## 4 Implementation of the Static Fuzzy Logic Controller (Static FLC)
### Implementation of the Fuzzy Inference Mechanism of the Static Fuzzy Logic Controller  
The proposed static FLC is mainly based on the Mamdani fuzzy approach. Through the fuzzification process, \(E_{k}\) and \(dE_{k}\) are fuzzified, and all aforementioned fuzzy sets are passed through the FIM. During this process of designing and developing the fuzzy rule base of the system, the input fuzzy sets belonging to \(E_{k}\) and \(dE_{k}\) are mapped into a different fuzzy set \(U_{k}\) (output fuzzy set (angular speed of the motor) with seven linguistic variables). This fuzzy relation is based on the MAX-MIN composition. In the fuzzy domain, the linguistic variables are \(E\), \(dE\) and \(U\), with the linguistic values as shown in Figure 4, then the fuzzy sets can be represented as shown in (5)-(7):  
\[E=\left\{\frac{\mu_{E}(NB)}{NB},\frac{\mu_{E}(NM)}{NM},\frac{\mu_{E}(NS)}{NS}, \frac{\mu_{E}(ZE)}{ZE},\frac{\mu_{E}(PS)}{PS},\frac{\mu_{E}(PM)}{PM},\frac{\mu_{ E}(PB)}{PB}\right\} \tag{5}\]  
\[dE=\left\{\frac{\mu_{dE}(NB)}{NB},\frac{\mu_{dE}(NM)}{NM},\frac{\mu_{dE}(NS)}{ NS},\frac{\mu_{dE}(ZE)}{ZE},\frac{\mu_{dE}(PS)}{PS},\frac{\mu_{dE}(PM)}{PM}, \frac{\mu_{dE}(PB)}{PB}\right\} \tag{6}\]  
\[U=\left\{\frac{\mu_{U}(NB)}{NB},\frac{\mu_{U}(NM)}{NM},\frac{\mu_{U}(NS)}{NS},\frac{\mu_{U}(ZE)}{ZE},\frac{\mu_{U}(PS)}{PS},\frac{\mu_{U}(PM)}{PM},\frac{ \mu_{U}(PB)}{PB}\right\} \tag{7}\]  
This fuzzy set mapping process is mainly based on the fuzzy-rule-based matrix, and the fuzzy rule base is based on fuzzy propositions (fuzzy statements), which are assigned to the fuzzy sets \(E_{k}\), \(dE_{k}\) and \(U_{k}\). If a fuzzy proposition \(\widetilde{P}\) is allocated to a fuzzy set, \(\widetilde{A}\), then the truth value \(T\left(\widetilde{P}\right)\) of this proposition can be expressed as (8):  
Figure 4: The non-singleton fuzzy sets (\(E_{k}\), \(dE_{k}\) and \(U\) (Speed)) of the proposed static FLC [105]. Figure 3: The experimental hardware setup of the developed static FLC [105]. \[T\Big{(}\widetilde{P}\Big{)}=\mu_{\widetilde{A}}(x) \tag{8}\]  
where \(0\leq\mu\leq 1\). Therefore, the degree of truth for the proposition \(\widetilde{P}\) is that \(\widetilde{P}:x\in\widetilde{A}\) is equal to the membership value of \(x\) in the fuzzy set \(\widetilde{A}\). In FLS, the fuzzy sets in the consequent and antecedent dimensions can make connections with the logical connectives (9), (10), (11) and (12). These connectives are shown by taking into account two propositions \(\widetilde{P}\) and \(\widetilde{Q}\), which are allocated to the fuzzy sets \(\widetilde{A}\) and \(\widetilde{B}\), respectively. Negation (absence):  
\[T\Big{(}\overline{\widetilde{P}}\Big{)}=1-T\Big{(}\widetilde{P}\Big{)} \tag{9}\]  
Disjunction (lack of correspondence): \(\widetilde{P}\vee\widetilde{Q}:x\) is \(\widetilde{A}\)**or \(\widetilde{B}\)**, then:  
\[T\Big{(}\widetilde{P}\vee\widetilde{Q}\Big{)}=\max\Big{(}T\Big{(}\widetilde{ P}\Big{)},T\Big{(}\widetilde{Q}\Big{)}\Big{)} \tag{10}\]  
Conjunction: \(\widetilde{P}\wedge\widetilde{Q}:x\) is \(\widetilde{A}\)**and \(\widetilde{B}\)**, then:  
\[T\Big{(}\widetilde{P}\wedge\widetilde{Q}\Big{)}=\min\Big{(}T\Big{(}\widetilde {P}\Big{)},T\Big{(}\widetilde{Q}\Big{)}\Big{)} \tag{11}\]  
Figure 2 shows the rule-based matrix of the developed static FLC based on \"if\" and \"then\" statements. This means if the fuzzy proposition \(\widetilde{P}\) implies the fuzzy proposition \(\widetilde{Q}\), then the fuzzy implication can be expressed as (12).