## 3 Flight Controller Design
### Position Integral Backstepping Controller Design  
Consider the under-actuated subsystem \(\mathcal{S}_{2}\) described in (20). Assume that the disturbances \(d_{x}\) and \(d_{\theta}\) satisfy  
\[ \begin{split}|d_{x}|&\leq d_{x}^{+}\\ |d_{x}|&\leq d_{x}^{+}\\ |d_{\theta}|&\leq d_{\theta}^{+}\end{split}  \tag{45}\]  
where \(\dot{d}_{x}^{+}\), \(d_{x}^{+}\), and \(d_{\theta}^{+}>0\) are known constants. Since the control force \(F\) has already been designed for the subsystem (19) to achieve the desired altitude \(z_{d}\), the control torque \(M\) should be further designed for the subsystem (20) to achieve the desired position \(x_{d}\) and to keep the attitude \(\theta\) stable. What follows is the illustration of the integral backstepping controller (IBC) design procedure used to highlight the tracking control problem for such cascade system (20). Following the similar manners presented in the previous section, define the position tracking errors \(e_{x1}\) and \(e_{x2}\) as follows:  
\[ \begin{split} e_{x1}&=x-x_{d}\\ e_{x2}&=\dot{e}_{x1}=\dot{x}-\dot{x}_{d}\end{split}  \tag{46}\]Then, the position tracking error dynamics are given by  
\[\dot{e}_{x1} =e_{x2} \tag{47}\] \[\dot{e}_{x2} =\frac{F_{8\theta}}{m}+\frac{d_{x}}{m}-\ddot{x}_{d}\] (48) \[\ddot{\theta} =\frac{M}{J}+\frac{d_{\theta}}{J} \tag{49}\]  
For the above error dynamics, the position integral backstepping control algorithm is proposed as follows:  
**Theorem 2**. : _Consider the error dynamics (47)-(49). The closed-loop system is stable in the sense of bounded input bounded output (BIBO) if there exist the feasible solution of the matrices \(\mathbf{P}_{x}=\mathbf{P}_{x}^{T}>0\) and \(\mathbf{Q}_{x}=\mathbf{Q}_{x}^{T}>0\) satisfying the following LMIs:_  
\[\mathbf{P}_{x}\mathbf{A}_{x}+\mathbf{A}_{x}^{T}\mathbf{P}_{x}+ \mathbf{Q}_{x}<\mathbf{0} \tag{50}\] \[\mathbf{Q}_{x}- \begin{bmatrix}\frac{F_{x2}^{2}}{m^{2}k_{21}}p_{2 3x}^{2}&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix} >0 \tag{51}\]  
_where the closed-loop system matrix_  
\[\mathbf{A}_{x}= \begin{bmatrix}0&1&0\\ 0&0&1\\ -k_{ix}&-k_{px}&-k_{dx}\end{bmatrix}  \tag{52}\]  
_with the control gains \((k_{px},k_{ix},k_{dx})>0\), and the matrix \(\mathbf{P}_{x}\) composed by the following specific structure:_  
\[\mathbf{P}_{x}= \begin{bmatrix}P_{1x}&\mathbf{P}_{2x}\\ \mathbf{P}_{2x}^{T}&\mathbf{P}_{3x}\end{bmatrix}  \tag{53}\]  
_where \(p_{1x}>0\), \(\mathbf{P}_{2x}=[p_{22x},p_{23x}]\in\mathbb{R}^{1\times 2}\), \(\mathbf{P}_{3x}\in\mathbb{R}^{2\times 2}\). Then, based on the integral backstepping control algorithm, the control torque is designed as_  
\[M=J\big{(}\dot{\varphi}_{2}-\varsigma_{\theta}(\mathfrak{s}_{ \theta}-\varphi_{1})-k_{z2}\big{(}\dot{\theta}-\varphi_{2}\big{)}-w\,\text{ sign}\big{(}\dot{\theta}-\varphi_{2}\big{)}\big{)} \tag{54}\]  
_where_  
\[\varphi_{2} =\frac{1}{\varsigma_{\theta}}\Big{[}\dot{\varphi}_{1}-2\mathbf{e }_{x12}^{T}\mathbf{P}_{3x}\mathbf{g}_{x1}-k_{z1}(\mathfrak{s}_{\theta}- \varphi_{1})\Big{]} \tag{55}\] \[\varphi_{1} =\frac{m}{F}u_{x}\] (56) \[u_{x} =\ddot{x}_{d}-k_{px}e_{x1}-k_{ix}\int_{0}^{t}e_{x1}(\tau)\,d\tau- k_{dx}e_{x2}\] (57) \[\mathbf{g}_{x1} = \begin{bmatrix}0&\frac{F}{m}\end{bmatrix} ^{T}\] (58) \[\mathbf{e}_{x12} = \begin{bmatrix}e_{x1}&e_{x2}\end{bmatrix} ^{T} \tag{59}\]  
_with the control gains \(k_{z1}>0\), \(k_{z2}>0\), and_  
\[w=\frac{d_{\theta}^{+}}{J}+\eta,\quad\eta>0 \tag{60}\]  
_If the control law (54) is realizable, then the error of (47)-(49) will approach a region \(\varepsilon_{xz}\), i.e.,_  
\[\left\| \begin{bmatrix}\mathbf{e}_{x}\\ \mathfrak{s}_{\theta}-\varphi_{1}\end{bmatrix} \right\|\to\varepsilon_{xz}\quad \text{as}\quad t\to\infty \tag{61}\]_where_  
\[\varepsilon_{xz}=\frac{\lambda_{max}(\mathbf{P}_{xz})}{\lambda_{min}(\mathbf{Q}_{ xz})}\frac{d_{x}^{+}}{mk_{ix}} \tag{62}\]  
_and_  
\[\mathbf{e}_{x} =\Big{[}\int_{0}^{t}\!e_{x1}(\tau)\,d\tau-\tfrac{d_{x}}{mk_{ix}} \quad e_{x1}\quad e_{x2}\Big{]}^{T} \tag{63}\] \[\mathbf{P}_{xz} = \begin{bmatrix}\mathbf{P}_{x}&\mathbf{0}_{3\times 1}\\ \mathbf{0}_{1\times 3}&1\end{bmatrix}