# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 2 Modeling of Variable-Load Quadrotor
### Design of an Event-Triggered Cascade Active Disturbance Rejection Controller
, \tag{28}\]  
where \(\gamma_{z2}\) is designed as a fixed value. Due to the event triggered mechanism, the trajectory tracking error for the \(z\) channel is defined as  
\[e_{z}=\rho_{z1}-z_{d}. \tag{29}\]  
According to the virtual control variable \(h_{z}\) and tracking error, the virtual control variable \(U_{z}\) is designed as follows:  
\[U_{z}=k_{z2}(h_{z}-\hat{\beta}_{z1})-\hat{\beta}_{z2}+Z\rho_{z}, \tag{30}\]  
where \(k_{z2}\) is the controller parameter to be tuned. **Remark 2**. : _The cascaded structure decouples the system into two subsystems, enabling targeted disturbance rejection. The outer-loop ESO (23) estimates position-related disturbances, while the inner-loop ESO (27) handles velocity-level uncertainties. The dual-loop design enables optimal balance between transient response and noise suppression by adjusting bandwidth. It overcomes the inherent limitation of conventional single-ESO systems, which must find a compromise between dynamic performance and disturbance rejection. This approach ultimately improves the overall system robustness._The control design of the \(x\) and \(y\) channels in the position controller is similar to that of the \(z\) channel. The sampled values of virtual control variables \(U_{x}\), \(U_{y}\), and \(U_{z}\) and yaw angle \(\psi\) at the event-triggered instant \(t_{p}^{\psi}\) are utilized to compute the desired roll/pitch angles and control command \(U_{1}\). \[\phi_{d}=\arctan(\cos\theta(t_{q}^{\theta})\frac{U_{x}\sin\psi(t_{p}^{\psi})-U _{y}\cos\psi(t_{p}^{\psi})}{U_{z}}), \tag{31}\]  
\[\theta_{d}=\arctan(\frac{U_{x}\cos\psi(t_{p}^{\psi})+U_{y}\sin\psi(t_{p}^{\psi })}{U_{z}}), \tag{32}\]  
\[U_{1}=\frac{U_{z}}{cos\phi(t_{q}^{\psi})cos\theta(t_{q}^{\theta})}. \tag{33}\]  
The variables \(\sigma_{x},\sigma_{y}\), and \(\sigma_{z}\) are calculated based on the sampled value of the control variable \(U_{1}(t)\) at the event-triggered instant \(\tau_{q}^{1}\). The controller event-triggered mechanism is designed to  
\[ \begin{cases}\sigma_{1}(t)=U_{1}(\tau_{q}^{1}),\quad t\in[\tau_{q}^{1},\tau_ {q+1}^{1})\\ \sigma_{x}(t)=\sigma_{1}(t)(\cos\psi(t_{p}^{\psi})\sin\theta(t_{p}^{\theta}) \cos\phi(t_{p}^{\phi})+\sin\psi(t_{p}^{\psi})\sin\phi(t_{p}^{\phi}))\\ \sigma_{y}(t)=\sigma_{1}(t)(\sin\psi(t_{p}^{\psi})\sin\theta(t_{p}^{\theta}) \cos\phi(t_{p}^{\phi})-\cos\psi(t_{p}^{\psi})\sin\phi(t_{p}^{\phi}))\\ \sigma_{z}(t)=\sigma_{1}(t)cos\phi(t_{p}^{\psi})cos\theta(t_{p}^{\theta})\\ \tau_{q+1}^{1}=\inf\biggl{\{}t>\tau_{q}^{1}\biggr{|}\left|U_{1}(t)-\sigma_{1 }(t)\right|>M_{1}^{1}e^{-M_{2}^{1}\hat{\epsilon}_{t}}\biggr{\}}\end{cases} , \tag{34}\]  
where \(M_{1}^{1}\) denotes the threshold extremum factor and \(M_{2}^{1}\) represents the error sensitivity factor. The controller trigger threshold is designed as a time-varying value related to the error, and the trajectory estimation error \(\hat{\epsilon}_{t}\) is introduced into the design. \[\hat{\epsilon}_{t}=\sqrt{(\hat{\alpha}_{x1}-x_{d})^{2}+(\hat{\alpha}_{y1}-y_{ d})^{2}+(\hat{\alpha}_{z1}-z_{d})^{2}}. \tag{35}\]  
**Remark 3**. : _This study integrates trajectory tracking error into the event-triggered mechanism of the controller, employing a time-varying threshold that dynamically adjusts according to error magnitude. As tracking error grows, the triggering threshold decreases while update frequency rises. Conversely, diminishing error leads to increased threshold values and reduced triggering frequency._  
**Remark 4**. : _For the quadrotor with strongly coupled dynamics, trajectory tracking error should be computed comprehensively in the 3D space rather than independently. In the error equation (35), the controller cannot obtain the actual state values in real time due to the presence of event-triggered mechanisms. Therefore, the estimated state of the ESO is used for the calculation of the event-trigger mechanisms (24) and (28) and the trajectory tracking error._  
_This design stems from two key considerations.