## 3 Results and Discussion
### Active Control Analysis  
#### 3.2.1 Active Control System Composition and Algorithm Flow  
In the actual working condition, the amplitude ratio of the Optomechanical coupling system of the spatial filter changes, due to the change of environment, and the peak value of the output light angle response will deviate from the pointing threshold. To solve the pointing problem caused by the sudden change of environment, this paper actively adjusts the PI control parameters to control the peak value of the output light angle response within the pointing threshold. MATLAB data processing software is used to study the control effect of active control parameters \(g_{P},g_{I}\) and their joint control on the output light angle response peak value of the spatial filter system.  
As shown in Figure 12, the Optomechanical coupling active control system is composed of the Optomechanical system, exciter, sensor, control unit, power amplifier, and actuator. The specific algorithm flow is shown in Figure 13. The sensor detects the peak value of output light angle response in one cycle, and the detection result is compared with the target result. When the detection result is less than or equal to the target result, there is no need to start the actuator. When the detection result is greater than the target result, the deviation between the detection result and the target result is calculated. The controller is started to calculate the control parameter value with zero deviation according to the control algorithm, and then drives the actuator to work. The detection result is compared with the target result again. If the condition is met, the control requirement is met. The cycle control is continued until the result meets the requirement.  
According to Figure 12, the closed-loop transfer function can be expressed as  
\[G(s)=\frac{cLs^{2}+(cR+kL)s+kR}{mLs^{3}+(mR+cL)s^{2}+(cR+kL-K_{F}g_{P})s+(kR- K_{F}g_{I})} \tag{19}\]  
The characteristic equation of the system is  
\[mLs^{3}+(mR+cL)s^{2}+(cR+kL-K_{F}g_{P})s+(kR-K_{F}g_{I})=0 \tag{20}\]  
Figure 12: Optomechanical coupling active control system.  
According to Routh criterion, Routh table is  
\[\begin{array}{c|cc}s^{3}&mL&(cR+kL-K_{F}g_{P})&0\\ s^{2}&(mR+cL)&(kR-K_{F}g_{I})&0\\ s^{1}&\frac{(mR+cL)(cR+kL-K_{F}g_{P})-mL(kR-K_{F}g_{I})}{(mR+cL)}&0&0\\ s^{0}&(kR-K_{F}g_{I})&0&0\\ \end{array} \tag{21}\]  
The sufficient and necessary conditions for the stability of the closed-loop control system are as follows.  
1. The necessary condition for the stability of the system is that the coefficient of the characteristic equation is greater than zero. \[\left\{\begin{array}{c}cR+kL-K_{F}g_{P}>0\\ kR-K_{F}g_{I}>0\end{array}\right.\] (22)
2. System stability requires that the first column of the Routh table is greater than zero. \[\left\{\begin{array}{c}(mR+cL)(cR+kL-K_{F}g_{P})-mL(kR-K_{F}g_{I})>0\\ (kR-K_{F}g_{I})>0\end{array}\right.\] (23) Synthesis (1) and (2) the stability condition of the closed-loop system is \[0<g_{P}<\min\left\{\frac{cR+kL}{K_{F}}-\frac{mL(kR-K_{F}g_{I})}{K_{F}(mR+cL)},\frac{cR+kL}{K_{F}}\right\}\] (24) \[0<g_{I}<\frac{kR}{K_{F}}\] (25)  
As can be seen from Figure 13, by detecting the peak value of the output light angle response, the response peak value is compared with the pointing threshold value. If the threshold condition is not met, the output light angle response peak value is detected again by actively controlling the Optomechanical system through the corresponding relationship  
Figure 13: Algorithm flow of the spatial filter Optomechanical coupling active control system.  
between the output light angle response peak value and the control parameters, until the pointing threshold value is reached.