Review of Fast Computation Methods for Finite-State Predictive Control of Multi-Phase Drives

Citation: Marsal, E.; Arahal, M.R.; Satué, M.G.; Perales, M. Review of Fast Computation Methods for Finite-State Predictive Control of Multi-Phase Drives. Appl. Sci. 2024, 14, 11774. https://doi.org/10.3390/ app142411774 Academic Editor: Alessandro Lo Schiavo Received: 22 November 2024 Revised: 12 December 2024 Accepted: 15 December 2024 Published: 17 December 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Review Review of Fast Computation Methods for Finite-State Predictive Control of Multi-Phase Drives Esteban Marsal 1,† , Manuel R. Arahal 1,*,† , Manuel G. Satué 1and Manuel Perales 2 1Systems Engineering and Automation Department, Escuela Técnica Superior de Ingeniería, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain; [email protected] (E.M.); [email protected] (M.G.S.) 2 Electronic Engineering Department, Escuela Técnica Superior de Ingeniería, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain; [email protected] *Correspondence: [email protected] †These authors contributed equally to this work. Abstract: Optimizing the cost function in predictive control of multi-phase drives is computationally intensive. This poses a challenge since the required sampling period for drives falls within the microsecond range. Numerous methods have been proposed in the literature to address this computational demand. This paper reviews recent proposals for multi-phase drives of various kinds. A structured classification of the proposals is provided. Furthermore, an integrated framework is employed to cohesively present and connect previously unrelated methods. Key elements such as Allowed Control Set, inter-sample modulation, and weighting factor use are identified. New developments in multi-vector and single-vector approaches are discussed. Practical limitations for each approach are also considered. Keywords: computational reduction; fast computation; finite state model predictive control; induction motor; multi-phase drives; variable speed 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. Appl. Sci. 2024,14, 11774. https://doi.org/10.3390/app142411774 https://www.mdpi.com/journal/applsci Appl. Sci. 2024,14, 11774 2 of 18 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 α - β plane, is responsible for torque production, while others ( x - y planes) contribute to losses without generating torque. Finally, planes such as z1−z2 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 α - β 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 α - β 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. This poses a challenge for electrical drives, as the sampling period for stator current control needs to be in the microsecond range. Longer sampling intervals lead to less effective modulation of stator voltages and a reduced bandwidth [ 11 ]. Please note that until recently, switching frequency was a major concern for some FSMPC proposals. This concern is progressively being reduced thanks to the introduction of modern power devices based on SiC [12,13] and GaN technologies [14,15]. Various methods have been proposed in the literature to address this issue. This paper examines those approaches relevant to MPS, offering a structured classification of the different solutions. Additionally, an integrated framework is used to present previously unrelated methods. The following section introduces the FSMPC framework for MPS, focusing on stator current control. Section 3provides a classification of previous approaches based on factors that influence computational load. A critical assessment is then presented in Section 4. 2. MPC for Current Control of MPS In this section, the basic elements of FSMPC are reviewed. In the description that follows, a five-phase induction motor is used as a reference and to provide a concrete example. As discussed in the Introduction, MPC concepts have been widely applied in the field of variable-speed drives, with predictive stator current control being one of the most relevant approaches for MPS. The relevant components of this control scheme are depicted in Figure 1 and are detailed below. The reference example is based on the work presented in [16]. 2.1. Drive Control FSMPC for variable-speed drives builds upon the Indirect Field-Oriented Control (IFOC) scheme [ 17 ]. This approach incorporates an outer feedback loop for speed regulation and an inner feedback loop dedicated to stator current control, as shown in Figure 1. For MPS, phase stator currents are transformed using the Clarke projection into the α - β and x - y planes. This transformation enables the decoupling of control variables, simplifying the control strategy. According to the IFOC approach, the torque-generating currents should track a reference value set by the torque/speed control loop. The x - y current references Appl. Sci. 2024,14, 11774 3 of 18 are maintained at zero to minimize energy losses. In this manner, rotor flux and torque are regulated independently. The rotor flux target is given by i∗ d , while i∗ q serves as the reference for electrical torque. These references are projected onto the α - β plane via the