# Hierarchical Model Updating Method for Vector Electric-Propulsion Satellites
## 3 Method
### Least-Squares Method  
The range of telemetry variables is determined for the normalization of telemetry data and simulation data. The initial value of the parameter \(x_{0}\) to be adjusted is usually  
Figure 1: Hierarchical model updating diagram. given in a range \([x_{0,\min},x_{0,\max}]\); a Monte Carlo method is adopted to calculate the telemetry range \([y_{\min},y_{\max}]\) in this paper. For each \(x_{0}\), a random point is picked within its interval. All values make up a specific simulation condition. The simulation value \(\mathbf{y}\) is obtained after running, and their minimum and maximum could be calculated. Multiple simulation conditions are generated randomly in this way. By repeating this process, after each simulation conditions run, the range of all previous simulation conditions is counted; we call it the global minimum \(\mathbf{y}_{\min}\) and global maximum \(\mathbf{y}_{\max}\). The first condition  
\[y_{j,\min}=\min\Bigl{\{}y_{j}(t_{0}),y_{j}(t_{1}),\cdots,y_{j}\Bigl{(}t_{f} \Bigr{)}\Bigr{\}}_{1} \tag{41}\]  
\[y_{j,\max}=\max\Bigl{\{}y_{j}(t_{0}),y_{j}(t_{1}),\cdots,y_{j}\Bigl{(}t_{f} \Bigr{)}\Bigr{\}}_{1} \tag{42}\]  
For the \(r\)-th condition  
\[y_{j,\min}=\min\Bigl{\{}\min\Bigl{\{}y_{j}(t_{0}),y_{j}(t_{1}),\cdots,y_{j} \Bigl{(}t_{f}\Bigr{)}\Bigr{\}}_{r-1},y_{j,\min}\Bigr{\}} \tag{43}\]  
\[y_{j,\max}=\max\Bigl{\{}\max\Bigl{\{}y_{j}(t_{0}),y_{j}(t_{1}),\cdots,y_{j} \Bigl{(}t_{f}\Bigr{)}\Bigr{\}}_{r-1},y_{j,\max}\Bigr{\}} \tag{44}\]  
When there are enough working conditions, \(\mathbf{y}_{\min}\) and \(\mathbf{y}_{\max}\) tend to be two constant values, which is the value range of \(\mathbf{y}\). The least-squares method is used to update each group of variables. For the state variables and telemetry values in the \(k\)-th group as \(\mathbf{x}^{(k)}\) and \(\mathbf{y}^{(k)}\), the least-squares optimization objective is  
\[\min_{x_{0}^{(k)}}\,mse(k)=\frac{1}{2}\sum_{j=1}^{m}\sum_{i=0}^{f}\Bigl{[} \bar{y}_{j}^{(k),sim}(t_{i})-\bar{y}_{j}^{(k),lele}(t_{i})\Bigr{]}^{2} \tag{45}\]  
where \(\mathbf{x}_{0}^{(k)}\) is the state value at the time \(t_{0}\), which is adjusted directly. [PERSON]'s iteration method is usually used to solve the least-squares problems for nonlinear models. The initial value \(x_{0}^{(k,0)}\) of \(x_{0}^{(k)}\) is given, and the simulation value is \(\mathbf{y}^{(k,0),sim}\), which becomes \(\bar{\mathbf{y}}^{(k,0),sim}\) after normalization. Then, calculate the partial matrix of \(\mathbf{y}\) related to \(\mathbf{x}\) at point \(\mathbf{x}_{0}^{(k,0)}\), where \(\mathbf{y}\) should be normalized as well  
\[\mathbf{H}= \begin{bmatrix}\frac{\partial\bar{y}_{1}}{\partial x_{1}}&\frac{ \partial\bar{y}_{1}}{\partial x_{2}}&\cdots&\frac{\partial\bar{y}_{1}}{ \partial x_{n}}\\ \frac{\partial\bar{y}_{2}}{\partial x_{1}}&\frac{\partial\bar{y}_{2}}{\partial x _{2}}&\cdots&\frac{\partial\bar{y}_{2}}{\partial x_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\bar{y}_{m}}{\partial x_{1}}&\frac{\partial\bar{y}_{m}}{\partial x _{2}}&\cdots&\frac{\partial\bar{y}_{m}}{\partial x_{n}}\\ \end{bmatrix} _{x_{0}^{(k,0)}} \tag{46}\]  
For complex nonlinear models, it is common to use the difference quotient formula to displace the partial derivative, that is,  
\[\frac{\partial\bar{y}_{j}}{\partial x_{i}}\approx\lim_{\Delta x_{i}\to 0} \frac{\bar{\delta}_{j}(x_{j}+\Delta x_{i})-\bar{\delta}_{j}(x_{i})}{\Delta x_{ i}} \tag{47}\]  
where \(\bar{\delta}\) means the normalized simulation value, which is the same as Equation (3).