# Hierarchical Model Updating Method for Vector Electric-Propulsion Satellites
## 2 Problem Formulation
### Measurement Equations and Telemetry Variables  
The measurement equations for the GPS' measured position and velocity are  
\[\mathbf{r}_{GPS}=\mathbf{r}+N(\mu_{GPS,r},\sigma_{GPS,r}) \tag{26}\]  
\[\mathbf{v}_{GPS}=\mathbf{v}+N(\mu_{GPS,\varphi},\sigma_{GPS,r}) \tag{27}\]  
where \(\mathbf{r}_{GPS}=\left[r_{xGPS},r_{yGPS},r_{zGPS}\right]^{T}\) is the position measurement, and \(\mathbf{v}_{GPS}=\left[v_{xGPS},v_{yGPS},v_{zGPS}\right]^{T}\) is the velocity measurement.  
The error transmission process of the star sensor is shown in Reference [23], and the measurement is \(\mathbf{Q}_{StarSensor}\). According to \(\mathbf{r}_{GPS}\), \(\mathbf{v}_{GPS}\), and \(\mathbf{Q}_{StarSensor}\), the Euler angles \(\varphi_{m},\theta_{m},\psi_{m}\) can be calculated.  
The measurement equations for the gyro measuring angular velocity are  
\[\left\{\begin{array}{l}\mathbf{\omega}_{Gyro}=\mathbf{\omega}_{real}+\mu_{Gyro, \omega}+\varepsilon_{Gyro,\omega}\\ \mathbf{\omega}_{real}=\mathbf{r}_{install}\cdot\mathbf{\omega}\end{array}\right. \tag{28}\]  
where \(\mathbf{r}_{install}\) is the install vector of gyro, \(\sigma_{Gyro,\omega}\) is the variance of random error, and the measured angular velocity component is \(\mathbf{\omega}_{Gyro}=\left[\omega_{xGyro},\omega_{yGyro},\omega_{zGyro}\right]^{T}\).  
The state variables of the power subsystem are measured by the corresponding sensors, and the measurements follow a normal distribution:  
\[\theta_{i,SunSensor}=\theta_{SW_{i}}+N\big{(}\mu_{sensor,\beta_{SW}},\sigma_{ sensor,\beta_{SW}}\big{)},i=1,2 \tag{29}\]\[P_{Wing_{i,m}}=P_{Wing_{i}}+N\left(\mu_{sensor,P_{wing}},\sigma_{sensor,P_{wing}} \right),i=1,2 \tag{30}\]  
\[U_{Wing_{i,m}}=U_{Wing_{i}}+N\left(\mu_{sensor,U_{wing_{i}}},\sigma_{sensor,U_{wing }}\right),i=1,2 \tag{31}\]  
\[I_{Wing_{i,m}}=I_{Wing_{i}}+N\left(\mu_{sensor,U_{wing_{i}}},\sigma_{sensor,U_{wing }}\right),i=1,2 \tag{32}\]  
\[Q_{Battery,m}=Q_{Battery}+N\left(\mu_{sensor,Q_{Battery}},\sigma_{sensor,Q_{ Battery}}\right) \tag{33}\]  
\[P_{Battery,m}=P_{Battery}+N\left(\mu_{sensor,P_{battery}},\sigma_{sensor,P_{ battery}}\right) \tag{34}\]  
\[I_{Battery,m}=I_{Battery}+N\left(\mu_{sensor,I_{Battery}},\sigma_{sensor,I_{ Battery}}\right) \tag{35}\]  
\[U_{Battery,m}=U_{Battery}+N\left(\mu_{sensor,U_{Battery}},\sigma_{sensor,U_{ Battery}}\right) \tag{36}\]  
where \(N(,)\) is normal distribution, and \(\mu_{sensor},\sigma_{sensor}\) represent the mean and variance of the noise measured by the sensors, respectively.  
The measurement equation of the propulsion subsystem is  
\[P_{Xc,m}=P_{Xc}+N(\mu_{sensor,P},\sigma_{sensor,P}) \tag{37}\]  
where \(P_{Xc,m}\) is the pressure measurement of the xenon tank.  
The thermal control subsystem uses the temperature sensor to measure the temperature of the node, and the measurement equation is  
\[T_{i,m}=T_{i}+N(\mu_{sensor,T},\sigma_{sensor,T}) \tag{38}\]  
The data on board is transmitted to the ground through the telemetry remote control subsystem, and the transmission process is affected by the relative position of the receiving and sending terminal, performance, data compression, transmission loss, and other factors, the basic transport equation in logarithmic form is expressed as  
\[P_{Receiver}=P_{Sender}+G_{Sender}+G_{Receiver}-L_{Space}-L_{x} \tag{39}\]  
where \(P_{Receiver}\) is the power of the receiver antenna, \(P_{Sender}\) is the sender power, \(G_{Sender}\) is the gain of the sender antenna, \(G_{Receiver}\) is the gain of the receiver antenna,\(L_{Space}\) is the path loss, and \(L_{x}\) is the other loss; for a detailed calculation, see Reference [27].  
In summary, \(\mathbf{y}\) consists of  
\[\begin{array}{ll}\mathbf{y}=&[r_{xGPS},P_{yGPS},r_{xGPS},v_{xGPS},v_{yGPS},v_{ zGPS},\phi_{m},\theta_{m},\psi_{m},\omega_{xGyro},\omega_{yGyro},\omega_{zGyro},\\ &\theta_{SW1,SunSensor},\theta_{SW2,SunSensor},P_{Wing_{1,m}},P_{Wing_{2,m}},U _{Wing_{1,n}},U_{Wing_{2,n}},I_{Wing_{1,m}},I_{Wing_{2,m}},\\ &Q_{Battery},P_{Battery},U_{Battery},I_{Battery},P_{Xc,m},T_{i,m}]\end{array} \tag{40}\]