## 3 Improved Active Disturbance Rejection Control Strategy Design
### Design of an Auto Disturbance Rejection Controller  
#### 3.1.1 The TD Design  
The second-order nonlinear tracking differentiator (TD) system is employed to extract the desired electrical angle signal and the other anticipated mutation signals. This approach allows for a well-organized transition process. The desired electrical angle signal can be effectively filtered, leading to the extraction of a noise-suppressed version of the signal. The discrete expression for this second-order nonlinear TD system is as follows:  
\[\left\{\begin{array}{l}fh=than(x_{1},x_{2},r,h)\\ x_{1}(k+1)=x_{1}(k)+Tx_{2}(k)\\ x_{2}(k+1)=x_{2}(k)+T\cdot fh\end{array}\right. \tag{9}\]  
Figure 4: Block diagram of the improved ADRC.  
In the formula, \(\varepsilon\) is the error variable of the input; \(\lambda\) is the linear interval width of the _fall_ function, which is related to the range of system error. \(\alpha\) is the power of the exponential function. When the lumped disturbance \(\bar{d}=0\), the third-order nonlinear extended state observer is asymptotically stable near the zero point of the equilibrium point, and the differential equation of Equation (15) can be transformed into the following form:  
\[\dot{\widetilde{\varepsilon}}=-\hat{A}(\widetilde{\epsilon})\widetilde{ \varepsilon}=-\begin{bmatrix}\beta_{1}&-1&0\\ \beta_{2}\theta&0&-1\\ \beta_{3}\theta&0&0\end{bmatrix}\begin{bmatrix}\widetilde{\varepsilon}_{1} \\ \widetilde{\varepsilon}_{2}\\ \widetilde{\varepsilon}_{3}\end{bmatrix} \tag{15}\]  
We reference Definition [24]. If there is a matrix \(D\) whose main diagonal elements are all positive so that the matrix \(D\hat{A}(\widetilde{\varepsilon})\) is symmetric positive definite, then the zero solution for the observer error equation of Equation (14) is Lyapunov asymptotically stable. Take matrix \(D\) as  
\[D=\begin{bmatrix}d_{11}&d_{12}&d_{13}\\ -d_{12}&d_{22}&d_{23}\\ -d_{13}&-d_{23}&d_{33}\end{bmatrix} \tag{16}\]  
According to Formulas (15) and (16), we can obtain  
\[\mathrm{D}\hat{A}(\widetilde{\varepsilon})=\begin{bmatrix}D_{11}&-d_{11}&d_{1 2}\\ D_{21}&d_{12}&-d_{22}\\ D_{31}&d_{13}&d_{23}\end{bmatrix} \tag{17}\]  
\[\begin{cases}D_{11}=d_{11}\beta_{01}+d_{12}\beta_{02}\delta_{1}+d_{13}\beta_{ 3}\delta_{01}\\ D_{21}=-d_{12}\beta_{01}+d_{22}\beta_{02}\delta_{1}+d_{23}\beta_{3}\delta_{01} \\ D_{31}=-d_{13}\beta_{01}-d_{23}\beta_{02}\delta_{1}+d_{33}\beta_{3}\delta_{01} \end{cases} \tag{18}\]  
To simplify the derivation of stability, under the condition that the main diagonal elements of matrix \(D\) are positive, the conditions are set as follows:  
\[\begin{cases}d_{11}=1\\ d_{22}=d_{33}=\sigma\end{cases} \tag{19}\]  
Formula (19) is organized as follows:  
\[d_{23}=\frac{1+\varpi\beta_{01}^{2}+\varpi\beta_{01}\beta_{03}\vartheta+\varpi \beta_{02}\vartheta}{(\beta_{01}\beta_{02}-\beta_{03})\vartheta} \tag{20}\]  
In the above formula, \(\varpi\) is a positive number to be determined and is greater than 0, \(\vartheta=\mathit{f}al(\widetilde{\varepsilon}_{1})/\widetilde{\varepsilon}_{1}\). It is known from Formula (20) above that when the gain coefficient of the observer satisfies the stability condition (\(\beta_{01}\ \beta_{02}-\beta_{03}\)) > 0, the symmetric matrix \(D\hat{A}(\widetilde{\varepsilon})\) is positive definite, and the zero solution of the matrix is Lyapunov asymptotically stable.