# A Performance Evaluation Approach for Satellite Attitude Control System in Tracking Mode
## 4 Establishment of Performance Indicator Evaluation Model
### Improved Principal Component Weight Determination Method Based on Generalized Grey Numbers  
When evaluating the satellite attitude control system, the indicator weights are only related to the characteristics of the telemetry data itself, so it is more appropriate to use the objective weighting method to deal with the attitude control system work data. Principal component analysis (PCA) is a commonly used objective weighting method. Its fundamental concept involves transposing the initial indicator matrix, calculating several principal components along with their eigenvalues, and ultimately determining indicator weights by the product of variance contribution rates and principal components [23]. Let the set of evaluation indicators of the performance indicator system be \(X=\left\{x_{1},\cdots,x_{i},\cdots,x_{j},\cdots,x_{n}\right\}(i,j=1,2,\cdots,n)\), and the weights \(W=\left\{\omega_{1},\cdots,\omega_{i},\cdots,\omega_{j},\cdots,\omega_{n}\right\}\) are calculated by the principal component analysis method. The weight of \(j\) indicator can be expressed as follows [24]:  
\[\omega_{j}=\frac{\sum\limits_{i=1}^{n}\lambda_{i}g_{ji}}{\sum\limits_{i=1}^{n} \lambda_{i}} \tag{16}\]In Equation (16), \(\lambda_{i}\) represents the eigenvalue of the indicator covariance matrix, g represents the eigenvector of the covariance matrix and its corresponding eigenvector is \(\xi_{i}=[g_{1i},g_{2i},\cdots,g_{ni}]_{n\times 1}^{T}\), which can be obtained from the properties of eigenvalues and eigenvectors:  
\[ \begin{bmatrix}Cov(1,1)g_{1i}+Cov(1,2)g_{2i}+\cdots+Cov(1,n)g_{ni}\\ Cov(2,1)g_{1i}+Cov(2,2)g_{2i}+\cdots+Cov(2,n)g_{ni}\\ \cdots\\ Cov(n,1)g_{1i}+Cov(n,2)g_{2i}+\cdots+Cov(n,n)g_{ni}\end{bmatrix} = \begin{bmatrix} \lambda_{i}g_{1i}\\ \lambda_{i}g_{2i}\\ \cdots\\ \lambda_{i}g_{ni}\end{bmatrix}  \tag{17}\]  
In Equation (17), \(Cov(i,j)\) is the covariance between the indicators. The covariance is calculated as [24]:  
\[ \begin{array}{l}Cov(x_{i},x_{j})=E(x_{i},x_{j})-E(x_{i})\cdot E(x_{j})\\ =\left[E(a_{i},a_{j})-E(a_{i})\cdot E(a_{j})\right]+\left[E(a_{i},c_{j})-E(a_ {i})\cdot E(c_{j})\right]\cdot\gamma_{j}\\ +\left[E(a_{i},c_{i})-E(a_{i})\cdot E(c_{i})\right]\cdot\gamma_{i}+\left[E(c_{ i},c_{j})-E(c_{i})\cdot E(c_{j})\right]\cdot\gamma_{i}\gamma_{j}\end{array}  \tag{18}\]  
Therefore, the covariance matrix and eigenvectors based on generalized grey numbers can be calculated. Based on the correspondence of the matrices can be obtained:  
\[\lambda_{i}g_{ji}=Cov(j,1)g_{1i}+Cov(j,2)g_{2i}+\cdots+Cov(j,n)g_{ni} \tag{19}\]  
The expression for the indicator weights obtained from the principal component analysis can be converted into [24]:  
\[\omega_{j}=\frac{Cov(j,1)\sum\limits_{i=1}^{n}g_{1i}+Cov(j,2)\sum\limits_{i=1 }^{n}g_{2i}+\cdots+Cov(j,n)\sum\limits_{i=1}^{n}g_{ni}}{\sum\limits_{i=1}^{n} \lambda_{i}} \tag{20}\]  
In Equation (20), \(Cov(j,i)(i\
eq j)\) represents the moderator variable for the weight of the remaining indicators \(x_{j}\). The correlation coefficients and calculation results between the evaluation indicators of the satellite attitude control system for the daily pointing mode are shown in Table 1. The indicators of attitude control response time and average angular velocity during maneuvers exhibit relatively high correlation coefficients with the other indicators. In response to the issue of excessively large inter-indicator correlation coefficients in traditional principal component analysis (PCA) methods, this paper proposes an improved principal component weight calculation method based on the generalized grey number. The calculation expression of the improved principal component weights is as follows:  
\[\omega_{j}^{0}=|Cov(j,j)|\sum\limits_{i=1}^{n}\bigl{|}g_{ji}\bigr{|}+F(j,1) \sum\limits_{i=1}^{n}\bigl{|}g_{1i}\bigr{|}+\cdots+F(j,n)\sum\limits_{i=1}^{n} \bigl{|}g_{ni}\bigr{|} \tag{21}\]  
\[F(j,i)=\left\{ \begin{array}{l}(\max\bigl{|}Cov(j)\bigr{|}-\bigl{|}Cov(j,i) \bigr{|}/\bigl{|}Cov(j,i)\bigr{|})\\ 1\end{array} \right. \tag{22}\]