## 4 Simulation and Discussion
Additionally, small chattering problems on PEMFC voltage and current were observed when SMC acted, and small undershoots on PEMFC current were caused by tracking step-down reference voltage governed by adaptive controllers. Due to the load demand changing, the PEMFC operation point may vary to adapt for the converter desired output voltage or current. Figure 7a,b exhibited the simulation results regulated by the adaptive backstepping SMC and RBFNN, respectively. In the adaptive backstepping SMC, the settling time in current tracking control is 0.0015 s. The maximum overshoot is about 3% (0.12 A), and the maximum undershoot is about 2% (0.08 A). In addition, depending on RBFNN estimation, the overshoots and undershoots are almost eliminated. The external disturbance estimation was presented in Figure 8a where the load was rapidly changing from 5 \(\Omega\) to 3 \(\Omega\) at 1.6 s and 3 \(\Omega\) to 5 \(\Omega\) at 2.8 s as the Condition 5 depicted. The traditional adaptive law and RBFNN achieved resistance estimation; however, the traditional scheme led in large overshoot or undershoot of resistance that might postpone output reach to setpoint as illustrated in Figure 8b. In the adaptive backstepping SMC, the maximum overshoot is about 14.2% (1.7 V), and the maximum undershoot is about 12.5% (1.5 V). In the RBFNN, the maximum overshoot is about 5.8% (0.7 V), and the maximum undershoot is about 1.7% (0.2 V). Furthermore, to investigate the structural parameters variation caused by perturbations on the inductor and capacitor of the power supply, the traditional adaptive law and RBFNN are also compared. The simulation profiles of \(L\) and \(C\) were shown in Figure 8c-e, the estimations based on RBFNN matched the real profiles of both \(L\) and \(C\) well. Additionally, the output voltage tracking control in the cases of \(L\) or \(C\) variations via the proposed methods demonstrated good performance. The results of comparison tests under Conditions 1-5 are presented in Table 3.  
Figure 5: Simulation results in Condition 2: (**a**) by SMC; (**b**) by adaptive backstepping SMC; (**c**) by RBFNN; (**d**) inductor current controlled by different control methods, enlarged view at 4 s.  
Figure 6: Simulation results in Condition 3: (**a**) by SMC; (**b**) by adaptive backstepping SMC; (**c**) by RBFNN; (**d**) output voltage controlled by different control methods, enlarged view at 6 s.  
Figure 7: Simulation results in Condition 4: (**a**) by adaptive backstepping SMC; (**b**) by RBFNN. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Conditions**} & \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{**Settling Time (s)**} & \multicolumn{2}{c}{**RSME**} \\ \cline{3-6}  & & **Current** & **Voltage** & **Current** & **Voltage** \\ \hline \multirow{2}{*}{Condition 1} & ABSMC & 0.014 & / & 0.6317 & / \\  & RBFNN & 0.003 & / & 0.2836 & / \\  & ABSMC & 0.003 & / & 0.0782 & / \\  & RBFNN & 0.002 & / & 0.0232 & / \\ Condition 3 & ABSMC & / & 0.012 & / & 0.1279 \\  & RBFNN & / & 0.006 & / & 0.1216 \\ Condition 4 & ABSMC & 0.015 & / & / & 0.0284 \\  & RBFNN & 0.001 & / & / & 0.0284 \\ Condition 5 & ABSMC & / & 0.010 & 0.2018 & / \\  & RBFNN & / & 0.004 & 0.1794 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Control comparison results under Conditions 1–5.