## 4 Stability and Altitude Tracking Performance Analysis
We introduce a new error vector \(\xi_{x}=\big{[}\xi_{2x},\xi_{dx}\big{]}^{T}\), where \(\xi_{2x}=\bigg{(}x-x\bigg{)}/\varsigma_{2x}\), \(x=\mu,q\), together with Equation (7), the time derivative of \(\xi\) is given by  
\[\left\{ \begin{array}{l}\dot{\xi}_{2x}=\frac{1}{\varsigma_{2x}}\xi_{dx}-\frac {w_{1x}}{\varsigma_{2x}}\xi_{2x}\\ \dot{\xi}_{dx}=\dot{d}_{x}-\frac{w_{3x}}{\varsigma_{2x}}\xi_{2x}\end{array}  \right.,\ \ x=\mu,q, \tag{37}\]  
Consider the Lyapunov function as  
\[L_{\text{W\'{\'{e}}}}=\sum\limits_{x=\mu,q}L_{\text{W\'{e}}\xi_{x}}=\sum \limits_{x=\mu,q}\Big{(}\frac{1}{2}l_{1x}\xi_{2x}^{2}+\frac{1}{2}l_{2x}\xi_{dx} ^{2}-l_{3x}\xi_{2x}\xi_{dx}\Big{)}, \tag{38}\]  
According to mean-square inequality, it follows that  
\[\frac{1}{2}(l_{1x}+l_{3x})\xi_{2x}^{2}+\frac{1}{2}(l_{2x}+l_{3x})\xi_{dx}^{2} \geq L_{\text{W\'{e}}\xi_{x}}\geq\frac{1}{2}(l_{1x}-l_{3x})\xi_{2x}^{2}+ \frac{1}{2}(l_{2x}-l_{3x})\xi_{dx}^{2}, \tag{39}\]  
where the values of \(l_{1x},l_{2x},l_{3x},l_{4x}\) are consistent with the analysis in step 1, namely under the constrains of Equation (34). Then, it can be extended from \(L_{\text{W\'{e}}\xi_{x}}\) to \(L_{\text{W\'{e}}}\) as  
\[\frac{1}{2}\kappa_{1\mu}\Big{(}\xi_{2\mu}^{2}+\xi_{d\mu}^{2}\Big{)}+\frac{1}{2 }\kappa_{1\mu}\Big{(}\xi_{2\mu}^{2}+\xi_{d\mu}^{2}\Big{)}\geq L_{\text{W\'{e}} }\geq\frac{1}{2}\kappa_{2}\Big{(}\xi_{2\mu}^{2}+\xi_{d\mu}^{2}+\xi_{2\mu}^{2} +\xi_{d\mu}^{2}\Big{)}, \tag{40}\]  
where \(\kappa_{1x}=\max(l_{1x}+l_{3x},l_{2x}+l_{3x})\), \(\kappa_{2}=\min\Big{(}l_{1\mu}-l_{3\mu},l_{2\mu}-l_{3\mu},l_{1\mu}-l_{3\mu},l_{ 2\mu}-l_{3q}\Big{)}\). Depending on Equation (34), computing the time derivative of \(L_{\text{W\'{e}}}\) yields  
\[ \begin{array}{l}\dot{L}_{\text{W\'{e}}}\hskip-14.226378 pt\stackrel{{ \cdot}}{{=}}\dot{L}_{\text{W\'{e}}\xi_{\mu}}+\dot{L}_{\text{W\'{e}}\xi_{q}}= \sum\limits_{x=\mu,q}\Big{(}-\frac{l_{1x}w_{1x}-l_{3x}w_{3x}}{\varsigma_{2x}} \xi_{2x}^{2}-\frac{l_{3x}}{\varsigma_{2x}}\xi_{dx}^{2}+\Big{(}l_{2x}\xi_{dx}-l _{3x}\xi_{2x}\Big{)}\dot{d}_{x}\Big{)}\\ &\leq\sum\limits_{x=\mu,q}\bigg{(}-\frac{l_{1x}w_{1x}-l_{3x}w_{3x}}{\varsigma_{2 x}}\xi_{2x}^{2}-\frac{l_{3x}}{\varsigma_{2x}}\xi_{dx}^{2}+N_{x}\sqrt{l_{2x}^{2}+l_{3x}^{2} \|\xi_{x}\|}\bigg{)},\end{array}  \tag{41}\]  
Substituting the Equation (40) into Equation (41), one has  
\[\dot{L}_{\text{W\'{e}}}\leq-\frac{\kappa_{3}}{\varsigma_{2}}L_{\text{W\'{e}}}+2 \kappa_{4}\sqrt{\frac{L_{\text{W\'{e}}}}{\kappa_{2}}}, \tag{42}\]  
where \(k_{2}\) and \(k_{4}\) are design parameters subject to \(k_{2}>\frac{1}{2}\) and \(k_{4}>\frac{1}{2}\), \(\lambda_{V}=\min(2k_{1},2k_{2}-1,2k_{3},2k_{4}-1)\) and \(N_{V}=O\big{(}c_{2}^{2}\big{)}+O\big{(}c_{\mu}^{2\rho\gamma-2}\big{)}+O\big{(}c _{\theta_{\
u}}^{2\rho\gamma-2}\big{)}+O\big{(}c_{\theta_{\
u}}^{2\rho\gamma-2} \big{)}\). Consequently, by solving Equation (50), it yields  
\[L_{V4}(t)\leq\frac{N_{V}}{\lambda_{V}}+\Big{(}L_{V4}(0)-\frac{N_{V}}{\lambda_{ V}}\Big{)}e^{-\lambda_{V}t}, \tag{51}\]  
which indicates that all the signals in the closed-loop system are bounded. Moreover, the altitude tracking error can be depicted as  
\[|z_{h}|\leq\sqrt{2\Big{(}\frac{N_{V}}{\lambda_{V}}+\Big{(}L_{V4}(0)-\frac{N_{V }}{\lambda_{V}}\Big{)}e^{-\lambda_{V}t}\Big{)}}, \tag{52}\]  
Namely \(\lim\limits_{t\to\infty}|h-h_{d}|\leq\sqrt{\frac{2N_{V}}{\lambda_{V}}}\). Theoretically, by selecting the arbitrarily small values of \(c_{2\mu}\) and \(c_{2q}\) in the ESOs and the parameter \(\varepsilon\) in the differentiator for obtaining a small numerator while choosing relatively large controller law gain \(k_{i},i=1,\cdots,4\) in the denominator, it can be concluded that the system output tracking error can converge to a sufficiently small neighbourhood of zero ultimately with less regulation time. This completes the proof of Theorem 1.