## 3 Sensor Fault Diagnosis
### Single Sensor Fault  
Assuming that \(m\) measurable parameters are used to estimate engine performance, \(m\) Kalman filters are designed accordingly. Each filter isolates the measured value of one of the sensors, and the remaining \((m-1)\) measured values are used for state/output estimation. The structure is shown in Figure 7. Take the \(i\)th Kalman filter as an example. Its input contains the remaining \((m-1)\) measurement values except for the \(i\)th sensor. After the failure of the \(i\)th sensor, the estimated values of the other filters are deviated from the actual measured values due to the access of the fault sensor signal. When the deviation of all the measured values is added up in a certain way, a relatively significant change can be obtained, and the faulty sensor can be identified. Moreover, since the \(i\)th filter is not connected to the fault sensor, the error between its estimated value and the actual measured  
Figure 5: Estimation of degradation by AEMM. Figure 6: Estimation of measurements by AEMM. value is small so that it can be used for fault signal reconstruction. The Kalman filter for the \(i\)th sensor is:  
\[\left\{ \begin{array}{l}\left[\begin{array}{c}\dot{x}\\ \hat{h}\end{array}\right]=A_{aug}\left[\begin{array}{c}\dot{x}-x_{e}(\alpha) \\ \hat{h}\end{array}\right]+K^{i}\big{(}y^{i}-\hat{y}^{i}\big{)}\\ \\ \hat{y}^{i}-y^{i}_{e}(\alpha)=C^{i}_{aug}\left[\begin{array}{c}\dot{x}-x_{e}( \alpha)\\ \hat{h}\end{array}\right]\end{array} \right. \tag{13}\]  
where \(y^{i}\) is the subset of measurement parameters excluding the \(i\)th sensor, matrix \(C^{i}_{aug}\) is the submatrix of matrix \(C_{aug}\) after removing the \(i\)th row, and \(K\) is the gain of the filter. Define the residual between the filter input \(y^{i}\) and the optimal output estimate \(\hat{y}^{i}\):  
\[e^{i}=y^{i}-\hat{y}^{i} \tag{14}\]  
when the sensor is fault-free and the filtering tends to be stable, the residual should obey the multi-dimensional standard normal distribution. However, when a fault occurs, the residual error does not possess this property, and its expectation is often not zero. Define the following variables:  
\[WSSR^{i}=W^{i}_{r}\Big{(}e^{i}\Big{)}^{T}\Big{(}\sigma^{2}_{i}\Big{)}^{-1} \Big{(}e^{i}\Big{)} \tag{15}\]  
where \(WSSR\) is the weighted sum of squared residual. When there is no fault, \(WSSR\) obeys the \(\chi^{2}\)-distribution of \((m-1)\) degrees of freedom. \(W^{i}_{r}\) is the weight, and it can adjust the output of the filter. Its value will affect the diagnostic effect, resulting in misdiagnosis and missed diagnosis. Select sensor measurement parameters as \(y=(n_{l},n_{h},P_{l3},T_{l3},P_{l21},T_{l21},P_{l6},T_{l6})\). Compared with AEMM established previously, the four additional sensors are the total pressure and temperature at the fan outlet and the total pressure and temperature at the mixing chamber inlet. State quantity and health parameters remain unchanged. The noise matrices Q and R are reselected, and 8 Kalman filters are designed. The input of each filter is another 7 measured values. Set the efficiency of high-pressure turbine components to degrade by 2%, and obtain the steady-state deviations of 8 sensors under normal conditions: \(\Delta n_{l}=-1.98\%\), \(\Delta n_{h}=-1.93\%\), \(\Delta P_{l3}=-52.84\) Kpa, \(\Delta T_{l3}=-10.0\) K, \(\Delta P_{l21}=-0.28\) Kpa, \(\Delta T_{l21}=-2.35\) K, \(\Delta P_{l6}=-2.96\) Kpa, \(\Delta T_{l6}=21.18\) K. At 2 s, a certain bias signal is injected into the outlet temperature sensor of the high-pressure compressor (\(T_{3}\)). As can be seen from Figure 8, except for the fourth filter, the \(WSSR\) value of all the other filters underwent drastic abrupt changes and then gradually approached a stable value, while the \(WSSR\) value of the fourth filter remained at a low level. Therefore, it can be determined that the fourth sensor, the temperature sensor \(T_{l3}\), has failed. Then, a drift fault of 1 K/s is injected  
Figure 7: Schematic diagram of Kalman filter bank structure. into the sensor.