Park transformation, resulting in a reference stator current in the α - β space as I∗ sα−β=Di∗ d,i∗ q⊺ , where matrix Dis defined as D=cos θasin θa −sin θacos θa. (1) * iq ωm * * id ωm PI θa FSMPC ωm VSC 5 VSC 1 * iαβ u ii D ωm Allowed VV set Figure 1. Diagram of FSMPC for a five-phase IM. The rotor flux position θa is estimated as θa=Rωedt , where ωe=ωsl +Pω , with P being the number of pairs of poles of the IM and ωsl the slip. This last quantity is related to rotor flux linkage ( ψr ) as ωsl =iqs/(τrψr/Lm) , where τr is the rotor time constant and Lm the mutual inductance. This expression can be simplified so that direct stator current is used instead of rotor flux, as it is the usual approach for IFOC. With this, one obtains ωsl =iqs/(ids ˆ τr) , where ˆ τr is an estimation of the rotor time constant τr=Lr/Rr , with Lr being the rotor inductance and Rr the rotor resistance, respectively. As a result, the set point for stator current tracking I∗ s(k) has an amplitude given by I∗ s=qi∗2 d+i∗2 q . Finally, the α-βreferences can be expressed as I∗ sα(t) = I∗ ssin(ωet),I∗ sβ(t) = I∗ scos(ωet),I∗ sx(t) = 0, I∗ sy(t) = 0. (2) At this point, the role of FSMPC is the generation of stator currents by issuing voltages through a VSI. This process is influenced by the specific VSI configuration and the number of phases, as described below. 2.2. VSI States and Voltages In a power-converter-supplied drive, the phase voltages are determined by the VSI state. This state can be represented as a vector u=(Ka,Kb,· · · ,Ke)⊤ , where the values Kh indicate the state of the corresponding VSI switch for phase h . For a five-phase VSI, there are 32 possible configurations, ranging from u0=(0, 0, 0, 0, 0)⊺ to u31 =(1, 1, 1, 1, 1)⊺ . Each configuration results in a specific set of phase voltages. The stator voltages corresponding to each VSI state can be expressed as V(k) = VDCTMu(k) , where VDC is the voltage supplied to the DC-link, and Tand Mare derived from the VSI structure [18]. The stator voltages can be mapped to the α - β and x - y subspaces using the Clarke transformation. The resulting voltages in the αβxy coordinate system are referred to as voltage vectors (VVs). Figure 2illustrates the distribution of VVs for a five-phase VSI. Appl. Sci. 2024,14, 11774 4 of 18 31 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 0 β α 13 18 4 1 5 8 12 9 13 2 6 3 7 10 11 15 16 20 21 24 28 25 29 18 22 19 23 26 30 27 0 y x 17 14 31 Figure 2. Voltage vectors for a five-phase VSI. 2.3. Basic VV and Virtual VV As discussed in the previous sections, each VSI configuration generates a specific voltage in the α - β - x - y axes. This voltage is referred to as the basic Voltage Vector (VV). It is useful to describe the distribution of these basic VVs according to their magnitude in the α-βsubspace: 1. Large VV (LVV): These are VVs that have the largest magnitude in the α - β subspace, as shown in Figure 2. 2. Medium VV (MVV): These VVs are placed in the second largest corona. 3. Small VV (SVV): These VVs are placed in the smallest corona. 4. Null Voltage (ZVV): This refers to the VVs in the origin. FSMPC can select any of these basic VVs to be applied at each sampling period. Approaches that follow this strategy are known as single-vector FSMPC. However, it is also possible to release a sequence of basic VVs during a single sampling period. This results in a form of explicit inter-sample modulation. The average voltage generated over the sampling period can be interpreted as a Virtual VV or Synthesized VV [19]. While there are several variants of the Virtual Voltage Vector (VVV) technique, it is particularly useful to focus on VVV for x - y elimination. In this method, VVs are chosen such that their average x - y excitation is zero. For example, a Large VV applied during a subperiod TL followed by a Medium VV during TM is selected. The time durations TL and TM are chosen so that they add up to the sampling period TL+TM=Ts . Additionally, the average voltage in the α - β subspace over the sampling period is set to zero, Vxy = 0. This cancellation is achieved by pairing an LVV with an MVV positioned in the opposite part of the x-ysubspace. 2.4. Predictive Model In standard FSMPC, the control action is defined by the state of the VSI, denoted as u . In multi-vector approaches, the control action consists of a sequence of different VSI configurations applied during a sample period. For example, in the case of Virtual Voltage Vector (VVV) techniques, each control action corresponds to a specific VVV. For the following discussion, it may be simpler to focus on the single-vector approach. FSMPC employs a predictive model to relate predicted stator currents to potential control actions. This model is typically represented by a set of discrete-time state-space equations, as ˆ i(k+ 1 ) = (C(ω) + G)i(k) + BV(k) , where i contains the α , β , x , and y stator currents, ˆ i is the prediction of i , and ω is the angular speed. Matrices C and B are Appl. Sci. 2024,14, 11774 5 of 18 obtained from first principles applying time discretization with sampling time Ts . This gives C= (I+AcTs)and B=TsBc, where Ac=    a2−a40 0 a4a20 0 0 0 a30 0 0 0 a3     ,Bc=    c2000 0c20 