## 2 Problem Solve
### Tracking Experiment  
The controller design method proposed in this paper was used to conduct a tracking experiment. The experimental scenario is shown in Figure 15. This figure includes the following: (a) electro-optical equipment; (b) field of view; (c) computer operation interface; (d) motor drive circuit; (e) optical dynamic target.  
Next, we will perform the fitting using MATLAB (2023a) and compare the results with the actual tracking experiment data. For the controller design of the electro-optical equipment's azimuth axis, identification experiments were conducted to determine the rigid body model of the control object. The comparison between the fitted results and the actual data is shown in the Figure 16. Additionally, a frequency sweep test was performed on the electro-optical equipment, and the analysis revealed that the frequency of the first resonant mode of the electro-optical equipment is 102.6 rad/s (16.3 Hz).  
Figure 14: The influence of torque disturbance on the speed loop output.  
Figure 15: Tracking experimental environment. (a) electro-optical equipment; (b) Field of view; (c) Computer operation interface; (d) Motor drive circuit; (e) Optical dynamic target.  
The model of the rigid structure is  
\[P_{r}=\frac{13.27}{s}\frac{400}{s+400} \tag{28}\]  
where \(\frac{400}{s+400}\) is the transfer function of the current loop. Magnitude of the transfer function from test data is shown in Figure 17.  
In the experiment, three transfer functions need to be designed: the turntable speed loop \(C_{v}\), the turntable position loop \(C_{p}\), and the compensation loop \(K_{A}\). The controller for the turntable speed loop is as follows:  
\[C_{v}=\frac{2.92(1+10.94s)}{s(0.005s+1)} \tag{29}\]  
\[C_{p}=\frac{12.17(1+4.39s)}{s(0.01s+1)} \tag{30}\]  
\[K_{A}=\frac{1}{1+0.16s} \tag{31}\]  
The step responses of the turntable's speed loop and position loop are shown in Figures 18 and 19. The position loop uses image feedback with the camera's frame rate set to 50 Hz. According to the optical system of the tracking turntable, 1 pixel = 18 arcseconds = 0.005 degrees.  
From the step response figures, it can be analyzed that the overshoot of the equivalent composite control is higher, which also confirms the derivation in Section 2.4 that equivalent composite control reduces the system's stability margin. We performed fitting for the azimuth and elevation inputs in the electro-optical equipment. The inputs and fitting results of the equivalent composite control are shown in Figures 20 and 21.  
Figure 16: Comparison of fitted results with actual data.  
Figure 17: Magnitude of the transfer function from test data.  
Figure 19: Step response of the position loop.  
Figure 20: Azimuth angle input and fitting results.  
Figure 18: Step response of the speed loop.  
Figure 21: Elevation angle input and fitting results.  
Based on the error analysis, the equivalent composite control reduces the peak error to one-sixth of the original value, which is very close to the value of \(K_{A}\) used in this paper. This confirms the derivation in Section 2.2.  
The final results indicate that the frequency of the first resonant mode of the turntable is 16.2 Hz, which is a lightly damped mode. With the input signals given by Equations (32) and (33), the maximum values are as follows: azimuth angular velocity is 12.14 \({}^{\circ}\)/s, angular acceleration is 5.72 \({}^{\circ}\)/s\({}^{2}\), and angular jerk is 5.68 \({}^{\circ}\)/s\({}^{3}\); elevation angular velocity is 7.04 \({}^{\circ}\)/s, angular acceleration is 3.31 \({}^{\circ}\)/s\({}^{2}\), and angular jerk is 1.55 \({}^{\circ}\)/s\({}^{3}\). The system achieves target tracking with a peak accuracy of 64\({}^{\prime\prime}\) and 22\({}^{\prime\prime}\) (rms).