## 2 FSympscc of a Six-Phase Induction Drive Using VVV
The model uses the vector space decomposition approach to map the 6-phase space into sub-spaces \(\alpha-\beta\) (energy conversion) and \(x-y\) (non-torque producing). The predictive model often is presented in state space. The state space vector \(i=\big{(}i_{\alpha},i_{\beta}\big{)}^{\top}\) is obtained from measurements of phase stator currents. Control actions for the actual and next sampling periods \(u(k)\), \(u(k+1)\) are the voltages corresponding to the VVV. \[\hat{i}(k+2|k)=Ai(k)+B_{1}u(k)+B_{2}u(k+1)+G(k) \tag{3}\]  
where matrix \(A\) and vectors \(B_{1}\), \(B_{2}\) are obtained as \(A=(I+T_{s}A_{c})^{2}\), \(B_{1}=(I+T_{s}A_{c})T_{s}B_{c}\), \(B_{2}=T_{s}B_{c}\). Matrices \(A_{c}\) and \(B_{c}\) are defined as  
Figure 2: Voltage vectors, numbered from 0 to 63, for a six-phase VSI shown in \(\alpha-\beta\) plane (**left**) and \(x-y\) plane (**right**). The S, M, ML and L coronas have been highlighted in the \(\alpha-\beta\) plane. \[A_{c}= \begin{bmatrix}-a_{s2}&a_{m4}&0&0&a_{r4}&a_{l4}\\ -a_{m4}&-a_{s2}&0&0&-a_{l4}&a_{r4}\\ 0&0&-a_{s3}&0&0&0\\ 0&0&0&-a_{s3}&0&0\\ a_{s4}&-a_{m5}&0&0&-a_{r5}&-a_{l5}\\ a_{m5}&a_{s4}&0&0&a_{l5}&-a_{r5}\end{bmatrix} ,B_{c}= \begin{bmatrix}c_{2}&0&0&0 \\ 0&c_{2}&0&0\\ 0&0&c_{3}&0\\ 0&0&0&c_{3}\\ -c_{4}&0&0&0\\ 0&-c_{4}&0&0\end{bmatrix}  \tag{4}\]  
The coefficients used are: \(c_{1}=L_{s}L_{r}-L_{m}^{2}\), \(c_{2}=L_{r}/c_{1}\), \(c_{3}=1/L_{ls}\), \(c_{4}=L_{m}/c_{1}\), \(c_{5}=L_{s}c_{1}\), \(a_{s2}=R_{s}c_{2}\), \(a_{s3}=R_{s}c_{3}\), \(a_{s4}=R_{s}c_{4}\), \(a_{r4}=R_{r}c_{4}\), \(a_{r5}=R_{r}c_{5}\), \(a_{l4}=L_{r}c_{4}\omega_{r}\), \(a_{l5}=L_{r}c_{5}\omega_{r}\), \(a_{m4}=Mc_{4}\omega_{r}\) and \(a_{m5}=Mc_{5}\omega_{r}\), \(\theta=2\pi/6\). Vector \(G\) accounts for the dynamics resulting from rotor currents that are usually not measured [24]. Note that alternative methods can be found in the scientific literature [25]. The selection of \(u(k+1)\) in discrete time \(k\) is made by minimizing CF, which can incorporate a number of different terms to convert a multi-objective optimization problem into a one with only one objective [26]. The simplest CF penalizes the predicted control error \(\hat{e}=(i^{*}-\hat{i})\), where \(i^{*}\) represents the reference for the state space vector \(i\), and \(\hat{i}\) the prediction two steps ahead. More complex objective functions can be used; in particular, penalization of VSI commutations is achieved by computing the number of the switch changes \(SC\) produced in the VSI after configuration \(u(k)\) is changed to \(u(k+1)\) as  
\[SC(k+1)=\sum_{i=1}^{6}|S_{i}^{2}(k+1)-S_{i}^{1}(k+1)|+|S_{i}^{1}(k+1)-S_{i}^{2 }(k)|+|S_{i}^{2}(k)-S_{i}^{1}(k)| \tag{5}\]  
where \(S\) is the state of the VSI switches. It must be taken into account that during a sampling period, there are different VSI states corresponding to the VVV in use. These are denoted as \(S^{1}\) for the first sub-period and \(S^{2}\) for the second sub-period. Recall from the introduction that each sub-period uses a fraction of the sampling period. In the present case, the lengths of each sub-period are \(T_{1}=0.73T_{s}\) and \(T_{2}=0.27T_{s}\). With these considerations, the particular cost function \(J(k+2)\) that will be used is defined as  
\[J(k+2)=\|\hat{e}_{\alpha\beta}(k+2)\|^{2}+\lambda_{sc}SC(k+2) \tag{6}\]  
where \(\|.\|\) denotes vector modulus and \(\lambda_{sc}\) is the WF that allows one to give more or less importance to the penalization of switching changes over the current tracking error. The diagram in Figure 1 can be used with different sets of VVV. Several possibilities can be applied, as explained below. Figure 2 shows the voltage vectors, numbered from 0 to 63, for a six-phase VSI in the \(\alpha-\beta\) and \(x-y\) planes. Four coronas (or dodecagons) are clearly visible. These are usually identified by the module of their voltage vectors in the \(\alpha-\beta\) plane. For each VV, the attribute (S, M, ML or L) is derived from its modulus, comparing it to the modulus of all the other VVs.