## 2 Model Predictive Control for PMSM Drives  
Figure 1 presents the topology of the inverter and PMSM system, where a conventional three-phase two-level voltage source inverter (2L-VSI) is employed. The inverter output currents are denoted as \(i_{\text{ou}}\), \(i_{\text{ob}}\), and \(i_{\text{oc}}\), with the DC-link voltage denoted as \(V_{\text{dc}}\). The output voltages of the universal bridge, \(v_{\text{a}}\), \(v_{\text{b}}\), and \(v_{\text{c}}\), can be expressed as follows:  
\[ \begin{cases}v_{a}=\frac{25s_{-}S_{\text{b}}-S_{\text{c}}}{3}\cdot V_{\text{ dc}}\\ v_{b}=\frac{25s_{-}S_{\text{a}}-S_{\text{c}}}{3}\cdot V_{\text{dc}}\\ v_{c}=\frac{25s_{-}S_{\text{a}}-S_{\text{b}}}{3}\cdot V_{\text{dc}}\end{cases}  \tag{1}\]  
Here, \(S_{\text{a}}\), \(S_{\text{b}}\), and \(S_{\text{c}}\) are the switching states. There are eight switching states, each corresponding to the three-phase universal bridge voltage. By applying the Clarke transformation, the universal bridge voltages can be presented as eight basic voltage vectors at stationary \(\alpha\beta\) frame:  
\[\mathbf{v}_{n}= \begin{bmatrix}2/3&-1/3&-1/3\\ 0&-\sqrt{3}/3&\sqrt{3}/3\end{bmatrix} [v_{a}\,v_{b}\,v_{c}]^{\text{T}} \tag{2}\]  
The eight voltage vectors \(\mathbf{v}_{n}\) (\(n=0,1,\ldots,7\)) and switching states are shown in Figure 2.  
Figure 1: Topology of the inverter and PMSM system. The stator voltage equation of the PMSM is  
\[\mathbf{v}_{s}=R_{s}\mathbf{i}_{s}+\frac{d\mathbf{\psi}_{s}}{dt} \tag{3}\]  
where \(\mathbf{v}_{s}=[v_{a}\ v_{b}\ v_{c}]^{\mathrm{T}}\) and \(\mathbf{i}_{s}=[i_{\alpha}\ i_{\mathrm{ob}}\ i_{\mathrm{oc}}]^{\mathrm{T}}\) denote the stator winding voltage and current, respectively. \(R_{s}\) is the stator resistance. The flux linkage is defined by  
\[\mathbf{\psi}_{s}=L_{s}\mathbf{i}_{s}+\mathbf{\psi}_{f} \tag{4}\]  
where \(\mathbf{\psi}_{f}\) denotes the rotor flux linkage. By applying the Clarke and Park transformations, the PMSM voltage and flux linkage equations in the dq-axis can be given as follows:  
\[\mathbf{v}_{sd}=R_{s}\mathbf{i}_{sd}+\frac{d\psi_{d}}{dt}-\psi_{q}\omega_{e} \tag{5}\]  
\[\mathbf{v}_{sq}=R_{s}\mathbf{i}_{sq}+\frac{d\psi_{q}}{dt}-\psi_{d}\omega_{e} \tag{6}\]  
\[\psi_{d}=L_{d}\mathbf{i}_{sd}+\psi_{f} \tag{7}\]  
\[\psi_{q}=L_{q}\mathbf{i}_{sq} \tag{8}\]  
Based on (5)-(8), the PMSM continuous-time equation can be expressed as  
\[ \begin{bmatrix}\frac{d\mathbf{i}_{d}}{dt}\\ \frac{d\mathbf{i}_{sq}}{dt}\end{bmatrix} = \begin{bmatrix}-\frac{\mathbf{k}_{s}}{L_{d}}& \frac{\omega_{s}L_{d}}{L_{d}}\\ -\frac{\omega_{s}L_{d}}{L_{d}}&-\frac{\mathbf{k}_{s}}{L_{d}}\end{bmatrix}\begin{bmatrix} \mathbf{i}_{sd}\\ \mathbf{i}_{sq}\end{bmatrix} + \begin{bmatrix}\frac{1}{d}&0\\ 0&\frac{1}{L_{d}}\end{bmatrix}\begin{bmatrix}\mathbf{v}_{sd}\\ \mathbf{v}_{sq}\end{bmatrix} - \begin{bmatrix}0\\ \frac{\omega_{e}\psi_{f}}{L_{d}}\end{bmatrix}  \tag{9}\]  
To realize the FCS-MPC strategy, a discrete-time model is employed to predict the system output in the next control period. Based on the Euler approximation formula, the discretized prediction model of the PMSM can be expressed as  
\[ \begin{bmatrix}\mathbf{i}_{sd}^{\mathrm{P}}(k+1)\\ \mathbf{i}_{sq}^{\mathrm{P}}(k+1)\end{bmatrix} =\mathbf{A}_{\mathbf{d}} \begin{bmatrix} \mathbf{i}_{sd}(k)\\ \mathbf{i}_{sq}(k)\end{bmatrix} +\mathbf{B}_{\mathbf{d}} \begin{bmatrix}\mathbf{v}_{sd} (k)\\ \mathbf{v}_{sq}(k)\end{bmatrix} - \begin{bmatrix}0\\ \frac{T_{s}\omega_{e}\psi_{f}}{L_{d}}\end{bmatrix}  \tag{10}\]  
where the matrices \(\mathbf{A}_{\mathbf{d}}\) and \(\mathbf{B}_{\mathbf{d}}\) are defined as  
\[\mathbf{A}_{\mathbf{d}}= \begin{bmatrix}1-\frac{T_{s}R_{s}}{L_{d}}&\frac{T_{s }\omega_{s}L_{d}}{L_{d}}\\ -\frac{T_{s}\omega_{s}L_{d}}{L_{d}}&1-\frac{T_{s}R_{s}}{L_{d}}\end{bmatrix} ; \mathbf{B}_{\mathbf{d}}= \begin{bmatrix}\frac{T_{s}}{L_{d}}&0\\ 0&\frac{T_{s}}{L_{d}}\end{bmatrix}  \tag{11}\]  
In MPC strategy, a cost function is used to define the optimal input of the inverter:  
\[G= \begin{bmatrix}\mathbf{i}_{sd}^{*}(k+1)-\mathbf{i}_{sd}^{\mathrm{P}}(k+1)\end{bmatrix}  ^{2}+