## 4 Extensive Computer Simulations
### Discussion of Simulation Results  
For both systems, the threshold value is set to \(\gamma=7.88\) corresponding to PFA=0.0005 in the proposed selective FMS filtering. Figures 2 and 3 show test variables for uncertainty presence and absence declaration.  
For the F404 gas turbine aircraft engine system, Figure 4 shows estimation errors for the second state indicating turbine temperature for three filters, the primary FMS filter with \(M_{p}=20\), the secondary FMS filter with \(M_{s}=10\), and the IMS filter. For the electric motor system, Figure 5 shows estimation errors for the second state indicating rotational speed for three filters, the primary FMS filter with \(M_{p}=20\), the secondary FMS filter with \(M_{s}=10\), and the IMS filter.  
Figure 2: Test variable for the F404 gas turbine aircraft engine system.  
According to the declaration rule (16) using the test variable (15) and the threshold value \(\gamma\) = 7.88 corresponding to PFA = 0.0005, the proposed selective FMS filtering with two kinds of measurement windows provides the state estimate as shown in Figures 6-8 and Figures 9-11 which show comparisons with three other filters. In addition, time averaged values of root mean square (RMS) estimation errors are presented in Table 1 for 20 simulations. As shown in simulation results, the proposed selective FMS filtering with two kinds of measurement windows can be better than the primary FMS filter and the IMS filter in terms of error magnitude and error convergence on the interval where modeling uncertainty exists. In addition, the proposed selective FMS filtering can be better than the secondary FMS filter when there is no temporary model uncertainty or after temporary model uncertainty is gone. These observations on computer simulations show that the proposed selective FMS filtering can work well in temporarily uncertain systems as well as in certain systems.  
Figure 4: Estimation errors of primary FMS, secondary FMS and infinite memory structure (IMS) filters for the F404 gas turbine aircraft engine system.  
Figure 3: Test variable for the electric motor system.  
Figure 5: Estimation errors of the primary FMS, secondary FMS and IMS filters for the electric motor system.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **IMS (Kalman)** & **Primary FMS** & **Secondary FMS** & **Selective FMS** \\ \hline F404 Gas Turbine Aircraft Engine & 0.0876 & 0.0309 & 0.0307 & 0.0263 \\ Electric Motor & 0.0025 & 0.0023 & 0.0016 & 0.0015 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of mean of RMS estimation errors.  
Figure 6: Estimation errors of the proposed selective FMS and primary FMS filters for the F404 gas turbine aircraft engine system.  
Figure 8: Estimation errors of the proposed selective FMS and IMS filters for the F404 gas turbine aircraft engine system.  
Figure 7: Estimation errors of the proposed selective FMS and secondary FMS filters for the F404 gas turbine aircraft engine system.  
Figure 10: Estimation errors of the proposed selective FMS and secondary FMS filters for the electric motor system.  
Figure 9: Estimation errors of the proposed selective FMS and primary FMS filters for the electric motor system.