0 0 0 c30 000c3     (3) In the previous expressions, the coefficients used are a2=−Rsc2 , a3=−Rsc3 , a4=−LMc4ωr , which depend on machine parameters such as c1=LsLr−L2 M , c2=Lr/c1 , c3= 1 /Lls , c4=LM/c1 . Furthermore, term G represents the effect of the rotor currents, which are typically unmeasured. Various methods for estimating these currents have been proposed to improve predictions [20,21]. Using the above expressions, the two-step-ahead prediction of the stator currents can be expressed as ˆ i(k+2) = A(ω)ˆ i(k+1) + BV(k+1)(4) where A=C+Gis introduced for clarity. The predicted control error is then given by ˆ e(k+2) = I∗ s(k+2)−A(ω)ˆ i(k+1) + BV(k+1)(5) 2.5. Cost Function In a general MPC application, the CF incorporates various terms. The primary one constitutes a penalization of output variables from their references. Additional terms include penalization of control effort and other objectives. In FSMPC stator current control, typically, predicted control errors and VSI switching events are penalized in the following manner [22]: J(k+2) = ˆ e2 α−β(k+2) + λxy ˆ e2 x−y(k+2) + λsc∆S(k+1)(6) where ∆S(k+ 1 ) represents the number of switch transitions at the VSI when the configuration u(k) is changed to u(k+ 1 ) . For single-vector approaches, the value of ∆S(k+ 1 ) can be calculated as ∆S(k+1) = 5 ∑ i=1 |ui(k+1)−ui(k)|(7) With this cost function structure, the weighting factors (WFs) are the parameters λxy and λsc . The first parameter emphasizes the reduction of x - y currents, while the second helps minimize the switching frequency. 2.6. Optimization and Deployment In FSMPC, the control move for the k+ 1 period is determined at discrete time k as uopt =argminJ(k+ 2 ) . In earlier studies, the optimization phase of FCSMPC was performed by exhaustive search. This involved computing the value of J for each candidate or allowed control move to obtain the minimum. The deployment of the method in a DSP is relatively straightforward. A flowchart of the FSMPC algorithm is shown in Figure 3. It is interesting to note that the main blocks are the same regardless of the type of machine (induction, permanent magnet, etc.) and regardless of the number of phases. For any particular case, one needs just to adjust the model and a few parameters. Appl. Sci. 2024,14, 11774 6 of 18 32 Candidates to Optimal Control Action Current Prediction (k+1) for i = 1:32 Predictive Control (k+2) Cost Function (g) i = 32 g minimized Application of Optimal Control Action Yes No Figure 3. Flowchart for FSMPC of stator current. The results of FSMPC have been featured in many papers [ 4 ]. In the case of MPS, special attention must be given to the different planes in the VSD decomposition. As an example, check the waveforms in Figure 4, where the α - β components clearly follow their sinusoidal references and the x-ycomponents are regulated around zero. 0 0.005 0.01 0.015 0.02 −3 −2 −1 0 1 2 3 4 Time (s) Currents (A) iα *iαiβ *iβixiy Figure 4. Typical evolution of stator currents for FSMPC. Despite the flexibility of FSMPC, its practical application is hindered by a significant computational load, especially in MPS, due to the large number of possible control moves. As discussed in the Introduction, a high computational load leads to a reduced sampling frequency, which in turn limits the control bandwidth. Additionally, in the case of single-vector approaches, the longer duration of the applied voltages results in significant distortion from the ideal sinusoidal waveform. This issue prompted researchers to explore methods for reducing the time required for the optimization phase. The various approaches developed are categorized in the next section. Appl. Sci. 2024,14, 11774 7 of 18 3. Fast Computation Strategies Numerous FSMPC proposals for MPS address the issue of fast computation. Some of these approaches combine multiple strategies. For example, in the context of Virtual Voltage Vectors, techniques such as control set reduction, elimination of weighting factors, and cost function simplification can be found. To provide a meaningful classification, the following fundamental techniques are considered. 3.1. Reduction of the Allowed Control Set The candidate control moves considered during the optimization phase form the Allowed Control Set (ACS). In single-vector FSMPC, this set is typically a subset of the complete set of VSI configurations. In multi-vector FSMPC, the ACS represents a subset of all possible sequences. To maintain clarity, the single-vector case will be discussed first. In this approach, the ACS is often composed of two or more basic VV subsets (e.g., LVV, MVV, etc.). A specific example involves the combination of the LVV set and the ZVV set, introduced in [10]. Using this idea, ref. [ 23 ] introduces an advanced FCS-MPC. This method employs Virtual Voltage Vectors that automatically suppress third harmonic currents, simplifying the cost function by omitting harmonic terms. Furthermore, an Adaptive Control Set dynamically adjusts Voltage Vector amplitudes based on rotor speed, significantly enhancing performance at low speeds. To ensure stability, delay compensation predicts the system state for the next control interval. By achieving