## 5 Closed-Loop System's Stability Analysis  
In this section, we move on establishing the stability analysis of the closed-loop system. Toward that end, the error dynamics of the closed-loop system needs to be derived first. By combining Equation (10) with the equations from (23) to (46), we have the following formula. \[\dot{\mathbf{e}}_{1}=\mathbf{X}_{2}-\dot{\mathbf{P}}_{d}=(\mathbf{X}_{2}-\bar{\mathbf{X}}_{2})+\bar {\mathbf{X}}_{2}-\dot{\mathbf{P}}_{d}=-k_{1}\mathbf{e}_{1}+\mathbf{e}_{2} \tag{47}\]  
\[\dot{\mathbf{e}}_{2} =\mathbf{X}_{3}+f_{2}(\mathbf{X}_{2})+\mathbf{F}(\mathbf{X}_{4})-\hat{\mathbf{X}}_{2} \tag{48}\] \[=\mathbf{X}_{3}+f_{2}(\mathbf{X}_{2})-f_{2}(\hat{\mathbf{X}}_{2})-k_{2}(\hat{ \mathbf{X}}_{2}-\bar{\mathbf{X}}_{2}-\bar{\mathbf{\xi}}_{1})-\hat{\mathbf{X}}_{3}+\dot{\mathbf{X}}_{ 2f}-k_{3}\mathbf{\xi}_{1}+\Delta\mathbf{F}-\dot{\bar{X}}_{2}\] \[=-k_{2}(\mathbf{e}_{2}-\mathbf{\xi}_{1})+\dot{f}_{2}+k_{2}\frac{\mathbf{\theta }_{2}}{\omega_{1}}+\mathbf{\phi}_{3}-k_{3}\mathbf{\xi}_{1}+\Delta\mathbf{F}-\epsilon_{2}\]\[\dot{\mathbf{\epsilon}}_{4} =(\mathbf{X}_{5}-\mathbf{X}_{5})+\mathbf{X}_{5}-\mathbf{\hat{X}}_{4}=\mathbf{e}_{5}-k_{4} (\mathbf{\hat{X}}_{4}-\mathbf{X}_{4})=\mathbf{e}_{5}-k_{4}(\mathbf{e}_{4}+\mathbf{\hat{X}}_{4}) \tag{49}\] \[=-k_{4}\mathbf{e}_{4}+\mathbf{e}_{5}+k_{4}\frac{\mathbf{\phi}_{4}}{\omega_{2}^ {2}}\] \[\dot{\mathbf{\epsilon}}_{5} =\mathbf{X}_{6}+f_{5}(\mathbf{X}_{5})-f_{5}(\mathbf{\hat{X}}_{5})-k_{5}(\mathbf{ \hat{X}}_{5}-\mathbf{\bar{X}}_{5}-\mathbf{\bar{\zeta}}_{2})-\mathbf{\hat{X}}_{6}+\mathbf{\hat{X }}_{5f}+\mathbf{\hat{\zeta}}_{2}-\mathbf{\hat{X}}_{5}\] \[=\mathbf{\bar{X}}_{6}+\mathbf{\bar{\zeta}}_{5}-k_{5}(\mathbf{e}_{5}-\mathbf{\bar{ \zeta}}_{2})+k_{5}\mathbf{\bar{X}}_{5}-k_{6}\mathbf{\bar{\zeta}}_{2}+\Delta\mathbf{U}_{2}- \mathbf{e}_{5}\] (50) \[=-k_{5}(\mathbf{e}_{5}-\mathbf{\bar{\zeta}}_{2})+\mathbf{\bar{\zeta}}_{5}+k_{ 5}\frac{\mathbf{\phi}_{5}}{\omega_{2}}+\mathbf{\phi}_{6}-k_{6}\mathbf{\bar{\zeta}}_{2}+ \Delta\mathbf{U}_{2}-\mathbf{e}_{5}\]  
From Equations (47)-(50), the closed-loop system's error dynamics can be written as  
\[ \begin{cases}\dot{\mathbf{\epsilon}}_{1}=-k_{1}\mathbf{e}_{1}+\mathbf{e}_{2}\\ \dot{\mathbf{\epsilon}}_{2}=\mathbf{\phi}_{3}+\mathbf{\bar{f}}_{2}-k_{2}(\mathbf{e}_{2}-\mathbf{ \bar{\zeta}}_{1})+k_{2}\frac{\mathbf{\phi}_{2}}{\omega_{1}}-k_{3}\mathbf{\bar{\zeta}}_ {1}+\Delta\mathbf{F}-\mathbf{e}_{2}\\ \dot{\mathbf{\xi}}_{1}=-k_{3}\mathbf{\xi}_{1}+\Delta\mathbf{F}\\ \dot{\mathbf{\epsilon}}_{4}=-k_{4}\mathbf{e}_{4}+\mathbf{e}_{5}+k_{4}\frac{\mathbf{\phi}_{4}}{ \omega_{2}^{2}}\\ \dot{\mathbf{e}}_{5}=-k_{5}(\mathbf{e}_{5}-\mathbf{\bar{\zeta}}_{2})+k_{5}\frac{\mathbf{\phi}_ {5}}{\omega_{2}}+\mathbf{\bar{f}}_{5}+\mathbf{\phi}_{6}-k_{6}\mathbf{\bar{\zeta}}_{2}+ \Delta\mathbf{U}_{2}-\mathbf{e}_{5}\\ \dot{\mathbf{\xi}}_{2}=-k_{6}\mathbf{\bar{\zeta}}_{2}+\Delta\mathbf{U}_{2}\\ \dot{\mathbf{\vartheta}}=\mathbf{\omega}\mathbf{A}\mathbf{\vartheta}+\mathbf{\omega}_{1}\mathbf{\delta }_{2}(\mathbf{\bar{f}}_{2}+\mathbf{\bar{F}})+\mathbf{\delta}_{3}\mathbf{h}_{p}+\mathbf{\omega}_{2} \mathbf{\delta}_{5}\mathbf{\bar{f}}_{5}+\mathbf{\delta}_{6}\mathbf{h}_{\theta}\end{cases}  \tag{51}\]  
Theorem 2: _Consider the quadrotor dynamics as described in Equation (2), for the given sufficiently smooth trajectory command \(\mathbf{P}_{d}\) and heading angle \(\mathbf{\psi}_{d}\), if the controller gains \(k_{i}\) (\(i=1,...,6\)) and the observer bandwidth parameters \(\mathbf{\omega}_{1}\), \(\mathbf{\omega}_{2}\) are chosen such that the matrix \(\mathbf{\Pi}\) defined in the following is positive definite, then the ESO-based saturated trajectory tracking controller, including the virtual controllers (25), (33), (38), and (46) and the applied controllers (35) and (46), can guarantee the ultimate boundedness of all signals in the closed-loop and make the tracking errors arbitrarily small._  
\[\mathbf{\Pi} = \begin{bmatrix}\mathbf{\Pi}_{1}&\mathbf{\Pi}_{2}\\ \mathbf{\Pi}_{2}^{T}&\mathbf{\Pi}_{3}\end{bmatrix}  \tag{52}\] \[\mathbf{\Pi}_{1} =