## 4 Implementation of the Static Fuzzy Logic Controller (Static FLC)
### Implementation of the Fuzzy Inference Mechanism of the Static Fuzzy Logic Controller
Implication: \(\widetilde{P}\to\widetilde{Q}:x\) is \(\widetilde{A}\), **then**\(x\) is \(\widetilde{B}\), then:  
\[T\Big{(}\widetilde{P}\to\widetilde{Q}\Big{)}=T\Big{(}\widetilde{P}\vee \widetilde{Q}\Big{)}=\max\Big{(}T\Big{(}\overline{\widetilde{P}}\Big{)},T \Big{(}\widetilde{Q}\Big{)}\Big{)} \tag{12}\]  
Then, as in (12), this binary logic implication of the \(\widetilde{P}\) and \(\widetilde{Q}\) fuzzy sets can be expressed as: \(\widetilde{P}\to\widetilde{Q}\) is **IF \(x\)**\(\widetilde{A}\), **THEN**\(y\) is \(\widetilde{B}\); then, this is equivalent to the fuzzy relation \(\widetilde{R}\), where \(\widetilde{R}=\Big{(}\Big{(}\widetilde{A}\times\widetilde{B}\Big{)}\cup \Big{(}\overline{\widetilde{A}}\times Y\Big{)}\Big{)}\). Then, the membership function of the fuzzy relation \(\widetilde{R}\) can be expressed as in (13). \[dE=\left\{\frac{\mu_{dE}(NB)}{NB},\frac{\mu_{dE}(NM)}{NM},\frac{\mu_{dE}(NS)}{ NS},\frac{\mu_{dE}(ZE)}{ZE},\frac{\mu_{dE}(PS)}{PS},\frac{\mu_{dE}(PM)}{PM}, \frac{\mu_{dE}(PB)}{PB}\right\} \tag{6}\]0  
In Mamdani-type FLCs, this implication (fuzzy sets connective) can be easily expressed in the fuzzy rule base with the \"if\" and \"then\" statement form. The MAX-MIN composition [106] was utilized as the fuzzy relation for this developed static FLC, as mentioned above. If \(\widetilde{R}\) is a binary fuzzy relation on \(U\times V\) and \(\widetilde{S}\) is a binary fuzzy relation on \(V\times W\), then the MAX-MIN composition of \(\widetilde{R}\) and \(\widetilde{S}\) is a binary fuzzy relation on \(U\times W\) denoted by \(\Big{(}\widetilde{S}\circ\widetilde{R}\Big{)}\) and which is given by (14):  
\[dE=\left\{\frac{\mu_{dE}(NB)}{NB},\frac{\mu_{dE}(NM)}{NM},\frac{\mu_{dE}(NS)}{ NS},\frac{\mu_{dE}(ZE)}{ZE},\frac{\mu_{dE}(PS)}{PS},\frac{\mu_{dE}(PM)}{PM}, \frac{\mu_{dE}(PB)}{PB}\right\} \tag{6}\]1  
For example, the  
are discrete devices that are unable to manipulate fuzzy outputs. Therefore, this fuzzy output needs to be converted into a crisp output that is related to the fuzzy domain. A de-fuzzification mechanism was utilized in this developed static FLC to convert the fuzzy quantity to a precise quantity. The output of this developed static FLC was the logical union of the seven membership functions designed and developed in the universe of discourse of the output variable speed (\(U_{k}\)). In this case, the union of these seven memberships functions involved with the MAX operator, as described in Section 4.2, could be expressed as (16):  
\[dE=\left\{\frac{\mu_{dE}(NB)}{NB},\frac{\mu_{dE}(NM)}{NM},\frac{\mu_{dE}(NS)}{ NS},\frac{\mu_{dE}(ZE)}{ZE},\frac{\mu_{dE}(PS)}{PS},\frac{\mu_{dE}(PM)}{PM}, \frac{\mu_{dE}(PB)}{PB}\right\} \tag{6}\]2  
where \(k\) is the sample instant and \(j\in\mathbb{Z}^{+}\) and \(1\leq j\leq 7\). According to (16), after computing the logical sum to obtain an average output crisp value, the centroid method or the center of gravity method was used as in (17):  
\[dE=\left\{\frac{\mu_{dE}(NB)}{NB},\frac{\mu_{dE}(NM)}{NM},\frac{\mu_{dE}(NS)}{ NS},\frac{\mu_{dE}(ZE)}{ZE},\frac{\mu_{dE}(PS)}{PS},\frac{\mu_{dE}(PM)}{PM}, \frac{\mu_{dE}(PB)}{PB}\right\} \tag{6}\]3  
where \(U(x)\) denotes each membership function of the output fuzzy set (angular speed of the DC motor), x denotes the value of the linguistic variable, and \(X_{min}\) and \(X_{max}\) represent the minimum and maximum ranges of linguistic variables. However, Figure 1 shows that to minimize the rover travel distance (the rover reaction distance for an updated signal received from the controller), the proposed control mechanism should have the capability to respond to a higher frequency for setpoint trajectories. Therefore, Figure 2 shows that to overcome the drawbacks [105] of the developed static FLC, a metaheuristic dynamic FLC is integrated with the static FLC.