## 3 Flight Controller Design
### Position Integral Backstepping Controller Design
Consider the Lyapunov function candidate as follows:  
\[V_{1}=V_{x}+\frac{1}{2}z_{1}^{2} \tag{77}\]  
Taking the time derivative of (77) along the trajectories (74)-(75) and introducing (71), it follows that  
\[\dot{V}_{1} =2\mathbf{e}_{x}^{T}\mathbf{P}_{x}\dot{\mathbf{e}}_{x}+z_{1}\dot {z}_{1}\] \[=2\mathbf{e}_{x}^{T}\mathbf{P}_{x}(\mathbf{f}_{x}+\mathbf{g}_{x} \varphi_{1}+\mathbf{B}_{x}z_{1}+\mathbf{D}_{x})+z_{1}\big{(}\mathrm{c}_{ \theta}\dot{\theta}-\dot{\varphi}_{1}\big{)}\] \[=2\mathbf{e}_{x}^{T}\mathbf{P}_{x}(\mathbf{f}_{x}+\mathbf{g}_{x} \varphi_{1}+\mathbf{D}_{x})+2\mathbf{e}_{x}^{T}\mathbf{P}_{x}\mathbf{g}_{x}z_ {1}+z_{1}\big{(}\mathrm{c}_{\theta}\dot{\theta}-\dot{\varphi}_{1}\big{)}\] \[=2\mathbf{e}_{x}^{T}\mathbf{P}_{x}(\mathbf{A}_{x}\mathbf{e}_{x})+2 \mathbf{e}_{x}^{T}\mathbf{P}_{x}\mathbf{v}_{x}+2\mathbf{e}_{x}^{T}\mathbf{P} _{x}\mathbf{g}_{x}z_{1}+z_{1}\big{(}\mathrm{c}_{\theta}\dot{\theta}-\dot{ \varphi}_{1}\big{)} \tag{78}\]  
Consider the specific structure of \(\mathbf{P}_{x}\) shown in (53), and then (78) becomes  
\[\dot{V}_{1} =2\mathbf{e}_{x}^{T}\mathbf{P}_{x}(\mathbf{A}_{x}\mathbf{e}_{x}) +2\mathbf{e}_{x}^{T}\mathbf{P}_{x}\mathbf{v}_{x}+2 \begin{bmatrix}e_{x0}\\ \mathbf{e}_{x12}\end{bmatrix} ^{T} \begin{bmatrix}\mathbf{P}_{1x}&\mathbf{P}_{2 x}\\ \mathbf{P}_{x}^{T}&\mathbf{P}_{3x}\end{bmatrix}\begin{bmatrix}0\\ \mathbf{g}_{x1}z_{1}\end{bmatrix} +z_{1}\big{(}\mathrm{c}_{\theta}\dot{\theta}- \dot{\varphi}_{1}\big{)}\] \[=-\mathbf{e}_{x}^{T}\mathbf{Q}_{x}\mathbf{e}_{x}+2\mathbf{e}_{x}^{ T}\mathbf{P}_{x}\mathbf{v}_{x}+2\Big{(}\mathrm{c}_{x0}\mathbf{P}_{2x}\mathbf{g}_{x1}z_ {1}+\mathbf{e}_{x12}^{T}\mathbf{P}_{3x}\mathbf{g}_{x1}z_{1}\Big{)}+z_{1}\big{(} \mathrm{c}_{\theta}\dot{\theta}-\dot{\varphi}_{1}\big{)}\] \[=-\mathbf{e}_{x}^{T}\mathbf{Q}_{x}\mathbf{e}_{x}+2\mathbf{e}_{x}^{ T}\mathbf{P}_{x}\mathbf{v}_{x}+2\mathrm{e}_{x0}\mathbf{P}_{2x}\mathbf{g}_{x1}z_ {1}+z_{1}\Big{(}\mathrm{c}_{\theta}\dot{\theta}-\dot{\varphi}_{1}+2\mathbf{e}_{ x12}^{T}\mathbf{P}_{3x}\mathbf{g}_{x1}\Big{)} \tag{79}\]  
Design a virtual control input \(\dot{\theta}=\varphi_{2}\) like (55). Substituting (55) into (79) gives  
\[\dot{V}_{1} =-\mathbf{e}_{x}^{T}\mathbf{Q}_{x}\mathbf{e}_{x}+2\mathbf{e}_{x}^{ T}\mathbf{P}_{x}\mathbf{v}_{x}+2\mathrm{e}_{x0}\mathbf{P}_{2x}\mathbf{g}_{x1}z_ {1}-k_{z1}z_{1}^{2}\] \[=-\mathbf{e}_{x}^{T}\mathbf{Q}_{x}\mathbf{e}_{x}+2\mathbf{e}_{x}^{ T}\mathbf{P}_{x}\mathbf{v}_{x}+2\mathrm{e}_{x0}\mathbf{P}_{2x}\frac{F}{m}z_{1}-k_{z1}z_ {1}^{2}\] \[=- \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} ^{T}\underbrace{ \begin{bmatrix}\mathbf{Q}_{x}&\mathbf{M}_{2 }^{T}\\ \mathbf{M}_{2}&k_{z1}\end{bmatrix} }_{\triangleq\mathbf{Q}_{xz}} \begin{bmatrix} \mathbf{e}_{x}\\ z_{1}\end{bmatrix} + \begin{bmatrix}\mathbf{e}_{x}\\ \mathbf{e}_{1}\end{bmatrix} ^{T}\underbrace{ \begin{bmatrix}\mathbf{P}_{x}& \mathbf{0}_{3\times 1}\\ \mathbf{0}_{1\times 3}&1\\ \end{bmatrix} }_{\triangleq\mathbf{P}_{xz}} \begin{bmatrix}\mathbf{v}_{x}\\ 0\end{bmatrix} \] \[=- \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} ^{T}\mathbf{Q}_{xz} \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} + \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix} ^{T}\mathbf{P}_{xz} \begin{bmatrix}\mathbf{v}_{x}\\ 0\end{bmatrix}  \tag{80}\]  
Based on the Schur complement [32], \(\mathbf{Q}_{xz}>0\) is guaranteed if and only if  
\[\mathbf{Q}_{x}>0\quad\text{and}\quad\mathbf{Q}_{x}-\mathbf{M}_{2}^{T}k_{z1}^{- 1}\mathbf{M}_{2}>0 \tag{81}\]or, equivalently,  
\[\mathbf{Q}_{x}>0\quad\text{and}\quad\mathbf{Q}_{x}- \begin{bmatrix}\frac{\varepsilon ^{2}}{m^{2}k_{z1}}p_{23x}^{2}&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix} >0 \tag{82}\]  
In the sense of the worst-case design, replacing \(F\) in (82) with \(F_{max}\) yields (51). If there exist the feasible matrices \(\mathbf{P}_{x}\) and \(\mathbf{Q}_{x}\) to fulfill (50) and (51), as mentioned in (30), (80) is governed by  
\[\dot{V}_{1}\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}