Time-Optimal Model Predictive Control of Permanent Magnet Synchronous Motors in the Whole Speed and Modulation Range Considering Current and Torque Limits

Improving control dynamics and enabling maximum torque and power conversion for a given electrical drive are important target quantities of drive control algorithms. To utilize the electrical drive to its maximum extent during transient and steady-state operation, a time-optimal continuous-control-set model predictive flux control (CCS-MPFC) for permanent magnet synchronous motors (PMSMs) is proposed. This scheme considers torque and current limits as softened state constraints in the CCS-MPFC's optimization problem to prevent transient overcurrents as well as torque over- and undershoots during time-optimal operating point changes. Furthermore, the overmodulation range including six-step operation can be entered seamlessly to ensure maximum power conversion at high speeds. Fastest transients within the whole modulation range are enabled by a time-optimal harmonic reference generator. Here, the flux reference of the CCS-MPFC is complemented with a harmonic content that enables operation in the overmodulation region. Further, the reference is prerotated during transient operation to attain time-optimal control performance. Extensive simulative as well as experimental investigations for linearly and nonlinearly magnetized PMSMs show that, compared with state-of-the-art methods, time-optimal control performance in the whole modulation range without transient overcurrents as well as torque over- and undershoots can be achieved by the proposed control method.


I. INTRODUCTION
Permanent magnet synchronous motors (PMSMs) are often applied for drive applications when high control dynamics and high torque and power densities are required.To maximize control dynamics as well as torque and power density for a given maximum current of the PMSM and dc-link voltage of the inverter, the drive system, i.e., inverter and motor, must be utilized to its maximum extent during all operation modes. 1  During transient operation, the acceleration and jerk of the motor depend on the torque control dynamics.Therefore, 1 Transient and steady-state operation.
enhancing the torque response dynamic of drive controllers is an active and relevant research area in academia and industry.Here, control characteristics, such as torque overshoots and undershoots as well as overcurrents, are undesired.
During steady-state operation at the voltage limit (also known as flux-weakening region), the maximum torque is limited by the available fundamental voltage of the inverter output.To maximize the fundamental voltage at the motor terminals, it is essential to utilize the given inverter's dc-link voltage to its maximum extent by entering the overmodulation range up to six-step operation [1].Here, the dc-link voltage is fully utilized and the fundamental of the inverter's FIGURE 1. Linear and overmodulation range, see (18), as well as the definition of the elementary voltage vectors s n , switch positions of the inverter, and hexagon sectors [2].
output voltage can be increased by approx.10% compared with the linear modulation range, cf., Fig. 1.Hence, the ability of the controller to operate in the entire overmodulation range is essential to enable maximum power conversion of the drive.
The goal of this work is to investigate and propose a control method for PMSMs that achieve the previously mentioned properties.
1) Highest control dynamics without torque overshoots and undershoots as well as overcurrents during transient operation.
2) Maximum power conversion 2 in the flux-weakening region by entering the overmodulation range up to six-step operation during steady-state operation.

A. STATE-OF-THE-ART TECHNIQUES
A torque for PMSMs can be realized by various operating points, which are the specified combination of stator currents.This degree of freedom is usually exploited by an open-loop (i.e., feedfoward) operating point control (OPC) [3] to calculate the operating point with the smallest current magnitude for a given reference torque assuming steady-state operation.This operating point is fed as reference (current) to the closed-loop (i.e., feedback) controller, e.g., a proportionalintegral field-oriented control (PI-FOC).The PI-FOC [4] still represents the most popular drive control scheme, despite the availability of various innovative control approaches.However, neither the entire overmodulation range during steady-state operation nor best possible control dynamics can be realized with the PI-FOC in its standard formulation requiring proper dq-axis decoupling.Numerous methods exist that improve the control dynamics compared with PI-FOC and/or allow the operation in the entire overmodulation range during steady-state operation.These methods are described in the following, classifying into methods that improve the transient performance and methods that enable the operation in the overmodulation range during steady-state operation.Time-optimal controllers (TOC) [5], [6], [7], [8], [9], [10] reach the reference in the fastest possible way and, therefore, improve the control dynamics to its maximum extent, 2 For a given maximum motor current.considering the physical and technical limitations, e.g., input constraints and digital control delay.During transients, the time-optimal stator voltages (input trajectories) in the stator-fixed frame are constant, see [5], [6].During online operation, a set of nonlinear equations, cf., (26), must be numerically solved in order to determine the time-optimal stator voltages [5], [6], [9].Hence, control methods that do not solve the TOC problem, such as PI-FOC [4], direct torque control [11], conventional model predictive controllers (MPC) [12], [13], [14], [15], or deadbeat direct torque and flux control [16], may not reach the reference in the fastest way, resulting in time-suboptimal control behavior.However, actuating time-optimal voltages can lead to significant overcurrents and torque over-and undershoots, cf., [9,Fig. 3].To prevent these undesired characteristics, the TOC methods [8], [9] are considering current and torque limits but are restricted to the linear modulation range during steady-state operation.The deadbeat flux control (DBFC) proposed in [10] is to the best of authors' knowledge the only TOC method that can reach the entire overmodulation range up to six-step operation.Nevertheless, the DBFC [10] does not take into account current and torque limits leading to considerable undesired overcurrents and torque over-and undershoots.
Methods designed specifically for the operation at the voltage limit, i.e., six-step operation, such as voltage angle controllers [17], [18], [19] or the direct self-control [20], are typically only employed in the constant-power region, with a switchover to other controllers, e.g., PI-FOC, in the constanttorque region below the nominal speed.For highly dynamic applications, this may result in transition shocks, which is undesirable.Recently published methods [2], [10], [21], [22] solve that problem by enabling the operation in the whole speed and modulation range including a seamless transition to six-step operation with a single control law.Here, only the DBFC [10] attains TOC performance as mentioned in the previous paragraph.
Table 1 lists the properties regarding the drive's utilization during transient and steady-state operation for the previously mentioned methods, which are the most relevant for this work.

