## 4 Design of Tracking Controller for the AHV  
In this section, the flight controller can be developed through Theorem 2.  
According to (14) and (10), we define new states \(\mathbf{x}_{1}=\begin{bmatrix}V&\dot{h}\end{bmatrix}^{T}\), \(\mathbf{x}_{2}=\begin{bmatrix}\dot{V}&\dot{h}\end{bmatrix}^{T}\), and \(\mathbf{x}_{n}=\begin{bmatrix}\ddot{V}&\dot{h}\end{bmatrix}^{T}\), where \(n=3\), and the model of AHV Equation (10) is transformed to the following new model with uncertainties:  
\[\left\{\begin{array}{l}\dot{\mathbf{x}}_{1}=\mathbf{x}_{2}+\mathbf{F_{1}}\\ \dot{\mathbf{x}}_{2}=\mathbf{x}_{3}+\mathbf{F}_{2}\\ \dot{\mathbf{x}}_{3}=\mathbf{f}(\mathbf{x},t)+\mathbf{G}(\mathbf{x},t)\mathbf{u}+\mathbf{F}_{3}\end{array}\right. \tag{72}\]  
where  
\[\left\{\begin{array}{l}\mathbf{f}(\mathbf{x},t)=\begin{bmatrix}\ddot{V}_{0}&H_{0}^{ (4)}\end{bmatrix}^{T}\\ \mathbf{G}(\mathbf{x},t)=\begin{bmatrix}b_{11}&b_{12}\\ b_{21}&b_{22}\end{bmatrix}\\ \mathbf{u}=\begin{bmatrix}\beta_{c}&\delta_{e}\end{bmatrix}^{T}\end{array}\right. \tag{73}\]Considering the model of AHV with uncertainties given by Equation (72), the errors are defined as follows:  
\[\left\{\begin{array}{l}z_{1}=x_{1}-\alpha_{1}\\ z_{2}=x_{2}-\alpha_{2}\\ z_{3}=x_{3}-\alpha_{3}\end{array}\right. \tag{74}\]  
where  
\[\left\{\begin{array}{l}\mathbf{\alpha}_{1}=\begin{bmatrix}V_{d}&\dot{h}_{d}\end{bmatrix }^{T}\\ \mathbf{\alpha}_{2}=-k_{1}\mathbf{z}_{1}+\dot{\mathbf{\alpha}}_{1}-\widehat{\mathbf{F}}_{1}\\ \mathbf{\alpha}_{3}=-k_{2}\mathbf{z}_{2}+\dot{\mathbf{\alpha}}_{2}-\mathbf{z}_{1}-\widehat{\mathbf{ F}}_{2}\end{array}\right. \tag{75}\]  
where \(\mathbf{\alpha}_{1}\) is the command signal, \(V_{d}\) and \(\dot{h}_{d}\) are the velocity command and differential of altitude command, respectively; \(\mathbf{\alpha}_{2}\) and \(\mathbf{\alpha}_{3}\) are virtual control laws.  
According to (55) and (56), the tracking controller for AHV is designed as follows:  
\[\mathbf{u}=\mathbf{G}(\mathbf{x},t)^{-1}\Big{(}-2_{2}-c_{1}\dot{\mathbf{z}}_{1}-c_{2}\dot\mathbf{{z}}_{2}-\mathbf{f}(\mathbf{x},t)-\widehat{\mathbf{F}}_{3}+\dot{\mathbf{\alpha}}_{3}+\mathbf{u}_{ sw}\Big{)} \tag{76}\]  
where \(\mathbf{u}_{sw}=\begin{bmatrix}-D_{1,1}|s_{1}|^{\alpha_{1}}sgn(s_{1})-D_{1,2}|s_{ 1}|^{\beta_{1}}sgn(s_{1})-D_{1,3}s_{1}\\ -D_{2,1}|s_{2}|^{\alpha_{2}}sgn(s_{2})-D_{2,2}|s_{2}|^{\beta_{2}}sgn(s_{2})-D_ {2,3}s_{2}\end{bmatrix}\), and adaptive laws are designed as follows:  
\[\left\{\begin{array}{l}\widehat{\mathbf{F}}_{1}=\Gamma_{1}\mathbf{z}_{1}\\ \widehat{\mathbf{F}}_{2}=\Gamma_{2}\mathbf{z}_{2}\\ \widehat{\mathbf{F}}_{3}=\Gamma_{3}\mathbf{S}\end{array}\right. \tag{77}\]  
where \(\mathbf{S}=c_{1}\mathbf{z}_{1}+c_{2}\mathbf{z}_{2}+\mathbf{z}_{3}\).  
According to Equation (48), a Lyapunov function candidate is chosen as:  
\[V_{3}=\frac{1}{2}\sum_{j=1}^{2}k_{j}\widetilde{\mathbf{z}}_{j}^{T}\mathbf{z}_{j}+ \frac{1}{2}\sum_{j=1}^{3}\widetilde{\mathbf{F}}_{j}^{T}\Gamma_{j}^{-1}\widetilde{ \mathbf{F}}_{j}+\frac{1}{2}\mathbf{S}^{T}\mathbf{S} \tag{78}\]  
We defining the matrix as:  
\[\left\{\begin{array}{l}\mathbf{z}=\begin{bmatrix}\mathbf{z}_{1}&\mathbf{z}_{2}\end{bmatrix }^{T}\\ \mathbf{Q}=\begin{bmatrix}k_{1}&0\\ c_{1}&k_{2}+c_{2}\end{bmatrix}\end{array}\right. \tag{79}\]  
From Equation (71), we get:  
\[\dot{V}_{3}=-\mathbf{z}^{T}\mathbf{Q}\mathbf{z}-\sum_{m=1}^{2}(D_{m,1}|s_{m}|^{\gamma_{m}+ 1}+D_{m,2}|s_{m}|^{\beta_{m}+1}+D_{m,3}s_{m}^{2})\leq 0 \tag{80}\]