Model Predictive Direct Self-Control for Six-Step Operation of Permanent-Magnet Synchronous Machines

The optimization of voltage utilization can boost electric drives' power and torque capabilities, which is of particular interest in transportation applications. However, this is a weak point of most control approaches, like linear field-oriented control (FOC) methods using standard modulation schemes. This article introduces the model predictive direct self-control (MPDSC) strategy, a modification of direct self-control for its application in a digital implementation to achieve maximum voltage utilization (i.e., six-step operation) for permanent-magnet synchronous machines. This solution is suitable for highly utilized machines with heavy magnetic (cross-)saturation and low sampling to fundamental frequency ratio. The modifications include using load-angle regulation to control the selected operating point and model prediction to compensate for the actuation delay. The proposed system can achieve six-step operation with accurate torque control and robustness against disturbances and parameter estimation inaccuracies. Comprehensive simulation and experimental results demonstrate the performance of the proposed MPDSC while operating at the voltage constraint. In particular, a transient rise time of 2.3 ms to maximum torque, current reduction for equal torque–speed operating point of up to 18%, and maximum torque increase for equal current amplitude of up to 15% compared to the conventional FOC have been empirically observed.


I. INTRODUCTION
T HE performance of electric drives depends on physical constraints, the components' capabilities, and the control strategies employed. In particular, in automotive applications with a classical two-level voltage-source inverter (VSI), the dc supply (e.g., energy storage) limits the voltage available Manuscript   for control. Optimizing the utilization of this limited resource through appropriate control strategies can boost the torque and power capabilities of the drive without modifying the hardware. However, maximum voltage utilization is a weak point of most classical control approaches. Interior permanent-magnet synchronous machines (IPMSMs) are typically preferred in transportation applications due to their high torque, power density, power factor, and mechanical design, allowing for high-speed operation. The operation within the system constraints of such a machine contains two regions: maximum torque-per-current (MTPC) and flux-weakening operation, which culminates in maximum torque-per-flux (MTPF) operation [1], [2]. MTPC is applied at low speeds to generate the desired torque output with minimum ohmic losses. Increasing the speed leads to higher voltage demand due to the voltage induced by the magnetic flux linkage. After reaching the voltage limit, the desired torque can no longer be achieved with MTPC since some current must be invested in reducing the machine flux linkage (i.e., flux weakening operation) to decrease the induced voltage. With further reduction of the maximum allowed flux linkage, the maximum torque operation is no longer obtained at the current limit but at the MTPF locus. Fig. 1(b) shows the loci for MTPC and flux weakening (comprised of MTPF and current limit) paths for an exemplary IPMSM.
Six-step modulation must be applied to attain maximum voltage utilization, i.e., sequentially switching between the six active voltage vectors (counter-)clockwise during one fundamental period. This means that the degrees of freedom for control are severely limited, as the applied voltage has a fixed amplitude, and the control strategy can only choose when to switch to the next active voltage vector. Fig. 1(a) depicts in green the fundamental component of the voltage achieved by this method. No further voltage utilization improvements are possible beyond this strategy, regardless of the control method. The six-step modulation has other consequences regarding increased harmonic content on the current, voltage, torque, and flux linkage (cf., Fig. 13). The control structure cannot address these disadvantages, as it is an inherent property of the modulation strategy required to maximize the utilization of the voltage given the physical constraints of the VSI, i.e., any deviations would require an introduction of additional switching events, which would reduce the voltage utilization. As they are inevitable, the tradeoff between additional ripple and voltage utilization is not a role of the proposed controller but of an upper level strategy accounting for the drive's desired performance.
The industry's most established control strategy for electric motors is field-oriented control (FOC), which independently controls the d-and q-axes currents. For low-speed operation, the desired voltage can be synthesized on average over a switching period with standard approaches such as pulsewidth modulation (PWM) or space vector modulation (SVM). This is no longer the case when the voltage command goes beyond the limit of the linear modulation range. The unavailability of an arbitrary dq-voltage command of such amplitudes (cf., Fig. 1) reduces the practical degrees of freedom of the system in that region, i.e., the dq-currents cannot be independently controlled anymore.
Furthermore, the machines employed in traction applications have specific characteristics, as are very low inductance values including operating point-depending magnetic (cross-)saturation, operation at high electric fundamental frequencies (up to 1-2 kHz), and relatively high voltage and power rating. These factors can limit the application of overmodulation and six-step strategies, as highlighted for the relevant literature in Fig. 2, which will be discussed in the following.

