# Review of Fast Computation Methods for Finite-State Predictive Control of Multi-Phase Drives
## 4 Critical Analysis and Discussion
### Multi-Vector Approaches  
Multi-vector methods in FSMPC are characterized by the application of more than one basic VV per sampling period. Early studies combined a non-zero VV with a Null VV to produce an average voltage with a reduced magnitude compared to that of the basic non-zero VV. In this way, the modulus of the voltage is affected so that a closer approximation to \(V^{*}\) is found [62]. Subsequently, this concept was expanded to include combinations of two non-zero VVs, generating an average voltage with specific benefits for certain applications. \begin{table}
\begin{tabular}{l l l} \hline \hline \multirow{2}{*}{Single vector} & Regions & [40; 41; 42; 43; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61] \\ \cline{2-3}  & Others & [10; 24; 25; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37] \\ \hline \multirow{8}{*}{Multi-vector} & 2 VVVs & [62; 63; 64; 65; 66; 67] \\ \cline{2-3}  & 3 VVVs & [68; 69; 70] \\ \cline{1-1} \cline{2-3}  & 4 VVVs or more & [19; 23; 24; 25; 34; 38; 47; 71; 72; 73; 74; 75; 76; 77; 78] \\ \cline{1-1} \cline{2-3}  & Dead-beat & [46; 69; 79; 80; 81; 82; 83] \\ \cline{1-1} \cline{2-3}  & VVV+DB & [70; 84; 85; 86] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Taxonomy of FSMPC for MPS regarding computational reduction. Figure 5: Categorization of fast computing strategies for FSMPC. In MPS, certain combinations of basic VVs can yield a zero average voltage in the \(x\)-\(y\) plane, minimizing \(x\)-\(y\) excitation. This allows the \(x\)-\(y\) terms to be omitted from the cost function, simplifying the computation. Additionally, the set of Virtual VVs is smaller than the complete VV set, reducing the enumeration process needed to optimize the cost function. Further constraints on the VVV sequence can reduce the optimization search space even more [51]. The VVV approach has been applied to various types of multi-phase systems [63; 68]. For example, ref. [47] explores using adjacent VVVs and the vector opposite to the previous optimal one, reducing the exhaustive search to only five evaluations. This method is demonstrated on a two-level, five-phase PMSM system. In terms of experimental application, a limitation of VVV methods is that they manage \(x\)-\(y\) currents in open-loop. Factors such as switch dead time, machine asymmetries, and back-EMF harmonics can lead to unwanted \(x\)-\(y\) currents, increasing copper losses. Another issue concerns DC bus utilization, as combinations of basic VVs do not provide the same voltage range as individual basic VVs [87]. To address these concerns, [88] proposed a method using dual VVVs for \(\alpha\)-\(\beta\) control while regulating \(x^{\prime}\)-\(y^{\prime}\) currents without interference. Closely related to VVVs are multi-vector methods incorporating the dead-beat concept. These aim at generating an average voltage exactly matching \(V^{*}\) by combining basic VVs during a single sampling period. The work of [40] uses DB with hybrid-state exploration to reduce computational load and current ripple. In [89], a similar approach is used with asymmetric space vector PWM. The method in [71] applies two VVs per period, switching in just two phases and evaluating only six candidate VVs at each period. Ref. [90] extends the DB concept to a multi-variable approach controlling all variables within a single predictive loop. Further variations are explored in [48; 91; 92; 93; 94; 95]. Despite its advantages, the DB approach has limitations in terms of experimental application. First, the goal of exact cancellation of the control error is practically unachievable due to unmodeled dynamics and parameter variations. Furthermore, the synthesis of \(V^{*}\) from multiple VVs leads to the average deception problem because the optimal voltage acts only as an average over the sampling period [87]. This issue affects the trade-offs between \(\alpha\)-\(\beta\) tracking and \(x\)-\(y\) regulation, which are particularly significant in multi-phase systems, as \(x\)-\(y\) currents lead to additional losses [96; 72; 97].