## 5 System Modeling and Simulation
### System Modeling of Capture Condition
\begin{array}{l}e_{3}=R_{3}\cdot f_{3}\\ e_{7}=R_{7}\cdot f_{7}\\ e_{12}=R_{12}\cdot f_{12}\\ e_{17}=R_{17}\cdot f_{17}\\ e_{20}=R_{20}\cdot f_{20}\\ e_{26}=R_{26}\cdot f_{26}\\ \end{array} \right.. \tag{26}\]  
The characteristic equation of Inertia Element and Capacitive Element \(I_{2}\), \(C_{14}\), \(I_{16}\), \(C_{22}\), and \(I_{25}\) is shown as follows:  
\[\left\{ \begin{array}{l}f_{2}=\frac{1}{I_{2}}p_{2}\\ e_{14}=\frac{1}{C_{14}}q_{14}\\ f_{16}=\frac{1}{I_{6}}p_{16}\\ e_{22}=\frac{1}{I_{25}}q_{22}\\ f_{25}=\frac{1}{I_{25}}p_{25}\\ \end{array} \right.. \tag{27}\]  
The characteristic equation of Inertial Elements \(I_{6}\), \(I_{11}\), and \(I_{19}\) with differential causality is shown as follows:  
\[\left\{ \begin{array}{l}e_{6}=I_{6}\cdot f_{6}^{\prime}\\ e_{11}=I_{11}\cdot f_{11}^{\prime}\\ e_{19}=I_{19}\cdot f_{19}^{\prime}\\ \end{array} \right.. \tag{28}\]  
The characteristic equations of Converter \(TF_{1}\), \(TF_{2}\), \(TF_{3}\), and \(TF_{4}\) are shown as follows:  
\[\left\{ \begin{array}{l}e_{4}=\frac{1}{m_{1}}\cdot e_{5}\\ f_{5}=\frac{1}{m_{1}}\cdot f_{4}\\ \end{array} \right., \tag{29}\]  
\[\left\{ \begin{array}{l}e_{8}=\frac{1}{m_{3}}\cdot e_{10}\\ f_{10}=\frac{1}{m_{2}}\cdot f_{8}\\ \end{array} \right., \tag{30}\]  
\[\left\{ \begin{array}{l}e_{9}=\frac{1}{m_{3}}\cdot e_{18}\\ f_{18}=\frac{1}{m_{3}}\cdot f_{9}\\ \end{array} \right., \tag{31}\]  
\[\left\{ \begin{array}{l}e_{24}=m_{4}\cdot e_{23}\\ f_{23}=m_{4}\cdot f_{24}\\ \end{array} \right.. \tag{32}\]  
The characteristic equations of \(0\) Knot and \(1\) Knot are shown as follows:  
\[\left\{ \begin{array}{l}e_{2}=Se_{1}-e_{3}-e_{4}\\ f_{1}=f_{2}=f_{3}=f_{4}\\ \end{array} \right., \tag{33}\]  
\[\left\{ \begin{array}{l}e_{5}=e_{6}+e_{7}+e_{8}+e_{9}\\ f_{5}=f_{6}=f_{7}=f_{8}=f_{9}\\ \end{array} \right., \tag{34}\]  
\[\left\{ \begin{array}{l}e_{10}=e_{11}+e_{12}+e_{13}\\ f_{10}=f_{11}=f_{12}=f_{13}\\ \end{array} \right., \tag{35}\]  
\[\left\{ \begin{array}{l}e_{13}=e_{14}=e_{15}\\ f_{14}=f_{13}-f_{15}\\ \end{array} \right., \tag{36}\]  
\[\left\{ \begin{array}{l}e_{15}=e_{16}+e_{17}\\ f_{15}=f_{16}=f_{17}\\ \end{array} \right., \tag{37}\]  
\[\left\{ \begin{array}{l}e_{18}=e_{19}+e_{20}+e_{21}\\ f_{18}=f_{19}=f_{20}=f_{21}\\ \end{array} \right., \tag{38}\]  
\[\left\{ \begin{array}{l}e_{21}=e_{22}=e_{23}\\ f_{22}=f_{21}-f_{23}\\ \end{array} \right., \tag{39}\]\[\left\{ \begin{array}{l}e_{24}=e_{25}+e_{26}\\ f_{24}=f_{25}=f_{26}\end{array} \right.. \tag{40}\]  
By sorting out and simplifying the above equations, the state equation of the system can be obtained, which is shown as follows:  
\[\left\{ \begin{array}{l}p_{2}^{\prime}=\frac{m_{1}^{2}m_{2}m_{3}l_{2}}{E}Se_{ 1}-\frac{m_{1}^{2}m_{2}m_{3}R_{3}+m_{3}R_{12}+m_{2}R_{20}}{E}p_{2}-\frac{m_{1} m_{2}m_{3}l_{2}}{C_{14}E}q_{14}-\frac{m_{1}m_{2}m_{3}l_{2}}{C_{22}E}q_{22}\\ q_{14}^{\prime}=\frac{1}{m_{1}m_{2}l_{2}}p_{2}-\frac{1}{l_{16}}p_{16}\\ p_{16}^{\prime}=\frac{1}{C_{14}}q_{14}-\frac{R_{17}}{l_{16}}p_{16}\\ q_{22}^{\prime}=\frac{1}{m_{1}m_{2}l_{2}}p_{2}-\frac{m_{4}}{l_{25}}p_{25}\\ p_{25}^{\prime}=\frac{m_{4}}{C_{22}}q_{22}-\frac{R_{26}}{l_{25}}p_{25}\end{array} \right. \tag{41}\]  
where \(E=m_{1}^{2}m_{2}m_{3}l_{2}+m_{2}m_{3}l_{6}+m_{2}m_{3}R_{7}+m_{3}I_{11}+m_{2}I_ {19}\). From the above equation, constant-coefficient matrix \(A_{1}\) and control matrix \(B_{1}\) can be obtained in the following form:  
\[A_{1}=\left[ \begin{array}{cccc}-\frac{m_{1}^{2}m_{2}m_{3}R_{3}+m_{3}R_{12}+m_ {2}R_{20}}{E}&-\frac{m_{1}m_{2}m_{3}l_{2}}{C_{14}E}&0&-\frac{m_{1}m_{2}m_{3}l_{ 2}}{C_{22}E}&0\\ \frac{1}{m_{1}m_{2}l_{2}}&0&-\frac{1}{l_{16}}&0&0\\ 0&\frac{1}{C_{14}}&-\frac{R_{17}}{l_{16}}&0&0\\ \frac{1}{m_{1}m_{3}l_{2}}&0&0&0&-\frac{m_{4}}{l_{25}}\\ 0&0&0&\frac{m_{4}}{C_{22}}&-\frac{R_{26}}{l_{25}}\end{array} \right] \tag{42}\]  
\[B_{1}=\left[ \begin{array}{cccc}\frac{m_{1}^{2}m_{2}m_{3}l_{2}}{E}&0&0&0&0\\ \end{array}