## 3 Proposed Non-Linear Current Control Scheme of Grid Connected Inverter with LCL Filter
### Output Current Control  
Modeling of a three-phase grid-connected inverter with an LCL filter in dq frame results in a 2-inputs-2-outputs transfer function from inverter terminal voltages to grid-side currents, as given in (16),  
\[\left[ \begin{array}{c}I_{gd}\\ I_{gq}\end{array} \right]=\left[ \begin{array}{cc}G_{11}(s)&G_{12}(s)\\ G_{21}(s)&G_{22}(s)\end{array} \right]\left[ \begin{array}{c}v_{ld}\\ v_{ld}\end{array} \right] \tag{16}\]  
in which (\(v_{ld}\), \(v_{ld}\)) are the control inputs, (\(i_{gd}\), \(i_{gq}\)) are outputs, and the G matrix is the non-diagonal transfer function. Meanwhile, modeling of the system in an ABC frame results in a single input-single output transfer function for each phase of the system. In this regard, the frequency characteristics of a single LCL filter and the system in (16) are different, as shown in Figure 3. The resonance mode of the system is evident in both frequency characteristics. Considering the transfer function in (16), there is a tight coupling between two channels (channel 1: input 1 to output 1, channel 2: input 2 to output). Figure 4 shows the Gershgorin band graph of (16) with the parameter values of Table 2, which shows that the system is not column and row dominant. Also, based on the MIMO control system theory, the independent control design for channel 1 and channel 2 may lead to instability of the overall system, regardless of the resonance mode being dampened [29]. The high order of the system and the existence of the resonance mode are other factors that increase the difficulty of the current controller design. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **PM without \(T_{d}\)** & \(T_{td}\) (**\(\mathbf{\mu s}\)**) & \(W_{b-curr}\) (**rad/s**) \\ \hline \(\rho=3000\) & 86 & 140 & 1600 \\ \(\rho=2500\) & 91 & 168 & 1450 \\ \(\rho=2000\) & 99 & 214 & 1220 \\ \(\rho=1500\) & 113 & 259 & 980 \\ \(\rho=1000\) & 130 & 351 & 800 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Effect of the control parameter on PM and bandwidth of the current control loop. Figure 3: Frequency characteristics of (**a**) single phase LCL filter and (**b**) the system in dq frame (transfer function in (16)). Based on the multivariable control theory, it is possible to decrease the coupling between channels in MIMO systems using a pre-controller such that the system becomes dominantly diagonal (column dominant or row dominant). However, it is not possible to completely remove the coupling and make the system diagonal using the mentioned pre-controller. To remove the coupling terms completely, a non-linear control scheme is proposed in this paper, which simultaneously damps the resonance modes. This controller is based on the back-stepping method, which is categorized as a state-feedback approach. To start the controller design with this method, two variables are defined as \(n_{1}=-\omega i_{lq}+\frac{v_{lq}-v_{d}}{L_{1}}\) and \(n_{2}=\omega i_{ld}+\frac{v_{lq}-v_{d}}{L_{1}}\), and an error variable vector is also defined as given by (17):  
\[E=x-x^{ref} \tag{17}\]  
where,  
\[ \begin{split} x^{ref}=\left[\begin{smallmatrix}{i_{ld}}^{ref}& {i_{ld}}^{ref}&{i_{ld}}^{ref}&{v_{cd}}^{ref}& {v_{cq}}^{ref}&{i_{gd}}^{ref}&{i_{gq}}^{ref}\\ &{x}=\left[\begin{smallmatrix}{i_{ld}}&{i_{ld}}&{i_{ld}}&{v_{cd}}&{v_{cq}}&{i_{gd}}&{i_{gq}}\\ \end{smallmatrix}\right]\end{split}  \tag{18}\]  
The reference vector values are given in (19) which are obtained by solving the state Equations (6)-(11) in a steady state. Also, \({i_{gq}^{ref}}\) and \({i_{gd}^{ref}}\) are obtained by the reactive power and the DC-link voltage control loops, respectively. \[ \begin{split} v_{cq}^{ref}&=-L_{2}\cdot\omega\cdot{ i_{gd}^{ref}}&{v_{cd}^{ref}}=L_{2}\cdot\omega\cdot{i_{gd}^{ref}}+v_{gd}\\ {i_{lq}^{ref}}&=-C_{f}\cdot\omega\cdot{v_{cd}^{ref}}&{i_{ lq}^{ref}}=C_{f}\cdot\omega\cdot{v_{cq}^{ref}}\end{split}  \tag{19}\]  
Figure 4: Gershgorin band graph of transfer function (16) with the parameter values of Table 3.