## 2 Model and Nominal NDI Control
### Longitudinal Model  
The state equation of the longitudinal system is as follows:  
\[\left\{ \begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array} \right. \tag{1}\]  
where \(x\) is the system state variables, \(u\) is the system input, \(y\) is the system output, they satisfy:  
\[\left\{ \begin{array}{l}x=[\alpha,q,\theta]^{T}\\ u=\delta_{e}\\ y=[\dot{\alpha},\alpha,q,\dot{q}]^{T}\end{array} \right. \tag{2}\]  
According to the scientific report (Document ID 19910003392) of the NASA Langley Research Center, and references [2; 3; 4; 5], the system state matrices \(A\), \(B\), \(C\), and \(D\) are represented as follows:  
\[A= \begin{bmatrix}-Y^{\alpha}+\frac{\delta}{V_{*}}\sin\mu_{*}&1&-\frac{\delta} {V_{*}}\sin\mu_{*}\\ M_{2}^{\alpha}-M_{2}^{\dot{\alpha}}(Y^{\alpha}-\frac{\delta}{V_{*}}\sin\mu_{*} )&M_{2}^{\dot{\alpha}}+M_{2}^{\dot{\alpha}}&-M_{2}^{\dot{\alpha}}\frac{\delta }{V_{*}}\sin\mu_{*}\\ 0&1&0\end{bmatrix}  \tag{3}\]  
\[B= \begin{bmatrix}-Y^{\delta_{e}}\\ M_{2}^{\delta_{e}}-M_{2}^{\dot{\alpha}}Y^{\dot{\alpha}}\\ 0\end{bmatrix}  \tag{4}\]  
\[C= \begin{bmatrix}-Y^{\alpha}+\frac{\delta}{V_{*}}\sin\mu_{*}&1&-\frac{\delta }{V_{*}}\sin\mu_{*}\\ 1&0&0\\ 0&1&0\\ M_{2}^{\alpha}-M_{2}^{\dot{\alpha}}(Y^{\alpha}-\frac{\delta}{V_{*}}\sin\mu_{*} )&M_{2}^{\dot{\alpha}}+M_{2}^{\dot{\alpha}}&-M_{2}^{\dot{\alpha}}\frac{\delta }{V_{*}}\sin\mu_{*}\end{bmatrix}  \tag{5}\]  
\[D= \begin{bmatrix}-Y^{\delta_{e}}\\ 0\\ 0\\ 0\\ M_{2}^{\delta_{e}}-M_{2}^{\dot{\alpha}}Y^{\dot{\alpha}_{e}}\end{bmatrix}  \tag{6}\]  
where the aerodynamic and structural composite state parameters are expressed as  
\[Y^{\alpha}=\frac{P_{0}\cos\frac{\alpha_{0}}{57.3}+57.3C_{z,A}\times 0.5\rho V ^{2}S}{mV} \tag{7}\]  
\[Y^{\delta_{e}}=\frac{57.3C_{z,\delta_{e}}\times 0.5\rho V^{2}S}{mV} \tag{8}\]  
\[M_{z}^{\alpha}=\frac{57.3C_{m,\delta_{e}}\times 0.5\rho V^{2}Sb}{ly} \tag{9}\]  
\[M_{z}^{\delta_{e}}=\frac{57.3C_{m,\delta_{e}}\times 0.5\rho V^{2}Sb}{ly} \tag{10}\]\[M_{z}^{\dot{\alpha}}=\frac{C_{m,\dot{\alpha}}\times 0.5\rho VSb^{2}}{I_{y}} \tag{11}\]  
\[M_{z}^{\dot{\alpha}}=\frac{C_{m,\dot{\alpha}}\times 0.5\rho VSb^{2}}{I_{y}} \tag{12}\]  
The small disturbance linearization processing method must be used near a certain system equilibrium point, so the system equilibrium point must be obtained before applying this method. The equilibrium point of the system is the equilibrium point where the dynamic system makes the state in a stable state, which represents the state point of the system where the rate of change in the state vector is 0. For a dynamic system of an aircraft dynamics model, the balance point is the point at which the flight trajectory and attitude are kept stable, such as the state points of horizontal steady flight, steady hover, steady climb, etc. **Remark 1**. : _An equilibrium state of the system is chosen as follows: Sea-skimming altitude H = 2 m, Mach M = 15 Ma, flight speed at balance point V\(\_\)= 51.0441 m/s, track inclination at balance point \(\mu\)\(\_\),ide = 0, \(P_{0}\) = 15.1 kg, angle of attack at balance point \(\alpha_{0}\) = 3.25\({}^{\circ}\), \(\rho_{0}\) = 0.1249 kg\(\cdot\)s\({}^{2}\)/m\({}^{4}\), air density \(\rho\) = \(\rho_{0}\) g = 1.22 kg/m\({}^{3}\), g = 9.806 m/s\({}^{2}\)._  
The longitudinal model of the balance point is an ideal state, and the modeling error is on the ideal track inclination angle \(\mu\)\(\_\),ide, and the real system cannot completely ensure that the balance point track inclination angle \(\mu\)\(\_\) is zero. Therefore, this study assumes that there is an unknown modeling error \(\Delta\mu\)\(\_\) and satisfies:  
\[|\Delta\mu\)\(\_\)\(|\leq\)\(\mu\)\({}_{\text{max}}\)\(\leq\) 5\({}^{\circ}\)  
\[\mu\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\_\)\(\)\(\mu\)\(\_\)\(\_\)\(\_\)\(\_\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(\)\(  
**Remark 2**.