# In-the-Loop Simulation Experiment of Aero-Engine Fault-Tolerant Control Technology
## 2 Materials and Methods
### Establishment of Augmented State Space Model for Turbofan Engine  
The Kalman filter is a state observer, which can only modify the state quantity. Therefore, it is necessary to introduce the health parameters characterizing the degradation of gas path components and extend them to the state variables in order to correct the health parameters. The state space expression of the introduced health parameters is:  
\[\left\{ \begin{array}{l}\dot{x}=A\cdot x+B\cdot u+L\cdot h\\ y=C\cdot x+D\cdot u+M\cdot h\end{array} \right. \tag{11}\]  
The parameters are:  
\[\left\{\begin{array}{l}u=\left[ \begin{array}{cccc}\Delta W_{f}&\Delta A _{8}\end{array} \right]^{T}\\ \begin{array}{cccc}x=\left[\begin{array}{cccc}\Delta N_{h}&\Delta N_{l} \end{array}\right]^{T}\\ y=\left[\begin{array}{cccccccc}\Delta N_{h}&\Delta N_{l}&\Delta P_{13}& \Delta P_{23}&\Delta P_{3}&\Delta P_{5}&\Delta T_{23}&\Delta T_{3}&\Delta T_{5 }\end{array}\right]^{T}\\ h=\left[\begin{array}{cccccccc}\Delta W_{fan}&\Delta E_{fan}&\Delta W_{ com}&\Delta E_{com}&\Delta W_{lpt}&\Delta E_{lpt}&\Delta W_{lpt}&\Delta E_{lpt} \end{array}\right]^{T}\end{array} \right. \tag{12}\]  
\(\Delta W_{fan}\)\(\Delta E_{fan}\), \(\Delta W_{com}\), \(\Delta E_{com}\), \(\Delta W_{lpt}\), \(\Delta E_{lpt}\), \(\Delta W_{lpt}\) and \(\Delta E_{lpt}\) are the flow and efficiency degradation coefficients of the four rotating parts, respectively; \(\Delta\) represents the relative change. After the health parameter is extended to the state quantity, Equation (11) becomes:  
\[\left\{ \begin{array}{c}\left[\begin{array}{c}\dot{x}\\ \dot{h}\end{array}\right]=\left[\begin{array}{cc}A&L\\ 0&0\end{array}\right].\left[\begin{array}{c}x\\ h\end{array}\right]+\left[\begin{array}{c}B\\ 0\end{array}\right]\cdot u\\ y=\left[\begin{array}{cc}C&M\end{array}\right]\cdot\left[\begin{array}{c}x \\ h\end{array}\right]+D\cdot u\end{array} \right. \tag{13}\]  
The noise term is introduced into the obtained augmented state space model Equation (13):  
\[\left\{ \begin{array}{c}\dot{x}(t)=A_{2}\cdot x(t)+B_{2}\cdot u(t)+W(t)\\ y(t)=C_{2}\cdot x(t)+D_{2}\cdot u(t)+V(t)\end{array} \right. \tag{14}\]  
where \(W(t)\) is system noise and \(V(t)\) is Gaussian white noise, i.e., measurement noise, and \(x(t)\) is the augmented state variable. Combining the extended Kalman filter with the nonlinear model, the optimal estimation of the state quantity is obtained as follows:  
\[\left\{ \begin{array}{c}\dot{\hat{x}}(t)=A_{2}\cdot\hat{x}(t)+K\cdot\left[y(t )-\hat{y}(t)\right]\\ \dot{y}(t)=f(\hat{x}(t))\\ \dot{x}(t+1)=x(t)+\hat{x}(t)\end{array} \right. \tag{15}\]  
\[\left\{ \begin{array}{c}\overline{P}=A_{2}\cdot P\cdot A_{2}{}^{T}+Q\\ K=\overline{P}\cdot C_{2}{}^{T}\cdot\left(C_{2}\cdot\overline{P}\cdot C_{2}{}^ {T}+R\right)^{-1}\\ P=(I-K\cdot C_{2})\cdot\overline{P}\end{array} \right. \tag{16}\]  
In Equation (15), \(\hat{x}(t)\) is the optimal estimation value of state quantity, \(y(t)\) is the output value of real engine sensor, \(\hat{y}(t)\) is the predicted output value of airborne model, \(f(\hat{x}(t))\) is a component level nonlinear model, and K is a constant Kalman gain matrix. In Equation (16), P, Q, and R are covariance matrices of state, system noise, and measurement noise, respectively. Iterate Equation (16), and finally K will tend to a fixed value. The matrix can extract the correction information of ten state variables (high- and low-pressure speed, efficiency, and flow health parameters of four rotating parts) from the residuals of nine sensors, so as to estimate the accurate health parameters. The Kalman gain matrix K is:  
\[K=\left[