# Fault Parameter Estimation Using Adaptive Fuzzy Fading Kalman Filter
## 4 Numerical Simulations
### Fault Parameter Estimation for Linear System
#### 4.1.2 Fault Parameter Estimation  
In this section, a drop in the hydraulic pressure of the wind turbine pitch system is considered. Two cases of fault injection scenarios are assumed: (a) an abrupt fault case, and (b) an incipient fault case. The abrupt fault is injected at t = 5 s, and the incipient fault starts at t = 5 s and ends at t = 10 s with the continuously increasing fault indicator increases from 0 to 1. The following figures depict estimation results for each scenario. Figure 5a shows the estimation results for the abrupt fault scenario using the standard KF and AFKF. As shown in Figure 5a the standard KF cannot estimate the true value of the fault parameter. The estimation is biased and seems to diverge (green line). A small length of the window may induce an overestimate because the filter is sensitive to noise and uncertainties. AFKF estimates the true value of the parameter. See the results with the window size of 2. As compared to the existing AFKF, the proposed AFFKF shows a better performance not only in the quantitative metric but also in term of the convergence property. See Figure 5b. Figure 5c,d compare the errors in estimations with different approaches. In Figure 5d, the AFFKF presents the improved convergence property. Table 3 gives a summary of the performance index about the estimation results of the fault parameter.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Performance**} & \multicolumn{4}{c}{**Approaches**} \\ \cline{2-5}  & **Standard KF** & **AFKF (M = 2)** & **AFKF (M = 6)** & **AFFKF** \\ \hline \(J(\zeta)\) & 0.1132 & 0.1093 & 0.1444 & 0.0989 \\ \(J(\omega_{n})\) & 0.5672 & 0.6127 & 0.8848 & 0.4250 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of performance index for estimation of normal parameter.  
Figure 4: Results of normal parameter estimations: (**a**) standard Kalman filter and adaptive fading Kalman filter (AFKF), (**b**) the proposed AFFKF (denoted as Fuzzy-AFKF).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Performance**} & \multicolumn{4}{c}{**Approaches**} \\ \cline{2-5}  & **Standard KF** & **AFKF (M = 2)** & **AFKF (M = 6)** & **AFFKF** \\ \hline \(J(\zeta)\) & 0.4654 & 0.0741 & 0.0667 & 0.0454 \\ \(J(\omega_{n})\) & 5.9493 & 0.7809 & 0.4712 & 0.6259 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of performance index for estimation of abrupt fault parameter.  
Figure 6 depicts the estimation results for the incipient fault scenario. The standard Kalman filter fails to estimate the true value of the parameter, and even the damping factor diverges. AFKF estimates the true value of natural frequency quickly, although an overestimate occurs around t = 10 s. The proposed AFFKF shows the result similar to AFKF, but the performance index is slightly improved (Table 4).  
Figure 5: Results of estimation of abrupt fault parameter for linear system: (**a**) standard Kalman filter and AFKF, (**b**) the proposed AFFKF, (**c**) errors in estimation with standard Kalman filter and AFKF, (**d**) errors in estimation with the proposed AFFKF.  
Figure 6: Results of estimation of incipient fault parameter for linear system.