## 3 Design of Controller and Nonlinear Disturbance Observer
### Design of Longitudinal and Latitudinal Controllers
\begin{array}{cc}\mu_{4}(R_{4}(\psi,u_{1})T_{4}(x_{5})- \rho_{4}-\dot{x}_{31}-\dot{\xi}_{32})\\ -\mu_{4}k_{41}\dot{\xi}_{41}\end{array} \right], \tag{54}\]  
\[\dot{\xi}_{5}=\left[ \begin{array}{cc}\dot{\xi}_{51}&\dot{\xi}_{52}\end{array}  \right]^{T}=\left[ \begin{array}{cc}\mu_{5}(s_{6}+\chi_{51}+\xi_{6}-\dot{x}_ {5d}-\dot{\xi}_{51})\\ -\mu_{5}s_{51}\dot{\xi}_{51}\end{array} \right], \tag{55}\]  
\[\dot{\xi}_{6}=\left[ \begin{array}{cc}\dot{\xi}_{61}&\dot{\xi}_{62}\end{array}  \right]^{T}=\left[ \begin{array}{cc}\mu_{6}(f_{6}+g_{6}u_{\phi\phi}-\rho_{6}- \dot{x}_{51}-\dot{\xi}_{52})\\ -\mu_{6}k_{61}\dot{\xi}_{61}\end{array} \right], \tag{56}\]  
where \(\mu_{i}=k_{ij}+\frac{1}{2}|s_{i}|^{-1/2}\), \(i=3,4,5,6\), \(j=1,2\). We can select the virtual controls \(\alpha_{i},\ i=3,5\) for the recursive input, and control inputs \(u_{i},\ i=2,3\) as follows:  
\[\alpha_{3}=-k_{31}\zeta_{31}-k_{31}\zeta_{31}-\beta_{\xi,31}sig^{1/2}(\xi_{31})+ \dot{x}_{3d}+\zeta_{32}, \tag{57}\]  
\[\alpha_{5}=-k_{51}\zeta_{51}-k_{51}\zeta_{51}-\beta_{\xi,51}sig^{1/2}(\xi_{51})+ \dot{x}_{5d}+\zeta_{52}, \tag{58}\]  
\[u_{2}=T_{4}(x_{5})=R_{4}^{-1}(\psi,u_{1})(-k_{41}\zeta_{41}-k_{41}\xi_{42}+\dot{ \rho}_{4}+\chi_{32}-\dot{\xi}_{41} \tag{59}\]  
\[-\beta_{\xi,41}sig^{1/2}(\xi_{41})-\dot{\xi}_{41}^{-1}\eta_{4}^{-1}\xi_{3}\eta_{ 3}s_{4}+\dot{\xi}_{42}),\]\[u_{3}=u_{\phi\theta}=g_{6}^{-1}(-k_{61}\zeta_{61}-k_{61}\zeta_{61}-f_{6}+\hat{ \rho}_{6}+\chi_{52}-\xi_{61} \tag{60}\] \[-\beta_{\zeta,61}sig^{1/2}(\xi_{61})-\xi_{6}^{-1}\eta_{6}^{-1}\xi_{5 }\eta_{5}\xi_{6}+\xi_{62})\]  
where \(R_{4}^{-1}=\left[ \begin{array}{cc}\cos\psi&\sin\psi\\ \sin\psi&-\cos\psi\end{array} \right]^{-1}=\left[ \begin{array}{cc}\cos\psi& \sin\psi\\ \sin\psi&-\cos\psi\end{array} \right]\), \(\eta_{k}=(c_{k}+4\varepsilon_{k}^{2})\zeta_{k1}-2\varepsilon_{k}\zeta_{k2}\), \(c_{k}>0\) and \(\varepsilon_{k}>0,k=3,4,5,6\), are constants. Next, substituting these controls into the dynamics of the compensating signals given in (47) and repeating the procedures of the previous step, we obtain the following compact expressions for the super-twisting state vector:  
\[\dot{\zeta}_{j}=-\mu_{j}\left[ \begin{array}{cc}k_{j1}&-1\\ k_{j2}&0\end{array} \right]\left[ \begin{array}{c}\zeta_{j1}\\ \zeta_{j2}\end{array} \right]+\left[ \begin{array}{c}\mu_{j}s_{j}\\ 0\end{array} \right],j=3,5, \tag{61}\]  
\[\dot{\zeta}_{k}=-\mu_{k}\left[ \begin{array}{cc}k_{k1}&-1\\ k_{k2}&0\end{array} \right]\left[ \begin{array}{c}\xi_{k1}\\ \xi_{k2}\end{array} \right]+\mu_{k}\left[ \begin{array}{c}\widetilde{\rho}_{k} \\ 0\end{array} \right]+\left[ \begin{array}{c}-\eta_{k}^{-1}\mu_{k-1}\eta_{k-1} \xi_{k}\\ 0\end{array} \right],\ k=4,6, \tag{62}\]  
where \(\widetilde{\rho}_{k}=\rho_{k}-\hat{\rho}_{k},k=4,6\). The Lyapunov function candidate with (60) and (61) is defined as follows:  
\[V_{2}=\sum_{k=3}^{6}\frac{1}{2}\zeta_{k}^{T}P_{k}\zeta_{k}, \tag{63}\]  
where \(P_{k}\) denotes positive definite matrices defined as follows:  
\[P_{k}=\left[ \begin{array}{cc}c_{k}+4\varepsilon_{k}^{2}&-2\varepsilon_{k} \\ -2\varepsilon_{k}&1\end{array} \right],\ k=3,4,5,6.\]  
The time derivative of (63) leads to the following expression:  
\[ \begin{array}{l}\dot{V}_{2}\leq-\mu_{3}\zeta_{3}^{T}Q_{3}\zeta_{3}-\mu_{5} \zeta_{5}^{T}Q_{5}\zeta_{5}+\mu_{3}\eta_{3}\zeta_{3}+\mu_{5}\eta_{5}\zeta_{5}- \mu_{4}\zeta_{4}^{T}Q_{4}\zeta_{4}-\mu_{6}\zeta_{6}^{T}Q_{6}\zeta_{6}-\mu_{3} \eta_{3}\eta_{3}\zeta_{3}\\ -\mu_{5}\eta_{5}\zeta_{5}+\tilde{\rho}_{4}^{T}\mu_{4}\eta_{4}+\tilde{\rho}_{6} ^{T}\mu_{6}\eta_{6}\\ \leq-\sum\limits_{k=3}^{6}\mu_{k}\zeta_{k}^{T}Q_{k}\zeta_{k}+\delta_{4}+ \delta_{6},\end{array}  \tag{64}\]  
where \(\left|\tilde{\rho}_{4}^{T}\mu_{4}\eta_{4}\right|\leq\delta_{4},\ \left|\tilde{\rho}_{6}^{T}\mu_{6}\eta_{6}\right|\leq\delta_{6}\), and the positive definite matrix  
\[Q_{k}=\left[ \begin{array}{cc}2k_{k1}(c_{k}+4\varepsilon_{k}^{2})-4k_{k2} \varepsilon_{k}&(c_{k}+4\varepsilon_{k}^{2})-2\varepsilon_{k}k_{k1}+k_{k2}\\ c_{k}+4\varepsilon_{k}^{2}-2\varepsilon_{k}k_{k1}+k_{k2}&4\varepsilon_{k} \end{array}