## 4 Simulation
**: _The inflow data (6 January 2022 to 29 July 2022) of an actual WWTP shown in Figure 6 is used to test MFA-NFITSMC, indicating that the external disturbance to WWTP is random rather than periodic. The actual WWTP consists of three anoxic tanks and five aerobic tanks, with physical parameters of 45.35 \(\times\) 4.85 \(\times\) 5 m. It has three aeration fans, each with a flow rate of 90 m\({}^{3}\)/min, a lift of 5 m, and a power of 110 kW, and its maximum processing capacity is \(12,000\)\(m^{3}\)/day. In Case 2, there are three different kinds of reference trajectories. \(S_{05 ref}=2\)\(mg/L\) is the first time-invariant DO reference trajectory. The second is the time-varying DO reference trajectory \(S_{05 ref}\), including 2 mg/L from day 0-1, 2.3 mg/L from day 1-1.5, and 1.8 mg/L from day 1.5-2. Figure 6b shows the changes in DO concentration for the third reference trajectory \(S_{05 ref}\), which corresponds to the practical reference trajectory._  
Figure 7 shows the control effect of MFA-NFITSMC on DO concentration using actual data of a WWTP. The results show that the proposed method can effectively suppress nonperiodic disturbances in the WWTP system and has strong robustness. The three evaluation indicators of \(ISE\), \(IAE\), and \(Dev_{\max}\) in Table 4 verify that the proposed method \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Strategy** & \(ISE\) & \(IAE\) & \(Dev_{max}\) \\ \hline MFA-NFITSMC & 0.00014 & 0.0273 & 0.0083 \\ OS-ELM [29] & 0.00069 \({}^{\star}\) & 0.0475 \({}^{\star}\) & 0.0381 \({}^{\star}\) \\ PI + AT [30] & 0.0009 \({}^{\star}\) & 0.0490 \({}^{\star}\) & - \\ AFC [31] & 0.0012 \({}^{\star}\) & 0.0792 \({}^{\star}\) & 0.0198 \({}^{\star}\) \\ MPC [7] & 0.0026 \({}^{\star}\) & 0.0890 \({}^{\star}\) & 0.0781 \({}^{\star}\) \\ BFC [31] & 0.0049 \({}^{\star}\) & 0.1507 \({}^{\star}\) & 0.0578 \({}^{\star}\) \\ PID & 0.0078 & 0.1576 & 0.1425 \\ \hline \hline \end{tabular} \({}^{\star}\)The data originates from the corresponding articles.  
\end{table} 
Table 2: The comparison of the performance of various control strategies for an invariant trajectory in Case 1’s dry weather. Figure 6: Influence information and trajectory changes in a real WWTP. (**a**) Flow rate and concentrations; (**b**) Practical reference trajectory changes. has a good control performance, can accurately track time-varying reference trajectories, and has good dynamic performance. **Case 3. **_As key indicators in WWTP, \(S_{NO,2}\) and \(S_{O,5}\) need to be controlled within a reasonable range. However, controlling \(S_{NO,2}\)_ by adjusting the internal recycle flow rate can also cause significant interference with \(S_{O,5}\). _Traditional methods are difficult to effectively control \(S_{NO,2}\) and \(S_{O,5}\) simultaneously. To verify the control effect of the proposed method, two independent MFA-NFITSMC controllers are used to control the second unit nitrate nitrogen concentration and the fifth unit DO concentration in rainy whether. The reference trajectory of nitrate nitrogen concentration is set to \(S_{NO2 ref}=1\ mg/L\) and the reference trajectory of DO concentration is set to \(S_{OSref}=2\ mg/L\). The controller parameters for nitrate nitrogen are set to: \(m=0.5\), \(n=0.5\), \(\alpha=20\), \(\beta=30\), \(\phi_{c}(0)=0.0001\); other parameters are the same as the DO controller. Furthermore, \(u_{\min}=0\) and \(u_{\max}=92,230\) are assigned to the lower and upper bounds of the actuator constraints, respectively._  
The control effect of the proposed method on the DO concentration and the nitrate nitrogen concentration in the WWTP is shown in Figure 8. The outcomes demonstrate MFA-NFITSMC's good performance in the control of \(S_{NO,2}\) and \(S_{O,5}\), with a control error of \(\pm 0.05\) mg/L for \(S_{NO,2}\) and \(\pm 0.015\) mg/L for \(S_{O,5}\). Table 5 compares the three evaluation indicators of \(ISE\), \(IAE\), and \(Dev_{\max}\) with PID, SMC, and ASMC strategies.