\eta_{i}=2\times 10^{-6}\) \\ \hline Convergence factor 2 & \(\eta_{p}=6\times 10^{-6}\); \(\eta_{d}=2\times 10^{-6}\); \(\eta_{i}=2\times 10^{-6}\) \\ \hline Convergence factor 3 &Then, the relationship between the desired trajectory (vertical axis) and the actual trajectory results from each controller (horizontal axis) is plotted.