## 4 Application in Electro-Optical Tracking System
The DC torque motor parameters are shown in Table 1. The MEMS gyro is used to measure the angular speed, and the absolute rotary electro-optical encoder is applied to measure the angle value within the range of \([-180^{\circ},180^{\circ}]\). The MEMS gyroscope is applied to form an inertial closed loop, improving the characteristics of the controlled object and rejecting the nonlinearity, which also has been proved to be effective in stabilizing the system [30]. The position controlled object after the velocity closed loop is shown in Figure 8, in which the solid blue line indicates the measured curve, and the red dotted line represents the fitted curve. The fitted curve shows that it is the first-order inertial link series integral link. Applying the PTSTP control scheme to the electro-optical servo system, the parameters of the controlled object in Equation (30) are  
\[A =\left[ \begin{array}{cc}0&1\\ 0&-25.126\end{array} \right],B=\left[ \begin{array}{cc}0\\ 26.442\end{array} \right]\] \[C =\left[ \begin{array}{cc}1&0\end{array} \right],|u|\leqslant 92^{ \circ}/s.\]  
The parameter in the control law Equation (32) is \(k_{p}=11.876,k_{i}=6.15,y_{L}=7.75^{\circ},r_{0}=2.359^{\circ}\). The results of using the PTSTP control scheme for the WRSs in the electro-optical servo system is shown in Figure 9, in which the system position output is normalized. It can be seen that the overshoot can be made within \(\pm 2\%\). Figure 8: The Bode diagram of the controlled object. Figure 9: The wide-range signals (WRSs) set-point tracking normalization curve. Note that the settling time in Table 2 is the minimum time required for the system position signal to reach and remain within the final value of \(\pm 2\%\). Considering the limited accuracy of the encoder, the settling time of \(1^{\circ}\) is calculated using the minimum time required for the system position signal to reach and remain within the final values of \(\pm 10\%\). First, from subfigure (d) of Figures 10-12, when the system use PTOS, the saturation of current output from the actuator indicates that the windup phenomenon occurs from the beginning of the control process, which is the reason for the large position overshoot. However, the position curve of PTSTP is smoother than PTOS, and the windup phenomenon is avoided because of the addition of TP. As can be seen from subfigure (a) in Figure 10, the oscillation phenomenon occurs at the early stage of the control process, which is the reason for the operation when \(|r_{A}|\leqslant r_{0}+y_{L},r_{i}=r_{tp}\) in the control law Equation (32). It enhances the smoothness of the PTSTP control scheme applying to the single-integral series inertial objects in a certain extent. Second, as the amplitude of the input signal decreases, the overshoot of the PTOS algorithm becomes larger, even reaching an unacceptable level. However, the overshoot of PTSTP keeps near \(5\%\) in Table 2, and the settling time is obviously less than PTOS in Table 3. Although \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Signal Amplitude**} & \multicolumn{2}{c}{**Settling Time**} \\ \cline{2-3}  & **PTSTP** & **PTOS** \\ \hline \(r=10^{\circ}\) & 2.951 & 6.238 \\ \(r=5^{\circ}\) & 1.614 & 1.772 \\ \(r=1^{\circ}\) & 0.327 & 0.567 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Settling time comparison  
Figure 12: Comparison between PTSTP and PTOS tracking target position signal \(r=1^{\circ}\). \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Signal Amplitude**} & \multicolumn{2}{c}{**Overshoot**} \\ \cline{2-3}  & **PTSTP** & **PTOS** \\ \hline \(r=10^{\circ}\) & 5.211\% & 13.874\% \\ \(r=5^{\circ}\) & 5.023\% & 17.832\% \\ \(r=1^{\circ}\) & 5.116\% & 210.501\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Overshoot comparison the advantage of settling time is not particularly obvious when tracking \(5^{\circ}\) signal, from the overall performance point of view, PTSTP is more advanced than the PTOS control scheme.