## 3 Proposed Method
### Fault Feature Selection  
Methods for fault diagnosis based on the Kalman filter model can be mainly divided into the state chi-squared test, residual chi-squared test, and improvements on the basis of them. The state chi-squared test involves constructing a state propagator to recursively estimate another state according to prior information. However, since it lacks measurement updates, errors accumulate over time and sensitivity to fault reduces. On the other hand, the residual chi-squared test constructs a test statistic based on innovations generated by the model. Innovations \(r_{k}\) are calculated from the actual measured values \(\mathbf{z}_{k}\) and their estimation \(\overset{\wedge}{\hat{\mathbf{z}}}_{k/k-1}\):  
\[r_{k}=\mathbf{z}_{k}-\overset{\wedge}{\hat{\mathbf{z}}}_{k/k-1} \tag{19}\]  
When no faults exist in the air data system, \(r_{k}\) should follow a zero-mean Gaussian distribution as a white noise sequence, with a variance of \(A_{k}\):  
\[A_{k}=h_{k}P_{k/k-1}h_{k}^{\mathsf{T}}+R_{k} \tag{20}\]  
When faults occur, the predicted values of the measurements are no longer unbiased estimates, and they do not follow a zero-mean white noise distribution. Therefore, a test statistic \(s_{k}\) can be defined:  
\[s_{k}=r_{k}^{\mathsf{T}}A_{k}^{-1}r_{k} \tag{21}\]  
The threshold \(Th\) can be obtained through the quantiles of the chi-squared distribution. If \(s_{k}>Th\), it indicates a fault occurrence. However, this method performs poorly in diagnosing gradual faults. Due to the gradual nature of such faults, the changes of output at a given moment may not be significant, resulting in minimal variations in innovations. The fault output further influences the estimation at the next moment, keeping the innovation within a small range and making fault diagnosis much more challenging.  
Despite various research efforts for improvement, challenges such as the difficulty of determining thresholds, long diagnostic time, and the risk of misdiagnosis still remain. Furthermore, achieving fault sensor localization and fault-type identification after detecting faults still requires further in-depth research. Therefore, the data-driven approach is considered, which leverages the powerful feature extraction and classification capabilities of neural networks to achieve fast and accurate fault diagnoses.  
In the selection of fault characteristic sequences, based on the analysis of the residual chi-squared test, the innovation sequence can reflect fault information, but it still has limitations. Therefore, drawing inspiration from the idea of the state chi-squared test, state residual \(e_{k}\) is incorporated, which is the difference between the state \(x_{k}\) and the estimation \(\hat{\vec{x}}_{k/k-1}\):  
\[e_{k}=x_{k}-\hat{\vec{x}}_{k/k-1} \tag{22}\]  
After introducing the state residual sequence, it gradually increases over time when gradual faults occur, which effectively compensates for the insensitivity of the innovation sequence to gradual faults. Therefore, the final selection of sequences is as follows:  
\[\left\{\begin{array}{l}r_{k}=z_{k}-\hat{\vec{\lambda}}_{k/k-1}\\ e_{k}=\vec{x}_{k}-\hat{\vec{x}}_{k/k-1}\end{array}\right. \tag{23}\]  
The innovations and residuals both encompass the 6-dimensional parameters \([V,\alpha,\beta,\phi,\theta,\psi]^{\mathrm{T}}\) of the ADS, a total of 12 dimensions. These data are used as inputs for the neural network for subsequent fault diagnosis, localization, and identification.