# A Shadowed Type-2 Fuzzy Approach for Crossover Parameter Adaptation in Differential Evolution
## 5 Interval Type-2 Fuzzy Systems Controller  
Position control of a direct current (DC) motor is a circuit capable of controlling the position of any DC motor with the simple movement of a potentiometer that acts as a positioner, and it does so through proportional control [47; 48; 49; 50]. One of the purposes of this work is the optimization of the interval type-2 fuzzy controller, which we will call (IT2-FLC), which is structured as illustrated in Figure 6.  
Figure 5: Shadowed fuzzy system integrated with the DE algorithm (DE-ST2-FS). Figure 6: IT2-FLC controller. The fuzzy system is defined as: the inputs are the error \"Err\" and the error change \"C_Err\", the output corresponds to the voltage \"Voltage\", the \"Err\" input contains three MFs, the ends are of type \"_trapIT2 mf\"_, and the center is \"_triIT2 mf\"_. The equations that describe the knowledge of each function are shown in Equations (21) and (22). The second input \"C_Err\" contains five membership functions, the ends are of type \"_trapIT2 mf\"_, and the three central functions are of type \"_triIT2 mf\"_. Finally, the output \"Voltage\" consists of five membership functions, the ends are of type \"_trapIT2 mf\"_, and the three central functions are of type \"_triIT2 mf\"_.  
\[trapIT2 mf(\mathrm{x},[a_{1},b_{1},c_{1},d_{1},a_{2},b_{2},c_{2},d_{2},\alpha])\]  
\[\mu(x)= \begin{cases}\begin{array}{c}a_{1}<a_{2},b_{1}<b_{2},c_{1}<c_{2},d_{1 }<d_{2}\\ \mu_{1}(x)=\max\Bigl{(}\min\Bigl{(}\frac{x-a_{1}}{b_{1}-a_{1}},1,\frac{d_{1}- x}{d_{1}-c_{1}}\Bigr{)},0\Bigr{)}\\ \mu_{2}(x)=\max\Bigl{(}\min\Bigl{(}\frac{x-a_{2}}{b_{2}-a_{2}},1,\frac{d_{2}- x}{d_{2}-c_{2}}\Bigr{)},0\Bigr{)}\\ \mu(x)=\begin{cases}max(\mu_{1}(x),\mu_{2}(x))&\forall x\
ot\in(b1,c2)\\ 1&\forall x\in(b1,c2)\\ \mu(x)=min(\alpha,min(\mu_{1}(x),\mu_{2}(x)))\end{cases}\\ \end{array}\\ \underline{\mu}(x)=min(\alpha,min(\mu_{1}(x),\mu_{2}(x)))\end{cases}  \tag{21}\]  
\[\mu(x)= \begin{cases}\begin{array}{c}a_{1}<a_{2},b_{1}<b_{2},c_{1}<c_{2}\\ \mu_{1}(x)=\max\Bigl{(}\min\Bigl{(}\frac{x-a_{1}}{b_{1}-a_{1}},\frac{c_{1}-x}{ c_{1}-b_{1}}\Bigr{)},0\Bigr{)}\\ \mu_{2}(x)=\max\Bigl{(}\min\Bigl{(}\frac{x-a_{2}}{b_{2}-a_{2}},\frac{c_{2}-x}{ c_{2}-b_{2}}\Bigr{)},0\Bigr{)}\\ \mu(x)=\begin{cases}max(\mu_{1}(x),\mu_{2}(x))&\forall x\
ot\in(b1,b2)\\ 1&\forall x\in(b1,b2)\\ \underline{\mu}(x)=min(\mu_{1}(x),\mu_{2}(x))\end{cases}\\ \end{array}\\ \end{cases} \\ \end{table}
Table 1: IT2-FLC rules. The conducted experimentation is summarized in Table 3, which demonstrates the usage of the DE-ST2-FS proposal without applying noise to the controller. Subsequently, the performance of a type-2 controller in handling intervals was evaluated by introducing Gaussian random noise at three different levels: 0.3, 0.5, and 0.9. An analysis of Table 4 shows that the utilization of the controller with a noise level of 0.5 is better in terms of minimum error compared to the proposal without applying noise to the controller. On the other hand, it is also observed that, in comparison to the minimum error, a better result is obtained when a noise level of 0.9 is applied compared to a noise level of 0.7. Regarding the averages of the experimentation, they all have the same order, but it can be observed how the average decreases when the noise level increases. Figure 7 shows a comparison of the experimentation with the different noise levels. The analysis of Figure 7 shows us that with a higher noise level, the DE-ST2-FS proposal shown in this work obtains better results; however, not applying noise to the DE-ST2-FS controller has a similar behavior to DE-ST2-FS with noise 0.5, while DE-ST2-FS with noise 0.7 and DE-ST2-FS with noise 0.9 have very similar behaviors. Overall, a better result is observed with DE-ST2-FS with noise 0.9. The results of the minimum values obtained in Table 4 are represented in Figures 8-11, showcasing the four different experimental scenarios: DE-ST2-FS without noise, DE-ST2-FS with noise 0.5, DE-ST2-FS with noise 0.7, and DE-ST2-FS with noise 0.9.