## 4 A Fault Identification Model for a CPS Based on Grey Relational Analysis
### Establishment of an Identification Model for Fault-Influencing Factors  
According to the basic principle of grey correlation quantization proposed by Professor [PERSON] [37; 38; 39], it is necessary to determine the reference sequence and the comparison sequence before calculating the grey correlation degree. The reference consists of one or more data sequences that reflect the fault characteristics of the CPS. It can be composed of statistical data that characterize the fault characteristics of the CPS. Each set of data can reflect the changes in the fault of the CPS. The reference sequence of the identification model for the influencing factors of the CPS can be expressed as follows:  
\[Y_{i}=\big{[}y_{i}(1)\quad y_{i}(2)\quad\dots\quad y_{i}(k)\quad\dots\quad y_{ i}(n)\big{]} \tag{1}\]  
where \(Y_{i}\) is the ith reference sequence in the grey relational analysis model of the influencing factors of the CPS fault. \(i=1,2,\dots,I\); \(\big{[}y_{i}(1)\quad y_{i}(2)\quad\dots\quad y_{i}(k)\quad\dots\quad y_{i}(n) \big{]}\) are the specific expressions of the \(i\)th reference sequence in the grey relational analysis model for the influencing factors of the CPS fault and reflect the changing law of \(Y_{i}\). \(n\) is the number of working conditions studied. This is the number of CPSs studied in this research. The comparison sequence consists of data characterizing the influencing factors of the CPS fault. There are m influencing factors of the fault of the CPS being studied. The vector matrix of the factors affecting the fault under n conditions is as follows:  
\[X_{i}= \begin{bmatrix}x_{i1}\\ x_{i2}\\ \vdots\\ x_{im}\end{bmatrix}\begin{bmatrix}x_{i1}(1)&x_{i1}(2)&\dots&x_{i1}(n)\\ x_{i2}(1)&x_{i2}(2)&\dots&x_{i2}(n)\\ \vdots&\vdots&&\vdots\\ x_{im}(1)&x_{im}(2)&\dots&x_{im}(n)\end{bmatrix}  \tag{2}\]  
where \(X_{i}\) is the comparison sequence corresponding to the reference sequence \(Y_{i}\) in the grey correlation analysis model for the influencing factors of the CPS fault. The sequence matrix contains m vectors of influencing factors, where the \(j\)th term in the \(k\)th condition is \(x_{ij}(k)\), \(j=1,2,\dots,m\), and \(k=1,2,\dots,n\). Since the physical meanings or the initial values of the data sequences are different, to increase the comparability of the data and reduce the errors caused by the physical meaning and difference of the research factors, it is necessary to perform dimensionless processing on the above sequences. Commonly used dimensionless methods include data initialization, data averaging, and data minimization [40]. In this study, the data averaging method is used to carry out the dimensionless processing of the raw data of the reference sequence and the comparison sequence in a fault classification system for a concrete dam production system. For the raw data of the reference sequence, the dimensionless formula can be expressed as follows (\(\overline{y_{i}(k)}\) is the average value of the data):  
\[y_{i}^{\prime}(k)=\left.y_{i}(k)\right/\overline{y_{i}(k)} \tag{3}\]  
For the raw data of the comparison sequence, the dimensionless formula can be expressed as follows (\(\overline{x_{ij}(k)}\) is the average value of the data):  
\[x_{ij}^{\prime}(k)=\left.x_{ij}(k)\right/\overline{x_{ij}(k)} \tag{4}\]  
The goal of a correlation analysis is to determine the degree of association between the comparison sequence and the reference sequence for each factor. The greater the correlation, the greater the contribution of these factors to the corresponding category production system faults. The smaller the correlation, the smaller the contribution of these factors to the corresponding category production system faults.