## Appendix A Summary Fuzzy Logic
\[{}^{\alpha+}\mathrm{F}=\{\mathrm{x}\in\mathrm{X}:\mu_{\mathrm{F}}(\mathrm{x})>\alpha\} \tag{10}\]  
It is easy to see that:  
\[\mathrm{Core}(\mathrm{F})={}^{1}\mathrm{F};\mathrm{Supp}(\mathrm{F})={}^{0+} \mathrm{F} \tag{11}\]  
The crossover point of a fuzzy set F (defined on the set X) is a point x in X such that  
\[\mathrm{Crossover}(\mathrm{F})=\{\mathrm{x}\in\mathrm{X}:\mu_{\mathrm{F}}( \mathrm{x})=0.5\} \tag{12}\]  
Typical Membership Function: There are many methods of establishing membership functions to determine the membership grade of an x element on a fuzzy set F, such as: triangular membership function, trapezoidal form, Gaussian form, Sign form, Sigmoidal form, Campanulate form. In fuzzy logic control, the triangular and trapezoidal membership functions are often used. Operations on fuzzy sets: There are many operations on fuzzy sets, but in this paper we introduce three basic operations: unions, intersection of two fuzzy sets and fuzzy complementation. Consider two fuzzy sets A, B on the universe of discourse X. For a given element x of the X, these operations are introduced. Based on the definition of fuzzy union and intersection, there are many formulas that can be applied to calculate the membership function of the union or intersection of two fuzzy sets, such as the formula based on maximum rule, Lukasiewicz rule, direct sum formula, etc. Here, the formulas based on maximum and minimum rules are introduced:  
Unions (max rule):  
\[\mu_{\mathrm{A}\cup\mathrm{B}}(\mathrm{x})=\max\{\mu_{\mathrm{A}}(\mathrm{x}),\mu_{\mathrm{B}}(\mathrm{x})\}=\mu_{\mathrm{A}}(\mathrm{x})\vee\mu_{\mathrm{ B}}(\mathrm{x});\forall\mathrm{x}\in\mathrm{X} \tag{13}\]  
Intersection (min rule):  
\[\mu_{\mathrm{A}\cap\mathrm{B}}(\mathrm{x})=\min\{\mu_{\mathrm{A}}(\mathrm{x}),\mu_{\mathrm{B}}(\mathrm{x})\}=\mu_{\mathrm{A}}(\mathrm{x})\wedge\mu_{\mathrm{ B}}(\mathrm{x});\forall\mathrm{x}\in\mathrm{X} \tag{14}\]  
Complement:  
\[\mu_{\mathrm{A}}(\mathrm{x})=1-\mu_{\mathrm{A}}(\mathrm{x});\forall\mathrm{x}\in \mathrm{X} \tag{15}\]  
Linguistic variable: A linguistic variable is a variable that represents a physical quantity by words or sentences in natural or artificial language. A linguistic value is a natural language term which is derived using quantitative or qualitative reasoning. A fuzzy linguistic variable x will take values that are linguistic values. Thus, a variable can both be represented by physical values and linguistic values. For example, velocity variables can be represented by two value domains. The physical value domain:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]0  
The linguistic value domain:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]1  
Here, the universal set is the velocity set V, the fuzzy set \({}^{\mathrm{L}}\mathrm{V}\) is defined on set V. With a physical value x \(\in\mathrm{V}\), the membership function can be determined, corresponding to subsets of \({}^{\mathrm{L}}\mathrm{V}\), and represented by vector \(\mu\) as follow:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]2  
The above mapping is called the fuzzification process, in which the physical value of a variable is transformed into linguistic value. Fuzzy relations: A fuzzy relation R on Cartesian product of universal sets \(U_{1}\times U_{2}\times\ldots\times U_{n}\) is any fuzzy subset of \(U_{1}\times U_{2}\times\ldots\times U_{n}\). Thus, a fuzzy relation R is defined by a membership function  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]3  
If the Cartesian product is formed by just two sets \(U_{1}\times U_{2}\), the relation is called a binary fuzzy relation on \(U_{1}\times U_{2}\).