## 3 Flight Controller Design
### Proposed Analytic Form of the Proposed Position Integral Backstepping Controller
Taking the time derivative of (88) along the system trajectories (85)-(87) and introducing (118) and (125) leads to  
\[\dot{V} = 2\mathbf{e}_{x}^{T}\mathbf{P}_{x}\dot{\mathbf{e}}_{x}+z_{1} \dot{z}_{1}+z_{2}\dot{z}_{2} \tag{127}\] \[= \underbrace{2\mathbf{e}_{x}^{T}\mathbf{P}_{x}(\mathbf{A}_{x} \mathbf{e}_{x}+\mathbf{v}_{x})+z_{1}\Big{(}\mathrm{c}_{\theta}\dot{\varphi}_ {2}-\dot{\varphi}_{1}+2\mathbf{e}_{x}^{T}\mathbf{P}_{x}\mathbf{g}_{x}\Big{)}} _{\leq-\lambda_{min}(\mathbf{Q}_{xz})\bigg{\|} \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \bigg{\|}\bigg{(}\bigg{\|} \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \bigg{\|}-\varepsilon_{x0}\bigg{)}\] \[\leq -\lambda_{min}(\mathbf{Q}_{xz})\bigg{\|} \begin{bmatrix}\mathbf{e}_{x }\\ z_{1}\end{bmatrix} \bigg{\|}\bigg{(}\bigg{\|}\begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{\|}\bigg{\|}-\varepsilon_{x2}\bigg{)}-k_{z1}z_{1}^{2}\] \[+z_{2}\bigg{(}\frac{M}{f}+\frac{d_{\theta}}{f}-\dot{\varphi}_{2,0} -\bigtriangleup\varphi_{2,0}+m\frac{u_{x}F_{M}M}{\mathrm{c}_{\theta}F^{2}}+ \mathrm{c}_{\theta}z_{1}\bigg{)}\] \[\leq -\lambda_{min}(\mathbf{Q}_{xz})\bigg{\|} \begin{bmatrix}\mathbf{e}_{x }\\ z_{1}\end{bmatrix} \bigg{\|}\bigg{(}\bigg{\|} \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \bigg{\|}-\varepsilon_{x2}\bigg{)}-k_{z1}z_{1}^{2}\] \[+z_{2}\bigg{(}\frac{\mathrm{c}_{\theta}F^{2}+Jmu_{x}F_{M}}{J \mathrm{c}_{\theta}F^{2}}M+\frac{d_{\theta}}{f}-\dot{\varphi}_{2,0}-\bigtriangleup \varphi_{2,0}+\mathrm{c}_{\theta}z_{1}\bigg{)}\]Hence, design the control torque \(M\) as follows:  
\[M =\frac{J\varsigma_{\theta}F^{2}}{\varsigma_{\theta}F^{2}+Jmu_{x}F_{M}}( \dot{\varphi}_{2,0}-\varsigma_{\theta}z_{1}-k_{z2}z_{2}-w\,\text{sign}(z_{2}))\] \[=\frac{J\varsigma_{\theta}F^{2}}{\varsigma_{\theta}F^{2}+Jmu_{x} \frac{m_{\text{gs}}\mu_{z}}{J\varsigma_{\theta}{}^{2}}}(\dot{\varphi}_{2,0}- \varsigma_{\theta}z_{1}-k_{z2}z_{2}-w\,\text{sign}(z_{2})) \tag{128}\]  
Collating (128) gives the AIBC algorithm shown in (94). Substituting (94) into (127) yields  
\[\dot{V} \leq -\lambda_{min}(\mathbf{Q}_{xx})\left\| \begin{bmatrix}\mathbf{e}_{x }\\ z_{1}\end{bmatrix} \right\|\left(\left\| \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \right\|-\varepsilon_{xx0}\right)\] \[-k_{z1}z_{1}^{2}+z_{2}\left(\frac{d_{\theta}}{J}-\triangle\varphi _{2,0}-k_{z2}z_{2}-w\,\text{sign}(z_{2})\right)\] \[\leq -\lambda_{min}(\mathbf{Q}_{xz})\left\| \begin{bmatrix}\mathbf{e}_ {x}^{x}\\ z_{1}\end{bmatrix} \right\|\left(\left\| \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \right\|-\varepsilon_{xx0}\right)\] \[-k_{z1}z_{1}^{2}-k_{z2}z_{2}^{2}-wz_{2}\,\text{sign}(z_{2})+z_{2 }\left(\frac{d_{\theta}}{J}-\triangle\varphi_{2,0}\right) \tag{129}\]  
Assume that \(\triangle\varphi_{2,0}\) in (126) is bounded and satisfies  
\[|\triangle\varphi_{2,0}|\leq M^{+} \tag{130}\]  
Then, (129) can be rewritten as  
\[\dot{V} \leq -\lambda_{min}(\mathbf{Q}_{xz})\left\| \begin{bmatrix}\mathbf{e}_ {x}\\ z_{1}\end{bmatrix} \right\|\left(\left\| \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \right\|-\varepsilon_{xz}\right)-k_{z1}z_{1}^{2}-k_{z2}z_{2 }^{2}-w|z_{2}|+|z_{2}||\frac{d_{\theta}^{+}}{J}+M^{+}|\] \[\leq -\lambda_{min}(\mathbf{Q}_{xz})\left\| \begin{bmatrix}\mathbf{e}_ {x}\\ z_{1}\end{bmatrix} \right\|\left(\left\| \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \right\|-\varepsilon_{xz}\right)-k_{z1}z_{1}^{2}-k_{z2}z_{2 }^{2}-|z_{2}|\left(w-\frac{d_{\theta}^{+}}{J}-M^{+}\right) \tag{131}\]  
Therefore, by choosing \(w\) shown in (106), (131) becomes  
\[\dot{V}\leq-\lambda_{min}(\mathbf{Q}_{xz})\left\| \begin{bmatrix}\mathbf{e}_{x }\\ z_{1}\end{bmatrix} \right\|\left(\left\| \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \right\|-\varepsilon_{xx0}\right)-k_{z1}z_{1}^{2}-k_{z2}z_{2 }^{2}-\eta_{1}|z_{2}| \tag{132}\]  
As a result, the subsystem \(S_{2}(e_{x},z_{1},z_{2})\) is stable if  
\[ \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \
otin\mathcal{E}_{2} \tag{133}\]  
where  
\[\mathcal{E}_{2}=\left\{ \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} \in\mathbb{R}^{4}:\left\|