## 2 NMV Model Description  
This paper is concerned with the problems of robust switching control for NMVs. Illustrations of typical NMV modes based on X-24B configuration [28] are shown in Figure 1. The data come from an experimental aircraft model, and the winglets were used to adjust the lift-drag ratio, mean aerodynamic chord, and reference area. According to [28; 29], based on reasonable assumption, the nonlinear equations of the longitudinal motion are formulated as  
\[\left\{ \begin{array}{ll}\dot{V}=\dfrac{T\cos\alpha-D}{m}-g\sin\gamma\\ \dot{h}=V\sin\gamma\\ \dot{\gamma}=\dfrac{L+T\sin\alpha}{mV}-\dfrac{g\cos\gamma}{V}\\ \dot{a}=q-\dot{\gamma}\\ \dot{q}=M_{yy}/I_{yy}\\ \ddot{\eta}=-2\zeta\omega_{n}\dot{\eta}-\omega_{n}^{2}\eta+\omega_{n}^{2}\eta _{c}\end{array} \right. \tag{1}\]  
where \(V,h,\gamma,\alpha\), and \(q\) are flight velocity, flight altitude, flight path angle, angle of attack, and pitch angle rate, respectively. \(\omega_{n},\zeta\), and \(\eta_{c}\) represent natural frequency, damp ratio, and throttle setting, respectively. The lift, drag, thrust, and pitching moment are given as follows:  
\[\left\{ \begin{array}{ll}L=0.5\rho V^{2}sC_{L}\\ D=0.5\rho V^{2}sC_{D}\\ T=0.5\rho V^{2}sC_{T}\\ M_{yy}=0.5\rho V^{2}\bar{s}(C_{M}^{\alpha}+C_{M}^{q}+C_{M}^{\dot{\alpha}}) \end{array} \right. \tag{2}\]  
Figure 1: Illustrations of typical NMV modes based on X-24B configuration. where \(\rho,s,\) and \(\overline{c}\) stand for the air density, reference area, and mean aerodynamic chord, respectively. The coefficients \(C_{L},C_{D},C_{T},C_{M}^{\kappa},C_{M}^{q},\) and \(C_{M}^{\delta_{e}}\) are modeled by data fitting, and the approximations are given as follows:  
\[\left\{ \begin{array}{l}C_{L}=C_{L}^{\kappa}\alpha+C_{L}^{0}\\ C_{D}=C_{D}^{\kappa 2}\alpha^{2}+C_{D}^{\kappa}\alpha+C_{D}^{0}\\ C_{T}=C_{T}^{\eta}\eta+C_{T}^{0}\\ C_{M}^{\kappa}=C_{M}^{\kappa,A^{2}}\alpha^{2}+C_{M}^{\kappa,A}\alpha+C_{M}^{ \kappa,0}\\ C_{M}^{\delta_{e}}=c_{e}(\delta_{e}-\alpha)\end{array} \right. \tag{3}\]  
According to the nonlinear system theory, the state vector \(x=[V,\gamma,\alpha,\eta,h]\) can be defined. If there are no parameter uncertainties, the input-output feedback linearization method can be adopted for model (1), and we obtain  
\[ \begin{bmatrix}\ddot{V}\\ h^{(4)}\end{bmatrix} = \begin{bmatrix}F_{V}\\ F_{h}\end{bmatrix} + \begin{bmatrix}b_{11}&b_{12}\\ b_{21}&b_{22}\end{bmatrix}\begin{bmatrix}\eta_{c}\\ \delta_{e}\end{bmatrix}  \tag{4}\]  
where the control input \(u=[\eta_{c},\delta_{e}]^{\top}\), \(F_{V},F_{h},b_{11},b_{12},b_{21},b_{22}\) can be obtained as follows [28]:  
\[\left\{ \begin{array}{l}F_{v}=\frac{\omega_{1}\ddot{X}_{0}+\dot{x}^{\top} \Omega_{2}\dot{x}}{m}\\ F_{h}=3\ddot{V}\dot{\gamma}\cos\gamma-3\dot{V}\gamma^{2}\sin\gamma+3\dot{V} \ddot{\gamma}\cos\gamma-V\dot{\gamma}^{3}\cos\gamma-3V\dot{\gamma}\ddot{ \gamma}\sin\gamma+f(V,\gamma)\\ f(V,\gamma)=\frac{\left(\omega_{1}\ddot{X}_{0}+\dot{x}^{\top}\Omega_{2}\dot{x }\right)\sin\gamma}{m}+V\cos\gamma\Big{(}\dot{x}^{\top}\Pi_{2}\dot{x}+\pi_{1} \ddot{X}_{0}\Big{)}\\ b_{11}=\frac{\rho V^{2}sc_{g}\omega_{n}^{2}}{2m}\cos\alpha\\ b_{12}=-\frac{\rho V^{2}s\overline{c}c_{e}}{2 mI_{y}}(T\sin\alpha+D_{\alpha})\\ b_{21}=\frac{\rho V^{2}sc_{g}\omega_{n}^{2}}{2m}\sin(\alpha+\gamma)\\ b_{22}=\frac{\rho V^{2}s\overline{c}c_{e}}{2 mI_{y}}[T\cos(\alpha+\gamma)+L_{ \alpha}\cos\gamma-D_{\alpha}\sin\gamma]\end{array} \right. \tag{5}\]  
where \(\omega_{1}=\partial f_{1}(x)/\partial x\), \(f_{1}(x)=T\cos\alpha-D-mg\sin\gamma\), \(\Omega_{2}=\partial\omega_{1}/\partial x\), \(\pi_{1}=\partial f_{2}(x)/\partial x\), \(f_{2}(x)=(L+T\sin\alpha)/mV-g\cos\gamma/V\), \(\Pi_{2}=\partial\pi_{1}/\partial x\).