# Model Predictive Control for Six-Phase Induction Machines with Insight into Past Current Errors
## 2 The Six-Phase Induction Motor Drive  
The employed electric drive is composed of a double three-phase two-level voltage source converter (VSC) connected to an asymmetrical six-phase induction machine (IM) [31]. The IM stator windings have been configured with isolated neutral points, simplifying the complexity of the control scheme [32]. A general diagram of the system's topology is shown in Figure 1. Every VSC leg is formed by two complementary switches that determine the switching state of each of the six phases, adding up to \(2^{6}=64\) possible states (\(S_{ph}\)), where the subscript \"\(ph\)\" and \"\(j\)\" refer to the phase (\(a\), \(b\), and \(c\)) and the VSC it belongs to (1 or 2), respectively. The complementary switches behave in a binary manner, with \(S_{ph_{j}}=1\) resulting in the upper switch being ON and the lower switch being OFF. Consequently, \(S_{ph_{j}}=0\) entails the opposite behavior for upper and lower switches. The expression (1) defines the stator phase voltages (\(v_{ph_{j}}\)) obtained in the considered electric drive:  
\[ \begin{bmatrix}v_{a1}\\ v_{b1}\\ v_{c1}\\ v_{a2}\\ v_{b2}\\ v_{c2}\end{bmatrix} =\frac{V_{dc}}{3} \begin{bmatrix}2&-1&-1&0&0&0\\ -1&2&-1&0&0&0\\ -1&-1&2&0&0&0\\ 0&0&0&2&-1&-1\\ 0&0&0&-1&2&-1\\ 0&0&0&-1&-1&2\end{bmatrix}\begin{bmatrix}S_{a_{1}}\\ S_{b_{1}}\\ S_{c_{1}}\\ S_{a_{2}}\\ S_{b_{2}}\\ S_{c_{2}}\end{bmatrix} , \tag{1}\]  
where \(V_{dc}\) is the voltage provided by the dc-link. Even though expressing IM behavior in phase variables is a common practice, the use of different reference frames can facilitate control design. An example is the use of vector space decomposition (VSD) which, in the case of the six-phase machine, maps phase stator values onto three orthonormal subspaces with distinct physical meaning [33]. This useful transformation is achieved through the use of the Clarke matrix (2). The primary plane is the \(\alpha\)-\(\beta\) subspace, which is related to flux and torque production. The secondary plane is the \(x\)-\(y\) subspace, and it maps the harmonic components that produce copper losses in distributed winding machines as long as the spatial harmonics are negligible. The last product of the use of the Clarke transform are the zero-sequence currents which, due to the fact that the neutral points of the studied system are isolated, can be considered zero and thus ignored. \[[C]=\frac{1}{3} \begin{bmatrix}1&-\frac{1}{2}&-\frac{1}{2}&\frac{\sqrt{3}}{2}&- \frac{\sqrt{3}}{2}&0\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}&\frac{1}{2}&\frac{1}{2}&-1\\ 1&-\frac{1}{2}&-\frac{1}{2}&-\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&0\\ 0&-\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&\frac{1}{2}&\frac{1}{2}&-1\\ 1&1&1&0&0&0\\ 0&0&0&1&1&1\end{bmatrix} , \tag{2}\]  
\[[x_{a}\;\;x_{\beta}\;\;x_{x}\;\;x_{y}\;\;x_{z1}\;\;x_{z2}]^{T}\!\!=[C][x_{a1}\; \;x_{b1}\;\;x_{c1}\;\;x_{a2}\;\;x_{b2}\;\;x_{c2}]^{T} \tag{3}\]  
where \(x\) can represent the current, the voltage or the flux of the electric machine. Using standard assumptions [2], the employed six-phase IM can be modeled as follows:  
Figure 1: Topology of the asymmetrical six-phase IM with a double three-phase two-level VSC. \[v_{\alpha} = (R_{s}+L_{s}\frac{d}{dt})i_{\alpha s}+M\frac{di_{\alpha r}}{dt},\] \[v_{\beta} = (R_{s}+L_{s}\frac{d}{dt})i_{\beta s}+M\frac{di_{\beta r}}{dt},\] \[v_{x} = (R_{s}+L_{s}\frac{d}{dt})i_{xs},\] \[v_{y} = (R_{s}+L_{s}\frac{d}{dt})i_{ys},\] \[0 = (R_{r}+L_{r}\frac{d}{dt})i_{\alpha r}+M\frac{di_{\alpha r}}{dt}+ \omega_{r}L_{r}i_{\beta r}+\omega_{r}Mi_{\beta s},\] \[0 = (R_{r}+L_{r}\frac{d}{dt})i_{\beta r}+M\frac{di_{\beta r}}{dt}+ \omega_{r}L_{r}i_{\alpha r}-\omega_{r}Mi_{\alpha s},\] \[T_{e} = 3 pM(i_{\beta r}i_{\alpha s}-i_{\alpha r}i_{\beta s}),\]  
where \(L_{s}=L_{ls}+M\), \(L_{r}=L_{lr}+M\), \(M=3L_{m}\), and \(\omega_{r}=p\omega_{m}\) are \(p\), and \(\omega_{m}\) is the pole pairs number and the mechanical speed, respectively. Furthermore, the indices \(r\) and \(s\) denote rotor and stator values, while subscript \(l\) and \(m\) denote leakage and magnetizing inductance, respectively.