## 3 Parameter Design of Anti Disturbance Voltage Control System
### Parameter Design of Second-Order LADRC System  
From Equations (12) and (13), the transfer functions from controller output \(u\) to system output \(y\), and from controller output \(u\) to system input \(r\) can be obtained:  
\[\frac{U(s)}{Y(s)}=-\frac{(\beta_{1}I_{2}+\beta_{2}I_{1}+\beta_{3})s^{2}+(\beta _{2}I_{2}+\beta_{3}I_{1})s+\beta_{3}I_{2}}{b_{0}s[s^{2}+(\beta_{1}+I_{1})s+ \beta_{1}I_{1}+\beta_{2}+I_{2}]} \tag{14}\]  
\[\frac{U(s)}{R(s)}=\frac{I_{2}(s^{3}+\beta_{1}s^{2}+\beta_{2}s+\beta_{3})}{b_{0 }s[s^{2}+(\beta_{1}+I_{1})s+\beta_{1}I_{1}+\beta_{2}+I_{2}]} \tag{15}\]  
The equivalent two-degree-of-freedom control block diagram of second-order LADRC is shown in Figure 4, where \(C_{1}(s)\) is the feedback controller transfer function, \(C_{2}(s)\) is the tracker transfer function, and \(P(s)\) is the transfer function of the controlled flyback converter. The expressions for \(C_{1}(s)\) and \(C_{2}(s)\) are as follows:  
\[C_{1}(s)=\frac{(\beta_{1}I_{2}+\beta_{2}I_{1}+\beta_{3})s^{2}+(\beta_{2}I_{2} +\beta_{3}I_{1})s+\beta_{3}I_{2}}{b_{0}s[s^{2}+(\beta_{1}+I_{1})s+\beta_{1}I_{ 1}+\beta_{2}+I_{2}]} \tag{16}\]\[C_{2}(s)=\frac{I_{2}(s^{3}+\beta_{1}s^{2}+\beta_{2}s+\beta_{3})}{b_{0}s[s^{2}+( \beta_{1}+I_{1})s+\beta_{1}I_{1}+\beta_{2}+I_{2}]} \tag{17}\]  
For the second-order LADRC system, the feedback gain and observer control adopt a bandwidth tuning method, introducing LESO bandwidth \(\omega_{\rm o}\) and controller bandwidth \(\omega_{\rm c}\), so that  
\[\left\{ \begin{array}{l}I_{1}=2\omega_{\rm c},\\ I_{2}=\omega_{\rm c}^{2},\\ \beta_{1}=3\omega_{\rm o},\\ \beta_{2}=3\omega_{\rm o}^{2},\\ \beta_{3}=\omega_{\rm o}^{3\prime},\end{array} \right. \tag{18}\]  
Define bandwidth center \(\omega_{\rm f}=\sqrt{\omega_{\rm c}\omega_{\rm o}}\), bandwidth ratio \(\alpha=\omega_{\rm o}/\omega_{\rm c}\), and \(\gamma=\sqrt{\alpha}\). Convert Equations (16) and (17) to:  
\[C_{1}(s)=\frac{K_{1}}{\frac{1}{\omega_{1}^{2}}s^{2}+\frac{2\zeta_{1}}{\omega_ {1}}s+1} \tag{19}\]  
\[C_{2}(s)=\frac{\frac{1}{(\gamma\omega_{\rm f})^{3}}s^{3}+\frac{3}{(\gamma \omega_{\rm f})}s^{2}+\frac{3}{\gamma\omega_{\rm f}}s+1}{\frac{1}{\omega_{1}^ {2}}s^{2}+\frac{2\zeta_{1}}{\omega_{1}}s+1} \tag{20}\]  
where  
\[\left\{ \begin{array}{l}K_{1}=\frac{\omega_{\rm f}^{2}\gamma^{3}}{1+6\gamma^ {2}+3\gamma^{4}},\\ \zeta_{1}=\frac{2\gamma^{2}+3}{2\gamma\sqrt{3+6\gamma^{2}+\gamma^{4}}},\\ \zeta_{2}=\frac{\omega_{\rm f}\gamma^{2}}{2\sqrt{1+6\gamma^{2}+3\gamma^{4}}}, \\ \omega_{1}=\frac{\omega_{\rm f}\gamma^{2}}{\sqrt{3+6\gamma^{2}+\gamma^{4}}},\\ \omega_{2}=\frac{\omega_{\rm f}\gamma^{2}+6\gamma^{2}+3\gamma^{4}}{\gamma},\\ \omega_{\rm m}=\sqrt{\omega_{\rm 1}\omega_{\rm 2}}=\sqrt{\frac{1+6\gamma^{2}+3 \gamma^{4}}{3+6\gamma^{2}+\gamma^{4}}}\omega_{\rm f}\end{array} \right. \tag{21}\]  
Under different bandwidth ratios \(\gamma\) and controller bandwidths \(\omega_{\rm c}\), in order to maintain generality, the control variable gain \(b_{0}=1\) is used to obtain the Bode diagram of the equivalent feedback controller transfer function \(C_{1}(s)\) and the tracker transfer function \(C_{2}(s)\) of the system, as shown in Figure 5.  
From Figure 5a,b, it can be seen that controller \(C_{1}(s)\) is a series second-order lead correction unit, so the maximum phase angle of the system is relatively large. Overall, it shows that the system can provide a lead angle, and the phase compensation that can be provided is the smallest when the bandwidth ratio \(\gamma=1\), that is, the controller bandwidth \(\omega_{\rm c}=\omega_{\rm o}\) is the smallest. The larger the difference between \(\omega_{\rm c}\) and \(\omega_{\rm o}\), the greater the phase compensation; the phase angle between \(\gamma\) and \(1/\gamma\) is the same, and the farther away \(\gamma\) is from 1, the greater the phase compensation. The amplitude frequency characteristic curve has two turning frequencies, namely \(\omega<\omega_{\rm f}\) and \(\omega>\omega_{\rm f}\).