# Quadrotor Cascade Control System Design Based on Linear Active Disturbance Rejection Control
## 3 Controller Design
### Adaptive LADRC Inner Loop Control  
LESO is the core part of self immunity to perturbations, for estimating the total perturbation, taking internal and external perturbations and uncertainties as the state variables of the system. Defining \(z= \begin{bmatrix}z_{1}&z_{2}&z_{3}\end{bmatrix} ^{T}\) to estimate the total perturbation of the quadrotor system of the system gives:  
\[\left\{ \begin{array}{c}\dot{z}=H_{1}z+H_{2}u+H_{3}l\\ y=H_{4}z\end{array} \right. \tag{18}\]\[\text{where }H_{1}= \begin{bmatrix}0&1&0\\ 0&0&1\\ 0&0&0\end{bmatrix} ,H_{2}= \begin{bmatrix}0\\ b_{0}\\ 0\end{bmatrix} ,H_{3}= \begin{bmatrix}0\\ 0\\ 1\end{bmatrix} ,H_{4}= \begin{bmatrix}1\\ 0\\ 0\end{bmatrix} .\]  
\[\left\{ \begin{array}{c}\dot{z}=H_{1}z+H_{2}u+\varpi(y-\hat{y})\\ \dot{y}=H_{4}z\end{array} \right. \tag{19}\]  
where \(\varpi= \begin{bmatrix}3\tau_{0}&3\tau_{0}{}^{2}&\tau_{0}{}^{3}\end{bmatrix} ^{T}\), \(\tau_{0}\) is the bandwidth of LESO. Parameterizing the gain of the observer through the characteristic equation gives:  
\[\eta(s)=\left(s+\tau\right)^{3}=s^{3}+\tau_{1}s^{2}+\tau_{2}s+\tau_{3} \tag{20}\]  
Thus, LESO [29] can be defined as:  
\[ \begin{cases}\dot{z}_{1}=\dot{s}_{1}\\ \dot{z}_{2}=\dot{s}_{2}\\ \dot{z}_{3}=u\end{cases}  \tag{21}\]  
Further, the LADRC control rate is defined as:  
\[\text{M}=\text{M}_{0}-\dot{z}_{3}/b_{0} \tag{22}\]  
where  
\[\text{M}_{0}=K_{p}(u_{I}-\dot{z}_{1})+K_{d}\big{(}\dot{u}_{d}-\dot{z}_{2}\big{)} \tag{23}\]  
Define a control rate as follows:  
\[\dot{\chi}=K_{p}(u_{I}-\dot{z}_{1})+K_{d}\big{(}\dot{u}_{d}-\dot{z}_{2}\big{)}- \beta\chi \tag{24}\]  
**Remark 1**. : _In most cases, we use the empirical test method to adjust the parameters of LADRC. In this paper, \(\tau\) is the bandwidth of LESO, which determines the tracking speed of LESO. The larger the \(\tau\), the higher the accuracy of LESO estimation, but too much rwill introduce high frequency noise. \(b_{0}\) is the disturbance compensation, the larger the \(b_{0}\), the greater the disturbance resistance. Usually, good control is obtained by making \(\tau=50,\ b_{0}=1\). For the two parameters \(K_{p}\) and \(K_{d}\) of PD controller, usually we first make \(K_{d}=1\) and gradually increase\(K_{p}\)to obtain the desired response time before adjusting \(K_{d}\)._  
To improve the robustness of the control system, we introduce adaptive control to optimize the parameters \(K_{p}\) and \(K_{d}\) of the PD controller, which also presents a novel approach for proving the stability of the controller. **Theorem 1**. : _Assuming that each signal in the system is bounded and the tracking error converges to zero, the following adaptive rate is used to make the system stable considering the nonlinear system in this paper._  
\[\dot{\hat{K}}_{p}=\vartheta_{p}\big{[}\chi K_{p}(u_{I}-z_{1})\big{]}/\widetilde {K}_{p} \tag{25}\]  
\[\dot{\hat{K}}_{d}=\vartheta_{d}\big{[}\chi K_{d}\big{(}\dot{u}_{d}-z_{2}\big{)} \big{]}/\widetilde{K}_{d} \tag{26}\]  
To solve the problem that the adaptive rate will have zero points, the adaptive rate is rewritten as:  
\[\hat{K}_{p}=\int\limits_{0}^{T}\Bigl{\{}\vartheta_{p}\big{[}\chi K_{p}(u_{I} -z_{1})\big{]}/\widetilde{K}_{p}\Bigr{\}}dt+K_{p0} \tag{27}\]\[\hat{K}_{d}=\int\limits_{0}^{T}\Big{\{}\theta_{d}[\chi K_{d}(u_{d}-z_{2})]/\tilde{ k}_{d}\Big{\}}dt+K_{d0} \tag{28}\]  
where \(\hat{K}_{p}\) and \(\hat{K}_{d}\) as estimates of PD controller gain, \(\theta_{p}\), \(\vartheta_{d}\), \(\chi\) are the appropriate parameters. **Remark 2**. : _Usually, researchers adjust the gain \(K_{p}\) and \(K_{d}\) of the PD controller online by an empirical trial-and-error method, but online adjustment is difficult and may lead to deviations in parameter selection and thus tracking errors. The quadrotor is susceptible to internal and external disturbances during operation, which leads to system instability.