# Hierarchical Model Updating Method for Vector Electric-Propulsion Satellites
## 2 Problem Formulation
### State Equations and State Variables
\tag{12}\]  
where \(P_{MaxCharge}\) is the maximum charging power of the battery. The battery charges when the battery is not full and the output power is greater than the load power, and it discharges when the battery quantity is greater than the minimum quantity and the electric power is smaller than the load power. The process is  
\[P_{Battery}=\left\{ \begin{array}{ll}P_{Wing}-P_{Device}-P_{Loss},&Q_{MinBattery }<Q_{Battery}<Q_{MaxBattery}\\ 0&else\end{array} \right. \tag{13}\]  
where \(Q_{MinBattery}\) and \(Q_{MaxBattery}\) are the minimum and maximum battery capacity, respectively. The quantity change in the battery can be expressed as  
\[\frac{dQ_{Battery}}{dt}=\left\{ \begin{array}{ll}k_{Charge}\cdot P_{Battery},& P_{Battery}>0\\ k_{Discharge}\cdot P_{Battery},&P_{Battery}<0\end{array} \right. \tag{14}\]  
where \(Q_{Battery}\) is the remaining quantity, \(k_{Charge}\) is the charging coefficient, and \(k_{Discharge}\) is discharging coefficient. The battery voltage  
\[U_{Battery}=\left\{ \begin{array}{ll}U_{Charge},P_{Battery}>0\\ U_{Discharge},P_{Battery}<0\end{array} \right. \tag{15}\]  
where \(U_{Charge}\) is the charge voltage, and \(U_{Discharge}\) is the discharge voltage. The battery current is  
\[I_{Battery}=\frac{P_{Battery}}{U_{Battery}} \tag{16}\]The mass of xenon in the xenon tank is  
\[\frac{dm_{Xe}}{dt}=\dot{m}_{control} \tag{17}\]  
where \(\dot{m}_{control}\) is the mass rate controlled by the computer on board. Each thruster is connected to two xenon tanks by a pipeline with different regulating valves. Only when the thruster receives the fire command and the pipeline between the xenon tank and the thruster is open, the thruster is working. The electric thrust force generated is  
\[F_{T}=I_{sp}\cdot g\cdot\frac{dm_{Xe}}{dt} \tag{18}\]  
where \(I_{sp}\) is the specific impulse, and \(g\) is the gravitational acceleration. The gas state of the xenon satisfies the ideal gas equation of state. When the volume \(V_{Xe}\) is fixed, the pressure is  
\[P_{Xe}=\frac{R_{gas}T_{Xe}}{M_{Xe}V_{Xe}}\cdot m_{Xe} \tag{19}\]  
where \(M_{Xe}\) is the average molar mass of xenon, \(R_{gas}\) is the gas constant, and \(T_{Xe}\) is the temperature. The temperature field model is calculated by finite element mesh division of the satellite model with Ansys, and each grid represents a node. The temperature of node \(i\) can be calculated by the heat balance equation:  
\[\frac{dT_{i}}{dt}=\frac{q_{S}^{i}+q_{E}^{i}+q_{ER}^{i}+q_{Rad}^{i}+q_{Cond}^{ i}+P_{i}-q_{OS}^{i}-q_{IS}^{i}}{m_{i}c_{i}} \tag{20}\]  
where \(q_{S}^{i}\) is solar radiation received by node \(i\), \(q_{E}^{i}\) is The heat from the Earth's infrared radiation, \(q_{ER}^{i}\) is the Earth's albedo heat, \(q_{Rad}^{i}\) is the thermal radiation from other thermal nodes to node \(i\), \(q_{Cond}^{i}\) is heat conduction from other thermal nodes to node \(i\), \(P_{i}\) is total thermal power of all components in the range of node \(i\), \(q_{OS}^{i}\) is dissipate heat for the outer surface of node \(i\), \(q_{IS}^{i}\) is heat dissipation for the inner surface of node \(i\), \(m_{i}\) is mass of node \(i\), and \(c_{i}\) is specific heat capacity. The details are in Reference [25]. The thermal control unit model includes a temperature sensor and a heater. The power of the heater is  
\[P_{ht}=I_{ht}{}^{2}R_{ht}=\frac{U_{ht}{}^{2}}{R_{ht}} \tag{21}\]  
where \(I_{ht},U_{ht},R_{ht}\) are the current, voltage, and resistance of the heater, respectively. For the sensors, the error is assumed to follow a normal distribution, and the error characteristic is described by the mean value \(\mu_{sensor}\) and variance \(\sigma_{sensor}\), which are constant:  
\[ \begin{cases}\frac{dt_{sensor}}{dt}=0\\ \frac{d\sigma_{sensor}}{dt}=0\end{cases}  \tag{22}\]  
We include the variables \(\mu_{GPS,r},\sigma_{GPS,r}\) when the Global Positioning System (GPS) measures \(r\); the variables \(\mu_{GPS,r},\sigma_{GPS,r}\) when the GPS measures \(\
u\); and the variables \(H_{StarSensor},\sigma_{StarSensor}\) when the star sensor measures attitude.