# Quadrotor Cascade Control System Design Based on Linear Active Disturbance Rejection Control
## 3 Controller Design
### Adaptive LADRC Inner Loop Control
Therefore, this paper adopts an adaptive control method to adjust the gain \(K_{p}\) and \(K_{d}\) of the PD controller in time, which compensates the tracking error caused by the deviation and disturbance of the controller parameters, enhances the robustness of the controller, and also contributes to the stability of the position control and attitude control of the whole quadrotor UAV through adaptive control._  
**Proof of Theorem 1. ** Define a positive definite Lyapunov function as follows:  
\[V(t)=\frac{1}{2}S_{l}^{2}+\frac{1}{2}\chi^{2}+\frac{1}{2}\tilde{K}_{p}\theta_{ p}^{-1}\tilde{K}_{p}+\frac{1}{2}\tilde{K}_{d}\theta_{d}^{-1}\tilde{K}_{d} \tag{29}\]  
Derivation of Equation (29) gives:  
\[\dot{V}(t)=S_{l}\dot{S}_{l}+\chi\dot{\chi}+\widetilde{K}_{p}\theta_{p}^{-1} \dot{\widetilde{K}}_{p}+\widetilde{K}_{d}\theta_{d}^{-1}\dot{\widetilde{K}}_{d} \tag{30}\]  
Bringing Equations (14) and (24) into Equation (30) gives:  
\[ \begin{array}{rl}\dot{V}(t)&=S_{l}\Big{(}\alpha_{1}\dot{\delta}_{1}+\dot{ \delta}_{2}\Big{)}+\chi\big{[}K_{p}(u_{l}-\dot{z}_{1})+K_{d}(u_{d}-\dot{z}_{2} )-\beta\chi\big{]}+\widetilde{K}_{p}\theta_{p}^{-1}\dot{\widetilde{K}}_{p}+ \widetilde{K}_{d}\theta_{d}^{-1}\dot{\widetilde{K}}_{d}\\ &=S_{l}\Big{(}\alpha_{1}\dot{\delta}_{1}+\dot{\delta}_{2}\Big{)}+\chi K_{p}(u_ {l}-\dot{z}_{1})+\chi K_{d}(u_{d}-\dot{z}_{2})-\beta\chi^{2}+\widetilde{K}_{p} \theta_{p}^{-1}\dot{\widetilde{K}}_{p}+\widetilde{K}_{d}\theta_{d}^{-1}\dot{ \widetilde{K}}_{d}\\ &=S_{l}\Big{(}\alpha_{1}\dot{\delta}_{1}+\ddot{z}_{1}-\ddot{u}_{d}\Big{)}+ \chi K_{p}(u_{l}-\dot{z}_{1})+\chi K_{d}(u_{d}-\dot{z}_{2})-\beta\chi^{2}+ \widetilde{K}_{p}\theta_{p}^{-1}\dot{\widetilde{K}}_{p}+\widetilde{K}_{d} \theta_{d}^{-1}\dot{\widetilde{K}}_{d}\\ &=S_{l}\Big{(}\alpha_{1}\dot{\delta}_{1}+z_{3}+u_{l}-\ddot{u}_{d}\Big{)}+ \chi K_{p}(u_{l}-\dot{z}_{1})+\chi K_{d}(u_{d}-\dot{z}_{2})-\beta\chi^{2}+ \widetilde{K}_{p}\theta_{p}^{-1}\dot{\widetilde{K}}_{p}+\widetilde{K}_{d} \theta_{d}^{-1}\dot{\widetilde{K}}_{d}\end{array}  \tag{31}\]  
Bringing Equation (17) into (31) gives:  
\[\begin{array}{rl}\dot{V}(t)&=S_{l}\Big{(}\alpha_{1}\dot{\delta}_{1}+z_{3}- \alpha_{1}\delta_{1}-\alpha_{2}\dot{\delta}_{1}-z_{3}+\ddot{u}_{d}-\ddot{u}_{d }\Big{)}+\chi K_{p}(u_{l}-\dot{z}_{1})+\chi K_{d}(u_{d}-\dot{z}_{2})\\ &-\beta\chi^{2}+\widetilde{K}_{p}\theta_{p}^{-1}\dot{\widetilde{K}}_{p}+ \widetilde{K}_{d}\theta_{d}^{-1}\dot{\widetilde{K}}_{d}\\ &=-\alpha_{1}S_{l}^{2}+\chi K_{p}(u_{l}-\dot{z}_{1})+\chi K_{d}(u_{d}-\dot{z}_{ 2})-\beta\chi^{2}+\widetilde{K}_{p}\theta_{p}^{-1}\dot{\widetilde{K}}_{p}+ \widetilde{K}_{d}\theta_{d}^{-1}\dot{\widetilde{K}}_{d}\\ &\leq-\alpha_{1}S_{l}^{2}+\chi K_{p}(u_{l}-\dot{z}_{1})+\chi K_{d}(u_{d}-\dot{ z}_{2})-\beta\chi^{2}-\widetilde{K}_{p}\theta_{p}^{-1}\dot{\widetilde{K}}_{p}- \widetilde{K}_{d}\theta_{d}^{-1}\dot{\widetilde{K}}_{d}\\ &\leq-\alpha_{1}S_{l}^{2}+\chi K_{p}(u_{l}-\dot{z}_{1})+\chi K_{d}(u_{d}-\dot{ z}_{2})-\beta\chi^{2}-\chi K_{p}(u_{l}-z_{1})-\chi K_{d}(\dot{u}_{d}-z_{2})\\ &\leq-\alpha_{1}S_{l}^{2}-\beta\chi^{2}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\From Equation (35), we have:  
\[\dot{V}(t)<0 \tag{36}\]  
Therefore, according to Barbalat's Theorem it can be judged that the system is asymptotically stable and the system stability is proved. \(\square\)