an optimal balance between precision and computational efficiency, this approach is highly suitable for demanding applications such as electric vehicle propulsion. Likewise, an optimized MPC strategy is proposed in [ 24 ], focusing on overcoming the operational performance and computational challenges of traditional methods. The approach introduces a set of 24 Virtual Voltage Vectors uniformly distributed to improve control precision by offering a denser selection of candidates. A novel minimum error duty cycle calculation is included to optimize the relationship between the selected Voltage Vectors and reference currents in the d - q axes, minimizing tracking errors. To reduce computational effort, a multi-stage evaluation method is developed, simplifying the vector selection process and reducing the number of evaluations from 24 to 8 cycles. A preselection based on deviation in stator current and the change in stator current vector is proposed in [ 25 ]. The direction and magnitude of per-unit current vector is considered for a region identification procedure, resulting in a large reduction in computing needs. However, just d3-q3harmonic currents are considered. 3.2. Simplification of the CF and Search Method Another strategy to accelerate the computation of the control signal in FSMPC for MPS focuses on simplifying the optimization problem, achieved through two primary approaches. The first approach simplifies the cost function by reducing the number of terms, including methods that eliminate the need for a weighting factor (WF). The second streamlines the search process by limiting the exploration of the full set of possible control actions, effectively pruning the search tree. It is crucial to highlight that these strategies do not reduce the number of ACSs; instead, they optimize the selection process for efficiency. A method for six-phase PMSM is presented in [26], employing two groups of Virtual Vectors with varying magnitudes. This approach improves torque regulation and eliminates the need for weighting factor tuning in the cost function. Additionally, a one-step lookup table is introduced to exclude unnecessary Voltage Vectors, significantly reducing computation time. Other authors propose a Model-Free Predictive Control (MF-PC) algorithm that eliminates complex operations such as multiplications and parameter estimation, providing enhanced accuracy and robustness to parameter variations while maintaining low computational complexity. Ref. [ 27 ] demonstrates that the method can effectively manage load disturbances and parameter mismatches, achieving reduced harmonic distortion and Appl. Sci. 2024,14, 11774 8 of 18 torque ripple. It highlights the advantages of a parameter-independent control strategy that avoids the computational burden of traditional approaches while maintaining high precision. Ref. [ 28 ] introduces the ABC frame for multi-phase systems, which predicts control actions independently for each phase. By avoiding the need for frame transformations and weighting factors, this approach significantly reduces computational burdens. Simulations and experiments demonstrate its effectiveness, achieving comparable performance to traditional methods while improving efficiency, particularly in systems with multiple phases. Regarding WF elimination, it naturally occurs in Virtual Voltage Vector (VVV) approaches, as x - y currents do not need to be included in the cost function, and switching becomes more predictable. Other works explicitly perform WF elimination, such as in [29–31] . Additionally, some approaches focus on the design of parameter-free cost functions, as seen in [32,33]. Restrained Search Predictive Control (RSPC) is a technique presented in [ 34 ] that reduces computational complexity by limiting the number of inverter switching states that can be evaluated in each sampling cycle. Instead of exhaustively analyzing all possible switching combinations, RSPC imposes constraints that select only a subset of switching vectors, thereby reducing the number of phase changes in the inverter. This approach lowers switching losses and improves system efficiency. RSPC focuses on two main restrictions: selecting only those vectors that involve one or no commutations per cycle and preventing any inverter phase from switching in two consecutive cycles. These constraints facilitate real-time implementation, enhance drive efficiency, and reduce the maximum switching frequency, which is especially advantageous in high-power applications and complex multi-phase systems. Improved techniques of RSPC is presented in [ 35 ] for multi-phase drives, where the commutations of VSI are minimized, favoring the drive efficiency. On the other hand, ref. [ 36 ] presents a computationally efficient multi-step direct predictive torque control (M-DPTC) for surface-mounted permanent magnet synchronous motors (SPMSMs). The method uses a simplified predictive model and a novel optimization approach to reduce computational complexity, achieving logarithmic time growth instead of exponential. By utilizing lookup tables and addition operations, the algorithm avoids exhaustive computations, enabling real-time implementation