## 4 Fractional-Order Controller and Fractional-Order Disturbance Observer Design
### Longitudinal and Latitudinal Controllers Design
\begin{bmatrix}\cos\psi&\sin\psi\\ \sin\psi&-\cos\psi\end{bmatrix} ,\,s_{ij}>0,\,i=3,4,5,6,\,j=1,2,3,\] \[\varepsilon_{\sigma i}>0,\,i=3,4,5,6,\,\text{are constants, and }\,\dot{f}_{44}\,\,\text{and}\,\,\dot{\tau}_{46}\,\,\text{are the estimates of }f_{44}\,\,\text{and}\,\,\tau_{46},\,\,\text{respectively.}\]  
The Lyapunov function candidates are defined as follows:  
\[V_{3} =\frac{1}{2}\sigma_{3}^{2}, \tag{64}\] \[V_{4} =\frac{1}{2}\sigma_{4}^{2},\] (65) \[V_{5} =\frac{1}{2}\sigma_{5}^{2}, \tag{66}\]\[V_{6}=\frac{1}{2}\sigma_{6}^{2}. \tag{67}\]  
The fractional-order operations of (64), (65), (66), and (67) by using (56), (57), (58), and (59) along with the results of (60), (61), (62), and (63) yield to the following expression:  
\[ \begin{split} D^{\overline{\tau}_{1}}(V_{3}+V_{4})=\sigma_{3}D^{ \overline{\tau}_{1}}\sigma_{3}+\sigma_{4}D^{\overline{\tau}_{1}}\sigma_{4}\\ =&\sigma_{3}\big{(}e_{4}-\dot{x}_{4d}+\sigma_{3}+ \alpha_{3}\big{)}+\sigma_{4}(R_{4}(\psi,u_{1})T_{4}(x_{5})+\sigma_{4}+f_{4d}) \\ &\quad+\left|\sum_{k=1}^{\infty}\frac{\Gamma(1+\overline{\tau}_{1 })}{\Gamma(1-k+\overline{\tau}_{1})\Gamma(1+k)}D^{k}\sigma_{3}D^{\overline{ \tau}_{1}-k}\sigma_{3}\right|+\left|\sum_{k=1}^{\infty}\frac{\Gamma(1+ \overline{\tau}_{1})}{\Gamma(1-k+\overline{\tau}_{1})\Gamma(1+k)}D^{k} \sigma_{4}D^{\overline{\tau}_{1}-k}\sigma_{4}\right|\\ \leq&-s_{31}\sigma_{3}^{2}-s_{32}|\sigma_{3}|^{3/ 2}+\mu_{3}|\sigma_{3}|^{3/2}-s_{41}\sigma_{4}^{2}-s_{42}|\sigma_{4}|^{3/2}\\ &\quad-s_{43}\sigma_{4}^{2}/\left(|\sigma_{4}|+e_{\sigma 4}\right)^{-1}+\mu_{4}|\sigma_{4}|+\widetilde{f}_{4d}\\ \leq&-(s_{32}-\mu_{3})|\sigma_{3}|^{3/2}-(s_{42}- \mu_{4})|\sigma_{4}|^{3/2}-|\sigma_{4}|\Big{(}s_{43}|\sigma_{4}|/\left(|\sigma _{4}|+e_{\sigma 4}\right)^{-1}-\delta_{4d}\Big{)},\end{split}  \tag{68}\]  
\[ \begin{split} D^{\overline{\tau}_{1}}(V_{5}+V_{6})=\sigma_{5}D^{ \overline{\tau}_{1}}\sigma_{5}+\sigma_{6}D^{\overline{\tau}_{1}}\sigma_{6}\\ =&\sigma_{5}\big{(}e_{6}-\dot{x}_{5d}+\sigma_{5}+ \alpha_{5}\big{)}+\sigma_{6}(f_{6}+g_{6}u_{6}+\sigma_{6}+\tau_{46})\\ &+\left|\sum_{k=1}^{\infty}\frac{\Gamma(1+\overline{\tau}_{1})}{ \Gamma(1-k+\overline{\tau}_{1})\Gamma(1+k)}D^{k}\sigma_{5}D^{\overline{\tau}_ {1}-k}\sigma_{5}\right|+\left|\sum_{k=1}^{\infty}\frac{\Gamma(1+\overline{ \tau}_{1})}{\Gamma(1-k+\overline{\tau}_{1})\Gamma(1+k)}D^{k}\sigma_{6}D^{ \overline{\tau}_{1}-k}\sigma_{5}\right|\\ \leq&-s_{51}\sigma_{5}^{2}-(s_{52}-\mu_{5})|\sigma_{ 5}|^{3/2}-s_{61}\sigma_{6}^{2}-(s_{62}-\mu_{6})|\sigma_{6}|^{3/2}\\ &-s_{63}\sigma_{6}^{2}/\left(|\sigma_{6}|+e_{\sigma 6}\right)^{-1}+\widetilde{f}_{46}\\ \leq&-(s_{52}-\mu_{5})|\sigma_{5}|^{3/2}-(s_{62}-\mu _{6})|\sigma_{6}|^{3/2}-|\sigma_{6}|\Big{(}s_{63}|\sigma_{6}|/\left(|\sigma_{ 6}|+e_{\sigma 6}\right)^{-1}-\delta_{46}\Big{)},\end{split}  \tag{69}\]  
where \(\widetilde{f}_{4d}=f_{4d}-\tilde{f}_{4d}\), \(\widetilde{f}_{d6}=f_{d6}-\tilde{f}_{d6}\), \(\left|\widetilde{f}_{4d}\right|\leq\delta_{d4}\), \(\left|\widetilde{f}_{d6}\right|\leq\delta_{d6}\), \(\delta_{d4}\) and \(\delta_{d6}\) a re unknown constants. Selecting \(s_{33}\), \(s_{43}\), \(s_{53}\), and \(s_{63}\) to be satisfied by the following conditions:  
\[ \begin{split} s_{43}|\sigma_{4}|/\left(|\sigma_{4}|+e_{\sigma 4}\right)^{-1}\geq\delta_{4d},\\ s_{63}|\sigma_{6}|/\left(|\sigma_{6}|+e_{\sigma 6}\right)^{-1}\geq\delta_{d6},\end{split}  \tag{70}\]  
we obtain  
\[D^{\overline{\tau}_{1}}(V_{3}+V_{4})\leq-(s_{32}-\mu_{3})|\sigma_{3}|^{3/2}-(s _{42}-\mu_{4})|\sigma_{4}|^{3/2}, \tag{71}\]  
\[D^{\overline{\tau}_{1}}(V_{5}+V_{6})\leq-(s_{52}-\mu_{5})|\sigma_{5}|^{3/2}-(s _{62}-\mu_{6})|\sigma_{6}|^{3/2}. \tag{72}\]