## 2 Model of Air-Breathing Hypersonic Vehicle  
The motion equation of the AHV used in this paper can be found in several studies, such as [45; 46]. The longitudinal dynamics model of AHV is as follows:  
\[\left\{ \begin{array}{l}\dot{V}=\frac{T\cos\alpha-D}{m}-\frac{h}{r}\sin \gamma\\ \dot{\gamma}=\frac{L+T\sin\alpha}{mV}-\frac{h-V^{2}r}{Vr^{2}}\cos\gamma\\ \dot{\alpha}=q-\gamma\\ \dot{q}=\frac{M_{yy}}{1 yr}\\ \dot{h}=V\sin\gamma\\ \ddot{\beta}=-2\xi\omega\dot{\beta}-\omega^{2}\beta+\omega^{2}\beta_{c}\end{array} \right. \tag{1}\]  
where \(V\) and \(h\) are velocity and altitude, respectively; \(m\) is mass; \(\gamma\), \(\alpha\), and \(q\) are flight path angle, attack angle, and pitch angle rate, respectively; \(\mu\) and \(r\) are Earth's gravity constant and radial distance from Earth's center; \(M_{yy}\) and \(I_{yy}\) are pitch moment and rotation inertia; \(\beta\), \(\beta_{c}\), \(\xi\), and \(\omega\) are the state of the engine, throttle setting, damp ratio, and natural frequency, respectively. \(L\), \(D\), and \(T\) are lift, drag, and thrust, respectively. The \(M_{yy}\), \(L\), \(D\), and \(T\) are as follows:  
\[M_{yy}=\frac{1}{2}\rho V^{2}s\overline{c}\left(C_{M}^{\alpha}+C_{M}^{\beta}+C_ {M}^{\delta_{c}}\right) \tag{2}\]  
\[L=\frac{1}{2}\rho V^{2}sC_{L} \tag{3}\]  
\[D=\frac{1}{2}\rho V^{2}sC_{D} \tag{4}\]  
\[T=\frac{1}{2}\rho V^{2}sC_{T} \tag{5}\]  
where \(\rho\), \(s\), \(\overline{c}\), \(C_{L}\), \(C_{D}\), and \(C_{T}\) are air density, wing area, mean aerodynamic chord, lift coefficient, drag coefficient, and engine thrust coefficient, respectively. At the cruise phase, the coefficients are set as follows:  
\[\left\{ \begin{array}{l}C_{L}=0.6203\alpha\\ C_{D}=0.6450\alpha^{2}+0.0043378\alpha+0.003772\\ C_{T}=\left\{\begin{array}{l}0.02576\beta\\ 0.0224+0.00336\beta\\ \end{array}\right.\left(\beta\geq 1\right)\\ C_{M}^{\alpha}=0.035\alpha^{2}+0.036617\alpha+5.3261\times 10^{-6}\\ C_{M}^{\alpha}=\frac{\overline{c}}{2V}q\left(-6.796\alpha^{2}+0.3015\alpha-0. 2289\right)\\ C_{M}^{\alpha}=c_{e}(\delta_{e}-\alpha)\end{array} \right. \tag{6}\]  
where \(\delta_{e}\) and \(c_{e}\) are elevator deflection and elevator coefficient, respectively; \(C_{M}^{\alpha}\), \(C_{M}^{q}\), and \(C_{M}^{\delta_{c}}\) are moment coefficient due to attack angle, moment coefficient due to pitch rate, and moment coefficient due to elevator deflection, respectively. For the longitudinal model of AHV, the third derivative of \(V\) and the fourth derivative for \(h\) in Equation (1) are obtained as follows:  
\[\left\{ \begin{array}{l}\dot{V}=\frac{T\cos\alpha-D}{m}-\frac{\mu}{r^{2}}\sin \gamma=f_{1}(\vec{x})\\ \ddot{V}=\frac{\partial\dot{\vec{r}}(\vec{x})}{\partial\vec{x}}=\omega_{1}\dot{ \vec{x}}\\ \ddot{V}=\vec{x}^{\prime}\omega_{2}\dot{\vec{x}}+\omega_{1}\dot{\vec{x}}\end{array} \right. \tag{7}\]  
\[\left\{ \begin{array}{l}\dot{h}=V\sin\gamma\\ \ddot{h}=\dot{V}\sin\gamma+V\gamma\cos\gamma\\ \dddot{h}=\ddot{V}\sin\gamma+2\dot{V}\dot{\gamma}\cos\gamma-V\dot{\gamma}^{2} \sin\gamma+V\ddot{\gamma}\cos\gamma\\ h^{(4)}=\ddot{V}\sin\gamma+3\ddot{V}\dot{\gamma}\cos\gamma+3\dot{V}\ddot{\gamma} \cos\gamma-3\dot{V}\dot{\gamma}^{2}\sin\gamma\\ -3V\dot{\gamma}\dot{\gamma}\sin\gamma-V\dot{\gamma}^{3}\cos\gamma+V\dddot{ \gamma}\cos\gamma\end{array} \right. \tag{8}\]  
where  
\[\left\{ \begin{array}{l}\dot{\gamma}=\frac{L+T\sin\alpha}{m\dot{V}}-\frac{\mu -V^{2}r}{Vr^{2}}\cos\gamma=f_{2}(\vec{x})\\ \ddot{\gamma}=\pi_{1}\dot{\vec{x}}\\ \dddot{\gamma}=\pi_{1}\dot{\vec{x}}+\dot{\vec{x}}^{T}\pi_{2}\dot{\vec{x}}\end{array} \right. \tag{9}\]  
where \(\vec{x}=[V,\gamma,\alpha,\beta,h]^{T}\), \(\omega_{1}=\partial f_{1}(\vec{x})/\partial\vec{x}\), \(\pi_{1}=\partial f_{2}(\vec{x})/\partial\vec{x}\), \(\pi_{2}=\partial\pi_{1}/\partial\vec{x}\). Therefore, the linearized model of AHV is represented as follows:  
\[\left[\dddot{V}\quad h^{(4)}\right]^{T}=\left[\dddot{V}_{0}\quad H_{0}^{(4)} \right]^{T}+ \begin{bmatrix}b_{11}&b_{12}\\ b_{21}&b_{22}\end{bmatrix} \left[\beta_{c}\quad\delta_{e}\right]^{T} \tag{10}\]  
where  
\[\left\{