B. CONTRIBUTION
In this work, the time-optimal harmonic reference generator (TO-HRG) for a continuous-control-set model predictive flux control (CCS-MPFC) is proposed.The TO-HRG manipulates the flux linkage in such a way that TOC performance can be achieved by the CCS-MPFC in the whole speed and modulation range without transient overcurrents as well as torque over-and undershoots, cf., Fig. 2.This flux linkage reference manipulation enables the following.
1) TOC performance during transient operation in the entire modulation range.2) Allow a seamless transition in the entire modulation range including six-step operation.The TO-HRG makes use of the reference prerotation (RPR) method [9] and the harmonic reference generator (HRG) method [2] to obtain the respective advantages of the control methods [2] and [9], see Table 1.It should be noted that this fusion is not straightforward.The proposed TO-HRG is incorporated into a torque control scheme, cf., Fig. 3 .Thanks to the proposed TO-HRG, the overall scheme is able to fully exploit the potential regarding dc-link utilization and control dynamics compared with the state-of-the-art methods, cf., Table 1.

C. ARTICLE STRUCTURE
The rest of this article is organized as follows.Necessary fundamentals and the general control framework are explained in Section II.In Sections III and IV, the HRG proposed in [2] and the RPR proposed in [9] are summarized.Section V focuses on the TO-HRG.Simulative as well as experimental studies and comparisons in the complete speed and modulation range are presented in Sections VI and VII.Finally, Section VIII concludes this article.

II. GENERAL CONTROL FRAMEWORK
The overall control scheme shown in Fig. 3 and necessary fundamentals are briefly summarized in the following.

A. COORDINATE SYSTEMS
A physical vector quantity x, e.g., voltages u, currents i, and flux linkages ψ, can be represented in the stator-fixed three-phase abc, in the stator-fixed αβ, or in the rotor-fixed dq coordinate system.Assuming a vanishing zero-sequence component (x a + x b + x c = 0), the required transformations can be formulated as wherein ε denotes the electrical rotor angle of the PMSM.Reference variables are denoted as x * and estimated variables as x.

B. DISCRETE-TIME PMSM MODEL
The flux linkage-based control scheme requires a discretetime model to predict the flux linkage ψ αβ .In addition, models for predicting the current i dq [cf., (6)] and torque T [cf., (10)] are necessary to respect current and torque constraints.These PMSM models take into account nonlinear magnetization, i.e., (cross-)saturation effects, and are summarized in the following. 3A detailed derivation can be found in [9].However, it is worth to note that the models are based on the following assumptions and simplifications: 1) Motor windings are assumed to be sinusoidally distributed; 2) Changes of the stator resistance due to temperature or skin effect are neglected; 3) The total motor losses correspond to the ohmic losses.The differential equation of the flux linkages ψ αβ (Faraday's law of induction) for a PMSM can be described as follows: Here, u αβ represents the stator voltage, i αβ the stator current, and R s the ohmic stator resistance.By applying the forward Euler method to (2) with a sampling time T s , and the sampling index k, the discrete-time flux linkage prediction model is given by with the identity matrix I.
By transforming (3) to the dq frame and using the approximation with the differential inductance matrix the prediction model of the current i dq can be derived with ) wherein ω denotes the electrical rotor speed.
For compactness reasons, the parameter-varying matrices , and E ψ [k] in the following.
The PMSM's electromagnetic torque T can be calculated using with the pole pair number p.By linearizing (7) with respect to the current with the partial derivative calculated from (7), the torque prediction model evaluates to Exemplary voltage trajectory u α of the employed SVM scheme [28] with pulse durations t x of the switching vectors s x (see Fig. 1) and equally shared zero voltage vector pulse duration, i.e., t 1 = t 8 [2].
Here, the current model ( 6) was employed to calculate the current difference i dq for a given voltage u dq in (8).

