# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 2 Modeling of Variable-Load Quadrotor
### Cascade Active Disturbance Rejection Controller Based on Particle Swarm Optimization  
Recognizing the critical influence of controller parameters on cascaded ADRC system performance, this study develops an online parameter self-tuning method based on PSO, which effectively enhances the adaptability and robustness of the control strategy. Each cascade sub-controller has six parameters [\(l_{1}\), \(l_{2}\), \(l_{3}\), \(l_{4}\), \(k_{1}\), \(k_{2}\)] to be tuned. Thus, the entire inner-loop attitude and outer-loop position control system has a total of 36 adjustable parameters. The bandwidth method helps simplify parameter tuning. This approach utilizes two important parameters: the observer bandwidth \(\omega_{o}\) and the controller bandwidth \(\omega_{c}\). The parameters can then be systematically determined through the following bandwidth-based relationships:  
\[l_{1}=\omega_{o1}^{2},\quad l_{2}=2\omega_{o1},\quad l_{3}=\omega_{o2}^{2}, \quad l_{4}=2\omega_{o2},\quad k_{1}=\omega_{c}^{2},\quad k_{2}=2\omega_{c}. \tag{37}\]  
within the attitude control loop, the three attitude controllers exhibit control objectives of similar magnitude. Likewise, the three position controllers in the displacement control loop maintain consistent scale. This means that identical parameter sets can be employed for each control loop. The simplification process cuts the total tuning parameters from 36 down to 6. Among these, three parameters handle attitude control while the remaining three manage position control. This reduction maintains effective control performance. The parameters requiring adjustment consist of \(\omega_{oa1}\), \(\omega_{oa2}\), \(\omega_{ca}\), \(\omega_{op1}\), \(\omega_{op2}\), and \(\omega_{cp}\). **Remark 6**. : _In the controller parameterization process of the quadrotor, the six sub-controllers function as a coupled system. The quadrotor exhibits strong inherent coupling between its displacement and rotational attitude. This interdependence limits how individual sub-controllers can be tuned. Consequently, the six sub-controllers cannot be parameterized completely independently. Consider the case where the yaw angle \(\psi=0\). If the attitude controller performs poorly, it may fail to generate adequate roll angle. This deficiency directly impairs tracking performance along the \(x\)-axis. In such cases, optimizing only the \(x\)-channel controller parameters fails to effectively improve tracking performance._  
In PSO, each particle updates its position and velocity according to the following rules:  
\[V_{i}^{t+1}=wV_{i}^{t}+c_{1}r_{1}(P_{besti}-X_{i}^{t})+c_{2}r_{2}(G_{besti}-X_{ i}^{t}), \tag{38}\]\[X_{i}^{t+1}=X_{i}^{t}+V_{i}^{t+1}, \tag{39}\]  
where \(i\) denotes the \(i\)th particle in the swarm, \(V_{i}^{t}\) denotes the velocity of the \(i\)th particle at the \(t\)th iteration, \(X_{i}^{t}\) denotes the position at the \(t\)th iteration, \(P_{best}\) denotes the best position in the history of individual particles, \(G_{best}\) denotes the global best position, \(w\) denotes the inertia weight, \(c_{1}\) and \(c_{2}\) denote acceleration constants, and \(r_{1}\) and \(r_{2}\) are random numbers between 0 and 1. By iteratively updating the particle positions and velocities, the PSO can achieve effective exploration of the solution space. The inertia weight \(w\) is an important parameter that determines the effect of inertia on the historical velocity of the particle when updating the velocity. A linearly decreasing inertia weighting strategy is introduced to dynamically adjust \(w\), which is updated as follows:  
\[w=w_{ini}-\frac{t}{t_{max}}(w_{ini}-w_{fin}), \tag{40}\]  
where \(w_{ini}\) and \(w_{fin}\) are the initial and final values of the inertia weights, respectively; \(t\) denotes the current number of iterations; and \(t_{max}\) denotes the maximum number of iterations. The algorithm dynamically adjusts parameter \(w\) through linear decrement, maintaining balance between global and local exploration throughout the optimization process.