## 5 Experiments
### Controllers Optimization
#### 5.2.2 PID Control  
The PID controller (Proportional-Integral-Derivative) is a feedback control mechanism that is widely used in industrial control systems, for example, in [40; 41; 42]. The PID control algorithm uses three different parameters: the proportional (\(\mathrm{k_{P}}\)), the integral (\(\mathrm{k_{i}}\)), and the derivative (\(\mathrm{k_{d}}\)) gains. The proportional value depends on the current error. The integral depends on past errors, and the derivative is a prediction of future errors. The sum of these three actions is used to adjust the process by means of a control element, such as the position of a control valve or the power supplied to a heater. The general PID algorithm is expressed in Equation (30):  
\[r\left(t\right)=MV\left(t\right)=Kp\;e(t)+Ki\;\int_{o}^{t}e\left(t\right)\; \mathrm{d}t+Kd\;\frac{de(t)}{dt}. \tag{30}\]  
In this case, the PID angular position of the motor is used, the ST2 FHS and ST2 FDE methods are used for the optimization of this control problem, and the control objective is to minimize the settling time. The transfer function for this controller is expressed in Equation (31), and the structure of the control system is shown in Figure 17:  
\[C(s)=\;K_{p}+\frac{K_{i}}{s}+K_{d}s=\frac{K_{d}s^{2}+K_{p}s+K_{i}}{s}. \tag{31}\]  
Table 28 shows the best parameters obtained from the optimization with the HS, ST2 FHS, DE, and ST2 FDE methods and the time in which the objective of stabilizing the motor position is achieved. Figures 18 and 19 show the graphical representation of the simulation of these parameters for the HS and ST2 FHS, DE, and ST2 FDE methods, respectively. Table 29 shows the results for the statistical test.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\boldsymbol{\mu_{1}}\) & \(\boldsymbol{\mu_{2}}\) & **Z-Value** \\ \hline \multirow{2}{*}{ST2 FHS} & FLC without noise & HS & \(-6.23\) \\  & FLC with noise & HS with noise & \(-1.67\) \\ \hline \multirow{2}{*}{ST2 FDE} & FLC without noise & DE & \(-18.8799\) \\  & FLC with noise & DE with noise & \(-8.8627\) \\ \hline \hline \end{tabular}
\end{table}
Table 27: Results for the statistical test of the DC motor.  
Figure 17: Structure of the PID (Proportional-Integral-Derivative) DC motor.  
Figures 18 and 19 show visually similar results, Table 27 contains the results of the values of each PID parameter obtained when using the proposed method, and it can be noted that there is a difference in the settling time between the two methods.  
Table 29 presents the statistical test to validate the improvement of the proposed approach with respect to the conventional method.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & **Best Settling Time** \\ \hline PID & 21 & 500 & 0.15 & 0.0338 \\ HS & 600 & 12,000 & 6 & 0.00029 \\ DE & 900 & 18,000 & 9 & 0.00020 \\ STZFHS & 900 & 30,000 & 9 & 0.00020 \\ STZFDE & 1500 & 30,000 & 15 & 0.00012 \\ \hline \end{tabular}
\end{table}
Table 28: Results for the experiments with PID for the DC motor.  
Figure 19: Best results obtained from the DE and STZFDE methods.  
Figure 18: Best result obtained from the HS and STZFHS methods.  
The Z-values obtained in Table 29 demonstrate that the proposed approach obtains better performance with respect to the conventional approach.