C. GOPINATH-STYLE FLUX OBSERVER
Since the control variable (flux linkage) of the CCS-MPFC is not measured, its instantaneous value ψ αβ [k] must be observed.For this purpose, the well-known Gopinath-style flux observer [23], [24] is employed in this work.Here, the flux linkage estimate of a current model in the dq frame, i.e. current-to-flux linkage look-up tables (LUTs), are merged with the flux linkage estimate of a voltage model in the αβ frame, cf., (3).
In addition, the Gopinath-style flux observer (GFO) observer with its parameter insensitive voltage model is also able to determine the estimation error of the current model (flux linkage LUTs) [24].This offset can be added to the flux linkages estimated with the current model in other subfunctions of the proposed scheme, e.g., OPC, TO-HRG, and CCS-MPFC (see Fig. 3), to compensate for flux linkage estimation errors due to temperature [25], manufacturing deviations [26], and aging processes [27].

D. PULSE CLIPPING (PC)
The proposed control scheme uses the commonly applied space vector modulation (SVM) [28] with equally shared zero voltage vector pulse durations during one sampling period in the whole speed and modulation range to calculate the switching commands s abc,ccs based on u αβ,ccs , cf., Fig. 3.In the linear modulation region all neighboring elementary vectors s x of the mean voltage u αβ,ccs will be active for a certain time t x calculated by the SVM during one controller cycle T s , cf., Fig. 4.However, in the overmodulation range not all neighboring elementary vectors of u αβ,ccs should be active during one controller cycle T s if the voltage is saturated by the hexagonal constraint.To ensure this desired switching behavior during steady-state operation (i.e., if condition ( 29) is fulfilled) in the overmodulation range, the PC scheme proposed in [2] is applied.Here, a PC time threshold T c must be selected.If the time durations t x (cf., Fig. 4) of undesired pulses fall below T c , they will be clipped.With the resulting pulse pattern and switching commands s abc , the average voltage u αβ during a sampling period is calculated and fed to the GFO, see Fig. 3.

E. CONTINUOUS-CONTROL-SET MODEL PREDICTIVE FLUX CONTROL
In this article, the CCS-MPFC proposed in [9] min u αβ,ccs [k] is utilized with the hexagonal input constraints (11e) defined by wherein u DC denotes the dc-link voltage.The cost function is designed to reduce the flux linkage control error.Furthermore, the current (11c) and torque (11d) limits, which are defined in the following, must be fulfilled.The state constraints (11c) and (11d) are already mapped to the input space via the current (6) and torque (10) prediction models.A more detailed derivation of the state constraints (11c) and (11d) can be found in [9].Therefore, the derivation of these constraints defined by A I , b I , A T , and b T is only summarized in the following.Time-optimal control trajectories without state constraints can result in over-and undershoots of torque and currents, cf., [9,Fig. 3].To prevent the resulting transient overcurrents, a tangential linear approximation of a circular current constraint with radius I max,dyn > I max as tuning parameter is applied, see Fig. 5(a).In order to avoid nonmonotonic torque trajectories, d-currents greater than a chosen threshold i d,max are prevented by a linear inequality constraint, cf., Fig. 5(a).Both current constraints can be formulated as linear inequality constraints Moreover, torque over-and undershoots are prevented by the torque constraints shown in Fig. 5(b), which can be represented by To calculate A I , A T , b I , and b T that define the state constraints (11c) and (11d), the current (13) and torque constraints ( 14) must be formulated in terms of the optimization variable u αβ,ccs .This is achieved by inserting the current ( 6) and torque model ( 10) in ( 13) and ( 14) with The CCS-MPFC (11) proposed in [9] is intended to steer the flux linkage to its steady-state reference within the linear modulation range.Therefore, the resulting current i dq [k] or flux linkage ψ dq [k] stays always inside the isovoltage locus for the end of the linear modulation range (m = 0.907).Exemplary isovoltage loci for (m = 0.907) are depicted in the figures of Section VI.If the state of the PMSM (i dq [k] or ψ dq [k]) is within the isovoltage locus for m = 0.907, the state can be steered to each direction in the dq plane due to a remaining voltage reserve.Consequently, the torque constraint (16) can always be satisfied.In this work, the CCS-MPFC is extended to operation within the entire overmodulation range.If the state of the PMSM is outside the isovoltage locus for m = 0.907, the state cannot be continually steered to each direction in the dq plane due the increased induced voltage and, therefore, the torque constraint ( 16) can no longer be always fulfilled.For this reason, the torque constraint is disabled if the PMSM's state is located outside the linear modulation In order to guarantee feasibility of the optimization problem, the state constraints, i.e., current (11c) and torque limits (11d), were softened with a slack variable [29], [30].In this article, MATLAB's embedded quadratic programming (QP) solver of the MPC toolbox [31], [32] was selected to solve (11).However, any conventional QP solver can be employed to solve the optimization problem (11).In addition, a one-step prediction is carried out before the QP solver is executed in order to compensate for the delay of one sampling instant caused by the discrete-time implementation.