A. State of the Art
The problem of voltage utilization has been the subject of extensive research for industrial and transportation applications. Operating around and beyond the linear voltage limit introduces several issues that need to be considered to maintain the drive's controllability and performance. In the overmodulation region, there is increased ripple and noise in both controlled and uncontrolled variables, which increases the difficulty of control and sampling, and affects the performance [3], [4], [5], [6].
FOC strategies require a safety voltage margin for dynamic current control [4], [7], which is naturally unavailable at the voltage limit. The bandwidth of the controllers is compromised under these conditions, as well as with the use of filters to reduce the harmonic content of the measured signals [3]. Additionally, appropriate saturation and antiwindup strategies [7] are necessary. The cross-coupling between the currents increases with speed [7], and the decoupling within FOC does not work when leaving the linear modulation region. Also, after the base speed is reached, flux-weakening strategies are required, such as dynamic modulation control [8]. These strategies may conflict with each other when all simultaneously attempt to modify the current reference fed to the controllers [9]. If multiple operating modes address these issues, an appropriate transition between them must be ensured for stability and performance [4], [6].
Control strategies enhancing FOC for overmodulation and six steps exist in the literature employing induction machines (IMs) [5], [10] and permanent-magnet synchronous machines (PMSMs) [3], [9], [11]. Among the latter, only Kwon et al. [3] achieve six-step operation, by creating an equivalent voltage margin exploiting the induced voltage through an appropriate dynamic setting of the current references. Kwon et al. in [9] achieve high-voltage utilization operation through a similar principle. While Kim et al. [11] provide great insight into the compensation of SVPWM issues with a remarkably low number of samples per fundamental period, it does not include any discussion of how the controllers handle the challenging conditions during overmodulation (i.e., lack of voltage margin and voltage saturation) or how this affects the dynamic performance of the current control. More often, different control strategies are used in conjunction with FOC to avoid the issues discussed previously, such as load-angle control (LAC) [4], voltage angle control (VAC) [6], or modulating voltage-scaled controller (MVSC) [12]. The transition between control strategies is a critical feature that must be handled appropriately to avoid unwanted transients and maintain control of the drive.
Alternative control strategies, such as model predictive control (MPC), can be applied instead to control the machine throughout the speed range with a single method. MPC methods can typically operate with superior dynamics to FOC while accounting for the system constraints. Several MPC methods can achieve high-voltage utilization but do not reach six-step operation [13], [14], while a few do [15], [16], [17], [18]. Nevertheless, they generally require high processing power and high sampling to fundamental frequency ratio. During this investigation, other strategies have been presented, such as the dead-beat flux control (DBFC) proposed in [19] and the adaptation of direct self-control (DSC) in [20], capable of operating IPMSMs with six-step modulation.
From these strategies, those that are demonstrated in the literature to be capable of achieving six-step operation with PMSMs are compared in Fig. 2 regarding the challenges related to highly utilized machines. The minimum number of samples per fundamental period (min(T e /T s )) relates to the challenge of achieving robust control at high speeds when the measurements are only sampled and the actuating commands only updated a few times per electrical rotation, and with reduced degrees of freedom (inherent to six-step operation). The normalized current transient achieved in one sampling period when applying maximum voltage ( 2u DC T s 3L s I max ) relates to the dynamic behavior between sampling instants. Highly utilized machines have small inductances leading to a large current transient. Furthermore, these results consider nominal inductances, the maximum current transient would be even more significant in the high-saturation region. The strategy in [19] is excluded from this comparison for lack of machine information, although from the results provided, the conditions approach that of MPDSC .
The attempts to adapt the strategies discussed in Fig. 2, focusing on [3], to control the considered highly utilized IPMSMbased automotive drive in six steps were unsuccessful. 1 We do not claim that our efforts in this area have been exhaustive, and we have attributed the difficulties encountered to the combination of the more challenging control characteristics shown in Fig. 2 in green. Hence, there is a need for computational lightweight and robust six-step control schemes, which can apply to highly utilized IPMSM drives under these conditions. A solution proposal based on DSC is introduced in the following.

