# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 2 Modeling of Variable-Load Quadrotor
### Prescribed Performance Function and Error Transformation  
Actuator faults are considered part of the total disturbance in the ADRC framework. The ESO estimates and compensates for these faults along with other disturbances. To enhance the disturbance rejection capability while mitigating initial overshoot, prescribed performance functions are introduced to further optimize control performance.  
Consider the altitude \(z\)-channel as an example. A monotonically decreasing performance boundary function \(p_{z}(t)\) is defined to constrain the transient tracking error \(e_{z}=z-z_{d}\). The boundary function is  
\[p_{z}(t)=(p_{z0}-p_{z0})e^{-\mu_{z}t}+p_{z0}, \tag{15}\]  
where \(p_{z0}>0\) is the initial value of the prescribed performance function and \(p_{z0}=\lim_{t\to\infty}p_{z}(t)>0\) is the steady state value of the prescribed performance function, \(p_{z0}>p_{z0}\). The \(p_{z0}\) directly determines the ultimate precision of the system. The \(\mu_{z}\) is the rate factor for the speed of convergence, which determines how quickly the error boundary decays from its initial to steady-state value. A larger \(\mu_{z}\) accelerates convergence but may cause high-frequency oscillations. A smaller \(\mu_{z}\) produces smoother responses but extends transition time.  
If the system error \(e_{z}\) is satisfied,  
\[-\underline{\lambda_{z}}p_{z}(t)<e_{z}<\overline{\lambda_{z}}p_{z}(t), \tag{16}\]  
then the prescribed performance is satisfied. Here, \(\underline{\lambda_{z}},\overline{\lambda_{z}}\in(0,1]\) are the error boundary parameters to be designed. These parameters control the allowable range of tracking errors. A larger \(\lambda_{z}\) permits greater instantaneous errors but may reduce steady-state accuracy. A smaller \(\lambda_{z}\) imposes stricter error constraints but may increase control effort.  
The constrained transient error \(e_{z}\) is transformed into an unconstrained error variable \(\epsilon_{z}\) through conversion mapping. The error transformation function is designed as follows:  
\[e_{z}(t)=p_{z}(t)S_{z}(\epsilon_{z}), \tag{17}\]  
where \(S_{z}(\epsilon_{z})\) is a smooth, strictly monotone decreasing function satisfying \(S_{z}(-\infty)=-\underline{\lambda_{z}}\), \(S_{z}(+\infty)=\overline{\lambda_{z}}\). The expression for \(S_{z}(\epsilon_{z})\) is  
\[S_{z}(\epsilon_{z})=\frac{\overline{\lambda_{z}}e^{\epsilon_{z}}-\underline{ \lambda_{z}}e^{-\epsilon_{z}}}{e^{\epsilon_{z}}+e^{-\epsilon_{z}}}. \tag{18}\]  
Figure 4: A dual-loop control framework based on event-triggered mechanism for the variable-load quadrotor.  
We then define \(Z=\dfrac{e_{z}}{p_{z}}\), and, by calculating the inverse function of Equation (18), one can obtain an expression for \(e_{z}\) as  
\[e_{z}=\dfrac{1}{2}\ln\left(\dfrac{Z+\lambda_{z}}{\overline{\lambda_{z}}-Z}\right). \tag{19}\]  
The derivation of \(e_{z}\) leads to  
\[\dot{e}_{z}=r_{z}(\dot{e}_{z}-Z\dot{p}_{z}), \tag{20}\]  
where \(r_{z}=\dfrac{\overline{\lambda}_{z}+\underline{\lambda}_{z}}{2p_{z}(Z+ \underline{\lambda}_{z})(\overline{\lambda}_{z}-Z)}\).