## 3 Flight Controller Design
### Proposed Analytic Form of the Proposed Position Integral Backstepping Controller
\right\|\to\varepsilon_{xz0}\quad\text{as}\quad t\to\infty \tag{107}\]  
_where_  
\[\varepsilon_{xz0}=\frac{\lambda_{max}(\mathbf{P}_{xz})}{\lambda_{min}(\mathbf{ Q}_{xz})}\sqrt{\bigg{(}\frac{d_{x}^{+}}{mk_{ix}}\bigg{)}^{2}+\bigg{(}\frac{k_{dx}d_{x}^ {+}}{F_{min}}+\frac{k_{dz}d_{x}^{+}}{\varsigma_{\theta}+F_{min}}|\varphi_{1} |\bigg{)}^{2}} \tag{108}\]  
_and \(\theta^{+}\) is the upper bound of the attitude angle \(\theta\), that is,_  
\[|\theta|\leq\theta^{+} \tag{109}\]_It is assumed that the upper bound \(\theta^{+}<\pi/2\) during the flight mission._  
Proof of Theorem 3.: When realizing the IBC (54), it can be found from (55) that \(\dot{\varphi}_{2}\) requires second derivatives of \(\varphi_{1}\). From (56), it implies that the information of \(F\) is needed to construct \(\varphi_{1}\). In other words, to derive the analytic form of IBC (54), namely AIBC, it is necessary to compute the first and second derivatives of \(F\) and \(\varphi_{1}\), respectively. Next, construct \(\dot{\varphi}_{2}\) such that the analytic form of IBC (54) can be obtained. When performing the time derivatives, the appeared terms \(\dot{x}\), \(\dot{z}\), and \(\ddot{\theta}\) are replaced by the system dynamic equations (9)-(11). The terms correlated to the disturbance will be expressed as a unified variable starting with \(\triangle\), and the remaining terms can be compensated since it is state-dependent, which is denoted as the subscript \(0\). Finally, the AIBC is constructed in terms of state-dependent terms only. As mentioned above, to obtain the first derivative of \(F\) and \(\varphi_{1}\), differentiate \(u_{z}\) in (38) and \(u_{x}\) in (57) as follows:  
\[\dot{u}_{z} =\dot{u}_{z,0}+\triangle u_{z,0} \tag{110}\] \[\dot{u}_{x} =\dot{u}_{x,0}+\triangle u_{x,0}\]  
where \(\dot{u}_{z,0}\) is shown in (104), \(\dot{u}_{x,0}\) is shown in (105), and  
\[\triangle u_{z,0} =-k_{dz}\frac{d_{z}}{m} \tag{111}\] \[\triangle u_{x,0} =-k_{dx}\frac{d_{x}}{m},\]  
respectively. Thus, the time derivatives of \(F\) in (37) and \(\varphi_{1}\) in (97) are  
\[\dot{F} =\dot{F}_{0}+\triangle F_{0} \tag{112}\] \[\dot{\varphi}_{1} =\dot{\varphi}_{1,0}+\triangle\varphi_{1,0} \tag{113}\]  
where \(\dot{F}_{0}\) is shown as (103), \(\dot{\varphi}_{1,0}\) as shown in (99), and  
\[\triangle F_{0} =-\frac{k_{dz}d_{z}}{\varsigma_{\varrho}} \tag{114}\] \[\triangle\varphi_{1,0} =\frac{k_{dx}d_{x}}{F}-\frac{m\iota_{x}k_{dz}d_{z}}{\varsigma_{ \varrho}F^{2}}{=}\frac{k_{dx}d_{x}}{F}-\frac{k_{dz}d_{z}}{\varsigma_{\varrho }F}\varphi_{1} \tag{115}\]  
respectively. According to (109), \(\triangle\varphi_{1,0}\) satisfies  
\[|\theta|\leq\theta^{+} \tag{109}\]0  
Replacing \(\phi_{1}\) in (55) with \(\phi_{1,0}\), it follows that  
\[|\theta|\leq\theta^{+} \tag{109}\]1  
Substituting (117) into (79) gives  
\[|\theta|\leq\theta^{+} \tag{109}\]2 \[|\theta|\leq\theta^{+} \tag{109}\]3 \[=- \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} ^{T} \begin{bmatrix}\mathbf{Q}_{x}&\mathbf{M}_{2}^{T}\\ \mathbf{M}_{2}&k_{z1}\end{bmatrix}\begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} + \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} ^{T} \begin{bmatrix}\mathbf{P}_{x}&\mathbf{0}_{3\times 1}\\ \mathbf{0}_{1\times 3}&1\end{bmatrix}\begin{bmatrix}\mathbf{v}_{x}\\ \Delta\varphi_{1,0}\end{bmatrix} \]\[=-\begin{bmatrix}\mathbf{e}_{x}\\ \left[z_{1}\right]^{T}\mathbf{Q}_{xz}\begin{bmatrix}\mathbf{e}_{x}\\ \left[z_{1}\right]+\left[\mathbf{e}_{z}\right]^{T}\mathbf{P}_{xx} \begin{bmatrix} \mathbf{v}_{x}\\ \triangle\varphi_{1,0}\end{bmatrix} \\ \leq-\lambda_{min}\left(\mathbf{Q}_{xz}\right) \begin{bmatrix}\left[\mathbf{e}_ {x}\right]\\ \left[z_{1}\right]\end{bmatrix} \left( \begin{bmatrix}\left[\mathbf{e}_{x} \right]\\ \left[z_{1}\right]\end{bmatrix} -\varepsilon_{xz0}\right) \tag{118}\]  
where \(\varepsilon_{xz0}\) is shown in (108).