## 4 Application in Electro-Optical Tracking System  
The electro-optical tracking device is mainly composed of the main control, servo system, sensors, data acquisition and forwarding unit. After receiving the target position signal sent by the control center unit, the servo system drives the motor to perform set-point tracking. As the system pointing error decreases, the target will gradually appear on the image sensor. The schematic diagram of the servo system is shown in Figure 6. It consists of the azimuth axis and the elevation axis, and is fixed on the ground. The servo system achieves the purpose of fast, smooth, and accurate set-point tracking by controlling the azimuth and elevation angles of the device. As the azimuth axis and the elevation axis have the same control mode, and the control method of the positive direction and the opposite direction of each axis is exactly the same, this paper takes the azimuth axis forward control as an example to analyze the control method of the electro-optical tracking system. Thereby, the range of the servo system target position signal is \((0^{\circ}\), \(180^{\circ}]\). To improve the rigidity of the inner loop [3], the microelectromechanical system's (MEMS) gyroscope is installed in the electro-optical servo system for speed measurement, which uses the speed signal to achieve a speed closed-loop. Therefore, the controlled object is an integral element connecting with a first-order inertia element, such as in Equation (30). \[ \begin{cases}\dot{x}=Ax+Bu_{t}\\ y=Cx\end{cases}  \tag{30}\]  
with  
\[ \begin{split} x=\begin{pmatrix}y\\ v\end{pmatrix},A=\left[\begin{array}{cc}0&1\\ 0&-\frac{1}{T}\end{array}\right],B=\left[\begin{array}{cc}0\\ \frac{b}{T}\end{array}\right],\\ C=\left[\begin{array}{cc}1&0\end{array}\right],|u|\leqslant u_{\max},\end{split} \]  
where \(b\), \(T\) are the parameters of the controlled object. Then, the control scheme PTSTP is applied to the electro-optical servo system, and its expression for input section is  
Figure 6: The schematic diagram of electro-optical tracking system. \[r_{i}= \begin{cases}r_{tp},&|r_{A}|\leqslant r_{0}+y_{L}\\ r&|r_{A}|>r_{0}+y_{L}.\end{cases}  \tag{31}\]  
To ensure the consistency of the system type number when applying the linear controller, the proportional differential controller of Equation (26) becomes the proportional integral controller of Equation (32). The control section is  
\[ \begin{split}& u_{t}=\left\{\begin{array}{ll}u_{p\_small},&r_{i}=r_{tp }\\ u_{p\_wide},&r_{i}=r\end{array}\right.,\\ & u_{p\_small}=k_{p}e{+}k_{i}\int e\,\mathrm{d}t,\\ & u_{p\_wide}=\operatorname{sign}\left(u\right)\cdot\min\left\{u_{\max},|u_{ out}|\right\},\\ & u_{out}=\left\{\begin{array}{ll}k_{p}e{+}k_{i}\int e\,\mathrm{d}t,&|e| \leq y_{L}\\ \operatorname{sign}\left(e\right){\cdot}u_{\max},&|e|>y_{L}\end{array}\right., \\ & y_{L}=\frac{u_{\max}}{k_{p}}\left(u_{\max}=v_{\max}\right), \end{split}  \tag{32}\]  
where \(k_{p},k_{i}\) represent the proportional coefficient and the integral coefficient, respectively, and are adjusted according to system stability and dynamic response [29]. As the motor speed is limited in the real system, it is tentatively set \(r_{0}\) to the angle at which the system speed is increased from \(0\) to \(v_{\max}\). Comparing the control law of single integral and double integral, it is found that Equation (31) has one small difference with Equation (25). When the amplitude of input signal is slightly bigger than \(y_{L}\), the control law of \(u_{p\_wide}\) will bring some chattering to the system, and the operation of _when_\(|r_{A}|\leqslant r_{0}+y_{L},r_{i}=r_{tp}\) will increase the elastic space of \(r_{0}\). The experimental platform of the electro-optical servo system is shown in Figure 7, which consists of a MEMS gyroscope, an encoder signal processing board, and a permanent magnet DC torque motor.