III. HARMONIC REFERENCE GENERATOR
In the overmodulation range, inevitable current and flux harmonics occur during steady-state operation due to the hexagonal voltage limit.These harmonics deteriorate controllers during steady-state operation that are using constant flux linkage or currents references in the dq coordinate system that do not contain these harmonics, cf., [21,Figs. 17 and 18].Applying an HRG prevents this deterioration by calculating and adding these harmonics to the mean reference under the assumption of a steady state with the help of voltage trajectories synthesized by an overmodulation scheme and a motor model.
Since the HRG proposed in [2] is part of the TO-HRG proposed in this work, the HRG proposed in [2] is summarized in the following.A detailed derivation and computational time-efficient implementation can be found in [2].
The inputs and outputs of the HRG are shown in Fig. 6.In addition to the flux linkage reference ψ * αβ , the HRG also provides a reference voltage u * αβ as output.This reference voltage is calculated with the help of an overmodulation scheme and corresponds to the voltage that must be applied during steady-state operation to follow the flux linkage reference.It is worth to note that this reference voltage will not be actuated directly but used by the PC scheme to detect which pulses of s abc,ccs should be suppressed, cf., Fig. 3, to achieve the desired switching behavior of the inverter during stationary operation in the overmodulation region.

A. FLUX LINKAGE REFERENCE
The flux linkage reference ψ * αβ depends on the rotor angle ε, the modulation index m, and the mean current reference i * dq (i.e., the harmonic-free fundamental current) calculated by an OPC, cf., Fig. 3.In this work, the OPC (higher level open-loop torque controller) presented in [3] is applied to calculate the operating point i * dq following the maximum torque per current (MTPC) 4 and maximum torque per voltage (MTPV) strategies.To compute the modulation index, the required mean voltage must be calculated.Here, (17) corresponds to the current difference equation ( 6) under the assumption of a steady state, and the mean flux linkage reference ψ * dq (i * dq ) is given by current-to-flux LUTs with i * dq as input.Then, the modulation index evaluates to If linear modulation is sufficient for the given operating point, i.e., m[k] < 0.907), no further calculations are required by the HRG and the mean flux linkage reference ψ * dq [k] without a harmonic content transformed to the αβ frame is fed to the CCS-MPFC as reference.However, if the modulation index is in the overmodulation range, the HRG must calculate the flux linkage reference, using voltage trajectories generated by an overmodulation scheme.In this work, the overmodulation scheme proposed in [1] is applied to generate voltage trajectories for a given fundamental voltage.In Fig. 7, the voltage values as a function of the modulation index m and the angle of the fundamental voltage are shown.
The flux linkage reference is calculated with Faraday's law of induction [cf., (2)] The integration constant C must be chosen in such a way that the mean of the flux linkage reference for one electrical revolution vanishes.Furthermore, the ohmic voltage drop is neglected and later incorporated by scaling and rotating the flux reference.By substituting the integration variable of ( 21) the flux linkage reference with omitted ohmic stator resistance can be calculated A graphical representation of ( 23) is depicted in Fig. 8. Instead of a time-consuming calculation of the integral terms of ( 23) during online operation utilizing the LUTs from Fig. 7, the integral terms as 2-D LUTs with m and ∠u αβ,f (upper integration limit) as inputs are utilized.Finally, the ohmic voltage drop of the stator resistance is incorporated by scaling and rotating the flux linkage reference (cf., Appendix in [2])

B. REFERENCE VOLTAGE
To ensure a desired switching behavior of the inverter during steady-state operation in the overmodulation range, e.g., six-step operation, the applied PC strategy proposed in [2] is utilized.The PC scheme requires a reference voltage u * αβ as input to suppress possible additional pulses contained in s abc,ccs , see Fig. 3.This reference voltage evaluates to and corresponds to the stationary (open-loop) voltage that would be actuated by the overmodulation scheme [1], see Fig. 8.

