# Multi-Phase Stator Current Tracking with Gradual Penalization of Commutations
## 3 Proposal
### Preliminaries  
In the following, some preliminary material is presented to clarify the proposal. First, the idea of gap is defined. Given a measurement vector \(y\) and a reference vector \(r\) (both in \(\alpha-\beta-x-y\) space), the gap is the predicted two-step ahead control error if no control action is issued (i.e., if a null VV is selected as control action). In this paper, a weighted metric is used to account for the effect of parameter \(\lambda_{xy}\) of the CF. As a result, the gap is defined as  
\[g_{\alpha\beta xy}=r(k+2)-\hat{y}^{F}(k+2), \tag{14}\]  
where \(\hat{y}_{F}(k+2)\) is the free-response weighted two-step ahead prediction computed as  
\[\hat{y}_{W}^{F}(k+2)=\begin{pmatrix}A\hat{y}_{\alpha-\beta}(k+1)\\ \sqrt{\lambda_{xy}}A\hat{y}_{x-y}(k+1)\end{pmatrix}. \tag{15}\]  
Notice that the weighted metric needs the introduction of factor \(\sqrt{\lambda_{xy}}\) affecting the \(x-y\) components of vector \(\hat{y}_{W}^{F}(k+2)\).  
With this metric, the regions of gap space corresponding to a particular VV can be defined by hyper-planes. This is very convenient because, given a hyper-plane and a point, it is very easy to compute their relative position. Every pair of VVs defines a plane, which, in turn, produces two subspaces. The VVs are marked as \(VV_{i}\) and \(VV_{j}\). The plane (II) passes through the mid-point of the segment that links \(VV_{i}\) and \(VV_{j}\). The plane is defined using the orthogonal vector \(n\). It is easy to see that any point \(g\) in one subspace produces a scalar product \(g^{\intercal}n>0\). Conversely, any point \(g\) in the other subspace produces a scalar product \(g^{\intercal}n<0\). Thus, for any given point, its associated subspace is determined by taking the sign function of the scalar product by \(n\).  
In a general case, many hyper-planes can be needed to determine the region where a particular control action is optimal. One of the findings of this work is that just three hyper-planes are needed for each hyper-quadrant. So, after computing the actual hyper-quadrant (\(Q\in\{1,2,\ldots,16\}\)), the three hyper-planes produce just eight regions. On each one, one VV provides the optimal solution in terms of the \(\alpha-\beta\) vs. \(x-y\) trade-off. So, the optimal VV is found by means of a region determination procedure without exhaustive optimization.  
The issue of commutation frequency is handled by the proposal by adding an extra layer where the switch changes required by the VV (obtained by the region-based selection) are compared with the cost of not commuting, i.e., the value \(J_{0}=J(k+2)\) when \(u(k+1)=u(k)\). This cost is easily computed as  
\[J_{0}=\|\hat{e}_{\alpha\beta}(k+2)\|^{2}+\lambda_{xy}\|\hat{e}_{xy}(k+2)\|^{2} \tag{16}\]  
In this way, commutations on the VSI are issued only when the cost of not doing so (\(J_{0}\)) is high enough. The balance between tracking quality and commutation frequency is treated with flexibility by means of parameter \(\lambda_{sc}\), as the allowed cost can be defined in terms of \(\lambda_{sc}\) and \(SC\). Since \(SC\) can take just six values (corresponding to \(SC=0\) commutations to \(SC=5\) commutations), a simple table can provide the permissible cost.