B. Direct Self-Control (DSC)
In [21], Depenbrock introduced DSC for its application on IMs with continuous-time control. This strategy generates the switching pulses according to the comparison between the lineto-line flux linkage and an externally generated reference, as shown in Fig. 3. The flux linkage reference amplitude dictates the frequency of the fundamental component of the flux when no zero-voltage pulses are introduced. Once a given line-to-line axis reaches the flux linkage reference, the appropriate switching is triggered to maintain that flux linkage at the reference value.
The system does not react to a change in the flux linkage reference value until the next switching event, i.e., the six-step voltage sequence is always followed without modulation between switching states. This way, the control strategy integrates pulse generation, removing the need for voltage margin to ensure controllability. The controlled variables are the line-to-line (or β-axes) flux linkages in the stationary reference frame. This avoids the complications for control that harmonic distortion introduces in the synchronous reference frame variables during the six-step operation.
The DSC structure has been explored for six-step operation in the literature since [21]. Research has focused almost exclusively on its application to IM control [22], [23], [24], including different inverter configurations [24] and modifications to the original control strategy [23]. Steimel [22] remarks the superior performance of DSC against synchronous pulse patterns in various issues (e.g., peak currents, motor losses, torque ripple, and dc bus current harmonics) due to its superior voltage and switching frequency utilization. Steimel [22] also notes the greater robustness of DSC against input voltage disturbances.
DSC-called methods have been applied to PMSMs [25], although their working principles deviate from the original control strategy in [21]. An example of DSC six-step operation control for an IPMSM is achieved in [20] by the combined action of modifying the estimated flux linkage according to the reference load angle and a torque control scheme for improved high-speed operation. However, the system proposed is not tested for a highly utilized machine, nor at very high speeds with low samples per fundamental period (cf., Fig. 2). Flux control is executed with hysteresis comparators in discrete time, which introduces flux control errors that would become more significant at high speeds. Likely due to this issue, substantial deviations are noticeable between current peak values in the results shown operating in six steps in steady state. This article addresses these issues and deviates from [20] in its core control principles. Nevertheless, both employ similar ancillary strategies (e.g., flux estimation via Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. an observer) and share their conceptual approach, i.e., to achieve torque control based on the load angle.

C. Contribution
The contributions of this article, introducing the model predictive direct self-control (MPDSC) strategy, can be summarized as follows.
1) Adaptation of the DSC strategy, nested with load-angle regulation, for control of the six-step operation in a highly utilized IPMSM in the high-speed flux-weakening range including MTPF. 2) Model-based flux linkage prediction to compensate for the digital sampling delay to achieve low sampling rates (e.g., for low-cost embedded hardware implementation).

3) Compensation of the inherent six-step ripple in load angle
for the improved controller performance. 4) Experimental demonstration of the proposed strategies with a machine intended for a real traction drive under realistic frequency, speed, and power conditions, proving the method applies to highly utilized drives, including magnetic saturation. Maximizing the utilization of the available voltage is only beneficial when the voltage limit restricts the operation (i.e., in the flux weakening range). Therefore, the proposed MPDSC approach is intended only for operation in the flux weakening range so that six-step modulation is feasible. An example of the benefits of the proposed control strategy can be seen in Fig. 1(c), which shows the maximum curves for torque and power in steady state with six-step modulation, achievable by MPDSC, and with linear modulation region, to which conventional FOC + PWM/SVM is limited to.

D. Notation and Structure
Hereafter bold characters (x, X) represent vectors or matrices, respectively, a bar (x ) represents the average value of a variable, a star (x * ) represents a reference variable, and a hat ( x ) represents an estimated variable. All figures show normalized variables in relation to the machine's maximum current, dc bus voltage, base speed, nominal torque, and permanent-magnet flux linkage, respectively (cf., Table III).
The rest of this article is organized as follows. Section II describes the modeling of the drive for the analysis of the six-step operation. Section III focuses on the structure of the proposed MPDSC. Section IV presents simulation results for said structure dealing with torque reference tracking. Section V describes the experimental setup to validate the proposed strategy and presents experimental results exemplifying the proposed strategy's steady state, transient operation, and disturbance rejection capabilities. Finally, Section VI concludes this article.

