## 2 Integrated Modeling
### Structural Subsystem  
The structural subsystem starts with physical coordinates in second-order form in the time-domain. By transforming the differential equation into a state-space equation, the physical coordinates convert to mode coordinates, and the output is transformed into a first-order form so that it can be linked to other subsystems. The advantages of this approach are as follows. As a complex system with multiple degrees of freedom, a satellite is a Multiple-Input Multiple-Output (MIMO) system. A state-space equation can be simplified by the mode selection matrix and becomes a Single-Input Single-Output (SISO) system, and because it is transformed based on the dynamic differential equation, it satisfies observability and controllability. Establish satellite dynamic differential Equation (4):  
\[M\ddot{x}+C\dot{x}+Kx=T_{d}+T_{c} \tag{4}\]  
where \(M\) is the mass matrix, \(C\) is the damping matrix, \(K\) is the stiffness matrix, and \(x\) is the displacement of nodes in physical coordinates. Based on the finite element model and using the mode superposition method, Equation (4) is transformed from physical coordinates to mode coordinates form:  
\[\ddot{q}+2\xi\omega\dot{q}+\omega^{2}q=\varphi^{\rm T}(T_{d}+T_{c}) \tag{5}\]  
where \(q\) is the mode coordinates, the first \(r\) states are taken (the three translational rigid-body modes are not considered), \(\xi\) is the diagonal mode damping matrix, \(\omega\) is the diagonal frequency matrix, and \(\varphi^{\rm T}\) is the mode shape matrix normalized by the mass matrix. Equation (6) is Equation (5) described in a state-space equation. \[\left[ \begin{array}{c}\dot{\eta}\\ y_{att}\\ y_{jit}\\ \end{array} \right]=\left[ \begin{array}{ccc}A&B_{T_{c}}&B_{T_{d}}\\ C_{att}&D_{T_{d}\_att}&D_{T_{c}\_att}\\ C_{jit}&D_{T_{d}\_jit}&D_{T_{c}\_jit}\\ \end{array} \right]\left[ \begin{array}{c}\eta\\ u_{T_{d}}\\ u_{T_{c}}\\ \end{array} \right] \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Harmonic** \\ **Factor** \\ \end{tabular} } & \multirow{2}{*}{\(C_{ij}\)} & \multirow{2}{*}{\(p_{j}\)} & \multicolumn{2}{c}{**At \(-\)817.5 rpm**} & \multicolumn{2}{c}{**At 974.4 rpm**} & \multicolumn{2}{c}{**At 1136.5 rpm**} \\ \cline{3-8}  & & & **Frequency** & **Amplitude** & **Frequency** & **Amplitude** & **Frequency** & **Amplitude** \\  & & **(Hz)** & **(N\(\cdot\)mm)** & **(Hz)** & **(N\(\cdot\)mm)** & **(Hz)** & **(N\(\cdot\)mm)** \\ \hline
6.86 & \(2.61\times 10^{-7}\) & 2.43 & 93.47 & \(-1.54\times 10^{-4}\) & 111.41 & \(2.05\times 10^{-4}\) & 129.94 & \(3.08\times 10^{-4}\) \\
9.07 & \(2.49\times 10^{-7}\) & \(-1.20\) & 123.58 & \(-3.03\times 10^{-4}\) & 147.40 & \(1.03\times 10^{-4}\) & 171.80 & \(1.16\times 10^{-4}\) \\
13.19 & \(1.26\times 10^{-7}\) & \(-1.44\) & 179.71 & \(-6.18\times 10^{-5}\) & 214.21 & \(1.05\times 10^{-4}\) & 249.84 & \(1.64\times 10^{-4}\) \\
16.35 & \(2.90\times 10^{-7}\) & 0.64 & 222.77 & \(-1.40\times 10^{-4}\) & 265.52 & \(2.35\times 10^{-4}\) & 309.70 & \(3.91\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Harmonic-related parameter. Figure 2: Campbell diagram of disturbance torque. where the state vector \(\eta=\left[ \begin{array}{cc}\dot{q}&q\end{array} \right]^{T}\), \(y_{att}\) and \(y_{it}\) are the control output and the micro-vibration response output, respectively, \(u_{T_{d}}\) and \(u_{T_{c}}\) are the control input and the micro-vibration disturbance input, respectively. The system matrix expression is shown in Equation (7),  
\[\left[ \begin{array}{cccc}A&B_{T_{c}}&B_{T_{d}}\\ C_{att}&D_{T_{d}\_att}&D_{T_{c}\_att}\\ C_{jit}&D_{T_{d}\_jit}&D_{T_{c}\_jit}\end{array} \right]=\left[ \begin{array}[] {cccc}-\omega^{2}&-2\xi\omega&\phi^{T}\alpha_{T_{d}}&\phi^{T}\alpha_{T_{c}}\\ 0&I&0&0\\ 0&\phi\beta_{T_{d}}&0&0\\ 0&\phi\beta_{T_{d}}&0&0\end{array} \right] \tag{7}\]  
where \(\alpha\) and \(\beta\) are the mode selection matrices. The transmission characteristics of satellite structural dynamics in the time-domain can be obtained by Equation (6).