## 2 Conventional Finite Control Set-Model Predictive Control for Two-Level Voltage Source Inverter  
### Three-Phase Two-Level Inverter  
Figure 1 depicts the simplest configuration of a three-phase two-level voltage source inverter. The inverter comprises one leg per phase, with each leg containing two switches, presenting a total of six switches. Each switch has two possible states, ON and OFF. The switches within the same phase (e.g., \(\mathrm{S_{a}}\) and \(\mathrm{S_{a}}^{\prime}\)) operate complementarily, i.e., when one switch is ON, its corresponding switch must be OFF. This complementary operation prevents short circuits during the switching transitions. When the ON state is represented as '1', and the OFF state is represented as '0', the switching state for each leg is defined as follows:  
\[S_{a}+S_{a}^{\prime}=1S_{b}+S_{b}^{\prime}=1S_{c}+S_{c}^{\prime}=1 \tag{1}\]Owing to these switching characteristics, each phase can output two voltage levels (+V\({}_{\text{dc}}\)/2 or -V\({}_{\text{dc}}\)/2). The three-phase AC output voltage of the inverter is determined by a combination of the switching states across all the phase legs. The output voltage generation process can be explained using the space-vector pulse width modulation (SVPWM) of the three-phase inverters. The SVPWM represents the three-phase output voltage as a single-space vector in a complex plane. The inverter selects the specific voltage vectors that exist within this space vector. This approach enables the inverter to generate three-phase AC outputs with various frequencies and magnitudes.  
A two-level voltage source inverter has eight possible switching states, corresponding to eight output voltage vectors. The output voltage vectors generated by each switching-state combination are as follows:  
\[\mathbf{V}_{abc}=\begin{bmatrix}\mathbf{v}_{an}\\ \mathbf{v}_{bn}\\ \mathbf{v}_{cn}\end{bmatrix}=V_{dc}\begin{bmatrix}(S_{a}-\frac{1}{2})\\ (S_{b}-\frac{1}{2})\\ (S_{c}-\frac{1}{2})\end{bmatrix} \tag{2}\]  
where \(\mathbf{V}_{abc}\) denotes the output voltage vector; \(\mathbf{v}_{\text{an}}\), \(\mathbf{v}_{\text{bn}}\), and \(\mathbf{v}_{\text{cn}}\) denote the output voltages of each phase; \(V_{\text{dc}}\) denotes the DC link voltage; and \(S_{\text{a}}\), \(S_{\text{b}}\), and \(S_{\text{c}}\) denote the switching states of each phase.  
The eight output voltage vectors comprise six active voltage vectors and two zero-voltage vectors. All the active voltage vectors exhibit an identical magnitude of 2V\({}_{\text{dc}}\)/3 but vary in phase. Since the two zero-voltage vectors exhibit different switching states but have identical values, the inverter effectively comprises seven distinct voltage vectors. The proper utilization of these zero-voltage vectors significantly affects the inverter efficiency and output quality [36; 37; 38].  
Figure 2 depicts the space voltage vectors corresponding to the switching states of the two-level inverter.  
The output voltage vector, \(\mathbf{V}_{abc}\), can be expressed in the \(\alpha\)-\(\beta\) coordinate system through Clarke transformation as follows:  
\[\begin{bmatrix}\mathbf{V}_{\alpha}\\ \mathbf{V}_{\beta}\end{bmatrix}=\frac{2}{3}\begin{bmatrix}1&\frac{1}{2}&-\frac{1}{ 2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{bmatrix}\begin{bmatrix}\mathbf{v}_{ \text{an}}\\ \mathbf{v}_{\text{bn}}\\ \mathbf{v}_{\text{cn}}\end{bmatrix} \tag{3}\]  
These voltage vectors are used in the MPC to predict future output voltages and determine the optimal switching states.  
Figure 1: Three-phase two-level inverter configuration with R-L-back EMF.