## 2 Problem Solve
### Tracking Performance of Equivalent Composite Control  
The controllers \(K_{P}(s)\) for the position loop and \(K_{v}(s)\) for the velocity loop in Figure 3 are designed in the form of Equation (5). Here, both the velocity and position loops are Type II systems, capable of achieving error-free tracking of the first derivative of the input signal. To facilitate subsequent formula derivations, calculations are simplified accordingly:  
\[K_{P}(s)=\frac{k_{p}(1+b_{1}s)}{s} \tag{7}\]  
\[K_{v}(s)=\frac{k_{v}{}^{\prime}(1+b_{2}s)}{s} \tag{8}\]  
Figure 3: Equivalent composite control architecture. Figure 2: Servo system of electro-optical equipment based on-axis tracking. The function of the controlled object \(G_{v}(s)\) is represented as follows:  
\[G_{v}(s)=\frac{C_{T}}{Js(1+a_{1}s)} \tag{9}\]  
where, \(k_{v}{}^{\prime}=Jk_{v}/C_{T}\) represents the true open-loop gain of the velocity loop, \(k_{p}\) is the open-loop gain of the position loop, \(C_{T}\) denotes the torque coefficient of the motor, and \(J\) is the moment of inertia of the tracking frame. The compensator controller \(K_{A}(s)\) is represented as shown in Equation (10), where \(a_{2}\) is the time coefficient of the compensator controller (\(a_{2}<1\)). It is essential to emphasize that although \(K_{A}(s)\) takes the form of a first-order low-pass filter, its significance does not reside in noise filtering, similar to the controller design approach in Reference [19]:  
\[K_{A}(s)=\frac{1}{1+a_{2}s} \tag{10}\]  
Based on the formulas above, we can derive the open-loop transfer function of the system, which is given specifically by the following formula:  
\[ \begin{split}& W_{Po}(s)=K_{P}(s)G_{p}(s)=\frac{k_{p}}{s^{3}a_{2}} \frac{1+d_{1}s+d_{2}s+d_{3}s}{1+c_{1}s+c_{2}s+c_{3}s}\\ & B(s)=1+(b_{1}+b_{2}+a_{2})s+[b_{1}b_{2}+a_{2}(b_{1}+b_{2})]s^{ 2}+a_{2}b_{1}b_{2}s^{3}\\ & A(s)=1+\frac{(a_{2}k_{v}b_{2}+1)}{a_{2}k_{v}}s+\frac{(a_{1}+a_{ 2})}{a_{2}k_{v}}s^{2}+\frac{a_{1}}{k_{v}}s^{3}\end{split}  \tag{11}\]  
From Equation (11), we observe that equivalent composite control elevates the order of the control system's open-loop transfer function. At this point, the velocity quality factor and acceleration quality factor of the electro-optical equipment are both represented as \(\infty\), while the jerk quality factor increases to \(1/a_{2}\) times the original acceleration quality factor. From a holistic design perspective, equivalent composite control can be viewed as a simpler implementation method of a Type III system. However, it is important to note that the compensator loop leads to changes in the pole locations of the closed-loop transfer function \(W_{Vc}\) of the velocity loop. \[W_{Vc}=\frac{v_{O}(s)}{v_{I}(s)}=\frac{K_{V}(s)G_{V}(s)}{[1+[1-K_{A}(s)]K_{V}( s)G_{V}(s)]} \tag{12}\]  
Examining the composite control structure depicted in Figure 4, the closed-loop transfer function of the position loop with the introduction of feedforward compensation channel can be expressed as shown in Equation (13):  
\[W_{Pc}(s)=\frac{\theta_{o}(s)}{\theta_{i}(s)}=\frac{[K_{F}(s)+K_{P}(s)]G_{P}( s)}{1+K_{P}(s)G_{P}(s)} \tag{13}\]  
where \(K_{F}(s)=s/(1+a_{2}s)\), the introduction of the feedforward compensation channel does not alter the poles of the closed-loop transfer function \(W_{Pc}(s)\). Figure 4: Composite control architecture. The open-loop transfer function of the composite control structure can be obtained using Equation (14):  
\[{W_{Po}}^{\prime}(s)=\frac{W_{Pc}(s)}{1-W_{Pc}(s)}=\frac{[K_{F}(s)+K_{P}(s)]G_{P}( s)}{1-K_{F}(s)G_{P}(s)} \tag{14}\]  
The expression for the open-loop transfer function \({W_{Po}}^{\prime}(s)\) of the composite control is given by  
\[{W_{Po}}^{\prime}(s)=\frac{k_{p}}{a_{2}s^{3}}\frac{B^{\prime}(s)}{A(s)}\quad{W _{Po}}^{\prime}(s)=\frac{k_{p}}{a_{2}s^{3}}\frac{1+{d_{1}}^{\prime}s+{d_{2}}^{ \prime}s+{d_{3}}^{\prime}s}{1+c_{1}s+c_{2}s+c_{3}s} \tag{15}\]  
Comparing Equation (10) with Equation (14), we observe that equivalent composite control and composite control have the same order and gain.