## 3 Flight Controller Design
### Position Integral Backstepping Controller Design
\] (64) \[\mathbf{Q}_{xz} = \begin{bmatrix}\mathbf{Q}_{x}&\mathbf{M}_{2}^{T}\\ \mathbf{M}_{2}&k_{z1}\end{bmatrix} \] (65) \[\mathbf{M}_{2} = \begin{bmatrix}-\tfrac{F}{m}p_{23x}&0&0\end{bmatrix} . \tag{66}\]  
Proof of Theorem 2.: Consider the system state \(s_{\theta}\) as the virtual control input \(\varphi_{1}\) of the inner systems (47) and (48). Design the virtual control \(\varphi_{1}\) like the form of (56):  
\[s_{\theta}=\varphi_{1}=\frac{mu_{x}}{F} \tag{67}\]  
where \(u_{x}\) is defined in (57). Replacing \(s_{\theta}\) in (48) with the virtual control (56) produces the following virtual error system:  
\[\dot{e}_{x1} =e_{x2} \tag{68}\] \[\dot{e}_{x2} =-k_{px}e_{x1}-k_{ix}\int_{0}^{t}\!e_{x1}(\tau)\,d\tau-k_{dx}e_{x 2}+\frac{d_{x}}{m}\]  
Introduce the extended state \(e_{x0}\) as  
\[e_{x0}=\int_{0}^{t}\!e_{x1}(\tau)\,d\tau-\frac{d_{x}}{mk_{ix}} \tag{69}\]  
Then, the virtual error system (68) becomes  
\[\dot{\mathbf{e}}_{x}=\mathbf{A}_{x}\mathbf{e}_{x}+\mathbf{v}_{x} \tag{70}\]  
where \(\mathbf{e}_{x}=[e_{x0},\mathbf{e}_{x12}^{T}]^{T}=[e_{x0},e_{x1},e_{x2}]^{T}\) is the extended position tracking error vector, \(\mathbf{v}_{x}=[-\frac{d_{x}}{mk_{ix}},0,0]^{T}\) is the extended disturbance vector, and the closed-loop system matrix is revealed in (52). By properly designing the control gain \((k_{px},k_{ix},k_{dx})\), Theorem 1 indicates that there exists a Lyapunov function \(V_{x}=\mathbf{e}_{x}^{T}\mathbf{r}_{x}\mathbf{e}_{x}\) such that  
\[\dot{V}_{x}=2\mathbf{e}_{x}^{T}\mathbf{P}_{x}(\mathbf{A}_{x}\mathbf{e}_{x}- \mathbf{v}_{x})\leq-\lambda_{min}(\mathbf{Q}_{x})\|\mathbf{e}_{x}\|(\| \mathbf{e}_{x}\|-\varepsilon_{x}) \tag{71}\]  
where the matrices \(\mathbf{P}_{x}=\mathbf{P}_{x}^{T}>0\) and \(\mathbf{Q}_{x}=\mathbf{Q}_{x}^{T}>0\) are the feasible solution pair of the Lyapunov equation, shown as (50). The error boundary \(\varepsilon_{x}\) is  
\[\varepsilon_{x}=\frac{2\lambda_{max}(\mathbf{P}_{x})}{\lambda_{min}(\mathbf{ Q}_{x})}\frac{d_{x}^{+}}{mk_{ix}} \tag{72}\]  
Equation (71) implies that the magnitude of the extended position tracking error vector \(\mathbf{e}_{x}\) will approach a region described by \(\varepsilon_{x}\). However, \(s_{\theta}\) is coupled by the system state variable, not a control input. In other words, the virtual control \(\varphi_{1}\) in (56) can be realized if \(s_{\theta}=\varphi_{1}\) can be fulfilled. Towards this aim, define the error variable \(z_{1}\) as follows:  
\[z_{1}=s_{\theta}-\varphi_{1} \tag{73}\]Then, the system \((\mathbf{e}_{x},\theta)\) is transformed into the domain of \((\mathbf{e}_{x},z_{1})\):  
\[\dot{\mathbf{e}}_{x} =\mathbf{f}_{x}+\mathbf{g}_{x}\varphi_{1}+\mathbf{g}_{x}z_{1}+ \mathbf{D}_{x} \tag{74}\] \[\dot{z}_{1} =\mathrm{c}_{\theta}\dot{\theta}-\dot{\varphi}_{1} \tag{75}\]  
where  
\[\mathbf{f}_{x}= \begin{bmatrix}e_{x1}\\ e_{x2}\\ 0\end{bmatrix} ,\quad\mathbf{g}_{x}= \begin{bmatrix}0\\ \mathbf{g}_{x1}\end{bmatrix} = \begin{bmatrix}0\\ 0\\ \frac{F}{m}\end{bmatrix} ,\quad\mathbf{D}_{x}= \begin{bmatrix}-\frac{d_{x}}{m \dot{x}_{ix}}\\ 0\\ \frac{d_{x}}{m}-\dot{x}_{d}\end{bmatrix}  \tag{76}\]  
Apparently, once the error variable \(z_{1}\) converges to zero, it means that the term \(\mathrm{s}_{\theta}\) converges to the desired virtual control input \(\varphi_{1}\). Then, the error vector \(\|\mathbf{e}\|\) will converge to a region described in (72). Follow the standard steps of the integral backstepping control algorithm, treat \(\dot{\theta}\) in (75) as the virtual control input \(\varphi_{2}\), and then design a proper \(\varphi_{2}\) such that the system \((\mathbf{e}_{x},z_{1})\) is stable.