# Quadrotor Cascade Control System Design Based on Linear Active Disturbance Rejection Control
## 4 Simulation  
The simulation analysis of quadrotor for multiple trajectory tracking is performed using MATLAB/Simulink. Assume that the initial position is \( \begin{bmatrix}0&0&0\end{bmatrix} ^{T}m\), the initial posture is \( \begin{bmatrix}0&0&0\end{bmatrix} ^{T}rad\). The parameters of the quadcopter system are shown in Table 1, the parameters of the LADRC control system are shown in Table 2, and the parameters of the SMC&A-LADRC control system are shown in Table 3. In order to reduce the parameters that can be adjusted by LADRC, we have given the parameter values of \(r,h,b_{0},\omega_{0}\) by an empirical trial-and-error method. Figure 4: Trajectory tracking curve of a quadrotor UAV in ideal conditions. Figure 5: Position tracking curve of a quadrotor UAV in ideal conditions. (**a–c**) are the tracking curves for each of the three position control channels. The purpose of simulation case 1 is to test whether the control system proposed in this paper is effective. The trajectory tracking curve of a quadrotor in 3D coordinate system is given in Figure 4. It is clear from the figure that the SMC&A-LADRC control scheme proposed in this paper can respond quickly and track the desired signal accurately and steadily. The tracking curves of position channel \(x\), \(y\) and \(z\) are given in Figure 5a-c, respectively. In-depth analysis can be obtained that all three position channels achieve stable tracking at 0.8 s with error control of 0.001% and no significant overshoot. The tracking curves of the attitude channel \(\theta\), \(\phi\) and \(\psi\) are given in Figure 6a-c, respectively, and it can be seen that the control strategy proposed in this paper can be quickly to the input signal, which proves that the SMC&A-LADRC control scheme is reasonable and effective. Figure 6: Attitude tracking curve of a quadrotor UAV in ideal conditions. (**a–c**) are the tracking curves for each of the three attitude control channels. **Simulation Example 2**. : _This simulation example is designed to test the robust performance of SMC&A-LADRC with wind disturbance as shown in Equation (38) and input signal as shown in Equation (37) in reference [30]. The position control response curve under perturbation is shown in Figure 7, and Figure 8 represents the error comparison graph between LADRC and SMC&A-LADRC._  
\[\left\{ \begin{array}{l}\Theta_{x}(t)=10\cos(0.5t-3.0403)+7\sin \left(0.7t+\frac{\pi}{3}\right)\\ \Theta_{y}(t)=6\sin(0.5t-1)+6\cos(0.8t)\\ \Theta_{z}(t)=5\sin(0.6t+3.08)\\ \Theta_{\theta}(t)=13\cos(0.6t-0.4)\\ \Theta_{\phi}(t)=4\sin(0.5t-0.1)+\cos(0.1t-0.65)\\ \Theta_{\psi}(t)=2\sin(0.3t+1)\\ \end{array} \right. \tag{38}\]  
Figure 7: Position control response curves under disturbances. (**a**–**c**) are the tracking curves for each of the three position control channels. The simulation case 2 is designed to test the robustness of SMC&A-LADRC. In order to achieve the UAV flying on the optimal path, strong wind disturbance is added to six channels in this paper. As shown in the analysis of Figure 7, SMC&A-LADRC reaches the equilibrium at 0.8 s and the tracking curve is smooth and fits the desired signal, while LADRC has obvious fluctuations in the process of tracking curve. Figure 8 shows the comparison of the tracking error between SMC&A-LADRC and LADRC position channels, and it is obvious from the figure that the error of SMC&A-LADRC is almost zero after reaching equilibrium, which indicates that the change in strong wind disturbance has little effect on the quadrotor system, and thus shows that the proposed SMC&A-LADRC method has a high robustness, and it can be used in the presence of wind disturbance and internal UAV perturbations; therefore, the quadrotor can be made to fly in the optimal path and achieve energy optimization. Figure 8: Comparison chart of the error of LADRC and SMC&A-LADRC. (**a–c**) are the errors of each of the three position control channels. **Simulation Example 3**.