## 2 System Description and Current Controller
### Integral-Resonant State Feedback Control  
Figure 3 represents the integral-resonant state feedback current control based on the linear quadratic regulator (LQR) for three-phase inverter connected with the grid inductance \(L_{g}\). To ensure asymptotic reference tracking as well as disturbance rejection for the harmonics in the orders of 6 th and 12 th in the SRF, the integral and resonant control terms are augmented in the state feedback control. In the discrete-time state-space, the integral and resonant terms are expressed as [16; 31]:  
\[\left[ \begin{array}{c}x^{d}_{i}(k+1)\\ x^{d}_{i}(k+1)\end{array} \right]=\mathbf{A}_{\text{cl}}\left[ \begin{array}{c} x^{q}_{i}(k)\\ x^{d}_{i}(k)\end{array} \right]+\mathbf{B}_{\text{cl}}\epsilon(k) \tag{10}\]  
\[\left[ \begin{array}{c}\xi^{d}_{1h}(k+1)\\ \xi^{d}_{2h}(k+1)\\ \xi^{d}_{1h}(k+1)\\ \xi^{d}_{2h}(k+1)\end{array} \right]=\mathbf{A}_{\text{ch}}\left[ \begin{array} []{c}\zeta^{d}_{1h}(k)\\ \zeta^{d}_{2h}(k)\\ \zeta^{d}_{1h}(k)\\ \zeta^{d}_{2h}(k)\end{array} \right]+\mathbf{B}_{\text{ch}}\epsilon(k)\text{ for }h\ =\ 6,\ 12 \tag{11}\]  
where \(\epsilon=[E^{q}\,E^{d}]^{T}=\mathbf{r}-\mathbf{C}_{d}\mathbf{x}\) is the current error vector, \(\mathbf{r}=[ \begin{smallmatrix}q^{q\ast}_{2}&i^{d\ast}_{2}\end{smallmatrix} ]^{T}\) is the reference current vector, and system matrices \(\mathbf{A}_{\text{cl}}\), \(\mathbf{B}_{\text{cl}}\), \(\mathbf{A}_{\text{cl}}\), and \(\mathbf{B}_{\text{ch}}\) are expressed as:  
\[\mathbf{A}_{\text{cl}}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right],\ \mathbf{B}_{\text{cl}}=\left[ \begin{array}{cc}T_{s}&0\\ 0&T_{s}\end{array} \right]\]  
Figure 2: Block diagram of the direct current control based on capacitor current damping. The state equations in (10) and (11) are augmented as:  
\[\mathbf{x}_{c}(k+1)=\mathbf{A}_{au}\mathbf{x}_{c}(k)+\mathbf{B}_{au}\varepsilon(k) \tag{12}\]  
where \(\mathbf{x}_{c}=\left[\mathbf{x}_{0}\,\mathbf{x}_{6}\,\mathbf{x}_{12}\right]^{T}\) is the entire state vector for the integral and resonant terms with \(\mathbf{x}_{0}=\left[x_{i}^{q}\,x_{i}^{q}\right]^{T}\), \(\mathbf{x}_{h}=\left[\zeta_{1h}^{q}\,\zeta_{2h}^{q}\,\zeta_{1h}^{d}\,\zeta_{2h }^{d}\right]^{T}\) for \(h=6,\,12\),  
\[\mathbf{A}_{au}=\left[ \begin{array}{cc}\mathbf{A}_{ci}&\\ &\mathbf{A}_{c6}\\ &&\mathbf{A}_{c12}\end{array} \right],\ \mathbf{B}_{au}=\left[ \begin{array}{c} \mathbf{B}_{ci}\\ \mathbf{B}_{c6}\\ \mathbf{B}_{c12}\end{array} \right].\]  
By using Equations (4), (5) and (10)-(12), the entire system model is combined as follows:  
\[\left[ \begin{array}{c}\mathbf{x}(k+1)\\ \mathbf{x}_{c}(k+1)\end{array} \right]=\left[ \begin{array}{cc}\mathbf{A}_{d} &0_{6\times 10}\\ -\mathbf{B}_{au}\mathbf{C}_{d}&\mathbf{A}_{au}\end{array} \right]\left[ \begin{array}{c}\mathbf{x}(k)\\ \mathbf{x}_{c}(k)\end{array} \right]+\left[ \begin{array}{c}\mathbf{B}_{d}\\ 0_{10\times 2}\end{array} \right]\mathbf{u}(k)+\left[ \begin{array}{c}\mathbf{ D}_{d}\\ 0_{10\times 2}\end{array} \right]\mathbf{e}(k)+\left[ \begin{array}{c}0_{6\times 2} \\ \mathbf{B}_{au}\end{array} \right]\mathbf{r}(k) \tag{13}\]  
\[\mathbf{y}_{s}(k)=\left[ \begin{array}{cc}\mathbf{C}_{d}&0_{2\times 10} \end{array} \right]\left[ \begin{array}{c}\mathbf{x}(k)\\ \mathbf{x}_{c}(k)\end{array} \right] \tag{14}\]  
where \(0_{m\times n}\) is the zero matrix with appropriate dimension. Considering the augmented system, the state feedback control is designed as:  
\[\mathbf{u}(k)=\left[ \begin{array}{cc}\mathbf{K}_{x}&\mathbf{K}_{C}\end{array}  \right]\left[ \begin{array}{c}\mathbf{x}(k)\\ \mathbf{x}_{c}(k)\end{array} \right] \tag{15}\]  
Figure 3: Block diagram of the LQR (linear quadratic regulator) -based integral-resonant state feedback current controller. where \(\left[ \begin{array}{cc}\mathbf{K}_{x}&\mathbf{K}_{C}\end{array} \right]\) is a set of state feedback gains with \(\mathbf{K}_{C}=\left[ \begin{array}{cc}\mathbf{K}_{i}&\mathbf{K}_{H}\end{array} \right]\).