# Model Predictive Control for Six-Phase Induction Machines with Insight into Past Current Errors
## 1 Introduction  
Multi-phase electric machines present an interesting solution for industrial applications when high power and reliability are required [1, 2, 3]. Even though the early development of these systems was thwarted by their inherent complexity in terms of control, the progress of digital signal processors has alleviated this problem [4]. Better power distribution, fault-tolerant capabilities, and lower torque ripple are some of the advantages that these machines present over their three-phase counterparts [5]. An example of this fact is that the impact of multi-phase systems on the field of electric vehicles is not limited to academic studies [6, 7, 8]. Companies such as BMW, Mercedes, and Audi have incorporated propulsion systems with multi-phase solutions into some of their high-end products [9, 10, 11]. Regarding the regulation of multi-phase electric drives, different control techniques have been extended from three-phase systems. Such is the case with indirect rotor field-oriented control [12, 13], direct torque control [14, 15], and finite control set model predictive control (FCS-MPC) [16, 17, 18] for both induction and permanent magnet synchronous machines [19, 20]. This last regulation scheme has gained traction in the last decade thanks to its high usage of the dc-link voltage, fast dynamic response, and inherent flexibility in adding specific control goals in a simple manner [4]. Despite these benefits, FCS-MPC presented a serious limitation in its early development, showing poor current quality for low values of stator impedance when only one switching state is applied per control action [4]. One possible approach to improve the current quality is the use of multi-vector solutions, which have been explored for systems with three phases [21], but also five [21; 22; 23], six [4; 24; 25], and nine phases [26], among others. By adequately composing new control actions from applying multiple switching states in one sampling period, currents in the secondary subspace can be significantly reduced, thus enhancing the machine performance. Nevertheless, this solution is not without its faults. First of all, applying a higher number of switching states comes at a dual cost: a higher switching frequency and a lower usage of the dc-link voltage [4; 27]. Furthermore, the reduction in currents in the secondary plane comes in part from the higher switching rate. Smaller sampling times inherently lead to an increase in switching frequency, and consequently, the difference in performance between a multi-vector solution and a standard FCS-MPC could reduce. This fact can be especially notable if the control actions of the standard FCS-MPC are pre-selected to ensure a low production of secondary currents [4]. Due to both the advantages and disadvantages of using multi-vector solutions, these schemes coexist with FCS-MPCs that only use one switching state. Some of the single-switching state regulation schemes focus on alternative control goals, such as enhancing current tracking [28], decreasing the VSC switching frequency [27; 29] or mitigating common-mode voltages [30]. These objectives are pursued through the modification of the cost function that governs the selection of control actions, which is the source of the FCS-MPC's flexibility. Nevertheless, regardless of the number of switching states applied per control period, when FCS-MPC is trying to reduce current errors, the cost function is only focused on minimizing the total error. This leaves the door open for the individual growth of the different current errors as long as the sum is somewhat compensated in the present control cycle. This behavior could present a blind spot in the control's ability to keep all errors low. Particularly, the sensitivity of the secondary subspace could lead to the accumulation of current errors through many cycles, worsening the system's performance. The present work contains two contributions (C1 and C2) that, to the best of the authors' knowledge, are novel.