# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 4 Stability Analysis  
### Convergence Analysis of Extended State Observer  
We define the estimation error of ESO as \(\bar{\alpha}_{i}=\alpha_{i}-\bar{\alpha}_{i}\), \(\bar{\beta}_{i}=\beta_{i}-\hat{\beta}_{i}\), \((i=1,2)\). Combining Equations (22), (23), (26), and (27), the dynamic equation of the estimation error system can be obtained as  
\[ \begin{cases}\dot{\bar{x}}_{1}=\dot{\alpha}_{1}-\dot{\bar{x}}_{1}=-l_{1}\bar{ \alpha}_{1}+\bar{\alpha}_{2}-l_{1}(\rho_{1}-\alpha_{1})\\ \dot{\bar{x}}_{2}=\dot{\alpha}_{2}-\dot{\bar{x}}_{2}=-l_{2}\bar{\alpha}_{1}+ \omega_{1}-l_{2}(\rho_{1}-\alpha_{1})\end{cases} , \tag{45}\]\[ \begin{cases}\hat{\beta}_{1}=\hat{\beta}_{1}-\hat{\beta}_{1}=-l_{3}\hat{\beta}_{1}+ \hat{\beta}_{2}-l_{3}(\rho_{2}-\beta_{1})\\ \hat{\beta}_{2}=\hat{\beta}_{2}-\hat{\beta}_{2}=-l_{4}\hat{\beta}_{1}+\omega_{2 }-l_{4}(\rho_{2}-\beta_{2})\end{cases} . \tag{46}\]  
Defining \(\delta_{1}=\rho_{1}-\alpha_{1}\), \(\delta_{2}=\rho_{2}-\beta_{1}\), the system (45), (46) can be written as  
\[ \begin{cases}\dot{\hat{x}}=A_{1}\tilde{x}-B_{1}\delta_{1}+B_{3}\dot{\omega}_{1 }\\ \dot{\hat{\beta}}=A_{2}\tilde{\beta}-B_{2}\delta_{2}+B_{3}\dot{\omega}_{2} \end{cases} , \tag{47}\]  
where  
\[\dot{\hat{x}}= \begin{bmatrix}\dot{\hat{x}}_{1}\\ \dot{\hat{x}}_{2}\end{bmatrix} ,\quad\dot{\hat{\beta}}= \begin{bmatrix}\dot{\hat{ \beta}}_{1}\\ \dot{\beta}_{2}\end{bmatrix} ,\quad A_{1}= \begin{bmatrix}-l_{1}&1\\ -l_{2}&0\end{bmatrix} ,\quad A_{2}= \begin{bmatrix}-l_{3}&1\\ -l_{4}&0\end{bmatrix} \]  
\[B_{1}= \begin{bmatrix}l_{1}\\ l_{2}\end{bmatrix} ,\quad B_{2}= \begin{bmatrix}l_{3}\\ l_{4}\end{bmatrix} ,\quad B_{3}= \begin{bmatrix}0\\ 1\end{bmatrix} .\]  
Before the proof, we give an assumption:  
**Assumption 4**. : _All external disturbances and unmodeled parts of the system and their derivatives are bounded by \(\|\dot{\omega}_{i}\|\leq\overline{\omega}_{i}\), where \(\overline{\omega}_{i}>0\) is a positive constant._  
**Theorem 1**. : _For the variable-load quadrotor system with actuator faults (14), the ESOs are designed as (23) and (27), with event-triggered mechanisms (24) and (28). Proper selection of observer gains \(l_{i}\) (\(i=1,2,3,4\)) guarantees that all estimation errors converge to an arbitrarily small compact set._  
Proof of Theorem 1.: We define the Lyapunov function as  
\[V_{1}=\tilde{\alpha}^{T}P_{1}\tilde{\alpha}+\tilde{\beta}^{T}P_{2}\tilde{ \beta}, \tag{48}\]  
where the matrices \(P_{1}\) and \(P_{2}\) are symmetric positive definite matrices and satisfy \(A_{i}^{T}P_{i}+P_{i}A_{i}=-I\) (\(i=1,2\)). The derivative of \(V_{1}\) is  
\[ \begin{split}\dot{V}_{1}&=\dot{\tilde{x}}^{T}P_{1} \tilde{\alpha}+\tilde{\alpha}^{T}P_{1}\dot{\hat{x}}+\dot{\tilde{\beta}}^{T}P_{ 2}\tilde{\beta}+\tilde{\beta}^{T}P_{2}\tilde{\beta}\\ &=(A_{1}\tilde{\alpha}-B_{1}\delta_{1}+B_{3}\dot{\omega}_{1})^{T} P_{1}\tilde{\alpha}+\tilde{\alpha}^{T}P_{1}(A_{1}\tilde{\alpha}-B_{1}\delta_{1}+B_{3} \dot{\omega}_{1})\\ &+(A_{2}\tilde{\beta}-B_{2}\delta_{2}+B_{3}\dot{\omega}_{2})^{T} P_{2}\tilde{\beta}+\tilde{\beta}^{T}P_{2}(A_{2}\tilde{\beta}-B_{2}\delta_{2}+B_{3} \dot{\omega}_{2})\\ &=\tilde{\alpha}^{T}(A_{1}^{T}P_{1}+P_{1}A_{1})\tilde{\alpha}-2 \tilde{\alpha}^{T}P_{1}B_{1}\delta_{1}+2\tilde{\alpha}^{T}P_{1}B_{3}\dot{ \omega}_{1}\\ &+\tilde{\beta}^{T}(A_{2}^{T}P_{2}+P_{2}A_{2})\tilde{\beta}-2 \tilde{\beta}^{T}P_{2}B_{2}\delta_{2}+2\tilde{\beta}^{T}P_{2}B_{3}\dot{ \omega}_{2}.\end{split}