## 2 On-Board Adaptive Model
### On-Board Adaptive Equilibrium Manifold Model  
Referring to the conventional linear state variable model with degradation, the following EMM with degradation is established:  
\[\left\{ \begin{array}{l}\dot{x}=A(\alpha)(x-x_{e}(\alpha))+L(\alpha)h\\ y-y_{e}(\alpha)=C(\alpha)(x-x_{e}(\alpha))+M(\alpha)h\end{array} \right., \tag{9}\]  
where \(h\) is the engine health parameter, representing the degradation of the flow/efficiency of each engine component. \(\alpha\) is the scheduling variable. Since the health parameters are independent of the scheduling variables, they are not scheduled by them, while the coefficient matrices \(L\) and \(M\) are related to the engine operating points and are scheduled by the scheduling variables. Considering that performance degradation is a slow development process during the service period of the engine, it does not require high dynamic characteristics and focuses  
Figure 4: EMM vs CLM—output quantity. Figure 3: EMM vs CLM—input quantity. on its steady-state value. Therefore, based on the EMM without degradation, the matrix expressions are retained, and the health parameters are changed in the component level model to cause changes in state quantities and output quantities. The degradation coefficient matrices \(L(\alpha)\) and \(M(\alpha)\) are obtained by the changing steady-state values identification to form the final EMM with degradation. An EMM with degradation is not enough to be called an adaptive model. It needs to have the ability to update health parameters (degradation) to match with the actual engine running state. For this reason, a Kalman filter is introduced to estimate the starter degradation from the measurable output deviation containing the measured noise. Firstly, model (9) is written into a standard form. For this reason, the health parameters are extended to state quantities. \[\left\{ \begin{array}{l}\left[\begin{array}{c}\dot{x}\\ \dot{h}\end{array}\right]=A_{aug}\left[\begin{array}{c}x-x_{e}(\alpha)\\ h\end{array}\right]+\omega\\ y-y_{e}(\alpha)=C_{aug}\left[\begin{array}{c}x-x_{e}(\alpha)\\ h\end{array}\right]+v\end{array} \right., \tag{10}\]  
where \(A_{aug}=\left[ \begin{array}{cc}A(\alpha)&L(\alpha)\\ 0&0\end{array} \right]\) and \(C_{aug}=\left[ \begin{array}{cc}C(\alpha)&M(\alpha)\end{array} \right]\). \(\omega\) and \(v\) are the system noise matrix and measurement noise matrix respectively, and the covariance matrices are \(Q\) and \(R\). Then, the Kalman filter uses the probability distribution of the measured value and the optimal estimate to obtain the optimal state estimate:  
\[\left\{ \begin{array}{l}\left[\begin{array}{c}\dot{x}\\ \dot{h}\end{array}\right]=A_{aug}\left[\begin{array}{c}\hat{x}-x_{e}(\alpha) \\ \hat{h}\end{array}\right]+K(y-\hat{y})\\ \hat{y}-y_{e}(\alpha)=C_{aug}\left[\begin{array}{c}\hat{x}-x_{e}(\alpha) \\ \hat{h}\end{array}\right]\end{array} \right., \tag{11}\]  
where \(\hat{x}\), \(\hat{y}\), \(\hat{h}\) are estimated values of state quantity, output quantity and degradation quantity. \(K\) is called the Kalman filter gain, and \(K=PC_{aug}R^{-1}\). \(P\) is the solution of the following Riccati equation:  
\[A_{aug}P+PA_{aug}^{T}-PC_{aug}^{T}R^{-1}C_{aug}P+Q=0. \tag{12}\]  
Thus, the AEMM is formed. The ability of health parameter estimation of this model is verified at the design point below. Set high-pressure turbine component efficiency in degrading by 2%, and the steady-state deviation of the output of the CLM and the output without degradation is respectively \(\Delta n_{l}=-1.98\%\), \(\Delta n_{h}=-1.93\%\), \(\Delta P_{3}=-52.84\) Kpa and \(\Delta T_{3}=-10.0\) K. The steady-state output is added with Gaussian noise and fed into the Kalman filter to obtain the optimal estimation of health parameters and measured values, as shown in Figures 5 and 6. It can be seen that the Kalman filter can accurately estimate the efficiency degradation of the high-pressure turbine components at 2% within 1.5 s, and the estimation accuracy of other health parameters is also within 0.1%.