## 4 Output Current Control
### PI Controller Design  
Commonly, the sample period is short and the time constant of the controlled plant is much larger than the sample period. Therefore, the digital PI controller can be designed in the s-domain. Figure 7 shows the output current controller model in the s-domain. The PI regulator \(G_{c}(s)\) can be written as follows:  
\[G_{c}(s)=K_{p}+\frac{K_{i}}{s}. \tag{13}\]  
Figure 6: Digital control model of the output current controller. Figure 7: Output current controller model in the s-domain. After the current is sampled, a sampling period delay \(T_{sp}\) exists, which is shown in Figure 4. Therefore, the calculation and transport delay can be written as follows:  
\[G_{dc}(s)=e^{-T_{sp}s}, \tag{14}\]  
where the sampling period \(T_{sp}=T_{s}/2\). In the s-domain, the sampling switch can be modeled as \(1/T_{sp}\)[24]. The modulator can be modeled by zero-order holder (ZOH) [25], which can be expressed as:  
\[H(s)=\frac{1-e^{-T_{sp}s}}{s}\approx T_{sp}e^{-0.5T_{sp}s}. \tag{15}\]  
Commonly, the cutoff frequency of the LC filter is larger than the bandwidth frequency of the ARSI. Therefore, the filter inductance is much lower than the load inductance, namely \(L_{f}<<L_{o}\). The output impedance can be written as follows:  
\[G_{1}(s)=\frac{i_{cf}(s)}{v_{o}(s)}=\frac{s^{2}L_{o}C_{f}+sR_{o}C_{f}}{s^{3}L_ {f}C_{f}L_{o}+s^{2}L_{f}C_{f}R_{o}+s(L_{o}+L_{f})+R_{o}}\approx\frac{sC_{f}}{ s^{2}L_{f}C_{f}+1} \tag{16}\]  
\[G_{2}(s)=\frac{i_{o}(s)}{i_{cf}(s)}=\frac{1}{s^{2}L_{o}C_{f}+sR_{o}C_{f}}. \tag{17}\]  
The system open-loop transfer function can be calculated as follows:  
\[G(s)=\frac{i_{o}(s)}{i_{o}^{s}(s)-i_{o}(s)}=(K_{p}+\frac{K_{i}}{s})\cdot\frac {K_{pwm}e^{-1.5T_{sp}s}}{(s^{2}L_{f}C_{f}+1)(sL_{o}+R_{o})+sK_{pwm}K_{cf}e^{- 1.5T_{sp}s}C_{f}(sL_{o}+R_{o})}. \tag{18}\]  
The closed-loop transfer function can be written as follows:  
\[T(s)=\frac{i_{o}(s)}{i_{o}^{s}(s)}=\frac{G_{dc}(s)G_{c}(s)G_{P}(s)}{1+K_{cf} G_{dc}(s)G_{T}(s)+G_{dc}(s)G_{c}(s)G_{P}(s)}. \tag{19}\]  
In order not to reduce the bandwidth of the ARSI, the cutoff frequency of the LC filter is designed to be two times larger than the system bandwidth\(f_{c}\). Therefore, the influence of the filter capacitor can be ignored when calculating the magnitude of the loop gain at \(f_{c}\) and frequencies lower than \(f_{c}\). The low-frequency controller characteristics will still be dominated by the series filter inductance. Thus, the transfer Equation (18) can be approximated as follows:  
Figure 6:0  
Figure 8 shows the Bode plots of the approximation model (20) and the precise model (18). It demonstrates that the cutoff frequency of the system will not be influenced by using the approximation model. When the crossover frequency and phase margin are specified as \(\omega_{c}\) and \(\varphi_{m}\) respectively, the parameters of the PI controller can be obtained:  
\[\left\{ \begin{array}{l} K_{p}=\dfrac{\sqrt{\left(L_{o}\omega_{c }\right)^{2}+R_{o}^{2}}}{ K_{pvm}\sqrt{1+\dfrac{1}{\tan^{2}(-\dfrac{\pi}{2}+1.5\omega_{c}T_{sp}+ \phi_{m}+\arctan\dfrac{\omega_{c}L_{o}}{R_{o}})}}\\ K_{i}=\dfrac{\omega_{c}K_{p}}{\tan(-\dfrac{\pi}{2}+1.5\omega_{c }T_{sp}+\phi_{m}+\arctan\dfrac{\omega_{c}L_{o}}{R_{o}})}\end{array} \right.. \tag{21}\]  
If \(K_{p}>>K_{i}/\omega_{c}\), Equation (21) can be approximated as:  
\[\left\{ \begin{array}{l} K_{p}=\dfrac{\sqrt{\left(L_{o}\omega_{ c} \right)^{2}+R_{o}^{2}}}{ K_{pvm}}\\ K_{i}=\dfrac{\omega_{c}K_{p}}{\tan(-\dfrac{\pi}{2}+1.5\omega_{c }T_{sp}+\phi_{m}+\arctan\dfrac{\omega_{c}L_{o}}{R_{o}})}\end{array} \right.. \tag{22}\]  
Figure 9 shows the Bode plot of the ARSI with different PI parameters. The circuit parameters are listed in Table 2. When \(K_{p}>>K_{i}/\omega_{c}\), the crossover frequency \(\omega_{c}\) is determined by \(K_{p}\). The larger \(K_{p}\), the larger \(\omega_{c}\) and the faster the dynamic response. Regarding \(K_{i}\), it will influence the low-frequency gain, which can influence the stable precision of the system.