II. DRIVE MODEL AND OPERATION CONSTRAINTS IN A SIX-STEP MODE
The drives in automotive applications typically employ highly utilized IPMSMs that exhibit significant magnetic (cross-)saturation. Such a machine can be described in the stator reference frame as where R s is the stator resistance and Ψ abc , u abc , and i abc are the flux linkage, the voltage, and the current in the abc-reference frame, respectively. The transformation from three-phase coordinates (abc) to the stationary orthogonal reference frame (αβ) to the rotor reference frame (dq) is calculated with where θ e is the rotor electrical angle and † depicts the matrix pseudoinverse. In the rotor reference frame, the machine equations are given by (4)-(7) as follows: T is the stator flux linkage, ω e is the electric rotational frequency, J is the rotation matrix, and L dq,Δ is the matrix of differential inductances [26]. The machine inductances strongly depend on the operating point, as evidenced by Fig. 4, and can further depend on temperature and speed due to iron losses. The electromagnetic torque can be calculated from the flux linkage and current in the dq-reference frame as where p is the machine pole pair number. A one-to-one mapping exists between operating points defined by the dq-current and the dq-flux linkage (cf., Fig. 4). Therefore, the torque can also be controlled through the selection of flux linkage amplitude and load angle, as given by Fig. 5 and  where Ψ s is the flux linkage amplitude, δ is the load angle, θ s is the flux linkage electric angle in stationary coordinates, and θ e is the rotor electric angle (i.e., the direction of magnetization of the rotor). This also applies for highly utilized machines with magnetic saturation effects. The MTPF point determines the maximum torque achievable by modifying the load angle for a certain flux linkage amplitude. The dc bus voltage, the rotational speed, and the voltage utilization limit the maximum achievable flux linkage amplitude.

A. Six-Step Operation
According to (1), neglecting the resistive voltage drop, the flux linkage derivative entirely depends on the stator voltage. The voltage has a fixed amplitude during the six-step operation, and its phase varies discretely between the six active voltage directions. This results in the hexagonal flux linkage path observed in Fig. 6, where Ψ r is the rotor flux linkage vector, Ψ s is the stator flux linkage vector, and Ψ h is the hexagonal flux linkage path amplitude. The consideration of resistive voltage drop would distort this shape to a small extent. However, the following approximations remain valid, especially for highly utilized motors with minimal stator resistance values.
The DSC strategy can maintain control of the hexagonal flux linkage path amplitude following the six-step voltage sequence. From the instant shown in Fig. 6, different flux linkage paths are possible that achieve this while increasing (green), maintaining (blue), or reducing (red) the hexagon path size.
Since the voltage amplitude does not change, the tip of the flux linkage space vector travels at a constant rate through the hexagonal path [cf., (1)]. The average flux linkage rotational frequency can be approximated based on the size of the hexagon as The load angle is dynamically modified depending on flux linkage amplitude in six-step operation as This provides a mechanism for six-step operation control with a single degree of freedom. The flux linkage amplitude reference fed to the flux controller should ensure stability, accurate loadangle reference tracking during steady-state operation, control of transients, and rejection of disturbances caused by changes in speed, machine parameters, or other sources.

B. Inverter Model
The following implementation is intended for a typical VSI, as represented in Fig. 7. The inverter and its drivers will introduce other nonlinear behavior to the system, such as inaccessible operating modes (e.g., short pulse removal), forward and resistive voltage drop, actuation delays, and required safety interlock time. These issues, particularly the time delays, must be accounted for in the switch-time calculation and voltageestimation systems to allow optimal MPDSC performance. The implemented compensation of these nonlinearities follows the same principles explained in [27], less the nonlinear switching transients.

III. MPDSC STRUCTURE
The MPDSC strategy generates switching time commands through a model predictive flux control (MPFC) strategy based on comparing the machine flux linkage in the stator reference frame with an externally generated reference. A load-angle controller (LAC) 2 generates this flux linkage amplitude reference. A suitable operating point strategy selects the load-angle reference to achieve the desired torque. The LAC and the MPFC require flux linkage estimation. Fig. 7 shows the proposed structure for implementing MPDSC.

