# Disturbance Observer-Based Anti-Shock Controller for Laser Beam Steering Systems
## 3 Anti-Shock Controller for FSM System
### DOB-Based Anti-Shock Controller  
The design of the DOB-based anti-shock controller, which aims to compensate for disturbances, is simple to implement because a DOB-based controller can be obtained simply by attaching a DOB to the original FSM control system. Theoretically, the DOB can be designed as an extended structure in the feedback control loop. The DOB-based controller rejects disturbances by injecting a compensation value into the control input; this compensation value is equal to the difference between the commanded control input to the plant and the plant output filtered by the inverse of the nominal plant \(P_{n}(s)\), i.e., \(P_{n}{}^{-1}(s)\), as illustrated in Figure 6. However, because \(P_{n}{}^{-1}(s)\) generally contains pure differential terms, it is not often physically realizable. Therefore, it is common to use a \(Q\) filter in conjunction with \(P_{n}{}^{-1}(s)\). This \(Q\) filter typically has the characteristics of a low-pass filter (LPF) with a direct current (DC) gain of unity. In this study, a binomial \(Q\) filter is used as the LPF. The higher the filter order, the better the realized performance. A high-order filter could adversely affect the controller performance owing to phase lag. In addition, the bandwidth of the \(Q\) filter influences the system response. As the bandwidth of the \(Q\) filter increases, the speed of rejecting undesirable disturbances increases. Thus, the \(Q\) filter is important for securing the performance of the DOB. Figure 6 schematically illustrates the DOB-based anti-shock controller for FSM control systems. The Open-loop transfer function of a continuous system is expressed as follows [34]:  
\[Q_{mn}(s)=\frac{\sum_{i=1}^{n}a_{mi}(\tau s)^{i}}{\left(\tau s+1\right)^{m}}, \tag{4}\]  
where \(\tau\) is a time constant, \(a_{mi}\) denotes the binomial coefficients calculated as \(\left(m!/m-i\right)\): \(i!\), \(m\) is the denominator order, and \(n\) is the numerator order (\(m\geq n\)). Generally, the robustness of the DOB improves as the denominator order increases. For example, if a \(Q_{31}\) filter is used instead of a \(Q_{20}\) filter, the robustness of the DOB system will improve. In addition, control system stability shares a trade-off relationship with the bandwidth of the \(Q\) filter. Therefore, the bandwidth of the \(Q\) filter must be selected experimentally to maximize the stability of the control system. For this reason, this study uses the binomial \(Q_{20}\) and \(Q_{31}\) filters in the proposed DOB-based controller for FSM systems, as illustrated in Figure 7. This is because the stability of the control system can be estimated from the residual control error signal (CES), which is the experimental result. The use of \(Q_{20}\) and \(Q_{31}\) filters effectively improves the disturbance-suppression performances of the FSM control system. Moreover, the disturbance-suppression performances of the FSM control system are effective in the low-frequency region with bandwidths of 15 and 30 Hz. The Open-loop transfer functions of the aforementioned \(Q_{20}\) and \(Q_{31}\) filters with bandwidths of 15 and 30 Hz are expressed as follows [34]:  
\[Q_{20}(s) =\frac{1}{\tau^{2}s^{2}+2\tau s+1}=\frac{1}{4.6\times 10^{-5}s^{2}+1.4\times 10^{-2}s+1}\left[15\,\mathrm{Hz}\right] \tag{5}\] \[Q_{20}(s) =\frac{1}{\tau^{2}s^{2}+2\tau s+1}=\frac{1}{1.2\times 10^{-5}s^{2}+6.8 \times 10^{-3}s+1}\left[30\,\mathrm{Hz}\right]\] (6) \[Q_{31}(s) =\frac{3\tau s+1}{\tau^{3}s^{3}+3\tau^{2}s^{2}+3\tau s+1}=\frac{5.1\times 10^{-2}s+1}{4.9\times 10^{-6}s^{3}+8.7\times 10^{-4}s^{2}+5.1\times 10^{-2}s+1} \left[15\,\mathrm{Hz}\right]\] (7) \[Q_{31}(s) =\frac{3\tau s+1}{\tau^{3}s^{3}+3\tau^{2}s^{2}+3\tau s+1}=\frac{2.6 \times 10^{-2}s+1}{6.1\times 10^{-7}s^{3}+2.2\times 10^{-4}s^{2}+2.6\times 10^{-2}s+1} \left[30\,\mathrm{Hz}\right] \tag{8}\]As mentioned previously, the loop gain of the control system should be high in the frequency domain of dynamic disturbances.