## 3 Proposed Discontinuous Model Predictive Control with Zero Vector Selection
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Vector Categories**} & \multirow{2}{*}{**Vector Subtypes**} & \multirow{2}{*}{**Vector States**} & \multicolumn{2}{c}{**Switching State**} & \multicolumn{2}{c}{**Stationary Reference**} \\  & & & & **(a, b, c)** & **Frame Voltage Vector** \\ \cline{3-6}  & & & \(S_{1}\) & \(S_{2}\) & \(V_{a}\) & \(V_{\beta}\) \\ \hline \multirow{8}{*}{Actual voltage vector} & Zero & \(V_{0}\) & & 000 & 0 & 0 \\ \cline{3-6}  & vector & \(V_{7}\) & & 111 & 0 & 0 \\ \cline{3-6}  & & \(V_{1}\) & & 100 & \(\frac{2}{3}V_{dc}\) & 0 \\ \cline{3-6}  & & \(V_{2}\) & & 110 & \(\frac{1}{3}V_{dc}\) & \(\frac{1}{\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{3}\) & & 010 & \(-\frac{1}{3}V_{dc}\) & \(\frac{1}{\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & vector & \(V_{4}\) & & 011 & \(-\frac{2}{3}V_{dc}\) & 0 \\ \cline{3-6}  & & \(V_{5}\) & & 001 & \(-\frac{1}{3}V_{dc}\) & \(-\frac{1}{\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{6}\) & & 101 & \(\frac{1}{3}V_{dc}\) & \(-\frac{1}{\sqrt{3}}V_{dc}\) \\ \hline \multirow{8}{*}{Multi virtual voltage vector} & \multirow{8}{*}{P-type Multi vector} & \(V_{m1}-P\) & 000 & 100 & \(\frac{1}{3}V_{dc}\) & 0 \\ \cline{3-6}  & & \(V_{m2}-P\) & 000 & 110 & \(\frac{1}{6}V_{dc}\) & \(\frac{1}{2\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{m3}-P\) & 000 & 010 & \(-\frac{1}{6}V_{dc}\) & \(\frac{1}{2\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{m4}-P\) & 000 & 011 & \(-\frac{1}{3}V_{dc}\) & 0 \\ \cline{3-6}  & & \(V_{m5}-P\) & 000 & 001 & \(-\frac{1}{6}V_{dc}\) & \(-\frac{1}{2\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{m6}-P\) & 000 & 101 & \(\frac{1}{6}V_{dc}\) & \(-\frac{1}{2\sqrt{3}}V_{dc}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Switching states of the inverters used in the proposed method. Figure 6: Zero-voltage vector selection based on offset voltage polarity. Figure 7 depicts a vector diagram of the multi-virtual voltage vectors based on the offset polarity. Figure 8 depicts the detailed switching states and generation process of P-type and N-type multi-vectors based on the selected zero-voltage vector. The reference voltage for discontinuous switching can be calculated by adding the offset voltage to the voltage vector determined using Equation (8) as follows:  
\[\mathbf{v}_{ref.x}(k+1)=\mathbf{v}_{x}^{*}(k+1)+\mathbf{v}_{offset}(k+1)\ \ (x=a,\ b,\ c) \tag{11}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vector** & **Vector** & **Vector** & \multicolumn{2}{c}{**Switching State**} & \multicolumn{2}{c}{**Stationary Reference**} \\
**Categories** & **Subtypes** & **Vector** & \multicolumn{2}{c}{**(a, b, c)**} & \multicolumn{2}{c}{**Frame Voltage Vector**} \\ \cline{4-6}  & & & \(\mathbf{S_{1}}\) & \(\mathbf{S_{2}}\) & \(\mathbf{V_{x}}\) & \(\mathbf{V_{\beta}}\) \\ \hline \multirow{9}{*}{N-type Multi virtual vector} & \(V_{m1}-N\) & \(111\) & \(100\) & \(\frac{1}{6}V_{dc}\) & \(0\) \\ \cline{3-6}  & & \(V_{m2}-N\) & \(111\) & \(110\) & \(\frac{1}{6}V_{dc}\) & \(\frac{1}{2\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{m3}-N\) & \(111\) & \(010\) & \(-\frac{1}{6}V_{dc}\) & \(\frac{1}{2\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{m4}-N\) & \(111\) & \(011\) & \(-\frac{1}{3}V_{dc}\) & \(0\) \\ \cline{3-6}  & & \(V_{m5}-N\) & \(111\) & \(001\) & \(-\frac{1}{6}V_{dc}\) & \(-\frac{1}{2\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{m6}-N\) & \(111\) & \(101\) & \(\frac{1}{6}V_{dc}\) & \(-\frac{1}{2\sqrt{3}}V_{dc}\) \\ \cline{3-6}  & & \(V_{m7}\) & \(011\) & \(100\) & \(\frac{1}{2}V_{dc}\) & \(\frac{\sqrt{3}}{2}V_{dc}\) \\ \cline{3-6}  & & \(V_{m8}\) & \(010\) & \(101\) & \(0\) & \(V_{dc}\) \\ \cline{3-6}  & & \(V_{m9}\) & \(110\) & \(001\) & \(-\frac{1}{2}V_{dc}\) & \(\frac{\sqrt{3}}{2}V_{dc}\) \\ \cline{3-6}  & & \(V_{m10}\) & \(100\) & \(011\) & \(-\frac{1}{2}V_{dc}\) & \(-\frac{\sqrt{3}}{2}V_{dc}\) \\ \cline{3-6}  & & \(V_{m11}\) & \(101\) & \(010\) & \(0\) & \(-V_{dc}\) \\ \cline{3-6}  & & \(V_{m12}\) & \(001\) & \(110\) & \(\frac{1}{2}V_{dc}\) & \(-\frac{\sqrt{3}}{2}V_{dc}\) \\ \hline \hline \end{tabular}
\end{table}