## 3 Disturbance-Rejection Control Algorithm for FSM System  
### IMP-Based Controller  
To enhance the tip-tilt control performance when dynamic disturbances are applied to the target application, a control system can be designed in which an IMP-based controller is attached to the feedback loop of the base control. Furthermore, the IMP-based control algorithm is available for the rejection of sinusoidal disturbances, and they can be simply  
Figure 4: Disturbance–attenuation performance of FSM control system. Figure 3: Open–loop transfer function of the proportional–integral (PI) controller in FSM control system. extended to the case when multiple sinusoidal components are present. As illustrated in Figure 5, a feedback controller with IMP can be canceled out faster and more accurately than in an unmodified tip-tilt controller. In addition, the proposed IMP-based controller modifies the AFC algorithm more easily, considering the frequencies of various disturbances. However, when the rotational position exceeds the predefined threshold in the presence of nonperiodic disturbances, such as external shock, the feedback loop through the IMP-based controller is opened, as depicted in Figure 5, because the control performance of the system for nonperiodic disturbances may degrade. Therefore, IMP-based controllers are typically used in control systems as nonlinear controllers. The transfer function of the IMP-based controller is denoted \(C_{IMP}\). The continuous-time type and discrete-time type are denoted by \(C_{IMP}(s)\) and \(C_{IMP}(z)\), respectively, and can be obtained as follows:  
\[C_{IMP}(s)=g\frac{\cos(\varnothing)s-sin(\varnothing)\omega}{s^{2}+\omega^{2}}, \tag{4}\]  
\[C_{IMP}(z)=g\frac{\cos(\varnothing)z-cos(\omega T_{s}+\varnothing)z}{z^{2}- cos(\omega T_{s})z+1}, \tag{5}\]  
where sampling time, periodic frequency, adaptive gain, and control gains of the IMP-based controller are denoted as \(T_{s}\), \(\omega\), \(g\), and \(\phi\), respectively. This presented that the IMP-based controller is input/output equivalent to a linear time-invariant controller, even though the whole adaptive structure is described by linear time-varying differential equations. It is important to note that this transfer function has a zero at \(s=-\omega tan(\phi)\). For the cancellation of periodic disturbances, the control input of an IMP-based controller has the following form in discrete time:  
\[u(k)=\hat{a}(k)cos(\omega T_{s}k)+\hat{b}(k)sin(\omega T_{s}k), \tag{6}\]  
The discrete-time update laws for the elimination of the disturbance are as follows:  
\[\hat{a}(k)=\hat{a}(k-1)+ge(k)cos(\omega T_{s}+\varnothing), \tag{7}\]  
\[\hat{b}(k)=\hat{b}(k-1)+ge(k)sin(\omega T_{s}+\varnothing), \tag{8}\]  
where update laws parameter \(e(k)\) is the control error, and \(g\) is the adaption gain of the IMP-based controller. The adaption gain g can be selected experimentally. If it is large, the magnitude of the open-loop transfer function at the disturbance frequency is larger, and the magnitude of the disturbance reduction performance is smaller. Moreover, the control performance of the system can be extremely unstable in other frequency regions because the sensitivity transfer function of the control system in those regions may attenuate in accordance with the Bode integral theorem. Therefore, the fundamental design of the IMP-based controller must be such that the control algorithm can be attenuated reliably. The disturbances whose frequencies correspond to those at which the control gain of the  
Figure 5: Schematic–diagram of FSM control system with IMP. IMP are infinite. In addition, the IMP-based control gain \(\phi\) at \(\omega\) (periodic frequency) is obtained as follows:  
\[\phi=\angle\frac{P(z)}{1+P(z)C(z)}, \tag{9}\]  
where \(P(z)\) is the designed FSM control system, and \(C(z)\) is the PI controller.