## 3 Fuzzy Fault Diagnosis Method
### Fuzzy Signal Processing Algorithm
\[A\circ B=\langle A_{L}\circ B_{L},A_{U}\circ B_{U}\rangle=\left\langle\mathop{ \vee}\limits_{i=1}^{n}(\underline{a}_{i}\wedge\underline{b}_{i}),\mathop{ \vee}\limits_{i=1}^{n}(\overline{a}_{i}\wedge\overline{b}_{i})\right\rangle \tag{7}\]  
When generalizing to the fuzzy signal set on any domain \(U\), suppose the fuzzy signal set \(X=\langle X_{L},X_{U}\rangle\) of the approximate space \((U,R)\). The fuzzy signal sets \(A=\langle A_{L},A_{U}\rangle\) and \(B=\langle B_{L},B_{U}\rangle\) on \(X\) belong to \(F(U)\). The inner product of \(A\) and \(B\) is shown in Formula (8). \[A\circ B=\langle A_{L}\circ B_{L},A_{U}\circ B_{U}\rangle=\left\langle\mathop{ \vee}\limits_{x\in X_{L}}(A_{L}(x)\wedge B_{L}(x),\mathop{\vee}\limits_{x\in X _{U}}(A_{U}(x)\wedge B_{U}(x))\right\rangle \tag{8}\]  
The dual operation of the inner product is an outer product. Suppose the fuzzy signal set \(X=\langle X_{L},X_{U}\rangle\) of the approximate space \((U,R)\). The fuzzy signal sets \(A=\langle A_{L},A_{U}\rangle\) and \(B=\langle B_{L},B_{U}\rangle\) on \(X\) belong to \(F(U)\). The outer product of \(A\) and \(B\) is shown in Formula (9). \[A\times B=\langle A_{L}\times B_{L},A_{U}\times B_{U}\rangle=\left\langle \mathop{\wedge}\limits_{x\in X_{L}}(A_{L}(x)\lor B_{L}(x)),\mathop{\wedge}\limits_{x \in X_{U}}(A_{U}(x)\lor B_{U}(x))\right\rangle \tag{9}\]Define the \"remainder\" operation on the closed interval \([0,1]\). For \(\forall\alpha_{1},\alpha_{2}\in[0,1]\), define the remainder of \(\alpha=\langle\alpha_{1},\alpha_{2}\rangle\) as \(\alpha^{c}=\langle\alpha_{1}^{c},\alpha_{2}^{c}\rangle=\langle 1-\alpha_{1},1- \alpha_{2}\rangle\). So, there is Formula (10). \[\left\{ \begin{array}{l}(A\times B)^{c}=A^{c}\circ B^{c}\\ (A\circ B)^{c}=A^{c}\times B^{c}\end{array} \right. \tag{10}\]  
Formula (9) is proved as follows:  
\[\begin{array}{rl}(A\times B)^{c}&=\left\langle 1-\underset{x\in X_{L}}{ \wedge}(A_{L}(x)\lor B_{L}(x)),\;1-\underset{x\in X_{U}}{\wedge}(A_{U}(x)\lor B _{U}(x))\right\rangle\\ &=\left\langle\underset{x\in X_{L}}{\vee}(1-(A_{L}(x)\lor B_{L}(x))),\; \underset{x\in X_{U}}{\vee}(1-(A_{U}(x)\lor B_{U}(x)))\right\rangle\\ &=\left\langle\underset{x\in X_{L}}{\vee}((1-A_{L}(x))\wedge(1-B_{L}(x))), \;\underset{x\in X_{U}}{\vee}((1-A_{U}(x))\wedge(1-B_{U}(x)))\right\rangle\\ &=\left\langle\underset{x\in X_{L}}{\vee}(A_{L}^{c}(x)\wedge B_{L}^{c}(x)), \;\underset{x\in X_{U}}{\vee}\left(A_{U}^{c}(x)\wedge B_{U}^{c}(x)\right) \right\rangle=\left\langle A_{L}^{c}\circ B_{L}^{c},\;A_{U}^{c}\circ B_{U}^{c }\right\rangle=A^{c}\circ B^{c}\\ &\text{Similar can prove }(A\circ B)^{c}=A^{c}\times B^{c}.\\ &\text{For }A=\langle A_{L},A_{U}\rangle\in F(U)\text{, let }\overline{a}=\left\langle \underset{x\in X_{L}}{\vee}A_{L}(x),\;\underset{x\in X_{U}}{\vee}A_{U}(x) \right\rangle\text{, }\underline{a}=\left\langle\underset{x\in X_{L}}{\wedge}A_{L}(x),\; \underset{x\in X_{U}}{\wedge}A_{U}(x)\right\rangle\text{. Where }\overline{a}\text{ and }\underline{a}\text{ are, respectively, called the upper approximation and the lower approximation of the closeness degree of }A\text{ on the fuzzy signal set. Suppose \(A,\;B\in F(U)\) are the set on fuzzy signal set \(X=\langle X_{L},X_{U}\rangle\). Let \((A,B)=(A\circ B)\wedge(A\times B)^{c}\), then the following Formula (12) is established. \[\left\{ \begin{array}{l}0\leq(A,B)\leq 1\\ (A,B)=(B,A)\\ (A,A)=\overline{a}\wedge\langle 1-\underline{a}\rangle,(U,\varphi)=0\\ A\subseteq B\subseteq C\Rightarrow(A,C)\leq(A,B)\wedge(B,C)\end{array} \right. \tag{12}\]  
Especially when \(\overline{a}=1\), \(\underline{a}=0\), then \((A,A)=1\). According to Formula (12) and the definition of the fuzzy closeness vector function. Suppose fuzzy signal set \(A,B\in F(U)\) on fuzzy signal set \(X=\langle X_{L},X_{U}\rangle\), then the product fuzzy closeness vector function of \((A,B)=(A\circ B)\wedge(A\times B)^{c}\) is shown in Equation (13). \[Nm(A,B)=(A\circ B)\wedge(A^{c}\circ B^{c}) \tag{13}\]