## 4 A Fuzzy PSO-Based Control Design
### Third Simulation Analysis: The Fuzzy Sliding Mode Controller Implementation  
This part is dedicated to evaluating the performance of the designed PSO fuzzy control technique. A step change desired trajectory was considered. A second-order under \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & \(\dot{\mathbf{E}}(\mathbf{t})\) & & & & & \\ \hline \multirow{8}{*}{\(E(t)\)} & & _NB_ & _NM_ & _NS_ & \(Z\) & _PS_ & _PM_ & _PB_ \\  & _NB_ & _PB_ & _PB_ & _PB_ & _PB_ & _PM_ & _PM_ & _PM_ \\  & _NM_ & _PB_ & _PB_ & _PB_ & _PM_ & _PS_ & _PS_ & _PS_ \\  & _NS_ & _PM_ & _PM_ & _PM_ & _PS_ & \(Z\) & \(Z\) & \(Z\) \\  & \(Z\) & _PS_ & _PS_ & _PS_ & \(Z\) & _NS_ & _NS_ & _NS_ \\  & _PS_ & \(Z\) & \(Z\) & \(Z\) & _NS_ & _NM_ & _NM_ & _NM_ \\  & _PM_ & _NS_ & _NS_ & _NS_ & _NM_ & _NB_ & _NB_ & _NB_ \\  & _PB_ & _NM_ & _NM_ & _NM_ & _NB_ & _NB_ & _NB_ & _NB_ \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fuzzy controller gains tuning rules. Figure 11: Gaussian membership function for \(\dot{E}(t)\). damped linear model was exploited to set the reference input for generating the desired trajectory \(y_{d}\) as well as its first derivative \(\dot{y}_{d}\):  
\[y_{d}(t)=\Bigg{(}1-\frac{\varsigma\omega_{n}t}{\sqrt{1-\varsigma^{2}}}\times \bigg{(}\varsigma\sin(\omega_{n}dt)+\bigg{(}\sqrt{1-\varsigma^{2}}\bigg{)}\cos (\omega_{n}dt)\bigg{)}\Bigg{)}r(t) \tag{56}\]  
The simulation parameters were specified as follows: \(\omega_{n}=5 rad/\min\) and \(\varsigma=0.9\). The simulation illustrates in Figure 12.  
For the PSO technique, the varying parameters \(w_{k}\), \(L_{1}\), and \(L_{2}\) are considered as the crucial features that impact the algorithm convergence characteristics [54]. The steadiness between global examination and local searchability is controlled by the weight of inertia. While global exploration is preferred by a larger inertia weight, local searchability is preferred by a smaller weight of inertia. The adopted parameters pertaining to the PSO algorithm are displayed in Table 3.  
Note that for the PSO algorithm, the simulations were repeats with other turning parameters, and each time, results are updated. Finally, it should be noted that the algorithm by PSO has been executed extensively. Figure 13 shows the process of minimizing the PSO coast function to determine the optimal fuzzy controller gains of the reactor system. Figure 14 illustrates the perfect tracking behavior improved SMC and Fuzzy PSO. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Particle number & 10 \\ \(L_{1}\) & 2 \\ \(L_{2}\) & 2 \\ \(R_{1}\) and \(R_{2}\) & Random \\ Iteration & 20 \\ \(w_{\min}\) & 0.3 \\ \(w_{\max}\) & 1.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters and settings of PSO algorithm. Figure 12: Desired trajectory and control PSO. Figure 12 illustrates the tracking control performance and the impressive dynamic behavior of the designed strategy. Indeed, despite the severe step deviation on the desired reference input, the process output trajectory was under-damped with an attenuated rising time and settling. The steady-state accuracy of the response is quite satisfactory onward the reference trajectory alteration. To assess the control scheme synthesized in this paper, a comparative analysis is established here to examine the performance of the fuzzy PSO regulator across the analytical SMC-based I/O FBL controller. The outcomes of the simulation analysis are depicted in Figures 14 and 15, which represent the system's output dynamics as well as the resultant control inputs, respectively. Figure 12 illustrates the excellent dynamic performance of the advanced fuzzy logic SMC-I/O FBL controller evading attenuated chattering behavior. The synthesized control signal is represented in Figure 15. It appears that the excessive chattering dynamic is quite mitigated in assessment with the numerical simulation outcomes illustrated in Figure 15 for the several control inputs.