# Re-Optimized Design for the ADR-Based Dual-Loop Controller of 100 m Class Aperture Radio Telescope
## 2 Multiple-Motor-Driven Antenna Servo System Modeling
\[M\ddot{q}+\mathbb{C}\dot{q}+Kq=F(u), \tag{5}\]  
\[L=E_{\rm k}-E_{\rm P}, \tag{6}\]  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{j}}\right)-\frac{\partial L }{\partial\dot{q}_{j}}+\frac{\partial D_{\rm b}}{\partial\dot{q}_{j}}=F_{j}(u), \tag{7}\]  
where \(M\), \(\mathbb{C}\), \(K\), \(F\) denote the system's mass matrix, damping matrix, stiffness matrix, and the generalized input forces, respectively, and \(j,j=1,2,3,4\), represents the number of the generalized coordinate components. Simplifying Equation (7), the following three equations can be obtained:  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)=\frac{d}{dt}\frac {\partial E_{\rm k}}{\partial q}=\left[ \begin{array}{cccc}J_{\rm d}&0&0&0\\ 0&J_{\rm b}&0&0\\ 0&0&n\left(J_{\rm t}+R^{2}kJ_{\rm w}\right)&0\\ 0&0&nK\bar{J}_{\rm m}\end{array} \right]\left[ \begin{array}{c}\theta_{\rm d} \\ \theta_{\rm b}\\ \theta_{\rm t}\\ \theta_{\rm g}\end{array} \right]=M\dot{q}, \tag{8}\]  
\[\frac{\partial L}{\partial q}=-\frac{\partial E_{\rm P}}{\partial q}=\left[ \begin{array}{cccc}K_{\rm s}&-K_{\rm s}&0&0\\ -K_{\rm s}&(K_{\rm s}+nK_{\rm b})&-nK_{\rm b}&0\\ 0&-nK_{\rm b}&n\left(K_{\rm b}+R^{2}kK_{\rm m}\right)&-nkRK_{\rm m}\\ 0&0&-nRkK_{\rm m}&nkK_{\rm m}\end{array} \right]\left[ \begin{array}{c}\theta_ {\rm d}\\ \theta_{\rm b}\\ \theta_{\rm t}\\ \theta_{\rm g}\end{array} \right]=Kq, \tag{9}\]  
\[\frac{\partial D_{\rm b}}{\partial\dot{q}}=\left[ \begin{array}{cccc}B_{\rm s }&-B_{\rm s}&0&0\\ -B_{\rm s}&B_{\rm s}+nB_{\rm b}&-nB_{\rm b}&0\\ 0&-nB_{\rm b}&n\left(B_{\rm b}+R^{2}kB_{\rm m}\right)&-nkRB_{\rm m}\\ 0&0&-nkRB_{\rm m}&nkB_{\rm m}\end{array} \right]\left[ \begin{array}{c}\theta_ {\rm d}\\ \theta_{\rm b}\\ \theta_{\rm t}\\ \theta_{\rm g}\end{array} \right]=\mathbb{C}\dot{q}, \tag{10}\]  
The input vector of the system is  
\[F(u)=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ nk\end{array} \right]T_{m}=\left[ \begin{array}{c}0\\ 0\\ 0\\ nkK_{\rm t}\end{array} \right]i=\Re u, \tag{11}\]  
where \(T_{\rm m}\) is the motor's output torque, \(K_{\rm t}\) is the torque constant, \(\Re\) is the input matrix, and \(i\) is the input current. The system's dynamic equation can be as shown in Equation (12) by rectifying. \[\ddot{q}=-M^{-1}\mathbb{C}-M^{-1}Kq+M^{-1}\Re u, \tag{12}\]  
Choosing \(x=\left[ \begin{array}{cc}\dot{q}&q\end{array} \right]^{T}\) as the system states and letting \(C\) be the output matrix, and \(y=Cx\) be the system outputs, then the state equation of the antenna servo system can be denoted as follows:  
\[\dot{x} = \left[ \begin{array}{cc}-M^{-1}\mathcal{C}&-M^{-1}K\\ I&0\end{array} \right]x+\left[ \begin{array}{c}M^{-1}\Re\\ 0\end{array} \right]u, \tag{13}\] \[= Ax+Bu\] \[y = Cx\]  
The azimuthal system's system matrix \(A\) and the input matrix \(B\) of the radio telescope are shown in Equation (14). \[A=\left[ \begin{array}{ccccccccc}-\frac{R_{\rm t}}{I_{\rm d}}&\frac{\rm F_{\rm d}}{I_ {\rm d}}&0&0&-\frac{K_{\rm t}}{I_{\rm d}}&\frac{K_{\rm t}}{I_{\rm d}}&0&0\\ \frac{R_{\rm t}}{I_{\rm b}}&-\frac{(R_{\rm t}+R_{\rm b})}{I_{\rm b}}&\frac{nR_{ \rm t}}{I_{\rm b}}&0&\frac{K_{\rm t}}{I_{\rm b}}&-\frac{(K_{\rm t}^{2}+R_{\rm t })}{I_{\rm b}}&\frac{nR_{\rm t}}{I_{\rm b}}&0\\ 0&\frac{R_{\rm t}}{I_{\rm b}}&-\frac{(R_{\rm t}^{2}+R_{\rm t})}{I_{\rm b}}& \frac{R_{\rm t}R_{\rm b}}{I_{\rm b}}&0&\frac{K_{\rm t}}{I_{\rm b}}&-\frac{(R_{ \rm t}^{2}+R_{\rm t}+R_{\rm b})}{I_{\rm b}}&\frac{nR_{\rm t}}{I_{\rm b}}\\ 0&0&\frac{R_{\rm t}}{I_{\rm m}}&-\frac{R_{\rm t}}{I_{\rm m}}&0&0&\frac{R_{\rm t }}{I_{\rm m}}&-\frac{R_{\rm t}}{I_{\rm m}}\\ 1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ \end{array} \right], \tag{14}\]  
where \(\bar{J}=J_{\rm t}+kR^{2}J_{\rm w}\). Equation (13) shows that the simplified antenna azimuthal model in Figure 1 is an eighth-order system. When the number \(k\) of motors in each driving arm is one, the model shown in Equation (14) becomes the single-motor-driven model, as shown in [19].