# Review of Fast Computation Methods for Finite-State Predictive Control of Multi-Phase Drives
## 1 Introduction  
Model Predictive Control (MPC) was initially introduced in the 1970s to handle complex dynamics, such as those found in chemical processes [1]. Its early applications were primarily for slow processes due to the high computational demand of MPC compared to PID control. The demand is largely due to the optimization stage. In said stage, a cost function (CF) assigns penalties to anticipated deviations of variables from their set points, as well as to additional terms (e.g., control effort). For linear systems with specific assumptions, this optimization process can be relatively straightforward. However, for general non-linear dynamics, the optimization becomes more complex, requiring significant computational time [2]. In the 1980s, as more advanced computing devices became available, the MPC method was proposed for controlling drives [3]. Typical control configurations for AC drives involve an inner loop for current regulation and an outer loop for rotor flux and speed control, with the predictive controller potentially handling both roles. The more challenging task is replacing the inner loop, as it involves a faster dynamics response [4]. A particular predictive setup is accomplished by executing direct digital control of the Voltage Source Inverter (VSI) [5]. In this arrangement, the VSI state is directly set by the MPC, eliminating intermediate modulation stages. This configuration is known as Finite-State MPC (FSMPC) or Finite-Control-Set MPC (FCSMPC) [6]. Here, the VSI state effectively becomes the control signal, and the primary term of the cost function can be specified in terms of either stator currents or electric torque and rotor flux. The first option is of interest in this context, as it has been favored in multi-phase research. The FSMPC approach has been the subject of research for the control of drives of different kinds, including induction, permanent magnet, reluctance, and other systems. In this review, the focus is put on MPS regardless of the kind of machine. It is worth recalling that electric multi-phase systems (MPS) offer certain benefits over conventional three-phase systems. In drives, multi-phase motors exhibit reduced torque oscillations, lower DC-link harmonics, enhanced fault tolerance, and improved phase power distribution [7]. Owing to these advantages, multi-phase drives have attracted research interest for applications like renewable energy systems, vehicle traction, and ship propulsion [8]. In MPS, the collection of phases is simplified by using the Vector Space Decomposition (VSD) method [9]. This projection maps phase quantities onto multiple planes. One plane, referred to as the \(\alpha\)-\(\beta\) plane, is responsible for torque production, while others (\(x\)-\(y\) planes) contribute to losses without generating torque. Finally, planes such as \(z_{1}-z_{2}\) might be absent in most configurations since homopolar currents typically do not flow. Through this decomposition, the FSMPC cost function can be defined in terms of the \(\alpha\)-\(\beta\) and (termed \(x\)-\(y\)) currents as primary objectives [10]. One significant reason for the recent expansion of MPC applications is the adaptability it provides. This adaptability arises from two main aspects. First, the use of a model allows MPC to be applied in challenging scenarios. These scenarios include multi-variable cases, existence of disturbances, delays, and non-linear behaviors [2]. The second factor is the use of a cost function to guide control action selection. The cost function can incorporate numerous terms and constraints, making it possible to address multiple objectives. In the case of FSMPC for MPS, the primary objectives are \(\alpha\)-\(\beta\) tracking and \(x\)-\(y\) minimization. Additional objectives, such as penalizing VSI switching frequency, are also possible [10]. However, the optimization of the CF on a sample-by-sample basis requires considerable computational time.