## 2 On-Board Adaptive Model  
### Equilibrium Manifold Model  
Equilibrium manifold model (EMM) is a model developed from the linearization model [15]. Consider the following smooth nonlinear system:  
\[\left\{ \begin{array}{l}\dot{x}=f(x,u)\\ y=g(x,u)\end{array} \right., \tag{1}\]  
where \(f(x,u)\) and \(g(x,u)\) are smooth functions. Do Taylor expansion at any equilibrium point and abandon the higher-order terms, and a linear model of the original system can be obtained:  
\[\left\{ \begin{array}{l}\dot{x}=A(x-x_{e})+B(u-u_{e})\\ y-y_{e}=C(x-x_{e})+D(u-u_{e})\end{array} \right.. \tag{2}\]  
The set of all equilibrium points in a system constitutes an equilibrium family \(\Gamma_{e}\) and \(\Gamma_{e}\) can be considered an equilibrium manifold. \[\Gamma_{e}=\{(x_{e},u_{e},y_{e})|f(x_{e},u_{e})=0,y_{e}=g(x_{e},u_{e})\}. \tag{3}\]  
These equilibria can be scheduled with a variable of the same dimension as the control quantity \(u\), which is defined as the scheduling variable \(\alpha\). \(\alpha\) is composed of relevant variables of the current working point and can be expressed as \(\alpha=p(x,y,u)\). Then we can get:  
\[\left\{ \begin{array}{l}x_{e}=x_{e}(\alpha)\\ u_{e}=u_{e}(\alpha)\\ y_{e}=y_{e}(\alpha)\end{array} \right.. \tag{4}\]  
Similarly, the state matrix and control matrix of the linearized model of the system at each equilibrium point can also be expressed as the matrix scheduled with \(\alpha\), so as to obtain the dynamic model of the system near the equilibrium manifold:  
\[\left\{ \begin{array}{l}\dot{x}=A(\alpha)(x-x_{e}(\alpha))+B(\alpha)(u-u_{e} (\alpha))\\ y-y_{e}(\alpha)=C(\alpha)(x-x_{e}(\alpha))+D(\alpha)(u-u_{e}(\alpha))\end{array}  \right.. \tag{5}\]  
To make the linearized model family (5) a nonlinear model, it is necessary to establish the relationship between scheduling variables \(\alpha\) and the current working point. Considering the algebraic relationship between \(y\) and \(x\), \(u\), only the following design is needed:  
\[\alpha=p(x,u). \tag{6}\]  
By combining Equations (5) and (6), a nonlinear model scheduled by \(x\) and \(u\) can be obtained, that is, an equilibrium manifold model. It can be seen that the EMM is derived from the linearized model family (5) and the scheduling relation (6). It has the characteristic of variable parameter of linear parameter varying (LPV) model whose coefficient matrix all changes with the working state, and the introduction of Equation (4) makes the model always change near the equilibrium manifold, which overcomes the deficiency of steady-state error in LPV model. In terms of the structure of the model, the EMM can be regarded as a Taylor expansion model at the equilibrium point of real-time change. The Equation (6) essentially establishes a mapping of the current operating point to the Taylor expansion point on the equilibrium manifold. Figure 2 shows the three mapping modes of equal input \(u=u_{e}(\alpha)\), equal partial state \(x_{i}=x_{ie}(\alpha)\), and the orthogonal expansion \(\sqrt{\left(x_{e}(\alpha)-x\right)^{2}+\left(u_{e}(\alpha)-u\right)^{2}}\). The Equation (6) can be obtained from these mappings. The three designs have different EMMand different errors from Equation (1). Determine the value of the scheduling variable utilizing equal input shown in Figure 2 and select fuel flow \(W_{f}\) and nozzle area \(A_{8}\) as the scheduling variables. Then, there is \(W_{fe}(\alpha)=W_{f}\), \(A_{8e}(\alpha)=A_{8}\) and \(\alpha=\left(W_{f},A_{8}\right)\). Represent \(A(\alpha)\) as \(\left[ \begin{array}{cc}a_{11}(\alpha)&a_{12}(\alpha)\\ a_{21}(\alpha)&a_{22}(\alpha)\end{array} \right]\) and the EMM of the above process can be expressed as:  
\[\left\{ \begin{array}{l}\dot{n}_{h}=a_{11}(\alpha)(n_{h}-n_{he}(\alpha))+a_{1 2}(\alpha)(n_{l}-n_{le}(\alpha))\\ \dot{n}_{l}=a_{21}(\alpha)(n_{h}-n_{he}(\alpha))+a_{22}(\alpha)(n_{h}-n_{le}( \alpha))\\ \alpha=u=\left(W_{fe},A_{8e}\right)\end{array}