A. Model Predictive Flux Control (MPFC)
The proposed MPFC is a modification of the strategy presented in Section I-B for its implementation in discrete time for processor-based applications with a low sampling rate. The MPFC follows the algorithm represented in Fig. 9, where k is the discrete-time sampling index. An integral part of the solution is the use of flux prediction based on the machine model (1), which enables minimal flux amplitude tracking errors with extremely small sampling to fundamental frequency ratio (6 < min(T e /T s ) < 7). Fig. 8 shows an example of such an operating point, depicting the variables involved and their interaction for switching command generation. This low number of samples per fundamental period is only demonstrated here through simulation results due to test bench constraints (load machine maximum speed), where the minimum achievable is 12.5. (1) is discretized following the explicit Euler method. Based on the final switching state command from the previous sampling period, the expected line-to-line flux linkage variation is calculated as

1) Flux Linkage Variation Estimation: Equation
where (i, j) are phases (a,b), (b,c), and (c,a) in order to provide the solution for each of the line-to-line axes, T s is the sampling period, and s i (k + 1) is the switching state in the phase i at the end of the momentary sampling period k. This consideration ensures that the estimation is accurate in the sampling period prior to switching. This solution is valid for operation above six samples per fundamental period, as seen in Fig. 8.

2) Flux Linkage Prediction:
To overcome the digital control delay, the MPFC requires a one-and two-step-ahead flux linkage prediction. This discrete-time prediction model can be realized as where Ψ L−L is the vector of estimated line-to-line flux linkages. This estimation is only accurate if Δ Ψ L−L is constant for the two following sampling periods, which is always the case before the flux linkage prediction triggers a switching event (cf., Section III-A4, Fig. 8). The reason is that the flux linkage variation in the relevant axis will not change significantly for the prior two active voltage vectors in the six-step sequence.

3) Time-to-Reference Calculation:
The time to reach the upper and lower references given the flux linkage prediction one step ahead and the expected line-to-line flux linkage variation is where Ψ * H is the hexagonal flux linkage amplitude reference for the line-to-line flux linkage (cf., Figs. 3 and 8).

4) Switching Criteria:
The switching criteria c(Ψ L−L ) determine whether, before the end of the next sampling period, switching is required to keep the flux linkage within the reference boundaries. They are generated based on the hysteresis comparison between the two-step-ahead flux linkage prediction and the flux linkage reference as given by These signals are equivalent to the switching pulse commands in the continuous-time implementation shown in Fig. 3, shifted (i.e., c(Ψ ij ) ↔ s j ). The initialization of these signals is performed according to the flux location (e.g., for the situation depicted in Fig. 6, with Ψ s (k) located between u 1 and u 2 , the initialization value would be c(Ψ ) L−L (k) = [1 0 1] T ).

5) Switch Timing:
The switching criteria select the appropriate switching time for each semiconductor to keep the lineto-line flux linkage in the reference boundary. Tables I and II summarize the selection process, where i is the relevant inverter leg, c i is the switching criteria used, and t i is the switching on and off time selected. The subindex H and L denote the higher and lower semiconductors. The output is, for each phase i, the vector Table II is structured following the sequence "keep OFF," "switch ON," "keep ON," and "switch OFF." Unless the switching command acts otherwise, at the beginning of each sampling period, the higher semiconductors are OFF, and the lower ones are ON. Feed-forward inverter nonlinearity compensation can be performed by subtracting the delay and interlock dead time from the switching time command  in the relevant scenarios to improve the accuracy of the switching time selection, and therefore, the flux linkage control.

