## Appendix A Summary Fuzzy Logic
A relation R between fuzzy sets A and B denotes composition of A and B  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]4  
Composition of fuzzy relations R and S: Let R, S, T be fuzzy relations on the Cartesian spaces \(X\times Y\), \(Y\times Z\), and \(X\times Z\), respectively, \(x\in X\), \(y\in Y\), \(z\in Z\). Union:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]5  
Intersection:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]6  
Complement:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]7  
Max-Min composition:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]8  
Max-Product composition:  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\]9  
Max-Average composition:  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]0  
Fuzzy rules and implications: The fuzzy implication (also known as fuzzy rule, fuzzy If-Then rule, or fuzzy conditional statement) is defined in form: \"_If x is A then y is B\"_, where A and B are two linguistic variables defined by fuzzy sets A and B on the universal sets X and Y, respectively. There are two well-known fuzzy implication rules, widely used in fuzzy system and fuzzy control:  
Min operation rule (Mamdani implication rule): The fuzzy membership function of fuzzy implication  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]1  
Product operation rule (Larsen):  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]2  
Fuzzy compositional rules: In this paper, the method Mamdani is introduced. The method Mamdani uses a minimum operator as a fuzzy implication operator, and max-min operator for the composition. Suppose fuzzy rule is Single-Input/Single-Output (SISO) Rule Base as R: if x is A then z is C, infer result C' for another rule if x is A' then z is C', where A, C, A' are known. When input data is a singleton such as A' = x0,\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]3  
When input data is fuzzy set \(A^{\prime}\),  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]4  
The \(\alpha\) is called the \"matching degree\", \"satisfaction degree\", or \"firing strength\". Multi-Input/Single-Output (MISO) Rule Base:  
\(R_{i}\): if \(x\) is \(A_{i}\) and \(y\) is \(B_{i}\) then \(z\) is \(C_{i}\), \(i=1,2,\ldots,n\). Where \(A_{i}\), \(B_{i}\), \(C_{i}\), \(i=1,2,\ldots,n\) and \(A^{\prime}\), \(B^{\prime}\) are known, the output \(C^{\prime}\) is obtained as follow:  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]5  
When input data are singletons such as \(x=x_{0}\) and \(y=y_{0}\),  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]6  
When input data are fuzzy sets, \(A^{\prime}\) and \(B^{\prime}\),  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]7  
Defuzzification: In practical applications, the control commands work by crisp value. Therefore it is necessary to defuzzify the result of the fuzzy inference. Defuzzification is the process where the mapping is done to convert the fuzzy results into crisp values. There are many methods for defuzzification technique. This paper introduces two well-known and widely used methods in fuzzy system and fuzzy control. Center of area method (COA):  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\]8  
where, the first formula for the case \(y\) is continuous; \(S\) is the Support (defined domain) of fuzzy set \(C^{\prime}\); \(n\) is the number of quantization levels of the output; \(C\) is a fuzzy set defined on the output dimension (\(y\)).