IV. REFERENCE PREROTATION
In order to solve the TOC problem to steer the PMSM's flux ψ αβ to its reference ψ * αβ in minimum time, without considering torque overshoots, undershoots, and overcurrents, the proposed TO-HRG adopts the RPR method.Since the RPR was already proposed in [9], it is only summarized in this article.A more detailed derivation and description of the RPR can be found in [9] while its inputs and outputs are shown in Fig. 9.
The following two properties for the solution of the TOC problem without considering torque overshoots, undershoots, and overcurrents can be found by applying Pontryagin's maximum principle [5], [6].
2) The time-optimal voltages u αβ (t ) are at their limit, i.e., saturated by the voltage hexagon.Actuating the time-optimal constant stator voltages u αβ during transient operation results in linear flux linkage trajectories ψ αβ (t ) if the minor effect of the ohmic voltage drop is neglected, see [9,Fig. 3b].To solve the TOC problem, the set of nonlinear equations has to be solved for u αβ and the time t rpr , that is needed to reach ψ * αβ [5], [6].The flux linkage reference ψ dq (i * dq [k]) in (26a) is computed with current-to-flux LUTs.Moreover corresponds to a circular voltage hexagon approximation of with the magnitude of the fundamental voltage during six-step operation, cf., (18).After a duration of t rpr , the linearly evolving predicted flux ψ αβ (t rpr , u αβ ) [right-hand side of (26a)] must match with the circularly evolving reference flux ψ * αβ (t rpr ) [left-hand side of (26a)].Furthermore, the time-optimal constant voltage u αβ must fulfill the magnitude criterion (26b).To solve (26) during online operation every sampling period, the computational time-efficient iterative numerical method investigated in [9] is applied.
Since the flux reference of the RPR is assumed to rotate on a circular path, cf., left-hand side of (26a), the control method with the RPR proposed in [9] is limited to the linear modulation range.
Actuating the time-optimal stator voltages calculated by solving (26b) can lead to undesired overcurrents and torque over-and undershoots, see [9, Fig. 3].These unintended control characteristics can be prevented by incorporating the TOC solution by prerotation the flux linkage reference (28) of the CCS-MPFC (11) that considers current and torque constraints with the prerotation angle ε rpr .This procedure results in TOC performance without overcurrents and torque over-and undershoots in the linear modulation range.In Fig. 10, exemplary simulative flux linkage trajectories with and without the RPR are depicted to show the transient control performance improvement by utilizing an RPR.Here, the time to reach the reference is decreased from 21 to 8 sampling periods.
Due to the digital implementation of the controller the minimum time t rpr to reach the reference ψ * αβ [k + 1] is T s .If t rpr is smaller than a selected threshold t thresh , steady-state control conditions are assumed.In this case, t rpr is set to T s if t rpr < t thresh then Condition ( 29) is evaluated before ψ * αβ is computed with (28).If ( 29) is fulfilled, the standard prerotation of the flux reference by one sampling instant ε rpr = ωT s is carried out.Hence, the obvious choice for the tuning parameter t thresh would be t thresh = T s .Due to the circular voltage hexagon approximation, slightly higher values, such as t thresh = 1.1T s . . .1.5T s , are recommended to distinguish between stationary and transient operation.

V. TIME-OPTIMAL HRG
In order to achieve TOC performance in the whole modulation region the TO-HRG is proposed in this work.Here, the HRG [2] summarized in Section III is merged with the RPR [9] summarized in Section IV.
The calculation steps that must be executed by the TO-HRG can be classified into the following three subroutines, cf., Fig. 11 ).In the following, these categories are explained in more detail.

A. CALCULATION OF THE HARMONIC FLUX REFERENCE CONTENT
The RPR can only be applied when the flux linkage reference is rotating circularly for a constant mean reference current i * dq .This is only true in the linear modulation region.Instead, additional harmonics occur in the overmodulation range that would lead to a noncircular shape of the flux linkage trajectory.Therefore, the RPR proposed in [9] cannot be applied directly in the overmodulation range.
Here, the flux linkage reference ψ * αβ (ε[k]) is computed with the HRG summarized in Section III as proposed in [2].

With
) is shown in Fig. 12 for visualization reasons only, but not required for the implementation of the TO-HRG.

C. CALCULATION OF THE FLUX AND VOLTAGE REFERENCE
With the prerotated angle ε[k] + ε rpr [k] as input of the HRG, cf., Fig. 11, the flux linkage reference ) for the CCS-MPFC (11) and the voltage reference u * αβ [k] (25) for the PC scheme are calculated.
The prerotated flux linkage reference ψ * αβ [k + 1] contains the harmonics caused by the voltage constraints in the overmodulation region and enables time-optimal control performance.Torque over-and undershoots as well as overcurrents are not taken into account by the TO-HRG, but are prevented by the state constraints of the CCS-MPFC.

VI. SIMULATIVE INVESTIGATION
The control scheme with the proposed TO-HRG is first investigated with a simulation.Here, the linearly magnetized (LM-)PMSM motor model with motor parameters listed in Table 2 is employed.Because sampling and controller are synchronized with the SVM, the current, torque, and flux ripples caused by the inverter switching are not visible in the following figures.
The transient control performance of the proposed control scheme is studied with torque step responses for different initial rotation angles ε(t = 0 s) and several stationary speeds.In addition, a comparison with the methods proposed in [2] and [10] is conducted.The simulation and controller settings are listed in Table 3.

