## 5 Experiments
### Results and Discussion  
To verify the vibration suppression effect and command tracking performance of the proposed method, a sine wave (1 Hz, 10\({}^{\circ}\)/s), which is a test command in engineering, is used as the angular velocity command. The experiments include five groups: A, B, C, D, and E. To simplify the description, the time-varying parameters \(u_{c}(t),u_{c}^{*}(t),u_{o}(t),u_{f}(t)\) are represented by \(u_{c},u_{c}^{*},u_{o},u_{f}\) in subsequent experiments. Group A uses the PI controller \(u_{c}\) with a heavy base support. Group B uses the PI controller \(u_{c}\) with a light base support and occurrence of vibrations. Group C uses the modified PI controller \(u_{c}^{*}\) with a light base support in the form of Equation (28) to suppress the vibrations. Group D uses the proposed controller \(u_{c}^{*}+u_{o}+u_{f}\) with a light base support in the form of Equation (27). Group E uses the controller \(u_{c}+u_{o}+u_{f}\) with a light base support. Vibrations still occur, and in some instances, are even more pronounced. To be able to perform a one-on-one comparison for the experiments, the same PI parameters have been retained for the five groups. The sine command, the response curves, and the relative angular velocity error of the five groups are shown in Figures 9-13. Furthermore, the root mean square errors (RMSEs) are shown in Figure 14.  
Figure 10: Group B: PI controller \(u_{c}\) with a light base support. (**a**) Sine command and relative angular velocity; (**b**) Relative angular velocity error. Figure 9: Group A: PI controller \(u_{c}\) with a heavy base support. (**a**) Sine command and relative angular velocity; (**b**) Relative angular velocity error. In view of Figures 9-14, the following conclusions can be drawn. 1. There is a significant nonlinear disturbance in the system, as demonstrated in Figures 9a-11a, especially when the angular velocity approaches zero. As a result, the system demonstrates a larger tracking error in Figures 9b-11b. 2. The base equivalent moment of inertia is lighter, and is more likely to cause system vibrations, as demonstrated in Figures 10a and 13a. 3. Comparing Figures 10a and 11a, the modified feedback controller effect is noticeable, and the vibrations are successfully suppressed. Figure 11: Group C: Modified PI controller \(u_{c}^{*}\) with a light base support. (**a**) Sine command and relative angular velocity; (**b**) Relative angular velocity error. Figure 12: Group D: Proposed controller \(u_{c}^{*}+u_{o}+u_{f}\) with a light base support. (**a**) Sine command and relative angular velocity; (**b**) Relative angular velocity error. Figure 13: Group E: Controller \(u_{c}+u_{o}+u_{f}\) with a light base support. (**a**) Sine command and relative angular velocity; (**b**) Relative angular velocity error. 4. In Figures 13a,b, without the modified feedback controller suppressing the vibrations, the composite controller will not suppress the vibrations, even when they are more pronounced. 5. Furthermore, the method proposed in this study shows great disturbance compensation and tracking performance improvements. The phase between the response and command in Figure 12a is ahead in comparison to Figure 11a. The relative angular velocity RMSE is reduced from 0.5165 \({}^{\circ}\)/s to 0.3610 \({}^{\circ}\)/s in Figure 14.  
To find the vibration frequency more intuitively, we applied Fourier transforms to find the frequency components of a signal buried in the noise. Based on the experiments described above, the relative angular velocity is selected as the input signal. Thereafter, the Fourier transform of the signal is computed, the frequency domain is defined, and the single-sided amplitude spectrum is plotted. The amplitude is dimensionless. The results are presented in Figure 15.  
From Figure 15, there is a peak of 26 Hz in Groups B and D, which has a negative effect on this system. If the modified controller (28) is used, the signal is redesigned.