# Fault Detection and Isolation for Redundant Inertial Measurement Unit under Quantization
## 2 Materials and Methods
### The Three-Channel Filter for FDI
#### 2.2.3 The Filter Design  
Based on the research and analysis of a low-pass filter, we put forward the constraint conditions that affect the fault detection performance: namely the cut-off frequency of the filter and its rising time. The cut-off frequency mainly affects false alarm rate and false isolation rate, while the rising time mainly decides the delay time of the FDI system. However, the cut-off frequency and the rise time are contradictory; therefore, we take the following compromise approach and theoretically analyze the parameters of the filter. The large amplitude of hard fault does not allow the original channel filter to have time delay; therefore, this channel does not need preprocessing. First-order channel filter uses the first-order low-pass filter (time constant: T)  
\[G(s)=1/(\mathrm{Ts}+1), \tag{9}\]  
The rise time of the system represents rapidity. \[T_{r}\approx 2.2\cdot\mathrm{T}, \tag{10}\]  
The low-pass filter can eliminate high-frequency noise so that only low-frequency signals can pass. Therefore, in order to achieve this goal, we make the cut-off frequency become small because when the signal frequency is smaller than the cut-off frequency, amplitude attenuation and phase attenuation are relatively small, and when the signal frequency exceeds the cut-off frequency, amplitude attenuation and phase attenuation both become quite large. Therefore, suppose there is an indication function:  
\[y(\mathrm{T})=k_{1}(2.2\mathrm{T})^{2}+k_{2}(\frac{1}{\mathrm{T}})^{2}, \tag{11}\]  
where \(k_{1}\) and \(k_{2}\) represent the weights of rising time and cut-off frequency, respectively, which are decided by the performance of the FDI system. The reasonable choice of \(k_{1}\) and \(k_{2}\) can make the value \(y(\mathrm{T})\) minimal. We obtain the following with the derivation of time constant T on both sides:  
\[y^{\prime}(\mathrm{T})=9.68k_{1}\mathrm{T}-k_{2}\frac{2}{\mathrm{T}^{3}}, \tag{12}\]  
Let \(y^{\prime}(\mathrm{T})=0\), and then we can obtain:  
\[\mathrm{T}=0.6742\times\sqrt[4]{\alpha}\alpha=k_{2}/k_{1}, \tag{13}\]  
Figure 4: The fault detection method based on the three-channel filter. where \(\alpha\) expresses the ratio of cut-off frequency to rising time in the first- order channel filter. At the same time, it also indirectly indicates the ratio of false alarm rate to delay time for medium fault. The transfer function of the critical damping system of the second-order channel filter can be defined as follows:  
\[G(s)=\frac{1}{(\mathrm{Ts}+1)(\mathrm{Ts}+1)}=\frac{1}{\mathrm{T}^{2}s^{2}+2 \mathrm{Ts}+1}, \tag{14}\]  
The time constant of the second-order channel filter should be determined by the fault tolerance time of its soft fault. As we know, the smaller the time constant is, the faster the second-order channel filter responses. Thus, we should select a reasonable time constant and make the channel filter have a short response time and a low cut-off frequency. The weights of the performance indices of the response time and the cut-off frequency are \(k_{3}\) and \(k_{4}\), respectively. The rising time of the critical damping system is  
\[T_{r}\approx 3.5\mathrm{T}, \tag{15}\]  
The cut-off frequency is inversely proportional to the time constant \(\mathrm{T}\). The index function of the second-order channel filter is  
\[y(\mathrm{T})=k_{3}(3.5\mathrm{T})^{2}+k_{4}(\frac{1}{\mathrm{T}})^{2}, \tag{16}\]  
The next step is the same as the first-order channel filter, and therefore we obtain  
\[\mathrm{T}=0.5292\times\sqrt[4]{\beta}\beta=k_{4}/k_{3}, \tag{17}\]  
According to the performance of the FDI method, we select \(k_{3}\) and \(k_{4}\) reasonably and obtain a proper time constant, where \(\beta\) expresses the ratio of cut-off frequency to rising time in the second-order channel filter. At the same time, it indirectly indicates the ratio of false alarm rate to delay time for soft fault.