A. INITIAL ROTOR ANGLE INVESTIGATION
In Fig. 13, the trajectories of current, torque, and flux linkage are depicted for uniformly distributed initial rotor angles ε 0 = ε(t = 0 s) ∈ {0, . .., π/3}.The flux linkage trajectory for ε 0 = 0 and ε 0 = π/3 rotated by an angle of −π/3 coincides due to the symmetry of the voltage hexagon, which results in the same dq current and torque trajectories.Since the voltage constraint (voltage hexagon) in the dq frame depends on the rotor angle ε, the current and torque trajectories generally vary for different ε 0 = ε(t = 0 s), see Fig. 13.Nevertheless, similar torque dynamics are attained without violating the dynamic current limit I max,dyn , see Fig. 13(a).Furthermore, torque over-and undershoots are prevented as long as i dq is within the linear modulation (m ≤ 0.907) isovoltage.When i dq enters the region in which overmodulation is required, the torque constraints (11d) are disabled and the controller is able to achieve maximum torque at the voltage limit (six-step operation).

B. INVESTIGATION OF THE SPEED DEPENDENCE
To investigate the control performance in the whole speed region, torque reference steps to maximum and minimum torque are commanded for different speeds.At standstill, the maximum and minimum torque can be realized without additional torque harmonics induced by the voltage limit.Here, mainly zero voltage vectors {s 1 , s 8 } are selected during steady-state operation.At the rated (mechanical) speed (n me = 3130 min −1 ), the rated motor operating point lies on the isovoltage for m = 1.Hence, six-step operation (m = 1) must be utilized to achieve maximum torque operation.Due to the ohmic voltage drop, a slightly reduced voltage is required for minimum torque operation compared with maximum torque operation.In this scenario, a modulation index of m ≈ 0.985 is sufficient to achieve the rated negative torque operating point at rated speed, which leads to additional switching instants in the range t ∈ [7 ms, 10 ms] during steady-state operation, see Fig. 14(a).Since the prerotation angle ε rpr depends on ω, different speeds n me lead to different torque and current trajectories, cf., trajectories for n me = {0, 3130} min −1 in Fig. 14.
For n me = 13000 min −1 , the intersections of the isovoltage (m = 1) and the MTPV curve are chosen by the OPC as operating points i * dq , see Fig. 14(b).Consequently, reduced torque magnitudes result compared with rated operation.Nevertheless, maximum possible torque magnitudes are obtained at the voltage limit thanks to the ability of the proposed control scheme to operate in six-step mode.Since i dq (t = 0 s) = 0 A is located outside the isovoltage (m = 1) for n me = 13000 min −1 , a torque undershoot is inevitable.However, the proposed control scheme ensures a minimum torque undershoot and prevents further torque undershoot within the linear modulation isovoltage (m = 0.907).

C. COMPARISON WITH STATE-OF-THE-ART METHODS
In this section, the proposed control scheme is compared with methods that enable a smooth transition from linear modulation to the entire overmodulation range.For this purpose, the CCS-MPFC with an HRG [2] and the DBFC proposed in [10] that achieves TOC performance are chosen.Since all three methods achieve the same steady-state control performance only the transient control performance is investigated.For clarity, the method presented in this work is referred to as TO-HRG and the method [2] as HRG for the experimental and simulative comparison sections of this article.
In Fig. 15, the trajectories of the TO-HRG, HRG, and DBFC for the challenging torque reference trajectory used for the speed dependency investigation, cf., Fig. 14, are depicted for an initial rotor angle ε(t = 0 s) = 0 at rated speed (n me = 3130 min −1 ).Although the trajectories of the TO-HRG for rated speed are already shown in Fig. 14, they are also shown in Fig. 15 to allow a better comparison with the HRG and DBFC methods.
Both, the TO-HRG and DBFC methods achieve TOC performance.The DBFC method does not take into account  current and torque constraints, which results in significant undesired overcurrents and torque over-and undershoots as well as nonmonotonic torque trajectories, see Fig. 15(a) for t ∈ [5 ms, 7 ms].The HRG and DBFC methods do not consider torque and current constraints.In contrast to the DBFC method however, the HRG is not able to achieve TOC performance, which still to undesired control e.g., overcurrents and torque undershoots, and also to reduced control dynamic.Only the proposed TO-HRG accomplish both, TOC performance without overcurrents and without increased torque over-and undershoots.

