# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 4 Stability Analysis
### Stability Analysis of Control System  
Theorem 1 ensures the convergence of estimation errors, proving the existence of a compact set, such that  
\[|\tilde{\pi}_{i}|\leq\iota_{a},\quad|\tilde{\beta}_{i}|\leq\iota_{\beta}, \tag{52}\]  
where \(\iota_{a}\) and \(\iota_{\beta}\) are small positive constants. We define the controller tracking error as \(e_{1}=\alpha_{1}-v_{1}\) and \(e_{2}=\beta_{1}-h\), and the unconstrained error variable \(\epsilon\) can be derived through Equation (19). Combining Equations (22), (25), (26), and (30) and the estimation error equations, the dynamics model of the tracking error system can be obtained as  
\[\dot{e}_{1} =\dot{\alpha}_{1}-\dot{\upsilon}_{1}=\alpha_{2}+h-v_{2}\] \[=k_{1}(v_{1}-\dot{\alpha}_{1})+v_{2}-\dot{\alpha}_{2}+\alpha_{2}- v_{2} \tag{53}\] \[=-k_{1}e_{1}+k_{1}\dot{\alpha}_{1}+\dot{\alpha}_{2},\] \[\dot{e}_{2} =\dot{\beta}_{1}-h=\beta_{2}+\sigma-h\] \[=k_{2}(h-\dot{\beta}_{1})-+\dot{h}-\dot{\beta}_{2}+\mathcal{Z} \dot{p}+\beta_{2}-\dot{h}+[U(\tau_{q})-U(t)]\] (54) \[=-k_{2}e_{2}+k_{2}\ddot{\beta}_{1}+\ddot{\beta}_{2}+\mathcal{Z} \dot{p}+\Delta U,\]  
where \(\Delta U=U(\tau_{q})-U(t)\) denotes the finite error introduced by the controller event-triggered mechanism. Theorem 2: _For the variable-payload quadrotor system with actuator faults (14), the ESOs (23) and (27) are designed with event-triggered mechanisms (24) and (28). The control system includes controller (30) with event-triggered mechanisms (34) and (36). With properly selected observer gains \(l_{i}\) (\(i=1,2,3,4\)) and controller gains \(k_{1}\), \(k_{2}\), all tracking errors are guaranteed to remain bounded._  
Proof of Theorem 2.: We define the Lyapunov function as  
\[V_{2}=\frac{1}{2}e_{1}^{2}+\frac{1}{2}e_{2}^{2}+\frac{1}{2}\epsilon^{2}, \tag{55}\]  
where \(e_{1}\) is the position tracking error, \(e_{2}\) is the velocity tracking error, and \(\epsilon\) is the unconstrained error transformed by a prescribed performance function. The derivation of \(V_{2}\) yields  
\[\dot{V}_{2}=e_{1}\dot{e}_{1}+e_{2}\dot{e}_{2}+\epsilon\dot{e}. \tag{56}\]  
From Equation (53), the following is obtained:  
\[e_{1}\dot{e}_{1}=-k_{1}\dot{e}_{1}^{2}+e_{1}(k_{1}\dot{\alpha}_{1}+\dot{\alpha }_{2}). \tag{57}\]From Equation (54), one gets  
\[\dot{e}_{1} =\dot{\alpha}_{1}-\dot{\upsilon}_{1}=\alpha_{2}+h-v_{2}\]0  
From Equation (20), it follows that  
\[\dot{e}_{1} =\dot{\alpha}_{1}-\dot{\upsilon}_{1}=\alpha_{2}+h-v_{2}\]1  
By [PERSON]'s inequality, we obtain  
\[ \begin{split}& e_{1}(k_{1}\tilde{\alpha}_{1}+\tilde{\alpha}_{2}) \leq\frac{e_{1}^{2}}{2\eta_{1}}+\frac{\eta_{1}(k_{1}\tilde{\alpha}_{1}+\tilde {\alpha}_{2})^{2}}{2}\leq\frac{e_{1}^{2}}{2\eta_{1}}+\eta_{1}(k_{1}^{2}a_{ \alpha}^{2}+t_{\alpha}^{2}),\\ & e_{2}(k_{2}\tilde{\beta}_{1}+\tilde{\beta}_{2})\leq\frac{e_{2}^ {2}}{2\eta_{2}}+\frac{\eta_{2}(k_{2}\tilde{\beta}_{1}+\tilde{\beta}_{2})^{2}}{2 }\leq\frac{e_{2}^{2}}{2\eta_{2}}+\eta_{2}(k_{2}^{2}t_{\beta}^{2}+t_{\beta}^{2 }),\\ & e_{2}Z\dot{p}\leq\frac{e_{2}^{2}}{2\eta_{3}}+\frac{\eta_{3}(Z \dot{p})^{2}}{2},\quad e_{2}\Delta U\leq\frac{e_{2}^{2}}{2\eta_{4}}+\frac{\eta _{4}\Delta U^{2}}{2},\\ &\epsilon r(e_{2}-Z\dot{p})\leq\frac{e^{2}}{2\eta_{5}}+\frac{\eta _{5}r^{2}e_{2}^{2}}{2}-\frac{\eta_{5}r^{2}(Z\dot{p})^{2}}{2}.\end{split}  \tag{60}\]  
where \(\eta_{i}(i=1,\cdots,5)\) are all positive constants. Organizing the above equations gives  
\[\dot{V}\leq-\kappa_{1}e_{1}^{2}-\kappa_{2}e_{2}^{2}-\kappa_{3}e^{2}+\Pi, \tag{61}\]  
where  
\[ \begin{split}&\kappa_{1}=k_{1}-\frac{1}{2\eta_{1}},\qquad\kappa _{2}=k_{2}-\frac{1}{2\eta_{2}}-\frac{1}{2\eta_{3}}-\frac{1}{2\eta_{4}}-\frac{ \eta_{5}r^{2}}{2},\qquad\kappa_{3}=r_{z}-\frac{1}{2\eta_{5}}\\ &\Pi=\eta_{1}(k_{1}^{2}a_{\alpha}^{2}+t_{\alpha}^{2})+\eta_{2}(k_ {2}^{2}t_{\beta}^{2}+t_{\beta}^{2})+\frac{\eta_{3}(Z\dot{p})^{2}}{2}+\frac{ \eta_{4}\Delta U^{2}}{2}-\frac{\eta_{5}r^{2}(Z\dot{p})^{2}}{2}.\end{split}