## 3 Improved Active Disturbance Rejection Control Strategy Design
### ADRC Regulation Design after the RBF Neural Network  
Active disturbance rejection control employs a nonlinear combination of error feedback to influence the input control variables, significantly impacting the controller's performance. By applying an intelligent control tuning method to the parameters of the PID controller, we introduce a neural network into the active disturbance rejection controller. Consequently, we design an active disturbance rejection controller based on this neural network approach. As illustrated in Figure 5, we utilize the radial basis function (RBF) neural network in this study due to its stability, reliability, and versatility. The three parameters (\(\beta_{01}\), \(\beta_{02}\), and \(\beta_{03}\)) in the extended state observer of the active disturbance rejection controller are optimized online, capitalizing on the RBF neural network's performance characteristics, including its rapid convergence speed and superior approximation. The radial basis function outputs the Gaussian function, which is expressed as follows:  
\[J_{j}=\exp\left(-\frac{\|I-D_{j}\|^{2}}{2\mu_{j}^{2}}\right)j=1,2,\cdots n \tag{23}\]  
In Formula (23), \(I=[I_{1},I_{2},\cdots,I_{n}]^{T}\) is the input vector, and \(D_{j}=[d_{11},d_{j2},\cdots d_{jn}]^{T}\) is the center vector of the \(j\) node (hidden neuron) in the network. \(\|\bullet\|\) denotes the distance between the input vector and the center vector and the distance between the input neuron and the Gaussian center vector in the hidden layer. \(\mu_{j}\) is the basis width parameter of the \(j\) hidden neuron node. The smaller \(\mu_{j}\) is, the smaller the radial basis width is, and the more selective the basis function is. In the control of a permanent magnet synchronous motor, the input vector of the identification neural network is  
\[I=[i_{q}(k),\theta(k),\theta(k-1)] \tag{24}\]  
Figure 5: ADRC observer parameter adaptive adjustment design structure. In addition, the performance index function is chosen to describe the size of the approximation error:  
\[E(k)=\frac{1}{2}e_{m}^{2}(k)=\frac{1}{2}[\theta_{e}(k)-\theta_{r}(k)]^{2} \tag{25}\]  
To speed up the convergence of \(E(k)\), a momentum term is introduced, \(\alpha\) is the momentum factor, and \(\eta\) is the learning rate, ranging from 0 to 1. Additionally, the iterative algorithm calculates the output weight, the node center value, and the node base width parameter of the network using the gradient descent method, as outlined below:  
1. Network output weighting quantity The network output weighting is \[O_{j}(k)=O_{j}(k-1)+\eta(\rho-\rho_{m})J_{j}+\alpha\big{(}O_{j}(k-1)-O_{ij}(k- 2)\big{)}\] (26) Among these, \(\rho_{m}\) is the neural network's output, and \(\rho\) is the actual rotation angle of the motor. 2. Node base width parameters The node base width parameter is \[\left\{ \begin{array}{l}\Delta\mu_{j}(k)=\eta[\rho(k)-\rho_{m}(k)]O_{j}j_{j} \frac{|l-D_{j}|^{2}}{\mu_{j}^{2}}\\ \mu_{j}(k)=\mu_{j}(k-1)+\Delta\mu_{j}(k)+\alpha[\mu_{j}(k-1)-\mu_{j}(k-2)] \end{array} \right.\] (27)
3. Node center value The node center value is \[\left\{ \begin{array}{l}\Delta d_{i}(k)=\eta[\rho(k)-\rho_{m}(k)]O_{j}\frac{ l-D_{j}}{\mu_{j}^{2}}\\ \mu_{j}(k)=\mu_{j}(k-1)+\Delta\mu_{j}(k)+\alpha[\mu_{j}(k-1)-\mu_{j}(k-2)] \end{array} \right.\] (28) The algorithm for the Jacobian matrix is \[\frac{\partial\rho(k)}{\partial i_{q}(k)}\approx\frac{\partial\rho_{m}(k)}{ \partial i_{q}(k)}=\sum_{j=1}^{m}O_{j}j_{j}\frac{d_{ji}-l(1)}{\mu_{j}^{2}}\] (29)  
The gradient descent method is employed to adjust the correction parameters \(\beta_{01}\), \(\beta_{02}\), and \(\beta_{03}\) in the ESO, and the control error is defined as \(e_{rr}(k)=\theta(k)-\theta_{m}(k)\), which are derr (the error differential), err (error), and dderr (the differential of the error differential), \(ee=[de_{rr},e_{rr},dde_{rr}]\):  
\[\left\{ \begin{array}{l}ee(1)=e_{rr}(k)-e_{rr}(k-1)\\ ee(2)=e_{rr}(k)\\ ee(3)=e_{rr}(k)-2e_{rr}(k-1)+e_{rr}(k-2)\end{array}