VII. EXPERIMENTAL INVESTIGATION
The experimental investigations were carried out on a laboratory test bench.The drive system under test consists of a highly utilized automotive interior PMSM with a two-level insulated-gate bipolar transistor (IGBT) inverter.A speedcontrolled induction machine connected to the test PMSM poses as load (motor), see Fig. 16.The datasheet parameters and flux linkages of the nonlinearly magnetized (NM-)PMSM are listed in Table 4 and shown in Fig. 17.
The proposed control scheme was developed in MAT-LAB Simulink and transferred a dSPACE DS1006MC rapidcontrol-prototyping system with the help of the automatic  C code generation environment of the MATLAB coder.Table 5 lists the test bench, inverter, and control parameters.All measurements have been collected with the dSPACE analogdigital converters synchronized with the control task and SVM.Consequently, current and flux linkage ripples caused by the inverter switching are not visible in the subsequent figures.Due to the limited bandwidth of torque sensors and the excitation of mechanical resonance frequencies, the airgap torque T shown in the figures of the experimental studies is estimated with (7) using flux linkages LUTs depicted in Fig. 17 and denoted with T .
Table 6 lists the turnaround times of the overall control scheme with its subfunctions (OPC, GFO, TO-HRG,

CCS-MPFC, and auxiliary functions). A low computational
of the proposed TO-HRG is reported here.The turnaround time of the auxiliary functions summarizes the time that is to execute SVM, PC scheme, coordinate transformations, and analog-digital conversion.Compared with the OPC and the GFO, the CCS-MPFC requires a variable number of iteration steps to solve the QP [9] resulting in varying turnaround time.Furthermore, the TO-HRG is deactivated in the linear modulation range.Hence, a turnaround time of (43.4-56.1 μs) results for the overall control scheme, cf., Table 6.
During steady-state control operation, the flux linkage reference must not be prerotated with the help of an RPR and no state-constraints are active.Therefore, the steady-state control performance of the proposed method is the same as the HRG method proposed in [2].Since steady-state investigations have been already performed thoroughly for the HRG method in [2], only highly dynamic transient experiments are performed in this work.

A. TORQUE STEP RESPONSE
In Figs.18 and 19, a torque step response to maximum torque for n me = 5600 min −1 with an initial rotor angle of ε(t = 0 s) = 0 is depicted.For visualization, both, the flux linkage ψ αβ and its reference ψ * αβ are transformed to the dq frame to allow a better recognition of the flux linkage control error compared with a visualization in the αβ frame.A slightly increased speed compared with the rated speed of n me = 5350 min −1 is ensured by the load motor such that the OPC always chooses operating points i * dq at the voltage limits, even for small variations of dc-link voltage and speed.At this slightly increased speed the modulation index is permanently in saturation (m = 1) and the six-step operation of the control scheme without additional intermediate pulses can be shown, see Fig. 18.Due to the torque transient of the test PMSM, the rotor speed is increased until the speed-controlled load motor reaches its steady state again, see Fig. 18(a).Furthermore, the dc-link voltage varies due to increased current demand of the test PMSM.Since both, the speed and the dc-link voltage vary, the operating point selected by the OPC is varying accordingly, cf., Fig. 18(a).This variation leads to inevitable additional oscillations of the current and a change in the diameter of the hexagonally shaped flux linkages during six-step operation, see Fig. 18(a).Nevertheless, the proposed control scheme is able to handle the provided challenging scenario with transiently varying speed and dc-link voltage without overcurrents and without torque overshoots as long as the state (current of flux linkage) of the PMSM is within the linear modulation isovoltage loci.
In addition, the current and flux linkage trajectories for a constant speed and constant dc-link voltage (t > 0.3 s) are highlighted in Fig. 19(a) and (c).These trajectories represent the steady-state control performance in this operating point.Here, the typical drop shape during six-step operation of the current in the rotor-fixed frame can be seen.Furthermore, the flux linkage coincides closely with its reference.

B. INVESTIGATION OF THE SPEED DEPENDENCE
The trajectories of torque, current, and switching vector with reference steps to maximum and minimum torque for different speeds including maximum speed are shown in Fig. 20.Furthermore, the variation of the isovoltage due to the varying dc-link voltage and speed caused by the torque transients is depicted in light green.
The speed of n me = 3000 min −1 leads to modulation indexes within the linear modulation range.Hence, no additional current, flux, or torque harmonics are induced by the voltage constraint during steady-state operation.Furthermore, the HRGs of the TO-HRG are inactive within the linear modulation range, and the proposed control scheme is equal to the TOC proposed in [9].Here, detailed simulative and experimental investigations including the low speed range and standstill can be found.
Because of the restricting isovoltage, the rated torque can no longer be attained for n me = {5600, 7000, 11 000} min −1 , cf., Fig. 20(a).Nevertheless, maximum and minimum torque yield for the corresponding speed is ensured.To enable maximum and minimum torque operation, six-step operation must be utilized for n me = {5600, 7000, 11 000} min −1 during steady-state operation.Except for a speed of n me = 5600 min −1 during t ∈ [8 ms, 10 ms], additional switching pulses occur.This happens because the transient from maximum to minimum torque transiently increases the dc-link voltage and decreases the speed.This leads to a short-time relaxation of the isovoltage locus whereby the rated generative operating point (intersection between MTPC and I max loci) is within the isovoltage.Consequently, operation at the voltage limit is no longer necessary and additional switching pulses take place.