B. Voltage and Flux Linkage Estimation
The control strategy requires knowledge of the machine flux linkage, for LAC and MPFC. The stator flux linkage is estimated using a Gopinath-style flux observer [28], which combines a current-mode estimation (6a) with a voltage-mode estimation (1) through a correction factor generated by a PI controller as where the superindex "Gop" refers to the output of the Gopinathstyle observer, and "I mod" refers to the output of the current mode. Appropriate coordinate transformations [cf., (2) and (3)] are applied to obtain the desired output variables (i.e., load angle and line-to-line flux linkage). The fade-over frequency between the current and voltage modes is chosen below 10% of the base speed. This ensures that the voltage model is dominant in the flux-weakening region, which introduces favorable behavior since it is essentially independent of parameter estimation accuracy (other than the stator resistance, whose effect is minor). Combined with the LAC loop, this ensures a stable and robust six-step operation.
The voltage estimation takes into account the reference switching times t * s,abc and the dc bus voltage u DC to calculate the phase voltage. Accounting for the distortion to those commands by the inverter drivers and the nonlinearities of the inverter itself (cf., Section II-B) significantly improves the accuracy. For this, predicting the two-step-ahead current is necessary to accurately account for the current polarity at the instant of switching, and therefore, the voltage during interlock dead time.
During six-step operation, the load angle presents an inherent ripple due to the stator flux linkage rotational frequency dependence on the location on the hexagon path. Considering a normalized hexagon with a corner amplitude equal to one as seen in Fig. 10, the normalized flux linkage components in αβ-coordinates can be calculated in the first hexagon sector (i.e., 0 ≤θ s,1 ≤ π 3 ) as whereψ α andψ β are the αβ-components of the normalized stator flux linkage,θ s is its angle,ω e is the rotational frequency of its fundamental component, andθ s,1 is the angle of its fundamental component. The angle of the normalized flux linkage can then be calculated asθ The inherent load-angle ripple results from the difference between the angle of the flux linkage,θ s , and that of its fundamental component,θ s,1 . This difference is independent of the operating point and can be generalized for calculation from the estimated stator flux linkage angle θ s as (23) or stored in an LUT to reduce the computational effort.
Removing this value from the estimated load angle before using this signal in the LAC prevents the controller from generating a control action (i.e., flux linkage amplitude reference) to compensate for this ripple. Moreover, due to aliasing effects and depending on the sampling frequency and rotational frequency, this ripple can be amplified through closed-loop control if it is not accounted for. This would cause unwanted low-frequency ripple in the rest of the machine variables. Thus, the load-angle ripple compensation allows for the discrete-time implementation of the proposed control strategy with arbitrarily chosen sampling frequency, within reasonable bandwidth boundaries.

C. Load-Angle Control (LAC)
A closed-loop torque controller could generate the flux linkage amplitude reference. However, the relation between load angle and torque is highly nonlinear and depends on the operating point (cf., Fig. 5). The operation around MTPF is particularly critical, as the dependence of torque on load angle changes sign. A torque controller that does not account for this effect would cause instability. A closed-loop LAC can be used instead (cf., Fig. 7). This way, the system's dynamic behavior does not change in the neighborhood of the MTPF operating point [cf., (12)]. Divergence from the correct MTPF operation would only introduce a steady-state torque deviation.
A combination of feed-forward control [estimating the steadystate flux linkage amplitude for six-step operation based on dc bus voltage and electric frequency as (11)] and feedback load angle control (through a PI controller with constant gains) generate the flux amplitude reference. Using such a structure provides a very robust strategy, as the controller adjusts the flux linkage amplitude following changes in the operating point to maintain the desired load angle, ensuring a stable six-step operation.
From the LAC perspective, the plant can be modeled as Fig. 11, which is linearized as where s is the Laplace complex frequency variable. The first term in (24) models the sampling-to-actuation delay T s . Then, the MPFC time response is considered, which is nonlinear as the line-to-line flux linkage evolves in a straight line with slope u DC . The initial location of the stator flux linkage when the reference changes also affects the response time, as the next reference will not be achieved until the next corner is reached. For its linearization, in the second term of (24), a PT 1 response is assumed, with a time constant equal to its worst-case scenario (i.e., reference change immediately after a switching event). Moreover, the dependence of the response time on u DC and Ψ * h can be modeled as a dependence on the speed operating point. The third term in (24) represents the flux-linkage-averagerotational-speed response to a change in hexagonal flux linkage amplitude, which can be modeled from (11) and is linearized around a specific speed-dc bus operating point as Finally, for controller tuning purposes, (24) is discretized with the Tustin method as where z is the z-transform complex frequency variable. It should be noted that all systems considered in Fig. 11  (24), (26) are independent of machine parameters. Therefore, so will the controller tuning. A controller design can be executed in a quasi-continuous way using (24) or directly in the discrete-time domain using (26). Furthermore, by accounting for (25), inverted, in the LAC as seen in Fig. 7, the operating-point dependence of the response with constant controller gains is significantly reduced. The sign of the speed must also be considered, as the change in the load angle has opposite polarity to changes in flux linkage amplitude depending on the direction of rotation.
The constant controller gains are tuned analytically for the relevant dc bus voltage and speed operating range, based on classic control tuning strategies (such as transfer function and Bode plot analysis) to ensure stability, limit the overshoot, and achieve maximum dynamic performance. Further trial and error tuning optimization was performed in simulation and experimentally, resulting in K P = −1.9 · 10 3 s −1 and K I = −6 · 10 4 s −2 .

