## 4 Fractional-Order Controller and Fractional-Order Disturbance Observer Design
### Fractional-Order Disturbance Observer Design  
The disturbance \(\boldsymbol{\rho}(t)\) of the following system (90) is assumed that \(|\boldsymbol{\rho}(t)|\) is bounded and there is an unknown constant \(\overline{\rho}\) such that \(|\boldsymbol{\rho}(t)|\leq\overline{\rho}\) is satisfied  
\[D^{\overline{7}_{1}}\boldsymbol{\sigma}_{o}=f_{o}+g_{o}\boldsymbol{u}_{o}+ \boldsymbol{\sigma}_{o}+\boldsymbol{\rho}, \tag{90}\]  
where \(\boldsymbol{\sigma}_{o}=[\sigma_{2}\ \sigma_{4}\ \sigma_{6}\ \sigma_{8}]^{T},f_{o}=[-g\ 0 \ f_{6}\ f_{8}]^{T},g_{o}=diag(\cos\phi\cos\theta/m_{0},R_{4},\ g_{6},\ g_{8}), \boldsymbol{u}_{o}=[u_{2}\ u_{4}\ u_{6}\ u_{8}]^{T},\boldsymbol{\sigma}_{o}=[ \omega_{2}\ \sigma_{4}\ \sigma_{6}\ \sigma_{8}]^{T}\), and \(\boldsymbol{\rho}=[f_{d2}\ f_{d3}\ \tau_{d6}\ \tau_{48}]^{T}\). Theorem 4.1: _The filtered error surface state \(\boldsymbol{\sigma}_{o}=[\sigma_{2}\ \sigma_{4}\ \sigma_{6}\ \sigma_{8}]^{T}\) and disturbances \(\boldsymbol{\rho}(t)\) can be estimated using the following fractional-order nonlinear disturbance observers:_  
\[ \begin{array}{l}D^{\overline{7}_{1}}\overset{\wedge}{\boldsymbol{\rho}}_{ o}=f_{o}+g_{o}\boldsymbol{u}_{o}+\boldsymbol{\sigma}_{o}+\hat{\boldsymbol{\rho}},\\ D^{\overline{7}_{1}}\overset{\wedge}{\boldsymbol{\rho}}=-\boldsymbol{ \Lambda}^{2}[\boldsymbol{\eta}_{\rho 1}\text{sinh}(c_{\rho 1}\boldsymbol{\eta}_{1}(\widetilde{\sigma}_{o}))+ \boldsymbol{\eta}_{\rho 2}\text{sinh}(c_{\rho 2}\boldsymbol{\Lambda}^{-1} \boldsymbol{\sigma}_{2}(\overset{\wedge}{\boldsymbol{\rho}}))],\end{array}  \tag{91}\]  
_where \(\widetilde{\boldsymbol{\sigma}}_{o}=\boldsymbol{\sigma}_{o}-\overset{\wedge}{ \boldsymbol{\sigma}_{o}}=[\widetilde{\sigma}_{2}\ \widetilde{\sigma}_{4}\ \widetilde{\sigma}_{6}\ \widetilde{\sigma}_{6}]^{T}\) denote the estimation error, \(\boldsymbol{\varphi}_{1}(\widetilde{\boldsymbol{\Theta}})=\kappa sig^{1/2}( \widetilde{\boldsymbol{\Theta}})+(1-\kappa)sig^{(2+v)/2}(\widetilde{ \boldsymbol{\Theta}})\), \(\boldsymbol{\rho}_{2}(\overset{\wedge}{\boldsymbol{\rho}})=sig^{1/2}( \overset{\wedge}{\boldsymbol{\rho}})\), \(\widetilde{\boldsymbol{\rho}}=\boldsymbol{\rho}-\overset{\wedge}{\boldsymbol{ \rho}}\), \(\overset{\wedge}{\boldsymbol{\rho}}\) and \(\overset{\wedge}{\boldsymbol{\rho}}\) denote the estimates of \(\boldsymbol{\sigma}_{o}\) and \(\boldsymbol{\rho}\), \(\boldsymbol{\Lambda}\), \(\boldsymbol{\eta}_{\rho 1},\boldsymbol{\eta}_{\rho 2}\),\(\boldsymbol{\epsilon}_{\rho 1}\), and \(\boldsymbol{\epsilon}_{\rho 1}\) is a diagonal constant matrix with \(diag(\epsilon_{\rho i})\geq 0\). One can then obtain that \(\overset{\wedge}{\boldsymbol{\sigma}}_{o}\rightarrow\boldsymbol{\sigma}_{o}\) and \(\overset{\wedge}{\boldsymbol{\rho}}\rightarrow\boldsymbol{\rho}\) as \(\boldsymbol{\Lambda}\rightarrow\infty\)._  
**Proof of Theorem 4.1. ** As \(\boldsymbol{\Lambda}\rightarrow\infty\), \(\left|D^{\overline{7}_{1}}\overset{\wedge}{\boldsymbol{\rho}}\right|=\left| \boldsymbol{\Lambda}^{2}[\boldsymbol{\eta}_{\rho 1}\text{sinh}(c_{\rho 1}\boldsymbol{\varphi}_{1}(\widetilde{\boldsymbol{\Theta}}))+ \boldsymbol{\eta}_{\rho 2}\text{sinh}(c_{\rho 2}\boldsymbol{\Lambda}^{-1} \boldsymbol{\varphi}_{2}(\overset{\wedge}{\boldsymbol{\rho}}))]\right|\) is assumed to reach an infinitely large value. This implies that the variation in \(\overset{\wedge}{\boldsymbol{\rho}}\) significantly exceeds \(f_{o}+g_{o}\boldsymbol{u}_{o}+\varpi_{o}\). Furthermore, it suggests that \(\underset{\boldsymbol{\Lambda}\rightarrow\infty}{\lim}D^{\overline{7}_{1}}(f_{o} +g_{o}\boldsymbol{u}_{o}+\varpi_{o}+\overset{\wedge}{\boldsymbol{\rho}})=D^{ \overline{7}_{1}}\overset{\wedge}{\boldsymbol{\rho}}\), \(\lim_{\boldsymbol{\Lambda}\rightarrow\infty}\boldsymbol{\Lambda}^{-1}(f_{o}+g_{o} \boldsymbol{u}_{o}+\boldsymbol{\sigma}_{o}+\overset{\wedge}{\boldsymbol{\rho}})= \boldsymbol{\Lambda}^{-1}\overset{\wedge}{\boldsymbol{\rho}}\). Hence, (91) holds according to Theorem 4.1 when \(f_{o}+g_{o}\boldsymbol{u}_{o}+\varpi_{o}+\hat{\boldsymbol{\rho}}\) is taken as \(\overset{\wedge}{\boldsymbol{\sigma}}_{o}\).