## 4 Fractional-Order Controller and Fractional-Order Disturbance Observer Design  
### Altitude Controller Design  
The altitude controller is designed to control the vertical displacement. The state equation of (8) can be described as follows:  
\[ \begin{array}{l}\dot{x}_{1}=x_{2},\\ \dot{x}_{2}=\frac{1}{m_{0}}\cos\phi\cos\theta u_{1}-g+f_{d2},\end{array}  \tag{36}\]  
where the altitude state variable \(x_{1}=z\), \(x_{2}=\dot{z}\), \(u_{1}=u_{z}\), and \(f_{d2}\) denotes the uncertainty. By introducing the command trajectory \(x_{1d}\) in the z-coordinate and defining the tracking errors as \(e_{1}=x_{1}-x_{1d}\) and \(e_{2}=x_{2}-\xi_{11}\), the fractional-order filtered error surfaces are introduced as follows:  
\[\sigma_{1}=D^{\gamma_{1}}e_{1}+D^{\gamma_{2}-1}k_{1}^{2}(b_{11}e_{1}+b_{12}sig ^{1/2}(e_{1}))-\xi_{11}, \tag{37}\]  
Figure 3: Comparative results for four fractional-order filtered error surfaces. \[\sigma_{2}=D^{7_{1}}e_{2}+k_{2}^{2}D^{7_{2}-1}(b_{21}e_{2}+b_{22}sig^{1/2}(e_{2}))- \xi_{12}, \tag{38}\]  
where \(k_{j}>0\), \(\kappa_{1j}>0\), \(b_{j1}>0\), \(b_{j2}>0\), \(j=1,2\), are constants and \(0<\gamma_{j}<1,\ j=1,2\). The state variables \(\xi_{11}\) and \(\xi_{12}\) in (33) and (34) are obtained from the dynamics of the compensating signals expressed as follows:  
\[D^{7_{1}}\xi_{11}=-\beta_{11}\xi_{11}+\xi_{12}+\zeta_{11}-\alpha_{1}-\lambda_{ 11}sig^{1/2}(\xi_{11}), \tag{39}\]  
\[D^{7_{1}}\xi_{12}=-\beta_{12}\xi_{12}-\xi_{11}-\lambda_{12}sig^{1/2}(\xi_{12}), \tag{40}\]  
where \(D^{7_{1}}=D^{1-7_{1}}\), \(\beta_{1j}>0\), and \(\lambda_{1j}>0,\ j=1,2\), are constants. Next, the output of the tracking differentiator \(\xi_{11}\) is obtained from the tracking differentiator dynamics described in Theorem 4,  
\[ \begin{array}{l}D^{7_{1}}\xi_{11}=\xi_{12},\\ D^{7_{1}}\xi_{12}=-\Lambda_{1}^{2}[c_{11}\sinh(r_{11}\varphi_{11}(\zeta_{11}- \alpha_{1}))+c_{12}\sinh(r_{12}\varphi_{12}(\zeta_{12})/\Lambda_{1})],\end{array}  \tag{41}\]  
where \(\varphi_{11}(\zeta_{11}-\alpha_{1})=\varepsilon_{1}sig^{1/2}(\zeta_{11}- \alpha_{1})+(1-\varepsilon_{1})sig^{(2+v_{1})/2}(\zeta_{11}-\alpha_{1}),\ \varphi_{12}(\zeta_{12})=sig^{1/2}(\zeta_{12})\), \(0<\varepsilon_{1}<1\), and \(0<v_{1}<1\) are constants. Taking the time derivative of (37) and (38) and applying \(D^{-\gamma_{1}}\) to both sides of (37) and (38), we obtain  
\[ \begin{array}{l}D^{7_{1}}\sigma_{1}&=\dot{e}_{1}+D^{7_{2}}k_{1}^{2}\Big{(}b _{11}e_{1}+b_{12}\,\text{sig}^{1/2}(e_{1})\Big{)}-D^{7_{1}}\xi_{11}\\ &=e_{2}+\zeta_{11}-\dot{x}_{1d}+D^{7_{2}}k_{1}^{2}\Big{(}b_{11}e_{1}+b_{12}\, \text{sig}^{1/2}(e_{1})\Big{)}+\beta_{11}\xi_{11}-\xi_{12}-\zeta_{11}+\alpha_ {1}+\lambda_{11}\,\text{sig}^{1/2}(\xi_{11})\\ &=e_{2}-\dot{x}_{1d}+\alpha_{1}+\alpha_{1},\end{array}  \tag{42}\]  
\[ \begin{array}{l}D^{7_{1}}\sigma_{2}&=\dot{e}_{2}+D^{7_{2}}k_{2}^{2}\Big{(}b _{21}e_{2}+b_{22}\,\text{sig}^{1/2}(e_{2})\Big{)}-D^{7_{1}}\xi_{12}\\ &=\frac{\cos\phi\cos\theta}{m_{0}}u_{1}-g+f_{d2}-D^{7_{1}}\zeta_{12}+D^{7_{2 }}k_{2}^{2}\Big{(}b_{21}e_{2}+b_{22}\,\text{sig}^{1/2}(e_{2})\Big{)}\\ &+\beta_{12}\xi_{12}+\dot{\xi}_{11}+\lambda_{12}\,\text{sig}^{1/2}(\xi_{12}) \\ &=\frac{\cos\phi\cos\theta}{m_{0}}u_{1}-g+\omega_{2}+f_{d2},\end{array}  \tag{43}\]  
where \(D^{7_{2}}=D^{7_{2}-\gamma_{1}}\), \(\sigma_{1}=D^{7_{2}}k_{1}^{2}(b_{11}e_{1}+b_{12}sig^{1/2}(e_{1}))+\beta_{11} \xi_{11}-\xi_{12}+\lambda_{11}sig^{1/2}(\xi_{11})\), and \(\omega_{2}=-D^{7_{1}}\zeta_{12}+D^{7_{2}}k_{2}^{2}(b_{21}e_{2}+b_{22}sig^{1/2}( e_{2}))+\beta_{12}\xi_{12}+\dot{\xi}_{11}+\lambda_{12}sig^{1/2}(\xi_{12})\). Moreover, \(\dot{\zeta}_{11}\) in (40) can be expressed into \(\dot{\zeta}_{11}=D^{7_{1}}\zeta_{12}\) by using a proper fractional-order operation and the relationship \(D^{7_{1}}\xi_{11}=\xi_{12}\) in (40).