# A Performance Evaluation Approach for Satellite Attitude Control System in Tracking Mode
## 4 Establishment of Performance Indicator Evaluation Model
### Implementation Steps of Grey-Target Decision Assessment Model  
In the process of performance evaluation for satellite attitude control systems, the sources of performance indicators vary and exhibit characteristics of uncertainty in the form of grey system data. Traditional methods of performance evaluation would lead to inaccurate results. The fundamental idea behind a grey-target decision is to compare the proximity of each indicator sequence to an ideal sequence, often referred to as the target center. This method is commonly used to address multi-objective decision-making or assessment problems [25; 26; 27; 28]. Therefore, the grey-target bull's-eye distance of the work process indicator sequences can be calculated, and the bull's-eye distance can be used to characterize the comprehensive evaluation value. This approach facilitates the ranking of the performance of different working processes. Based on grey-target decision theory, this paper adopts the improved principal component method based on the generalized grey number to determine the weights, and establishes the grey-target evaluation model to evaluate the performance of the satellite attitude control system tracking mode. The specific methodological steps are as follows: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & **Pointing** & **Relative** & \multirow{2}{*}{**Stability**} & **Response** & **Angular** & **Energy** \\  & **Accuracy** & **Accuracy** & & **Time** & **Velocity** & **Consumption** \\ \hline Principal component & 0.0529 & 0.0063 & 0.0216 & 0.5072 & 0.3382 & 0.0737 \\ Improved principal component & 0.1198 & 0.2147 & 0.1292 & 0.1837 & 0.1249 & 0.2277 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of weight determination results. (1) Process the satellite telemetry data, extract the data of the day-to-day pointing mode, and solve the performance indicator of each work process according to the definition of the indicator, and pay attention to the problem of the reference coordinate system when calculating. (2) Normalize the solved index data to obtain the standardized index matrix \(A_{m\times n}=(a_{ij})\), where \(a_{ij}\) is the \(j\) processed efficiency index of the \(i\) group of attitude control processes. (3) The standardized data are greyed out and expressed by the generalized standard grey number. (4) Calculate the weights of the indicators according to the improved principal component weight model based on the generalized grey number, and the weights are calculated as \(W=\{\omega_{1}^{\otimes},\omega_{2}^{\otimes},\cdots,\omega_{n}^{\otimes}\}\). (5) Compare the size of each indicator of different control processes to obtain the optimal vector, the bull's eye \(G(\otimes_{0})\). The comparison of the size of the grey number is realized by comparing the size of the mathematical expectation, for the grey numbers \(G(\otimes_{1})\) and \(G(\otimes_{2})\), the size relationship is:  
\[\left\{ \begin{array}{ccc}G(\otimes_{1})\textgreater{}G(\otimes_{2})&E(G( \otimes_{1}))\textgreater{}E(G(\otimes_{2}))\\ G(\otimes_{1})\textless{}G(\otimes_{2})&E(G(\otimes_{1}))\textless{}E(G( \otimes_{2}))\\ G(\otimes_{1})=G(\otimes_{2})&E(G(\otimes_{1}))=E(G(\otimes_{2}))\\ \end{array} \right. \tag{24}\]  
According to Equation (23), the center of the target can be compared to obtain the center of the target as:  
\[G(\otimes_{0})=\{\max(G(\otimes_{i1})),\max(G(\otimes_{i2})),\cdots,\max(G( \otimes_{in}))\} \tag{25}\]  
(6) Calculate the off-target distance for each attitude control process data \(d(G(\otimes_{i}),G(\otimes_{0}))\), and the center-of-target distance can be calculated according to Equation (25):  
\[d(G(\otimes_{i}),G(\otimes_{0}))=\sqrt{\omega_{1}^{\otimes}(G(\otimes_{i1})-G (\otimes_{01}))^{2}+\cdots+\omega_{n}^{\otimes}(G(\otimes_{in})-G(\otimes_{0 n}))^{2}} \tag{26}\]  
In Equation (26), \(G(\otimes_{i})\) represents the index sequence of a control process.