### Structure of the Three-Phase Grid-Connected Inverter Based on an LCL Filter and Active Modeling
By employing Pade approximation, as proposed, the LCL filter can be approximated to the same order as an L filter (a first-order system) for control design. The Pade approximant \(P_{N,M}(x)\) of a function \(f(x)\) is defined as a ratio of two polynomials with degrees \(M\) and \(N\), and it matches the Taylor series expansion of the function up to degree \(N\)+\(M\). Ignoring the grid voltage, the transfer function of the LCL filter from the inverter output voltage \(v_{\text{inv}}\) to the injected grid current \(i_{g}\) is as follows:  
\[H_{LCL}(s)=\frac{i_{g}}{v_{\text{inv}}}=\frac{1}{L_{c}L_{g}C_{f}s^{3}+\Big{(}r _{c}L_{g}C_{f}+r_{g}L_{c}C_{f}\Big{)}s^{2}+\Big{(}L_{c}+L_{g}+r_{c}r_{g}C_{f} \Big{)}s+r_{c}+r_{g}} \tag{5}\]  
The first-order Pade approximant \(P_{0,1}(x)\) in Formula (5), as given by [PERSON] et al. [40], can be equivalent to the control design model in the \(dq\) reference frame, represented by the Pade approximant \(P_{0,1}(x)\) using expression (6):  
\[H_{Pad}(s)=\left(\frac{i_{g}}{v_{inv}}\right)_{Pad}=\frac{1}{\Big{(}L_{c}+L_{g }+r_{c}r_{g}C_{f}\Big{)}s+r_{c}+r_{g}} \tag{6}\]  
The additional terms are minimal, and the expression is largely similar to the transfer function of an L filter, which corresponds to the sum of the inductances in the LCL filter (\(L_{c}+L_{g}\)) and the resistances (\(r_{c}+r_{g}\)). The primary reason this approximation works is that, in the low-frequency range, the filter capacitor can be neglected, causing the LCL filter to behave like an equivalent inductor \(L_{eq}=L_{g}+L_{c}\) with an associated series resistance \(r_{\text{eq}}=r_{g}+r\). However, this approximation is valid only when considering the magnitude response. Given that the exact mathematical model of the system is unknown and considering the external disturbance \(\omega\), the system dynamics of the LCL filter, according to the approximate model, can be framed within the ADRC framework as follows:  
\[\dot{y}=f(t,y,\omega)+bu \tag{7}\]  
In Equation (7) above, \(f(\cdot)\) represents the \"total disturbance\" that needs to be assessed and mitigated, encompassing both internal and external dynamics. Here, \(y\) denotes the grid current, \(b\) is a system parameter, and \(u\) is the control input applied to the inverter. This article simplifies the system in the s-domain by using frequency response analysis to study the performance and stability of the system. By applying the Laplace transform, we obtain Equations (8) and (9) as follows:  
\[C_{f}\frac{dv_{C}^{dq}}{dt}=i_{g}^{dq}-i_{C}^{dq}-j\omega C_{f}v_{C}^{dq} \tag{2}\]0  
\[C_{f}\frac{dv_{C}^{dq}}{dt}=i_{g}^{dq}-i_{C}^{dq}-j\omega C_{f}v_{C}^{dq} \tag{2}\]1Lastly, by substituting the ADRC-based control law, i.e., \(u(s)=G_{c}(s)(r(s)-H(s)y(s))\), into expression (7):  
\[ \begin{array}{l}y(s)=\frac{1}{s}(f(s)+bu(s))\\ =\frac{1}{s}(f(s)+bG_{c}(s)(r(s)-H(s)y(s)))\\ =\frac{(s+\beta_{1}+k_{p})}{(s+k_{p})(s^{2}+\beta_{1}+\beta_{2})}sf(s)+\frac{k _{p}}{s+k_{p}}r(s)\\ =G_{fy}(s)f(s)+G_{ry}(s)r(s)\end{array}  \tag{10}\]  
It is evident that the stability control of the closed-loop circuit depends solely on the observer bandwidth and the control bandwidth. Based on Equation (9) above, the control output comprises both the disturbance rejection term and the reference tracking component. By neglecting the disturbance\(f\)(s), the output can be simplified as shown in expression (11) below:  
\[G_{ry}(s)=\frac{y(s)}{r(s)}=\frac{k_{p}}{s+k_{p}} \tag{11}\]  
The tracking characteristic relies upon the controlling bandwidth \(k_{p}=-w_{c}\). Increasing the high-frequency gains also can improve the high-pass filtering.