## Appendix A Proof of Step 1 in Theorem 1  
This step is analysed base on different simplification modes of \(\text{sat}(\xi_{1x}/w_{2x}),x=\{\mu,q\}\) in the expressions of \(L_{V\xi_{x}}\) and \(\dot{L}_{V\xi_{x}}\), which includes four cases. _Case 1_: When \(\left|\xi_{1\mu}\right|\leq w_{2\mu}\) and \(\left|\xi_{1q}\right|\leq w_{2\mu}\) such that \(\text{sat}\Big{(}\xi_{1\mu}/w_{2\mu}\Big{)}=\xi_{1\mu}/w_{2\mu}\) and \(\text{sat}\Big{(}\xi_{1q}/w_{2q}\Big{)}=\xi_{1q}/w_{2q}\), then Equation (32) can be modified as  
\[ \begin{array}{rcl}L_{V\xi}&=&\sum\limits_{x=\mu,q}\Big{(}\frac{1}{2}\Big{(}1 _{1x}+\frac{l_{4x}}{w_{2x}}\Big{)}\xi_{1x}^{2}+\frac{1}{2}l_{2x}\xi_{dx}^{2}- l_{3x}\xi_{1x}\xi_{dx}\Big{)}\\ &\leq&\sum\limits_{x=\mu,q}\Big{(}\frac{1}{2}\Big{(}1_{1x}+\frac{l_{4x}}{w_{2x }}+l_{3x}\Big{)}\xi_{1x}^{2}+\frac{1}{2}(l_{2x}+l_{3x})\xi_{dx}^{2}\Big{)} \leq\frac{1}{2}\sum\limits_{x=\mu,q}\eta_{1x}\Big{(}\xi_{1x}^{2}+\xi_{dx}^{2} \Big{)},\end{array}  \tag{11}\]  
Figure 9: The output of the extended state observers (ESOs): **(a)** The compound disturbances and its estimation in the flight path angle loop and **(b)** the compound disturbances and its estimation in the pitch angular rate loop. where \(\eta_{1\mu}=\max\!\left(l_{1\mu}+\frac{l_{4\mu}}{w_{2\mu}}+l_{3\mu},l_{2\mu}+l_{3 \mu}\right)\) and \(\eta_{1q}=\max\!\left(l_{1q}+\frac{l_{4q}}{w_{2q}}+l_{3q},l_{2q}+l_{3q}\right)\). Considering _assumption 1_ as \(\dot{d}_{x}\leq N_{x}\) and taking \(\eta_{2x}=l_{2x}w_{1x}w_{3x}\frac{\zeta_{1x}^{3}}{\zeta_{2x}^{3}}-l_{3x}w_{3x} \frac{\zeta_{1x}^{2}}{\zeta_{2x}^{2}}-3l_{3x}w_{1x}^{2}\frac{\zeta_{1x}}{\zeta_ {2x}}+2l_{3x}w_{1x^{\prime}}^{2}\). Equation (35) can be rewritten as  
\[\begin{array}{ll}\dot{L}_{V\xi}&=\sum\limits_{x=\mu,q}\dot{L}_{V\xi_{x}}\leq \sum\limits_{x=\mu,q}\left(-\frac{1}{\zeta_{1x}}\!\left(\eta_{2x}\xi_{1x}^{2}+l _{3x}\xi_{dx}^{2}\right)+\left(l_{2x  
\[\dot{L}_{V\xi}\leq-\frac{1}{2\varsigma_{1\mu}}\Big{(}\eta_{2\mu}\xi_{1\mu}^{2}+l_{ 3\mu}\xi_{4\mu}^{2}\Big{)}-\frac{1}{2\varsigma_{1\mu}}\Big{(}\big{(}l_{1\mu}u_{1 }-l_{3\mu}w_{3\mu}\big{)}\xi_{1q}^{2}+l_{3\mu}\xi_{4\mu}^{2}+2\eta_{6q}\Big{(} \big{|}\xi_{1q}\big{|}-w_{2\mu}\Big{)}\Big{)}\leq-\eta_{10}L_{V\xi} \tag{100}\]  
where \(\eta_{10}=\min\Big{(}\frac{\eta_{2\mu}}{\varsigma_{1\mu}\eta_{1\mu}},\frac{l_{ 3\mu}}{\varsigma_{1\mu}\eta_{1\mu}},\frac{l_{1\mu}w_{1q}-l_{3\mu}w_{3\mu}}{ \varsigma_{1\mu}\eta_{5\mu}},\frac{l_{3\mu}}{\varsigma_{1\mu}\eta_{5\mu}},\frac {l_{3\mu}}{\varsigma_{1\mu}\eta_{5\mu}},\frac{2l_{6q}}{\varsigma_{1\mu}\eta_{ 5\mu}}\Big{)}\). _Case 4:_ When \(\big{|}\xi_{1\mu}\big{|}>w_{2\mu}\) and \(\big{|}\xi_{1q}\big{|}\leq w_{2\mu}\), by utilizing the intermediate results in Case 1 and Case 2 and with a similar procedure, the expressions of \(L_{V\xi}\) and \(\dot{L}_{V\xi}\) can be computed as  
\[L_{V\xi} =\frac{1}{2}l_{1\mu}\xi_{1\mu}^{2}+\frac{1}{2}l_{2\mu}\xi_{4\mu}^ {2}-l_{3\mu}\xi_{1\mu}\xi_{4\mu}+l_{4}\Big{(}\big{|}\xi_{1\mu}\big{|}-w_{2\mu} \Big{)}+\frac{1}{2}\Big{(}l_{1q}+\frac{l_{4\mu}}{w_{2\mu}}\Big{)}\xi_{1q}^{2} +\frac{1}{2}l_{2\mu}\xi_{4\mu}^{2}-l_{3q}\xi_{1q}\xi_{4\mu} \tag{101}\] \[\leq\frac{1}{2}\eta_{5\mu}\Big{(}\xi_{1\mu}^{2}+\xi_{4\mu}^{2}+ \big{|}\xi_{1\mu}\big{|}-w_{2\mu}\Big{)}+\frac{1}{2}\eta_{1q}\Big{(}\xi_{1q}^ {2}+\xi_{4q}^{2}\Big{)}\]  
\[\dot{L}_{V\xi}\leq-\frac{1}{2\varsigma_{1\mu}}\Big{(}\big{(}l_{1\mu}w_{1\mu}- l_{3\mu}w_{3\mu}\big{)}\xi_{1\mu}^{2}+l_{3\mu}\xi_{4\mu}^{2}+2\eta_{6q}\Big{(} \big{|}\xi_{1\mu}\big{|}-w_{2\mu}\Big{)}\Big{)}-\frac{1}{2\varsigma_{1\mu}} \Big{(}\eta_{2\mu}\xi_{1q}^{2}+l_{3\mu}\xi_{4\mu}^{2}\Big{)}\leq-\eta_{11}L_{V\xi} \tag{102}\]  
where \(\eta_{11}=\min\Big{(}\frac{l_{1\mu}w_{1\mu}-l_{3\mu}w_{3\mu}}{\varsigma_{1\mu} \eta_{5\mu}},\frac{l_{3\mu}}{\varsigma_{1\mu}\eta_{5\mu}},\frac{2l_{6q}}{ \varsigma_{1\mu}\eta_{5\mu}},\frac{\eta_{2q}}{\varsigma_{1\mu}\eta_{4\mu}}, \frac{l_{3\mu}}{\varsigma_{1\mu}\eta_{4\mu}}\Big{)}\).