## 4 Fractional-Order Controller and Fractional-Order Disturbance Observer Design
### Longitudinal and Latitudinal Controllers Design
Taking the time derivative of (52) and applying \(D^{-\gamma_{1}}\) to both sides of the resulting equation, we obtain \[D^{\overline{7}_{1}}\sigma_{3} =\dot{e}_{3}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(}b_{31}e_{3}+b_{32} sig^{1/2}(e_{3})\Big{)}-D^{\overline{7}_{1}}\xi_{31} \tag{56}\] \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\] \[=e_{4}-\dot{x}_{4d}+\varpi_{3}+\alpha_{3}\] \[D^{\overline{7}_{1}}\sigma_{4}= \dot{e}_{4}+D^{\overline{7}_{2}}k_{4}^{2}\Big{(}b_{41}e_{4}+b_{42 }\,\text{sig}^{1/2}(e_{4})\Big{)}-D^{\overline{7}_{1}}\xi_{42}\] (57) \[= R_{4}(\psi,u_{1})T_{4}(x_{5})+f_{d4}-D^{\overline{7}_{1}}\zeta_ {32}+D^{\overline{7}_{2}}k_{4}^{2}\Big{(}b_{41}e_{4}+b_{42}\,\text{sig}^{1/2}( e_{4})\Big{)}\] \[+\beta_{42}\xi_{42}+\xi_{41}+\lambda_{42}\,\text{sig}^{1/2}(\xi_ {42})\] \[= R_{4}(\psi,u_{1})T_{4}(x_{5})+\varpi_{4}+f_{44}\] \[D^{\overline{7}_{1}}\sigma_{5} =\dot{e}_{5}+D^{\overline{7}_{2}}k_{5}^{2}\Big{(}b_{51}e_{5}+b_{ 52}sig^{1/2}(e_{5})\Big{)}-D^{\overline{7}_{1}}\xi_{51}\] (58) \[=e_{6}+\zeta_{51}-\dot{x}_{5d}+D^{\overline{7}_{2}}k_{5}^{2}\Big{[} b_{51}e_{5}+b_{52}sig^{1/2}(e_{5})\Big{]}+\beta_{51}\xi_{51}-\xi_{52}-\zeta_{51}+ \alpha_{5}+\lambda_{51}sig^{1/2}(\zeta_{51})\] \[=e_{6}-\dot{x}_{5d}+\varpi_{5}+\alpha_{5},\] \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]0 (59) \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]1 where \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]2 \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]3 \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]4 \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]5  
Moreover, \(\dot{\zeta}_{31}\) and \(\dot{\zeta}_{51}\) in (56) and (59) can be expressed as \(\dot{\zeta}_{31}=D^{\overline{7}_{11}}\zeta_{32}\) and \(\dot{\zeta}_{51}=D^{\overline{7}_{11}}\zeta_{52}\), respectively, by a proper fractional-order operation and the relationship \(D^{\gamma_{1}}\dot{\zeta}_{k1}=\zeta_{k2}\) in (51). We can select the virtual controls \(a_{i},\ i=3,5\) for the recursive input, and control inputs \(u_{i},\ i=2,3\) as follows:  
\[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]6 \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]7 (61) \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]8 (62) \[=e_{4}+\zeta_{31}-\dot{x}_{1d}+D^{\overline{7}_{2}}k_{3}^{2}\Big{(} b_{31}e_{3}+b_{32}\,\text{sig}^{1/2}(e_{3})\Big{)}+\beta_{31}\xi_{31}-\xi_{32}- \zeta_{31}+\alpha_{3}+\lambda_{31}sig^{1/2}(\zeta_{31})\]9 (63) \[\text{where }R_{4}^{-1} = \begin{bmatrix}\cos\psi&\sin\psi\\ \sin\psi&-\cos\psi\end{bmatrix} ^{-1}=