## 4 Results and Discussion
### Test Systems
#### 4.2.3 Test System 3  
The first-order linear system is chosen for testing in order to confirm the superiority of the ALGWO-BPNN in the PID control process, and the model transfer function is indicated in (22). The value of the ub is 10, so the mapping range of the activation function is set to  
Figure 8: The results of convergence curves. (**a**) The convergence curves of F8 and (**b**) the convergence curves of F9 and (**c**) the convergence curves of F10 and (**d**) the convergence curves of F11. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Base Function**} & \multicolumn{1}{c}{**GWO**} & \multicolumn{1}{c}{**ALGWO**} & \multicolumn{1}{c}{**SOGWO**} & \multicolumn{1}{c}{**PSO**} & \multicolumn{1}{c}{**DE**} \\ \hline \multirow{3}{*}{F8} & STD & \(8.45\times 10^{2}\) & \(1.26\times 10^{3}\) & \(7.11\times 10^{2}\) & \(8.35\times 10^{2}\) & \(6.86\times 10^{2}\) \\  & AVE & \(-5.98\times 10^{3}\) & \(-5.98\times 10^{3}\) & \(-6.44\times 10^{3}\) & \(-6.80\times 10^{3}\) & \(-5.97\times 10^{3}\) \\ \multirow{3}{*}{F9} & STD & 0 & 0 & 0 & \(8.22\times 10^{0}\) & \(1.57\times 10^{1}\) \\  & AVE & 0 & 0 & 0 & \(3.47\times 10^{1}\) & \(2.57\times 10^{2}\) \\ \multirow{3}{*}{F10} & STD & \(1.49\times 10^{-15}\) & 0 & \(2.39\times 10^{-15}\) & \(3.59\times 10^{-6}\) & \(7.20\times 10^{-1}\) \\  & AVE & \(1.43\times 10^{-14}\) & \(4.44\times 10^{-15}\) & \(1.40\times 10^{-14}\) & \(2.56\times 10^{-6}\) & \(1.72\times 10^{-1}\) \\ \multirow{3}{*}{F11} & STD & 0 & 0 & 0 & \(7.30\times 10^{-3}\) & \(2.67\times 10^{0}\) \\  & AVE & 0 & 0 & 0 & \(7.10\times 10^{-3}\) & \(3.63\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison results of five algorithms on four groups of benchmark functions. 0 to 10. The control findings are shown in Table 6 and Figure 9. The best performance is demonstrated by the \(g(x)\) activation function. The convergence characteristics can be seen in Figure 10.  
\[G(s)=\frac{1.7}{320s+1} \tag{22}\]  
The variance and standard deviation of utilizing the ALGWO method are noticeably lower, as seen in Table 6, as compared to using the GWO algorithm. For instance, the PID control system utilizing the \(g(x)\) BPNN reduces the standard deviation from 13.5263 to 2.4216 and also lowers the average value by 1.5. As a result, it has improved resilience and average accuracy. The average value is decreased by roughly 20 after using the BPNN model, so the average accuracy is increased. The \(g(x)\) activation function has a better impact when utilized as an activation function. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & \(g(x)\) & \(t(x)\) & \(h(x)\) & **PID** \\ \hline \multirow{2}{*}{GWO} & AVE & 277.2945 & 295.4016 & 280.0701 & 301.2233 \\  & STD & 13.5263 & 13.4028 & 13.5383 & 0.2515 \\ \multirow{2}{*}{ALGWO} & AVE & 275.7396 & 280.0453 & 278.7984 & 301.0937 \\  & STD & 2.4216 & 12.3446 & 11.8726 & 0.2811 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The comparison of final fitness of algorithms on test system 3. Figure 9: The results of control effect. (**a**) The simulation results of BPNN-PID using \(g(x)\) and (**b**) simulation results of BPNN-PID using \(t(x)\) and (**c**) simulation results of BPNN-PID using \(h(x)\) and (**d**) simulation results of conventional PID. As shown in Figure 9, the controllers employing BPNN-PID outperform the PID controllers in reducing overshoot when the system reference value changes. As shown in Figure 10, the ALGWO algorithm is used to control the BPNN-PID control scheme in comparison to GWO, and it approaches the ideal solution at iteration 200 with a faster convergence rate, demonstrating ALWO's more deterministic global search capability and quicker discovery speed of high-dimensional optimal solutions. The optimization values utilizing the ALGWO algorithm are 275.7396 and 301.0937, respectively, in terms of optimization accuracy. It is shown that both in high-dimensional and low-dimensional scenarios, it is superior to GWO.