## 4 Simulation  
In Figure 1, the BSM1 employs the constructed MFA-NFITSMC to confirm the intended method's control effect. The BSM1 used the rainy and dry weather input data files, which came from the working groups of COST Actions 682 and 624. These files were downloaded as normative data to be used in the controller evaluation. Figure 3 shows the dynamic inflow data and key component concentrations on dry and rainy weather from the 7 th day to the 14 th day, including the influent flow \(Q_{i}\), the ammonia nitrogen \(S_{NH}\), and the readily biodegradable substrate \(S_{S}\). The operating period is two weeks, with a sampling interval of 15 min. The inflow data from the previous week is used to stabilize the system, while the dynamic data from the following week is used to test the controller performance. The MFA-NFITSMC parameters are set as follows: \(T=10^{-5}\) day, \(m=0.1\), \(n=0.1\), \(\alpha=10\), \(\beta=15\), \(q_{1}=3\), \(q_{2}=5\), \(\lambda=0.6\), \(\phi_{c}(0)=0.2\), \(\eta=0.8\), \(\mu=1\). For model-free adaptive integral terminal sliding mode control (MFA-ITSMC), simply set \(\beta=0\), and the other parameters are the same as above. Furthermore, \(u_{\min}=0\) and \(u_{\max}=360\) are assigned to the actuator constraints' upper and lower bounds, respectively. The evaluation of the system control effectiveness is mainly based on the following indicators  
\[\left\{ \begin{array}{l}ISE=\int_{t0}^{tf}e_{i}^{2}dt\\ IAE=\int_{t0}^{tf}\left|e_{i}\right|dt\\ Dev_{\max}=max|e_{i}|\end{array} \right. \tag{37}\]  
where the control system's response rate is indicated by the integral of square error (\(ISE\)), the transient response and appropriate damping are indicated by the integral of absolute error (\(IAE\)), and the stability of the control system is indicated by the maximal absolute deviation (\(Dev_{\max}\)), respectively. **Case 1. **_Usually, when the fifth unit's DO concentration is maintained around \(2\)\(mg/L\), the effluent quality meets the standard. Therefore, there are two types of reference trajectories in Case 1. \(S_{\text{O5 ref}}=2\)\(mg/L\) is the first time-invariant reference trajectory of DO. The second is the DO reference trajectory \(S_{\text{O5 ref}}\) that varies over time and includes \(2\)\(mg/L\) from days 7-9, \(1.9\)\(mg/L\) form days 9-11, \(2.2\)\(mg/L\) from days 11-12 and \(2\)\(mg/L\) from days 12-14. Considering the external interference of influent flow and component concentration in Figure 3, the controller performance was tested on dry and rainy weather and compared with other methods._  
The proposed method's control effect on the DO concentration in the WWTP is shown in Figures 4 and 5. The results show that the MFA-NFITSMC strategy can ensure the tracking of the DO concentration to the reference value under different weather conditions and reference trajectory; it has a fast response speed and strong anti-interference ability, and the control effect is better than PID and MFA-ITSMC methods. Tables 2 and 3 show the control effects of different control strategies on DO concentration on dry and rainy  
Figure 3: Dry and rainy weather influent. (**a**) Flow rate \(Q_{i}\); (**b**) Constituent concentrations \(S_{S}\) and \(S_{NH}\). weather using three evaluation indicators: \(ISE\), \(IAE\), and \(Dev_{\max}\). It can be seen that MFA-NFITSMC is better than other control strategies for all indicators. Therefore, the proposed method has a better tracking performance and is suitable for a WWTP with strong disturbances. Figure 4: DO control effect under different controllers in dry weather. (**a**) DO tracking effect of trajectory 1 in dry weather; (**b**) DO tracking effect of trajectory 2 in dry weather. Figure 5: DO control effect under different controllers in rainy weather. (**a**) DO tracking effect of trajectory 1 in rainy weather; (**b**) DO tracking effect of trajectory 2 in rainy weather. **Case 2.