## 3 Voltage Source Inverter Model  
This section provides a brief description of the dynamic model of the VSI and the filter used in the experimental setup for this study. The dynamic model of the filter is expressed in the \(dq\) synchronous reference frame by applying the Clarke and Park transformations (1) and (2):  
\[ \begin{bmatrix}v_{\text{c}\text{a}}\\ v_{\text{c}\text{g}}\end{bmatrix} = \begin{bmatrix}\frac{2}{3}&-\frac{1}{3}&- \frac{1}{3}\\ 0&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}\end{bmatrix}\begin{bmatrix}v_{\text{c} \text{a}}\\ v_{\text{c}\text{b}}\\ v_{\text{c}\text{c}}\end{bmatrix} , \tag{1}\]  
\[ \begin{bmatrix}v_{\text{c}\text{d}}\\ v_{\text{c}\text{d}}\end{bmatrix} = \begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}v_{\text{c}\text{a}}\\ v_{\text{c}\text{g}}\end{bmatrix} . \tag{2}\]  
In (2), the angle \(\theta\) is determined by integrating the reference frequency of capacitor voltage (\(\theta=\int\omega dt\)). The resulting dynamic model of the LC filter is described by the following equations [14, 26]:  
\[ \begin{split} L_{\text{f}}\frac{d\mathbf{i}_{\text{f}}}{dt}& =-R_{\text{f}}\mathbf{i}_{\text{f}}-j\omega L_{\text{f}}\mathbf{i}_{\text{f}}- \mathbf{v}_{\text{c}}+\mathbf{v}_{\text{s}},\\ C_{\text{f}}\frac{d\mathbf{v}_{\text{c}}}{dt}&=\mathbf{i}_{\text{f}}-j \omega C_{\text{f}}\mathbf{v}_{\text{c}}-\mathbf{i}_{\text{o}}.\end{split}  \tag{3}\]  
where \(\{L_{\text{f}},R_{\text{f}},C_{\text{f}}\}\) are the LC filter parameters, \(\mathbf{i}_{\text{f}}=[i_{\text{fd}}\ i_{\text{fd}}]^{\top}\) is the current vector at the inductor, \(\mathbf{v}_{\text{s}}=[v_{\text{sd}}\ v_{\text{sd}}]^{\top}\) is the inverter modulated voltage vector, and \(\mathbf{i}_{\text{o}}=[i_{\text{od}}\ i_{\text{od}}]^{\top}\) is the output current. Equation (3) can be expressed in a continuous state-space model, where the voltage and current are separated into their real and imaginary components:  
\[ \begin{split}\frac{d}{dt}\begin{bmatrix}i_{\text{fd}}\\ i_{\text{f}\text{q}}\\ v_{\text{cd}}\\ v_{\text{cd}}\end{bmatrix}&=\begin{bmatrix}-\frac{R_{\text{f}}}{L_{\text{f}}}& \omega&-\frac{1}{L_{\text{f}}}&0\\ -\omega&-\frac{R_{\text{f}}}{L_{\text{f}}}&0&-\frac{1}{L_{\text{f}}}\\ \frac{1}{C_{\text{f}}}&0&0&\omega\\ 0&\frac{1}{C_{\text{f}}}&-\omega&0\end{bmatrix}\begin{bmatrix}i_{\text{fd}} \\ i_{\text{f}\text{q}}\\ v_{\text{cd}}\end{bmatrix}+\begin{bmatrix}\frac{1}{L_{\text{f}}}&0\\ 0&\frac{1}{L_{\text{f}}}\\ 0&0\\ 0&0\end{bmatrix}\mathbf{v}_{\text{s}}+\begin{bmatrix}0&0\\ 0&0\\ -\frac{1}{C_{\text{f}}}&0\\ 0&-\frac{1}{C_{\text{f}}}\end{bmatrix}\mathbf{i}_{\text{o}},\\ \mathbf{y}&=\begin{bmatrix}\mathbf{0}_{2\times 2}&\mathbb{I}_{2\times 2}\end{bmatrix} \begin{bmatrix}i_{\text{fd}}&i_{\text{f}\text{q}}&v_{\text{cd}}&v_{\text{cq}} \end{bmatrix}^{\top}.\end{split}  \tag{4}\]  
The system of Equation (4) can be expressed in its compact form as follows:  
\[ \begin{split}\dot{\mathbf{x}}&=\mathbf{A}\mathbf{x}+\mathbf{B} \mathbf{u}+\mathbf{B}_{\text{p}}\mathbf{i}_{\text{o}},\\ \mathbf{y}&=\mathbf{C}\mathbf{x}.\end{split}  \tag{5}\]  
From (5), the system's discrete state-space representation is derived using a sampling period \(T_{\text{s}}\), as outlined in [14, 26]:  
\[ \begin{split}\mathbf{x}_{t+1}&=\mathbf{A}_{\text{d}} \mathbf{x}_{t}+\mathbf{B}_{\text{d}}\mathbf{u}_{t}+\mathbf{B}_{\text{pd}} \mathbf{i}_{\text{o},t},\\ \mathbf{y}_{t}&=\mathbf{C}\mathbf{x}_{t},\end{split}  \tag{6}\]  
where  
\[ \begin{split}\mathbf{A}_{\text{d}}&=e^{\mathbf{A}T_{\text{s}}},\\ \mathbf{B}_{\text{d}}&=(e^{\mathbf{A}T_{\text{s}}}- \mathbb{I}_{4\times 4})\mathbf{A}^{-1}\mathbf{B},\\ \mathbf{B}_{\text{pd}}&=(e^{\mathbf{A}T_{\text{s}}}- \mathbb{I}_{4\times 4})\mathbf{A}^{-1}\mathbf{B}_{\text{p}}.\end{split}  \tag{7}\]The discrete-time model in (6) serves as the foundation for constructing the prediction model required for MPC control. This work uses the multivariable frequency response of the model in (6) to establish design criteria for the closed-loop control.