## 2 Problem Solve
### Impact of Equivalent Composite Control on the Velocity Loop  
The impact of equivalent composite control on the velocity loop is illustrated in Figure 11. The new closed-loop transfer function is given by  
\[\begin{split}& W_{Vc}(s)=\frac{\dot{\theta}^{\prime}_{i}(s)}{ \dot{\theta}_{o}(s)}=\frac{1}{a_{2}s}\frac{1+(b_{2}+a_{2})s+b_{2}a_{2}s^{2}}{A( s)}\\ & A(s)=1+\left(\frac{b_{2}a_{2}k_{v}+1}{a_{2}k_{v}}\right)s+ \left(\frac{a_{1}+a_{2}}{a_{2}k_{v}}\right)s^{2}+\frac{a_{1}}{k_{v}}s^{3}\end{split} \tag{21}\]  
The output \(\dot{\theta}_{o}(s)\) of \(W_{Vc}(s)\) remains as velocity. However, to ensure that the low-frequency output matches the true desired input \(\dot{\theta}^{\prime}_{o}(s)=a_{2}s\dot{\theta}_{i}(s)\), the input must be \(\dot{\theta}_{i}(s)\). This implies that after employing equivalent composite control, if the electro-optical equipment still requires the feedforward of the carrier's attitude angular velocity \(\dot{\theta}_{d}(t)\) (such as the rotational speed of a vehicle, ship, or aircraft), the velocity signal must be differentiated and scaled down by a factor of \(a_{2}\). Equivalent composite control has two effects on the inertial stabilization platform:  
1. Considering \(a_{2}\in[0.134,0.25]\) and the amplification effect of differentiation (differencing) on high-frequency noise, equivalent composite control imposes quite high demands on the resolution of the velocity gyroscope.
2. It is necessary to design filters to suppress high-frequency noise. A high bandwidth cannot effectively suppress the high-frequency noise of the gyroscope, while a low bandwidth essentially filters out useful signals. \(\dot{\theta}_{d}(t)\) represents a typical random signal, and classical filters struggle to achieve good results.  
We analyze the effect of gyroscope noise \(n_{g}(t)\). The transfer function is given by Equation (22). The transfer function without compensation is shown in Equation (23). Comparing the introduction of time constants \(a_{2}\) and \(a_{3}\) in the two equations reveals that the pole changes induced by their introduction make the velocity output more sensitive to high-frequency noise.  
\[T_{1}(s)=\frac{1}{1+a_{3}s}\frac{1+(b_{2}+a_{2})s+b_{2}a_{2}s^{2}}{A(s)} \tag{22}\]  
\[{T_{1}}^{\prime}(s)=\frac{1+b_{2}s}{1+b_{2}s+\frac{1}{k_{v}}s^{2}+\frac{a_{1}  
constant \(a_{2}\) on the poles, equivalent composite control also increases the system's gain by a factor of \(1/a_{2}\) and adds an integral term. This makes the velocity output more sensitive to low-frequency noise from the current sensor.  
\[T_{2}(s)=-\frac{C_{T}}{a_{2}Jk_{v}}\frac{1+a_{2}s}{A(s)} \tag{24}\]  
\[{T_{2}}^{\prime}(s)=-\frac{C_{T}}{Jk_{v}}\frac{s}{1+b_{2}s+\frac{1}{k_{v}}s^{2 }+\frac{a_{1}}{k_{v}}s^{3}} \tag{25}\]  
The transfer functions for torque disturbance \(\tau(t)\) and output \(\dot{\theta}_{o}(t)\) are given by Equations (26) and (27), respectively. The impact is similar to that of current loop noise, and therefore, further elaboration is unnecessary here.  
\[T_{3}(s)=\frac{1}{a_{2}Jk_{v}}\frac{1+(a_{1}+a_{2})s+a_{1}a_{2}s^{2}}{A(s)} \tag{26}\]  
\[{T_{3}}^{\prime}(s)=\frac{s}{Jk_{v}}\frac{1+a_{1}s}{1+b_{2}s+\frac{1}{k_{v}}s ^{2}+\frac{a_{1}}{k_{v}}s^{3}} \tag{27}\]  
The simulated velocity results are depicted in Figures 12-14. With \(a_{2}=0.1415\) and \(a_{3}=0.0531\), the simulation outcomes align with the results derived from theoretical analysis.  
Figure 12: The influence of gyro noise on the speed loop output.  
Figure 13: The influence of hall sensor noise on the speed loop output.