D. Operating Point Strategy
The load-angle reference must be selected to achieve the desired output torque at the given operating conditions (dc bus voltage and speed). General operating point selection strategies acting within the flux-weakening region known from the dq-current plane could be applied after a transformation to the load angle-flux linkage amplitude plane using stored machine data. Hereafter, an operating point selection specific for MPDSC is proposed, as shown in Fig. 7, generating the loadangle reference from the torque and flux amplitude reference values.
The external torque reference is limited to avoid exceeding the MTPF or the current limit estimated through LUT T lim . The flux linkage amplitude of the current operating point is estimated from the flux linkage amplitude reference generated by the LAC. It is first fed to a low-pass filter (LPF) to decouple the load-angle reference generation from higher frequency behavior (cf., Fig. 7). This way, the reference load angle is isolated from the quick changes in flux linkage amplitude during torque transients. For example, in Fig. 13, the load-angle reference is barely affected by the changing flux linkage amplitude during the transient, and no delay in torque control is introduced. The flux linkage input to LUT δ * is also slowly adjusted when the speed or dc bus changes, and therefore, the required flux linkage amplitude changes.
To generate the data for LUT δ * , a machine characterization is performed while operating with MPDSC. A representative range of load-angle reference values is enforced at different flux amplitudes (selected during six-step operation through an appropriate choice of speed and dc bus voltage combinations). The steady-state values of load-angle reference, flux amplitude reference, dq-current, and measured torque are stored. This data are processed to relate the load-angle reference values to torque and flux linkage amplitude reference as Fig. 12. Details regarding data storage requirements are discussed in Section V. Based on the characterization data, the MTPC and MTPF loci are defined as Accurate characterization of the MTPF locus is important to guarantee the torque utilization of the drive. Accurate characterization of the MTPC locus is also required, as MPDSC is only intended for operation in the flux-weakening region. For general MTPC operation, other control strategies (e.g., FOC) would be more desirable. Further detailed information on these operating regions can be found in [1] and [2]. Generating the data for LUT δ * based on experimental data from the online operation of MPDSC has the benefit of embedding the nonlinear effects associated with the six-step operation, machine magnetic saturation, and estimation inaccuracies. This   TABLE III  PARAMETERS OF IPMSM, LOAD MACHINE, INVERTER, AND CONTROLLER allows for high torque control accuracy despite these issues. This reference generation method could be further enhanced by online parameter estimation (e.g., as [29]) or other system identification methods to account for parameter variation through the drive lifetime and according to temperature.

IV. SIMULATION RESULTS
The parameters of the drive model used in the simulation and the later experimental setup are summarized in Table III. The system's response to a torque step transient from T * = 0.2T n to maximum torque operation at ω = 2ω 0 is shown in Fig. 13.
The control achieves the desired change in torque while continuously following the six-step voltage sequence. Note that the operation goes momentarily beyond the MTPF locus without any control issues. The load-angle reference estimation accurately tracks the average value of the load angle, removing the inherent six-step ripple without adding any delay to the control. The load-angle reference is isolated from the flux linkage amplitude transient through the LPF. The torque dynamics follow the load-angle response very closely with a slight initial reduction due to the flux amplitude transient. The flux linkage amplitude reference is accurately achieved with one step delay once one line-to-line flux linkage can reach the reference (also at the lower limit, which is not shown in the figure).
The simulation and experimental results discussed in this article serve as practical proof of the system's stability. A formal verification (e.g., Lyapunov) would be possible only when applying severe simplifications due to nonlinearities in the system and control strategy, removing practical applicability.

