## 5 Case Study
### Identification and Calculation of Various Fault Factors
\(\overline{\Delta}=\frac{1}{3\times 4}\sum_{j=1}^{4}\sum_{k=1}^{3}\left|y^{ \prime}_{2}(k)-x^{\prime}_{j}(k)\right|=0.3488\), \(E_{\Delta}=\overline{\Delta}/\Delta_{max}=0.4986\), and since \(\Delta_{min}<3\overline{\Delta}\)= 0.9051, the data sequence is normal. Then, the range of the resolution coefficient can be determined as \(1.5E_{\Delta}\leq\rho\leq 2E_{\Delta}\), that is, \(0.8036\leq\rho\leq 1.0715\). In this research, \(\rho=1.75\) and \(E_{\Delta}=0.9375\). Substituting the above results into Equation (10), the grey correlation coefficient matrix for each influencing factor can be obtained, as follows:  
\[ \begin{bmatrix}\xi_{21}\\ \xi_{22}\\ \xi_{23}\\ \xi_{24}\end{bmatrix} = \begin{bmatrix}0.5626&0.6584&0.8660\\ 0.9351&0.8257&0.9363\\ 0.5400&0.5721&1.0000\\ 0.6477&0.5506&0.8559\end{bmatrix}  \tag{22}\]  
Substituting the above grey correlation coefficient matrix into Equation (5), the correlation coefficient \(r_{2j}\) of the human factor P, environmental factor E, mechanical component life factor M, and other factors O for a class B batch metering system fault can be obtained. \(j=1,2,3,4\), and \(r_{1}=[r_{21},r_{22},r_{23},r_{24}]=[0.6957,0.9081,0.7040,0.6847]\). It is obvious that \(r_{22}>r_{23}>r_{21}>r_{24}\). Therefore, the relationship between the environmental factor E and class B faults is large. It is indicated that class B faults are greatly affected by environmental factors. Corresponding measures should be taken to prevent the occurrence of a class B fault in the batch metering system. (3) The influence degrees of the four kinds of influencing factors on a class C fault, i.e., a pneumatic system fault, are studied. The reference sequence is \(y^{\prime}_{3}\) and the comparison sequences are \(x^{\prime}_{1}\), \(x^{\prime}_{2}\), \(x^{\prime}_{3}\), and \(x^{\prime}_{4}\). The coefficient matrix is given by:  
\[ \begin{bmatrix}y^{\prime}_{3}\\ x^{\prime}_{1}\\ x^{\prime}_{2}\\ x^{\prime}_{3}\\ x^{\prime}_{4}\end{bmatrix} = \begin{bmatrix}0.6098&1.4146&0.9756\\ 1.1642&1.0858&0.7500\\ 0.6818&1.2955&1.0227\\ 1.2149&0.9298&0.8554\\ 1.0051&0.8832&1.1117\end{bmatrix}  \tag{23}\]  
Substituting the coefficient matrix into Equation (9), the absolute difference matrix is obtained as follows:  
\[ \begin{bmatrix}\Delta_{31}\\ \Delta_{32}\\ \Delta_{33}\\ \Delta_{34}\end{bmatrix} = \begin{bmatrix}0.5544&0.3288&0.2256\\ 0.0721&0.1192&0.0471\\ 0.6051&0.4849&0.1202\\ 0.3953&0.5314&0.1361\end{bmatrix}  \tag{24}\]According to Equations (7) and (8), it is found that \(\Delta_{min}=0.0471\) and \(\Delta_{max}=0.6051\). Substituting the above absolute difference matrix into Equation (10), the mean of all absolute differences \(\overline{\Delta}\) can be obtained. Substituting the above results into Equation (10), the grey correlation coefficient matrix for each influencing factor can be obtained, as follows:  
\[ \begin{bmatrix}\xi_{31}\\ \xi_{32}\\ \xi_{33}\\ \xi_{34}\end{bmatrix} = \begin{bmatrix}0.5313&0.6712&0.7631\\ 0.9584&0.8886&1.0000\\ 0.5075&0.5678&0.8872\\ 0.6229&0.5429&0.8660\end{bmatrix}  \tag{25}\]  
Substituting the above grey correlation coefficient matrix into Equation (5), the correlation coefficient \(r_{3j}\) of the human factor P, environmental factor E, mechanical component life factor M, and other factors O for a class C fault, i.e., a pneumatic system fault, can be obtained. \(j=1,2,3,4\), and \(r_{1}=[r_{31},r_{32},r_{33},r_{34}]=[0.6552,0.9490,0.6542,0.6773]\). It is obvious that \(r_{32}>r_{34}>r_{31}>r_{33}\). Therefore, the relationship between the environmental factor E and class C faults is large. It is indicated that class C faults are greatly affected by environmental factors. Corresponding measures should be taken to prevent the occurrence of a class C fault in the pneumatic system. (4) The influence degrees of the four kinds of influencing factors on a class D fault are studied. For blender malfunction faults, the reference sequence is \(y^{\prime}_{4}\) and the comparison sequences are \(x^{\prime}_{1}\), \(x^{\prime}_{2}\), \(x^{\prime}_{3}\), and \(x^{\prime}_{4}\).