## 4 Simulation Research Results
### Simulation Experiments of the Monitoring System
Figures 8 and 9 present the BOD and DO distributions and changes in the gain coefficient for various cases of disturbance intensity which were characterized by different values of covariance matrices for system disturbances. In Figures 8a and 9a, the time waveforms of signals generated by the soft sensor \(\mathrm{BOD_{AA}}\) more accurately reflect the signals of the disturbed object as compared with the Kalman filter \(\mathrm{BOD_{KF}}\). In the case of DO, \(\mathrm{DO_{AA}}\) and \(\mathrm{DO_{KF}}\) signal distributions were very similar. Figures 8b and 9b show the gain coefficient distributions for BOD and DO, respectively. Changes in the value of these gains for BOD were several times higher than for DO. Figure 8: Distribution of BOD and DO signals (**a**) and adaptive algorithm gain coefficients (**b**) for matrices \(\mathbf{W}=[3,-2;-2,1]\) and \(\mathbf{V}=[0.1]\). Figure 9: Distribution of BOD and DO signals (**a**) and adaptive algorithm gain coefficients (**b**) for matrices \(\mathbf{W}=[6,-4;-4,2]\) and \(\mathbf{V}=[0.1]\). Based on the conducted simulations, tests results were obtained showing the advantage of the adaptive algorithm over the Kalman filter algorithm, mainly referring to the unmeasured signal of the object that is BOD. Regardless of the intensity of disturbance, better results were obtained with an adaptive algorithm. It should also be emphasized that the RMSE monitoring quality index for the DO deficit was an order of magnitude smaller in comparison with the BOD signal (see Figure 12). Better monitoring quality results from measurements of the DO deficit signal. An unexpected advantage of the Kalman filter measured by the RMSE indicator for the DO deficit results mainly from the assumption of zero initial values of coefficient gains in the adaptive algorithm. This setup of values  
Figure 14: Mean percentage error (MPE) quality indicators for BOD (**a**) and DO (**b**) signals generated. Figure 12: Root-mean-squared error (RMSE) quality indicators BOD (**a**) and DO (**b**) signals generated. Figure 13: Values of quality indicators for DO signal at different initial gain values. results in a slightly lower quality of monitoring, especially in the initial period of simulation, on the one hand, and on the other, great versatility of the adaptive algorithm. Assuming that the initial value of the gain coefficients in the adaptive algorithm was nonzero, this resulted in a significant improvement in the quality of the monitored DO signal, as shown in Figure 13.  
The MPE quality indicator used for BOD and deficit DO signals also indicated more successful results obtained with the adaptive algorithm in relation to the Kalman filter (see Figure 14). As previously mentioned, for the BOD signal, the quality of monitoring was almost twice as good. For the DO deficit signal, the MPE quality indicators did not differ as much as in the case of RMSE indicators and the values more clearly indicated a qualitative advantage for the adaptive algorithm. The results obtained in the conducted simulation experiments generally indicated a better quality of signal monitoring with an adaptive algorithm in comparison with the Kalman filter. Simulation experiments aimed at comparing the computational complexity of algorithms were carried out in the Matlab environment. Results of the tested algorithms expressed in the form of duration of calculations confirmed the lower complexity of the adaptive algorithm, as presented in Table 4.  
The results shown in Table 4 are mean values from 10 numerical experiments. The need of averaging calculation times resulted from the occurrence of random disturbances for the assumed parameters. These disturbances generated various extortions, and as a consequence, created various errors which resulted in different calculation times. The calculation times in the Kalman algorithm did not differ much, but in the adaptive algorithm, the values of these times showed larger differences. However, in each case, the adaptive algorithm showed better results by an order of magnitude.