## 3 Disturbance-Rejection Control Algorithm for FSM System
### IMP-Based Controller
As illustrated in Figure 6, according to the results of the experiment conducted to analyze uncompensated CES, periodic disturbances of several frequencies in the CES are generated primarily by the operating conditions. In particular, in the analyzed residual CES, periodic disturbance characteristics are generated during tip-tilt control owing to the vibration of the platform cooling fan. The power spectrum of the uncompensated CES shows that the disturbance occurs at integer multiples of the fundamental rotating frequency (i.e., 60 Hz) of the cooling fan. In addition, two specific peaks are formed at 65 and 71 Hz, and they are not harmonic frequency components of the fundamental rotating frequency of the cooling fan. These disturbance frequencies may be attributed to the mechanical resonances of the platform. Therefore, to eliminate periodic disturbances, such as the harmonic component resulting from rotation conditions of the cooling fan and mechanical resonance frequencies of the platform, the FSM control system is designed using an IMP-based  
As illustrated in Figure 4, the phase angles of the disturbance-attenuation performances at \(\omega_{1}\) = 60 Hz, \(\omega_{2}\) = 65 Hz, and \(\omega_{3}\) = 71 Hz are estimated as \(\phi_{1}\) = 80\({}^{\circ}\), \(\phi_{2}\) = 78\({}^{\circ}\), and \(\phi_{3}\) = 77\({}^{\circ}\), respectively. The IMP-based control input \(u(k)\) can be defined utilizing the control law expressed in Equation (6). These phase values for cancellation frequencies can be used from the adaptation law, which is expressed in Equations (7) and (8). As depicted in Figures 7 and 8, the adaptive gain g of the IMP-based controller should be selected experimentally to maximize the stability of the control system in this study, and \(g\) is set to 2 \(\times\) 10\({}^{-4}\). Consequently, the control performance of the IMP-based controller for disturbance attenuation is over 20 dB greater than that of the PI controller for a few frequency components. According to the experimental results of the PI controller for FSM control with and without the IMP-based AFC, the target rotational position was maintained successfully at 500 \(\upmu\)rad. Figure 6 shows that the power spectrum of the residual CES for the PI controller without the IMP-based controller was 7.97 \(\upmu\)rad. When we used the IMP-based controller at 60, 65, and 71 Hz in the FSM control system, the RMS values of the CES were 7.43, 7.17, and 6.05 \(\upmu\)rad, respectively. Additionally, we applied the IMP-based controller to multiple frequency components. When the IMP-based controller was used at 60 and 65 Hz in the FSM control system, the RMS value of the CES was 6.00 \(\upmu\)rad. Furthermore, when the IMP-based controller was used at 60 and 71 Hz in the FSM control system, the RMS of the CES was 5.41 \(\upmu\)rad, as illustrated in Figure 9a,b. These results indicated that the IMP-based controller was efficient at suppressing periodic disturbances in the FSM system, as shown in Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Specification**} & **PI Controller** & **IMP (60 Hz)** & **IMP (65 Hz)** & **IMP (71 Hz)** & **IMP (60/65 Hz)** & **IMP (60/71 Hz)** \\ \hline Only IMP & Control error (\(\upmu\)rad rms) & 7.97 & 7.43 & 7.17 & 6.05 & 6.00 & 5.41 \\ \hline \end{tabular}
\end{table} 
Table 2: Overall experimental results of FSM control system using IMP. Figure 8: Open–loop transfer function of IMP-based controller (at 60 and 71 Hz).