V. EXPERIMENTAL RESULTS
The MPDSC strategy has been evaluated in a laboratory test bench, shown in Fig. 14. The electrical drive system under test is a highly utilized IPMSM intended for a traction application. Its rated parameters are summarized in Table III, and Fig. 4 shows its magnetic saturation effects. Standard offline characterization procedures identified the motor parameters, leading to the data for the machine model employed in the simulation evaluation and LUT generation for control purposes (cf., Section III-D). The MPDSC requires storing ca. 4 700 data points out of which ca. 3 700 are also used by FOC, including control gains and LUTs. The optimization of data usage was not carried out in this investigation.
A two-level IGBT inverter (Semikron 3xSkiiP 1242GB120-4D) provides the stator voltage. The load is a speedcontrolled IM (Schorch LU8250M-AZ83Z-Z, maximum speed 12 000 min −1 ), mechanically coupled with the test motor. The control strategy was implemented in a dSPACE DS1006MC rapid-control-prototyping system. The dSPACE analog-todigital converters process the measurements synchronously with the control task at a fixed frequency of 10 kHz. Additionally, measurements from current probes (Iwatsu SS-270) and differential voltage probes (Bumblebee 1 000 V CAT III) were captured in an oscilloscope (Yokogawa DL850) to observe highfrequency phenomena. Finally, the testbench setup includes a torque transducer (HBM-T10FS), whose measurements are only relevant for steady-state analysis due to the limited bandwidth of the sensor and the mechanical resonance at the shaft connection.
The turnaround times of the different control parts have been evaluated by compiling multiple versions of the controller, selectively eliminating specific sections. 3 The results can be observed in Table IV. 4 The MPDSC involves a comparable execution time as FOC, with the most significant contributor being voltage estimation. This is due to the iteration performed to include an accurate two-steps-ahead current prediction for the correct estimation of the passive rectifier operation during  IV  TURNAROUND TIMES FOR REAL-TIME IMPLEMENTATION OF MPDSC (μ IS THE  MEAN VALUE AND σ IS THE STANDARD DEVIATION OF THE DATA COLLECTED) interlock-safety dead time, which could be further optimized to reduce the computational burden. Nevertheless, the overall computational burden for the MPDSC can be considered as lightweight as other standard control approaches, therefore, making it particularly suitable for low-cost microcontroller implementation in industrial or transportation applications. The line-to-line voltage measurement acquired with differential voltage probes shows that the six-step sequence is kept. A small error exists in flux linkage amplitude control, primarily due to slight inverter model and flux linkage prediction inaccuracies. This error introduces some disturbances in LAC, for which the controller compensates. The deviation in torque steady state from the measurement remains small (ca. 2%). Fig. 16 depicts the peak motoring and generating torque achievable by the MPDSC strategy and conventional FOC for a fixed current amplitude. The benefits of the higher voltage utilization are clear, as MPDSC can operate at a significantly higher torque (up to 15% increase) under the same operating conditions. Conversely, for achieving the same torque output a lower current amplitude (up to 18% reduction) would be required by MPDSC under the same conditions.

B. Torque-Transient Operation
Examples of the response to a change in torque reference with MPDSC and conventional FOC can be observed for two different speeds in the flux-weakening region in Figs. 18 and 19, for motoring and generating operation, respectively. During the transient, the frequency of the stator flux linkage is adapted through its amplitude to obtain the desired change in load angle. The dynamic capabilities are very satisfactory, achieving the desired torque with a short rise time (ca. 2.3 ms) from the applied reference step.   These sharp torque transients introduce a significant disturbance in the dc bus voltage, as evidenced by Fig. 17, as well as by the phase voltage measurement in Figs. 18 and 19. However, the control strategy shows robustness against this disturbance, with only a slight deviation of the torque operating point during the dc bus voltage transient (e.g., near 10 ms in Fig. 19).
These figures also depict the performance of FOC, limited to the linear modulation region, regarding torque response and measured current amplitude. The dynamic performance of MPDSC is superior to that of the implemented FOC. Moreover, the current to achieve the same final torque is smaller with MPDSC.

VI. CONCLUSION
This article introduces the MPDSC strategy. Through a combination of MPFC with LAC, a suitable operating-point selection, and robust flux linkage estimation, this strategy successfully achieves maximum voltage utilization through six-step operation in a highly utilized PMSM. The strategy can maintain a high dynamic and robust torque control during six-step operation and for a very low number of samples per fundamental period (close to 6). The computational effort for real-time implementation is comparable to that of conventional FOC. In the considered drive, the proposed strategy achieves a rise time to maximum torque operation of 2.3 ms, a current reduction for an equal torque-speed operating point of up to 18%, and a maximum torque increase for an equal current amplitude of up to 15% against conventional FOC limited to the linear modulation region. The benefits of such a strategy in terms of voltage utilization have been demonstrated in simulation and experimentally.
Future investigations will focus on issues kept out of the scope of this article. Additional simulation and experimental analyses have been performed, considering the fade-over between MPDSC and other control strategies (e.g., FOC), which enable the system to operate as their enhancement, facilitating its quick adoption by the industry. Moreover, the effects of detuning and incorrect parameters and flux linkage estimation have been studied, verifying the proposed control strategy's stability and robustness, and making it suitable for sensitive applications such as transportation.