## 3 Design of Controller and Nonlinear Disturbance Observer  
### Design of Altitude Controller  
The vertical displacement is determined by the altitude controller. The state equation for altitude dynamics in (8) can be expressed as follows:  
\[ \begin{split}\dot{x}_{1}=x_{2},\\ \dot{x}_{2}=\frac{1}{m}(\cos\phi\cos\theta)u_{1}-g-\rho_{2},\end{split}  \tag{22}\]  
where the altitude state variable \(x_{1}=z\), \(x_{2}=\dot{z}\), and \(u_{1}=u_{z}\). By introducing the command trajectory \(x_{1d}\) in the z-coordinate, the sliding mode surfaces to include the tracking error and new states are defined as follows:  
\[s_{1}=x_{1}-x_{1d}-\xi_{11}, \tag{23}\]  
\[s_{2}=x_{2}-\chi_{11}-\xi_{12}, \tag{24}\]where the sliding mode surfaces include the tracking error and the compensating signals. The state variables \(\xi_{11}\) and \(\xi_{12}\) in (23) and (24) are obtained from the dynamics of the compensating signals expressed as follows:  
\[ \begin{array}{l}\dot{\xi}_{11}=-k_{11}\xi_{11}+\xi_{12}+\chi_{11}-\alpha_{1}- \beta_{\zeta,11}sig^{1/2}(\xi_{11}),\\ \dot{\xi}_{12}=-k_{12}\xi_{12}-\xi_{11}-\beta_{\zeta,12}sig^{1/2}(\xi_{12}), \end{array}  \tag{25}\]  
where \(\beta_{\zeta,ij}>0,\ i=1,2,\ j=1,2\), denote constants. Next, the output of the tracking differentiator \(\chi_{11}\) is obtained from the tracking differentiator dynamics described in Theorem (3):  
\[ \begin{array}{l}\dot{\chi}_{11}=\chi_{12},\\ \dot{\chi}_{12}=-R_{1}^{2}[r_{11}\text{sinh}(a_{11}\varphi_{11}(\chi_{11},\alpha _{1}))+r_{12}\text{sinh}(a_{12}R_{1}^{-1}\varphi_{12}(\chi_{12}))],\end{array}  \tag{26}\]  
where \(\varphi_{11}(\chi_{11},\alpha_{1})=\kappa_{1}sig^{1/2}(\chi_{11}-\alpha_{1})+( 1-\kappa_{1})sig^{(2+\gamma_{1})/2}(\chi_{11}-\alpha_{1})\), \(\varphi_{12}(\chi_{12})=sig^{1/2}(\chi_{12})\), \(sig^{1/2}(\cdot)=\left|s_{i}\right|^{1/2}sign(\cdot)\). \(\alpha_{1}\) denotes the virtual control defined later, \(r_{1i}>0\) and \(a_{1i}>0\) are constants, and \(\kappa_{1}>1\) and \(0<\gamma_{1}<1\) denote constants. We define the following super-twisting state variable vectors to design the controller using the variables defined previously. \[ \begin{array}{c}\xi_{1}=\left[\begin{array}{cc}\xi_{11}&\xi_{12}\\ \end{array}\right]^{T}\\ =\left[\begin{array}{cc}k_{11}s_{1}+sig^{1/2}(s_{1})\\ -k_{12}\int_{0}^{t}\left[k_{11}^{2}s_{1}+\frac{3k_{11}}{2}sig^{1/2}(s_{1})+ \frac{1}{2}sign(s_{1})\right]d\tau\\ \end{array}\right],\end{array}  \tag{27}\]  
\[ \begin{array}{c}\xi_{2}=\left[\begin{array}{cc}\xi_{21}&\xi_{22}\\ \end{array}\right]^{T}\\ =\left[\begin{array}{cc}k_{21}s_{2}+sig^{1/2}(s_{2})\\ -k_{22}\int_{0}^{t}\left[k_{2}^{2}s_{2}+\frac{3k_{2}}{2}sig^{1/2}(s_{2})+ \frac{1}{2}sign(s_{2})\right]d\tau\\ \end{array}\right].\end{array}  \tag{28}\]  
The super-twisting state variable vectors consist of the sliding mode surfaces and signum functions. Using the definition in (23), the time derivative of the components of the state vectors in (27) and (28) can then be written as  
\[ \begin{array}{c}\dot{\xi}_{11}=k_{11}\dot{s}_{1}+\frac{1}{2}|s_{1}|^{-1/2} \dot{s}_{1}\\ =\mu_{1}(x_{2}-\dot{x}_{1d}-\dot{\xi}_{1})\\ =\mu_{1}(x_{2}-\dot{x}_{1d}+k_{11}\dot{\xi}_{11}-\dot{\xi}_{12}-\dot{\chi}_{11} +\alpha_{1}+\beta_{\zeta,11}sig^{1/2}(\xi_{11}))\\ =\mu_{1}(s_{2}-\dot{x}_{1d}+k_{11}\dot{\xi}_{11}+\alpha_{1}+\beta_{\zeta,11}sig^ {1/2}(\xi_{11})),\end{array}  \tag{29}\]  
\[ \begin{array}{c}\dot{\xi}_{12}=-\mu_{1}k_{12}\dot{\xi}_{11},\\ \dot{\xi}_{21}=k_{21}\dot{s}_{2}+\frac{1}{2}|s_{2}|^{-1/2}\dot{s}_{2}\\ =\mu_{2}\Big{(}\frac{1}{m}(\cos\phi\cos\theta)u_{1}-g-\rho_{2}-\chi_{12}+k_{12} \xi_{12}+\xi_{11}+\beta_{\xi,12}sig^{1/2}(\xi_{12})\Big{)},\end{array}  \tag{30}\]  
\[ \begin{array}{c}\dot{\xi}_{22}=-\mu_{2}k_{22}\dot{\zeta}_{21},\end{array}  \tag{31}\]  
where \(\mu_{1}=k_{11}+\frac{1}{2}|s_{1}|^{-1/2}\) and \(\mu_{2}=k_{21}+\frac{1}{2}|s_{2}|^{-1/2}\). We can select the virtual control from (29) and control input from (31) as follows:  
\[