# Fault Detection and Isolation for Redundant Inertial Measurement Unit under Quantization
## 3 Experiments Results and Analysis
### Experiments of Ramp and Square Wave Faults  
#### 3.3.1 Ramp Fault  
The ramp fault is 0.1 degree per hour. The FDI results are shown in Figures 11-13. The parameters of the first-order and second-order filters are 0.434 and 3.85 respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Fault Magnitude (\({}^{\circ}/\mathrm{h}\))** & **5** & **0.5** & **0.4** & **0.3** & **0.2** & **0.1** \\ \hline
**False alarm rate** & **0.000** & 0.010 & 0.030 & 0.130 & 0.500 & 0.860 \\
**False isolation rate** & **0.030** & 0.626 & 0.773 & 0.724 & 0.740 & 0.786 \\
**PCD1** & **0.010** & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
**PCD2** & **0.960** & 0.690 & 0.260 & 0.090 & 0.010 & 0.000 \\
**PCD3** & **0.000** & 0.300 & 0.700 & 0.690 & 0.320 & 0.050 \\
**PCD4** & **0.000** & 0.000 & 0.010 & 0.090 & 0.170 & 0.090 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Calculation results of Monte Carlo simulations.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Time Constant** & **3** & **3.85** & **5** & **8** & **10** & **20** \\ \hline
**False alarm rate** & 0.060 & **0.043** & 0.000 & 0.000 & 0.000 & 0.000 \\
**False isolation rate** & 0.277 & **0.050** & 0.060 & 0.040 & 0.020 & 0.000 \\
**PCD1** & 0.020 & **0.000** & 0.000 & 0.000 & 0.000 & 0.000 \\
**PCD2** & 0.880 & **0.760** & 0.680 & 0.100 & 0.000 & 0.000 \\
**PCD3** & 0.040 & **0.180** & 0.320 & 0.900 & 1.000 & 1.000 \\
**PCD4** & 0.000 & **0.000** & 0.000 & 0.000 & 0.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The relationship between time constant and the effect on detecting faults (step fault \(5^{\circ}/\mathrm{h}\) ).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Time (s)** & **First Order Filter** & **Second Order Filter** \\ \hline
0.020 & **0.434** & **3.850** \\
0.040 & 0.600 & 5.000 \\
0.060 & 0.650 & 6.000 \\
0.200 & 5.000 & 15.000 \\
0.400 & 8.000 & 20.000 \\
1.000 & 10.000 & 30.000 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameters recommended for fault detection and isolation (FDI) with the typical fault in different sampling times.  
Figure 11: Original channel filter. (**a**) Fault detection; (**b**) Fault isolation.  
Figure 12: First-order channel filter. (**a**) Fault detection; (**b**) Fault isolation.  
It is found that the ramp fault detection function of the original channel filter has never exceeded its threshold value in Figure 11, indicating that no fault has been detected. The Figure 12 shows that the fault detection function exceeds the threshold nearing 600 s, indicating that the first-order channel filter has detected a fault. However, its fault isolation function has been separated, nearing 600 s. Furthermore, Figure 13 shows that the fault detection function exceeds the threshold, also nearing 600 s, indicating that the second-order channel filter has detected a fault. Its fault isolation function has been separated nearing 600 s and the curve is smoother.  
Similarly, we take the second-order channel filter as an example (the Monte Carlo simulation 1000 times). The statistical results are shown in the following Table 6.