## 4 Stability and Altitude Tracking Performance Analysis
\tag{31}\]  
Consider the Lyapunov function as  
\[L_{V\xi_{x}}=\frac{1}{2}l_{1x}\xi_{1x}^{2}+\frac{1}{2}l_{2x}\xi_{dx}^{2}-l_{3x }\xi_{1x}\xi_{dx}+l_{4x}\int_{0}^{\xi_{1x}}\text{sat}\Big{(}\frac{\xi_{1x}}{ w_{2x}}\Big{)}d\xi_{1x},\;\;\;x=\mu,q, \tag{32}\]  
where \(l_{1x},l_{2x},l_{3x},l_{4x}\) are positive constants and taking the time derivative of \(L_{V\xi_{x}}\) along with the formula of (31), one has  
\[\begin{array}{l}\dot{L}_{V\xi_{x}}=\Big{(}l_{1x}\xi_{1x}-l_{3x}\xi_{dx}+l_{ 4x}\text{sat}\Big{(}\frac{\xi_{1x}}{w_{2x}}\Big{)}\Big{)}\Big{(}\frac{\xi_{1x} }{\zeta_{1x}}-\frac{w_{1x}\xi_{1x}}{\zeta_{1x}}w_{1x}w_{2x}\Big{(}\frac{1}{ \zeta_{2x}}-\frac{1}{\zeta_{1x}}\Big{)}\text{sat}\Big{(}\frac{\xi_{1x}}{w_{2x }}\Big{)}\Big{)}\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quadFor subsequent use, two compact sets are defined as  
\[ \begin{split} S_{1}&=\bigg{\{}\Big{[}\xi_{\mu}^{T},\xi_{ \eta}^{T}\Big{]}^{T}\in\mathbb{R}^{4}\bigg{|}L_{V\xi}\big{(}\xi_{\mu},\xi_{q} \big{)}=L_{V\xi_{\mu}}\big{(}\xi_{\mu}\big{)}+L_{V\xi_{\mu}}\big{(}\xi_{q} \big{)}\leq D_{1}\bigg{\}},\\ S_{2}&=\bigg{\{}\Big{[}\xi_{\mu}^{T},\xi_{\eta}^{T} \Big{]}^{T}\in\mathbb{R}^{4}\bigg{|}L_{V\xi}\big{(}\xi_{\mu},\xi_{q}\big{)} \leq D_{2}\bigg{\}},\end{split}  \tag{36}\]  
where \(D_{1}\) and \(D_{2}\) are positive with \(D_{2}=\max\limits_{|\xi_{1\mu}|\leq\varpi_{2\mu}|\xi_{1\mu}|\leq\varpi_{2\mu}} \big{(}L_{V\xi}\big{(}\xi_{\mu},\xi_{q}\big{)}\big{)}\) and \(D_{1}=\max\limits_{\big{[}D_{2},L_{V\xi}\big{(}\xi_{\mu}(0),\xi_{q}(0)\big{)} \big{]}}\). The proof for convergence of the observers will be split into two steps. _Step 1_: Here, we discuss the boundedness of the error vector \(\xi=\big{[}\xi_{\mu}^{T},\xi_{\eta}^{T}\big{]}^{T}\) for any initial \(\xi\in S_{1}-S_{2}\); there exists a time instant \(t_{1}\) when \(t>t_{1}\), \(\xi\in S_{2}\) is satisfied. Due to page limitations, a more detailed proof is presented in Appendix A, which is elaborated with four cases. _Step 2_: In this step, we prove the convergence of the estimation errors on the basis of the conclusion in Step 1. Since, when \(t>t_{1}\), \(\xi=\big{[}\xi_{\mu}^{T},\xi_{\eta}^{T}\big{]}^{T}\in S_{2}\) always holds, which means that \(\big{|}\xi_{1\mu}\big{|}\leq w_{2\mu}\) and \(\big{|}\xi_{1\mu}\big{|}\leq w_{2\mu}\).