## 3 Extended State Observer  
As only the vehicle's position and attitude are known, an ESO is devised to supply estimates of velocity, angular rate states, and the lumped disturbances. Following the principle of high-gain observer design [52], the ESO is formulated as  
\[\dot{\mathbf{X}}=\mathbf{A}_{0}\mathbf{\hat{X}}+\mathbf{\Phi}(\mathbf{\hat{X}})+\mathbf{\hat{U}}+\mathbf{ H} \begin{bmatrix}\mathbf{X}_{1}-\mathbf{\hat{X}}_{1}\\ \mathbf{X}_{4}-\mathbf{\hat{X}}_{4}\end{bmatrix}  \tag{8}\]  
Figure 1: Quadrotor configuration. with  
\[ \begin{split}&\hat{\mathbf{X}}=\begin{bmatrix}\hat{\mathbf{X}}_{1}&\hat{\mathbf{X}}_{2}& \hat{\mathbf{X}}_{3}&\hat{\mathbf{X}}_{4}&\hat{\mathbf{X}}_{5}&\hat{\mathbf{X}}_{6}\end{bmatrix} ^{T},\\ &\\ &\mathbf{\Phi}(\hat{\mathbf{X}})=\begin{bmatrix}\mathbf{0}&f_{2}(\hat{\mathbf{X}}_{2})&\mathbf{0}& \mathbf{0}&f_{5}(\hat{\mathbf{X}}_{5})&\mathbf{0}\end{bmatrix}^{T},\\ &\\ &\hat{\mathbf{U}}=\begin{bmatrix}\mathbf{0}&F(\hat{\mathbf{X}}_{4})&\mathbf{0}&\mathbf{0}&B_{5}\mathbf{ U}_{2}&\mathbf{0}\end{bmatrix}^{T},\\ &\\ &\mathbf{H}=\begin{bmatrix}3\omega_{1}\mathbf{I}_{3}&3\omega_{1}^{2}\mathbf{I}_{3}&\omega_ {1}^{2}\mathbf{I}_{3}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&3\omega_{2}\mathbf{I}_{3}&3\omega_{2}^{2}\mathbf{I}_{3}&\omega_ {2}^{3}\mathbf{I}_{3}\end{bmatrix}^{T}\end{split} \end{split}\]  
where \(\hat{\mathbf{X}}_{i}(i=1,...,6)\) is the estimated value of \(\mathbf{X}_{i}\), \(\mathbf{H}\) denotes the observer gain matrix, and \(\omega_{1},\omega_{2}\) are the ESO's bandwidth parameters. To begin the analysis of the ESO's stability, let us define the estimation error \(\bar{\mathbf{X}}\) as  
\[\bar{\mathbf{X}}=\mathbf{X}-\bar{\mathbf{X}}= \begin{bmatrix}\hat{\mathbf{X}}_{1}&\hat{\mathbf{X}}_ {2}&\hat{\mathbf{X}}_{3}&\hat{\mathbf{X}}_{4}&\hat{\mathbf{X}}_{5}&\hat{\mathbf{X}}_{6}\end{bmatrix}  ^{T}\]  
Then, the corresponding scaled estimation error is defined as  
\[\mathbf{\vartheta}= \begin{bmatrix}\mathbf{\vartheta}_{1}&\mathbf{\vartheta}_{2}&\mathbf{ \vartheta}_{3}&\mathbf{\vartheta}_{4}&\mathbf{\vartheta}_{5}&\mathbf{\vartheta}_{6}\end{bmatrix}  ^{T}= \begin{bmatrix}\hat{\mathbf{X}}_{1}\omega_{1}^{2}&\hat{\mathbf{X}}_{2}\omega_{1}& \hat{\mathbf{X}}_{3}&\bar{\mathbf{X}}_{4}\omega_{2}^{2}&\hat{\mathbf{X}}_{5}\omega_{2}& \bar{\mathbf{X}}_{6}\end{bmatrix} ^{T} \tag{9}\]  
By recalling Equations (7) and (8), we obtain the scaled estimation error dynamics as  
\[ \begin{split}&\dot{\mathbf{\vartheta}}=\mathbf{\omega}\mathbf{A}\mathbf{ \vartheta}+\omega_{1}\delta_{2}[f_{2}(\mathbf{X}_{2})-f_{2}(\hat{\mathbf{X}}_{2})+F( \mathbf{X}_{4})-F(\hat{\mathbf{X}}_{4})]+\delta_{3}\mathbf{h}_{p}+\omega_{2}\delta_{5}[f_ {5}(\mathbf{X}_{5})-f_{5}(\hat{\mathbf{X}}_{5})]+\delta_{6}\mathbf{h}_{\theta}\\ &=\mathbf{\omega}\mathbf{A}\mathbf{\vartheta}+\omega_{1}\delta_{2}(\hat{f}_{2}+ \mathbf{F})+\delta_{3}\mathbf{h}_{p}+\omega_{2}\delta_{5}f_{5}+\delta_{6}\mathbf{h}_{ \theta}\end{split}  \tag{10}\]  
where  
\[\mathbf{\omega}= \begin{bmatrix}\omega_{1}\mathbf{I}_{9}&\mathbf{0}\\ \mathbf{0}&\omega_{2}\mathbf{I}_{9}\end{bmatrix} ,\] \[\mathbf{A}= \begin{bmatrix}-3I_{3}&I_{3}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ -3I_{3}&\mathbf{0}&I_{3}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ -I_{3}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-3I_{3}&\mathbf{I}_{3}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-3I_{3}&\mathbf{0}&\mathbf{0}\end{bmatrix} ,\] \[\delta_{2}= \begin{bmatrix}\mathbf{0}&\mathbf{I}_{3}&\mathbf{0}&\mathbf{0}&\mathbf{0} &\mathbf{0}\end{bmatrix} ^{T},\,\delta_{3}= \begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{I}_{3}& \mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix} ^{T},\delta_{5}= \begin{bmatrix}\mathbf{0}&\mathbf{0}& \mathbf{0}&\mathbf{0}&\mathbf{I}_{3}&\mathbf{0}\end{bmatrix} ^{T},\] \[\delta_{6}= \begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{I}_{3} \end{bmatrix}