within short sample periods. Experimental and simulation results validate its effectiveness in reducing switching frequency and losses while maintaining steady and dynamic performance. A novel technique, introduced in [ 37 ], proposes a dual second-order MPC strategy. This approach enhances flexibility in controlling Voltage Vector direction and amplitude, reduces torque and current ripples, and improves steady-state performance. By applying second-order MPC to both speed and current loops, the system achieves faster response times and reduced overshoot compared to traditional PI control. The method employs four-vector space vector PWM (SVPWM) modulation, optimizing the current loop while maintaining minimal computation requirements, as validated through simulations and experiments. The approach presented in [ 38 ] explores two modified MPC strategies designed to enhance the control of asymmetric six-phase PMSMs. The first method integrates MPC within the conventional cascaded speed–current control loop, replacing the current control with MPC to achieve faster response times and reduce harmonic currents. The second method eliminates the cascaded loops entirely, consolidating both speed and current control into a single MPC, which improves the system’s dynamic performance and flexibility. Both approaches utilize Vector Space Decomposition (VSD) to isolate torque-producing components and suppress non-torque harmonics, thereby optimizing the Maximum Torque per Ampere (MTPA) operation and enhancing efficiency in multi-phase systems. An improved control scheme for five-phase PMSMs that enhances tracking accuracy and DC-link voltage utilization while reducing computational load is presented in [39]. It employs a double-vector method where two optimal voltage vectors are selected online Appl. Sci. 2024,14, 11774 9 of 18 during each sampling interval, paired with duty ratio optimization to improve tracking precision in two subspaces ( α - β and x - y ). By limiting the candidate vectors to only the five nearest ones in the reference sector, the method drastically reduces the computational requirements compared to traditional schemes, enabling practical real-time implementation without sacrificing performance. Another approach presents a hybrid exploration state method [ 40 ] that combines four-state and seven-state exploration techniques by dividing the space vector plane into regions based on the reference Voltage Vector’s magnitude and angle. Key benefits include reduced RMS current ripple compared to conventional methods; optimized computational effort, minimizing processing load versus the seven-state method; and improved switching frequency balance, ensuring efficient and stable motor control. This method offers a practical and efficient solution for applications demanding high-precision motor control. 3.3. Explicit MPC Reducing the computational load has been tackled by resorting to general optimization methods, some of which have proven effective in cases where control actions are quantized, as in FSMPC. In Explicit Model Predictive Control (EMPC) [ 41 ], a quadratic optimization problem is decomposed into multiple local subproblems, with their solutions computed offline. Implementing EMPC involves identifying the relevant subproblem (via region determination) and retrieving the appropriate control law from a precalculated table [ 42 , 43 ]. One of the main challenges with reported EMPC applications is the application of a general region-splitting methodology that lacks physical insight, leading to unnecessarily high complexity. Additionally, the inclusion of complex cost functions can make the EMPC problem more challenging. Another general technique is the Sphere Decoding Optimization algorithm [ 44 ]. Likewise, Laguerre function-based MPC with a modulator has been proposed for long-horizon problems [ 45 ]. However, these methods focus on long horizons and do not completely eliminate the need for combinatorial search. 3.4. Dead-Beat Approach Dead-beat control is enabled by direct digital control of dynamical systems. The basic idea is to apply a control action that, after one sampling period, will drive the error to zero. Its application to drives can be traced back to the 1980s [ 46 ]. In the case of FSMPC, using Equation (5), one can derive the voltage that, applied during the k+ 1 interval, would drive the error to zero at k+2. Denoting it by V∗, it turns out to be V∗=B−1i∗(k+2)−A(ω)ˆ i(k+1). (8) Voltage V∗ might not be one of the 32 VVs that the VSI can produce. Then it must be synthesized using various basic VVs applied at different times and for different subperiods during the whole sampling period. The basic VVs that need be applied can be found using trigonometric rules and lookup tables [ 47 , 48 ]. Different approaches have appeared using the DB concept. They have been applied to different types of drives and numbers of phases. 3.5. Region-Based Methods FSMPC based on regions can be seen as a particular case of EMPC where the regions are determined using physical insight. The idea can be put forward for the single-vector case using the following observation. 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