## 4 Control Methodology
### BP Neural Network Design  
The continuous time-varying wind field disturbance and the continuous reduction of the load mass will cause the control effect of the controller to deteriorate. LESO observes the system status and the total disturbance. However, in the face of continuous time-varying wind field disturbance and changing load quality, the fixed observer bandwidth will lead to larger observation errors and worse control effects. To solve this problem, this research employs BP neural network (BP) to adjust the bandwidth gain value of LESO online, enhance the observation performance of LESO, and compensate for adverse effects of wind disturbances and changes in load mass. **Remark 3**. : _LESO can observe each state variable and total disturbance of the quadrotor system according to the input and output of the system. However, due to the discretization of the observer, the bandwidth value of the LESO cannot exceed the sampling period of the system, otherwise it will lead to the divergence of the LESO observations. If the bandwidth value is small, the LESO observation accuracy will be low. Therefore, we introduce APC to compensate the observation error caused by bandwidth selection by dynamically adjusting the \(k_{p}\) and \(k_{d}\) values in the PD controller, so as to improve the control accuracy of LADRC and the dynamic performance of the system. This approach will make LADRC more adaptable and extensive._  
The structure of BP is shown in Figure 3.  
The output error is defined as  
\[e_{2}=x_{d}-x_{1}, \tag{27}\]  
where \(x_{d}\) is a given signal, \(x_{1}\) is the state variable of the system; derivation of \(e_{2}\) can be obtained in the following way:  
\[\dot{e}_{2}=\dot{x}_{d}-\dot{x}_{1}=\dot{x}_{d}-x_{2}. \tag{28}\]  
The input of BP neural network can be expressed as  
\[IN= \begin{bmatrix}e_{2}&\dot{e}_{2}\end{bmatrix} ^{T}. \tag{29}\]  
The input and output of the hidden layer are  
\[ \begin{cases}&H_{a}^{in}=\sum\limits_{a=1}^{6}\dot{f}_{a}^{in}\cdot IN\\ &H_{a}^{out}=\tan sig\big{(}H_{a}^{in}\big{)}\end{cases} \quad a=1,2,3,4,5,6. \tag{30}\]  
Figure 3: The structure of BP. The hidden unit is 6, and the excitation function adopts the tangent sigmoid transfer function:  
\[\tan sig(k)=\frac{2}{e^{-2k}+1}-1. \tag{31}\]  
**Remark 4**. : _BP neural network adjusts the bandwidth gain vector \(\left[ \begin{matrix}\beta_{1}&\beta_{2}&\beta_{3}\end{matrix} \right]\) of LESO in real time according to the error information of the system. When the system produces output error, the BP neural network will output and adjust \(\left[ \begin{matrix}\beta_{1}&\beta_{2}&\beta_{3}\end{matrix} \right]\) according to the system error information to reduce the observation error of LESO, thereby reducing the system output error and improving the control accuracy of the system._  
The BP neural network that adjusts the LESO bandwidth gain value is divided into three layers of network structure; the input signal of the BP is output error \(e_{2}\), the differential signal \(\dot{e}_{2}\) of the output error and the bias value \(k\). The number of neurons in the hidden layer is 6 to ensure the strong ability of BP neural network to approximate the error. The output layer contains three neurons, corresponding to the gain vector of LESO. The output of BP neural network can be expressed by the following Formula (34):  
\[y(k)=f\left(\sum_{i=1}^{l}\omega_{ik}g\left(\sum_{m=1}^{M}\omega_{mi}\upsilon_ {m}\right)\right). \tag{32}\]  
Since the BP neural network will normalize the input signal \(\left[ \begin{matrix}e_{2}&\dot{e}_{2}\end{matrix} \right]\), its output signal will also be mapped to the range \(\left(-1\quad 1\right)\), so the output of the BP cannot be directly used as the bandwidth gain value of LESO. To make the BP\(-\)ALADRC controller converge, the output is denormalized to meet the requirement of the gain vector \(\left[ \begin{matrix}\beta_{1}&\beta_{2}&\beta_{3}\end{matrix} \right]\) of LESO in the BP\(-\)ALADRC controller.