## Appendix A Summary Fuzzy Logic  
Unlike the crisp control that works based on a crisp logic, i.e., binary logic, the fuzzy controller works based on fuzzy logic, implementing many-valued or infinite-valued logic. The basis of crisp logic is that the crisp set has rigid boundaries while the fuzzy logic depends on fuzzy set with soft boundaries. The fuzzy set enables us to represent the real world easily and closely to human thinking. Crisp logic and fuzzy logic are mathematical languages, crisp sets and fuzzy sets include objects, elements, etc., and they are the basis for the mathematical operations. In the theories of fuzzy set, fuzzy logic develops a pretty large mathematical basis for performing mathematical operations on fuzzy sets. Accordingly, firstly, the fuzzy set elements are assigned a linguistic variable in which the linguistic values represented by the element's membership function on the fuzzy set. Based on the universe of discourse, by quantitative or qualitative inferences, the linguistic value of a linguistic variable is determined. In fuzzy control, this process is called fuzzification process. In order for fuzzy inference to be performed with linguistic variables on fuzzy sets, the tools are developed. Firstly, characteristic parameters of fuzzy set; membership function to determine membership grade of a variable of an element in a fuzzy set. The Inferences: fuzzy relations; fuzzy rule base; Compositional Rule of Inferences... In fuzzy control, these tools are used to perform two main functions: implication and composition of fuzzy rules to come to desired results. This process plays an important role in the fuzzy controller (Appendix B and Figure A1). The physical system only receives and performs commands by physical values, therefore, defuzzification is the process of converting linguistic values into physical (crisp) values based on fuzzy inferences. Accordingly, the fuzzy controller operates in a process with three above-mentioned modules (Appendix B). Fuzzy logic is a language, a mathematical language, like any other language we know. The elements on which fuzzy logic depend is called fuzzy set and it is also alternatively called the fuzzy algebra. Fuzzy logic is essentially a mixture of various mathematical languages to define another new language as fuzzy logic. Let X be a non-empty set (called the universal set or the universe of discourse or simply domain), which is a set that has all possible members of a particular domain. A Fuzzy Set F is determined in universal set X in which each element is an ordered pair (x, uF(x)), where: uF: X \(\rightarrow\) [0, 1]. Logical mapping uF is considered as membership function of fuzzy set F.  
Characteristic parameters of fuzzy set:  
The support (defined domain) of fuzzy set F (defined on the set X) is the crisp subset and defined as  
\[\text{Supp}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})>0\}\] (A1)  
The core (trusted domain) of a fuzzy set F (defined on the set X) is the crisp subset and defined as  
\[\text{Core}(\text{F})=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})=1\}\] (A2)  
The height (h) of fuzzy set F (defined on the set X) is the maximum membership function value attained by any point:  
\[\text{h}(\text{F})=\underset{\text{x}\in\text{X}}{\text{sup}}\text{u}_{\text{F }}(\text{F})\] (A3)  
An \(\alpha\)-cut of a fuzzy set F (defined on the set X) is a crisp set which its each membership function value in F are greater than or equal to \(\alpha\). \[{}^{\alpha}\text{F}=\{\text{x}\in\text{X}:\text{u}_{\text{F}}(\text{x})\geq \alpha\}\] (A4)A strong \(\alpha\)-cut of a fuzzy set F (defined on the set X) is a crisp set which its each membership function value in F are strictly greater than \(\alpha\).