## 3 Disturbance-Rejection Control Algorithm for FSM System
### IMP-Based Controller with DOB  
The design key of the IMP-based controller with DOB comes from the fact that it improves the disturbance-attenuation performance without redesigning the feedback controller. Theoretically, the IMP-based controller with DOB is designed for disturbance attenuation based on the nominal plant.  
In general, the DOB is to use the inversed plant model to cancel the disturbance to the control system and then modify the control input to compensate. Also, the low-pass \(Q\) filter \(Q(z)\) is applied to reduce high-frequency measurement noise and allow \(Q(z)\)\(P_{n}{}^{-1}(z)\)  
Figure 9: Experimental results of FSM control system: (**a**) IMP (at 60 and 65 Hz) and (**b**) IMP (at 60 and 71 Hz).  
to be realizable. Figure 10 presents a schematic diagram of the FSM control system using an IMP-based controller with DOB.  
In this study, the stability of the control system is related to the tunable coefficient and the order of the \(Q\) filter. Therefore, the tunable coefficient and order of the \(Q\) filter should be designed to improve the stability of the control system through the measurement data of the disturbance attenuation. Here, the \(Q_{20}\) and \(Q_{30}\) filters were utilized for the DOB-based controller for the FSM system, as illustrated in Figure 11. Via the analysis of the residual control-error signal (CES), which is the experimental result, the stability of the control system can be estimated. Consequently, utilizing \(Q_{20}\) and \(Q_{30}\) filters effectively improved the disturbance-suppression performances of the FSM control system. Moreover, the disturbance-suppression performances of the FSM motion system were effective in the low-frequency region with bandwidths of 15 and 30 Hz. The open-loop transfer functions of the aforementioned \(Q_{20}\) and \(Q_{30}\) filters with bandwidths of 15 and 30 Hz are expressed as follows:  
\[Q_{20}(s)=\frac{1}{\left(\tau s+1\right)^{2}}=\frac{1}{\tau^{2}s^{2}+2\tau s+1}, \tag{10}\]  
\[Q_{30}(s)=\frac{1}{\left(\tau s+1\right)^{3}}=\frac{1}{\tau^{3}s^{3}+3\tau^{2} s^{2}+3\tau s+1}, \tag{11}\]  
Figure 11: Cont.  
Figure 10: Schematic diagram of FSM control system using IMP-based controller with DOB.  
The attenuation control performance of the FSM control system should be as high as possible in the frequency band of the disturbance applied to the system but not so high that it degrades the system's stability. Therefore, the loop gain of the FSM control system utilizing the IMP-based controller with DOB was selected to be higher than the loop gain of the IMP-only-based controller. Furthermore, as you increase the denominator order and bandwidth of the \(Q\) filter, the attenuation ping response of the FSM control system utilizing DOB is better. As the control performance, the open-loop transfer functions of the overall FSM control system utilized the IMP-based controller with DOB are illustrated in Figures 12-15. The open-loop transfer function of the FSM control system utilized by the IMP with the DOB can be obtained as follows:  
\[G_{Total}(z)=\frac{(C_{IMP}(z)+C(z))P_{n}(z)+1-(1-Q(z))}{(1-Q(z))P_{n}(z)}, \tag{12}\]  
Figure 11: Open–loop transfer function \(Q\) filter for FSM control system: (**a**) \(Q_{20}\) and (**b**) \(Q_{30}\).  
Figure 12: Open–loop transfer function of IMP-based controller (at 60 and 65 Hz) with DOB (\(Q_{20}\) filter).  
Figure 14: Open–loop transfer function of IMP-based controller (at 60 and 71 Hz) with DOB (\(Q_{20}\) filter).  
Figure 13: Open–loop transfer function of IMP-based controller (at 60 and 65 Hz) with DOB (\(Q_{30}\) filter).