## 3 Fuzzy Fault Diagnosis Method
### Fuzzy Signal Processing Algorithm  
Define the fuzzy closeness vector function \(N\) on \(F(U)\). Suppose \(A,B,C\in F(U)\), \(L=[0,1]\), if the mapping \(N:F(U)\times F(U)\to L\times L\) satisfies the conditions:  
(1) \(N(A,B)=N(B,A)\Leftrightarrow N(A_{L},B_{L})=N(B_{L},A_{L})\), and \(N(A_{U},B_{U})=N(B_{U},A_{U})\);  
(2) \(N(A,A)=1\Leftrightarrow N(A_{L},A_{L})=N(A_{U},A_{U})=\langle 1,1\rangle \triangleq 0\), \(N(U,\varphi)=\langle 0,0\rangle\triangleq 0\), If \(A\
eq\varphi\), then \(N(A,A)>0\);  
(3) If \(A\subseteq B\subseteq C\), then \(N(A,C)\leq N(A,B)\wedge N(B,C)\Leftrightarrow N(A_{L},C_{L})\leq N(A_{L},B_{ L})\)\(\wedge B(B_{L},C_{L})\), and \(N(A_{U},C_{U})\leq N(A_{U},B_{U})\wedge N(B_{U},C_{U})\). Figure 5: Visualized results of fault diagnosis of the system. Then, \(N(A,B)\) is the fuzzy closeness vector function of the fuzzy signal sets \(A\) and \(B\); \(N\) is the fuzzy closeness vector function on \(F(U)\). According to the definition of the fuzzy closeness vector function and the actual needs of this article, two specific fuzzy signal mapping processing algorithms are given below. Difference algorithm: Define the difference fuzzy closeness vector function \(N_{d}\). If \(U=\{u_{1},u_{2},\cdots,u_{n}\}\), then the fuzzy closeness vector function of the difference between \(A\) and \(B\) is shown in Equation (4). \[N_{d}(A,B)\stackrel{{\Delta}}{{=}}1-\frac{1}{n}\sum\limits_{i=1}^ {n}|A(u_{i})-B(u_{i})| \tag{4}\]  
where \(A=\langle A_{L},A_{U}\rangle\) and \(B=\langle B_{L},B_{U}\rangle\) belong to Set \(F(U)\). \(A(u_{i})=\langle A_{L}(U_{i}),A_{U}(U_{i})\rangle\), \(B(u_{i})=\langle B_{L}(U_{i}),B_{U}(U_{i})\rangle\). The above content can be launched:  
\[ \begin{array}{ll}A(u_{i})-B(U_{i})&=\langle A_{L}(U_{i}),A_{U}(U_{i})\rangle -\langle B_{L}(U_{i}),B_{U}(U_{i})\rangle\\ &=\langle A_{L}(U_{i})-B_{L}(U_{i}),A_{U}(U_{i})-B_{U}(U_{i})\rangle\end{array}  \tag{5}\]  
The proof process of Formula (4) is as follows:  
\[ \begin{array}{l}(1)\;N(A,B)=1-\frac{1}{n}\sum\limits_{i=1}^{n}|A(u_{i})-B(u_ {i})|=1-\frac{1}{n}\sum\limits_{i=1}^{n}|B(u_{i})-A(u_{i})|=N(B,A),\\ (2)\;N(A,A)\;=\;1-\frac{1}{n}\sum\limits_{i=1}^{n}|A(u_{i})-A(u_{i})|=1,\;N(U, \varphi)\;=\;1-\frac{1}{n}\sum\limits_{i=1}^{n}|U(u_{i})-\varphi(u_{i})|\\ =1-1=0,\\ (3)\;\mbox{If }A\subseteq B\subseteq C\mbox{, then }|A(u_{i})-C(u_{i})|\geq|A(u_{i})-B(u_{i})|\mbox{ and }|A(u_{i})-C(u_{i})|\geq|B(u_{i})-C(u_{i})|,\; \;\;\mbox{therefore, }\;\;\;1-\frac{1}{n}\sum\limits_{i=1}^{n}|A(u_{i})-C(u_{i})|\leq 1- \frac{1}{n}\sum\limits_{i=1}^{n}|A(u_{i})-B(u_{i})|\;\;\;\mbox{ and }\\ 1-\frac{1}{n}\sum\limits_{i=1}^{n}|A(u_{i})-C(u_{i})|\leq 1-\frac{1}{n}\sum \limits_{i=1}^{n}|B(u_{i})-C(u_{i})|\mbox{. So, }N(A,C)\leq N(A,B)\wedge N(B,C).\end{array} \]  
If \(U\) is a closed domain \(\Omega\) on the real number domain, then the vector function of the fuzzy closeness of the difference between \(A\) and \(B\) is shown in Formula (6). \[N_{d}(A,B)=1-\frac{1}{|\Omega|}\int_{\Omega}|A(u)-B(u)|du \tag{6}\]  
Among them, \(|\Omega|\) is the measure of \(\Omega\), such as length, width, volume. The proof process of Formula (6) is similar to that of Formula (4), so it will not be proved any more. Product algorithm: Define the product fuzzy closeness vector function \(N_{m}\). Suppose finite domain \(U\), \(A=\langle A_{L},A_{U}\rangle\) and \(B=\langle B_{L},B_{U}\rangle\) belong to Set \(F(U)\). Where \(A_{L}=(\underline{a}_{1},\underline{a}_{2},\cdots,\underline{a}_{n})\) and \(A_{U}=(\overline{a}_{1},\overline{a}_{2},\cdots,\overline{a}_{n})\); \(B_{L}=(\underline{b}_{1},\underline{b}_{2},\cdots,\underline{b}_{n})\), \(B_{U}=(\overline{b}_{1},\overline{b}_{2},\cdots,\overline{b}_{n})\). The inner product of \(A\) and \(B\) on the fuzzy signal set is shown in Formula (7).