## 4 Simulation Research Results
### Simulation Experiments of the Monitoring System  
In other studies, particular attention has been paid to cases in which the most divergent distributions of signals are generated due to changes in model parameters. For such cases, correct functioning of the adaptive algorithm was received. The tests were carried out in accordance with the presented scheme (Figure 5), in which a model with preaccepted model parameters (\(A_{pm}\)) was used in the monitoring system signal generator. The mark \"real object\" with real parameters (\(A_{R}\)) in Figure 5 means different parameters in relation to the model used in the monitoring signal generator. For the approach thus constructed, errors (as the difference between \(\hat{x}\) and \(x_{pm}\) and \(x_{R}\)) were investigated for BOD and DO, respectively. In addition, cases of the same as well as various disturbances in the model and real object were assumed. The functional meaning of Figure 5 can be described with a logical set of rules presented below:  
IF (\(A_{R}=A_{pm}\)) THEN (generated object signals) = CORECT  
IF (\(A_{R}\
eq A_{pm}\)) THEN (generated object signals) = CORECT too. Figure 6 presents the distribution of errors generated with the participation of the same disturbances. The generated disturbances of the error signal for BOD and DO in both cases oscillated around the zero value, which means that the monitoring system functioned correctly despite inaccurate values of the model parameters. Figure 5: Schematic diagram of generating object signals with inaccurate model parameters. Simulation tests also included cases where nonidentical disturbances were applied to the preaccepted model and real object, as shown in Figure 7.  
For such cases, proper functioning of the monitoring system was also obtained. The presented cases indicate the algorithm's robustness to inaccuracy in identification of the model parameters' values. For the abovementioned case, the quantitative measure of quality of received signals measured with the MPE indicator (11) is presented in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{W_{1}}=\mathbf{W_{2}}\) **(Figure 5)** & **BOD** & **DO** \\ \hline \(\begin{array}{c}\text{\emph{MPE}}\ \epsilon_{pm}\\ \text{\emph{MPE}}\ \epsilon_{R}\end{array}\) & 70.2012 & 67.6716 \\ \(\begin{array}{c}\text{\emph{MPE}}\ \epsilon_{R}\end{array}\) & 20.9545 & 8.6493 \\ \hline \(\begin{array}{c}\text{\emph{W}}_{1}\
eq\mathbf{W_{2}}\ \text{\emph{(Figure 5)}}\end{array}\) & **BOD** & **DO** \\ \hline \(\begin{array}{c}\text{\emph{MPE}}\ \epsilon_{pm}\\ \text{\emph{MPE}}\ \epsilon_{R}\end{array}\) & 95.4833 & 69.0614 \\ \hline \(\begin{array}{c}\text{\emph{APE}}\ \epsilon_{R}\end{array}\) & 17.6248 & 7.9677 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean percentage error (MPE) indicators of BOD and DO signals errors resulting from inaccurately selected model parameters. Figure 6: Distribution of errors of BOD (**a**) and DO (**b**) signals resulting from inaccurately selected model parameters (the same extortions). Figure 7: Distribution of BOD (**a**) and DO (**b**) signals errors resulting from inaccurately selected model parameters (different extortions). The values of qualitative indicators for MPE monitoring in cases of identical and different extortion values affecting both the model and the real object for BOD and DO were comparable. It should be emphasized that the lower values of MPE obtained for DO resulted from the fact that the measurements referred to this signal specifically. Figures 8-11 show signal distribution generated by the soft sensor adaptive algorithm (index AA) and the Kalman filter algorithm (index KF) for different values of covariance matrices of the system (\(\mathbf{W}\)) and measurement (\(\mathbf{V}\)) disturbances as well as the changes in gain coefficients generated by the adaptive algorithm with these disturbances.