C. COMPARISON WITH STATE-OF-THE-ART METHODS
The results of the experimental comparison with the NM-PMSM depicted in Fig. 21 coincide with the simulative comparison with the LM-PMSMin the following.
1) Both the proposed TO-HRG and HRG methods [2] achieve the same steady-state control performance.
2) The TO-HRG enables fastest control performance without violating the state constraints.3) Since the flux linkage reference for the HRG method [2] is not prerotated, a reduced control dynamic results.4) Slight overcurrents can occur with the HRG method since state constraints are not taken into account.The DBFC method [10] investigated in the simulative comparison, cf., Section VI-C led to emergency shutdowns of the test bench because a current limit was exceeded.Therefore, only the TO-HRG and HRG methods are experimentally compared and depicted in Fig. 21.In [10], smaller steps of the reference torque of the DBFC are commanded resulting in less prominent torque over-and undershoots as well as overcurrents.Khatib et al. [10] proposed to either accept the undesired control behavior or to reduce the rate of change of the torque reference for reference steps with increased amplitude.By reducing the rate of change of the reference torque, TOC performance of the DBFC can no longer be attained.

VIII. CONCLUSION
In this article, the time-optimal CCS-MPFC [9] that considers current and torque limits but is limited to the linear modulation range during steady-state operation was fused with the CCS-MPFC [2] that utilizes an HRG to enable operation in the entire modulation range up to six-step operation.This fusion was enabled with the proposed TO-HRG.The resulting modulator-based control scheme was extensively investigated and compared with state-of-the-art methods in simulations and experiments.Here, the following main characteristics of the control scheme that fully utilizes the drive's potential during steady-state and transient operation become apparent.1) Transient processes, i.e., operating point changes, are performed in the fastest possible way without overcurrents and torque over-and undershoots.2) During steady-state operation at the voltage limit, maximum possible power and torque conversion of the drive is ensured due to the six-step operation ability of the control scheme.3) Seamless transition between modulation regions and operating points are enabled without controller reconfiguration.The potential of the proposed control scheme for other motor types, such as externally excited synchronous motors or induction motors, as well as its extension to multiphase motor and multilevel inverter applications shall be investigated in the future.

FIGURE 2 .FIGURE 3 .
FIGURE 2. Simplified proposed time-optimal control scheme containing the manipulation of the flux reference with the TO-HRG (prerotation and addition of harmonic content) and a CCS-MPFC with current and torque constraints.

FIGURE 6 .
FIGURE 6. Inputs and outputs of the HRG.

FIGURE 7 .
FIGURE 7. Voltage values of Holtz's overmodulation scheme [1] as a function of the fundamental voltage angle ∠u αβ,f and modulation index for voltage hexagon sector 2 [2].

FIGURE 9 .
FIGURE 9. Inputs and outputs of the RPR.

FIGURE 10 .
FIGURE 10.Exemplary transient flux linkage trajectories with and without the prerotation of the flux linkage reference.

FIGURE 11 .
FIGURE 11.Block diagram of the TO-HRG.

FIGURE 12 .
FIGURE 12. Illustration of the TO-HRG procedure.

FIGURE 14 .
FIGURE 14. Simulative trajectories for reference steps to maximum and minimum torque for different speeds n me and ε(t = 0 s) = 0. (a) Torque and switching vector trajectories.(b) Current trajectories.

FIGURE 16 .
FIGURE 16.Test bench with load motor and test PMSM.

FIGURE 17 .
FIGURE 17. Flux linkage maps of the test PMSM.

FIGURE 18 .
FIGURE 18. Experimental torque step response to maximum torque at a speed of n me = 5600 min −1 with ε(t = 0 s) = 0. (a) Full view.(b) Zoom in.

FIGURE 20 .
FIGURE 20.Experimental TO-MPC trajectories for maximum and minimum torque operation for different speeds n me with an initial rotor angle ε(t = 0 s) = 0. (a) Torque and switching vector trajectories.(b) Current trajectories.

FIGURE 21 .
FIGURE 21.Experimental trajectories of the proposed and HRG methods [2] for maximum and minimum torque operation at n me = 5600 min −1 with an initial rotor angle ε(t = 0 s) = 0. (a) Torque and switching vector trajectories.(b) Current trajectories.