Model Predictive Torque Control for Permanent- Magnet Synchronous Motors Using a Stator- Fixed Harmonic Flux Reference Generator in the Entire Modulation Range

To increase torque and power conversion of permanent-magnet synchronous motors (PMSMs) at high-speed operation, the dc-link voltage of the inverter must be fully utilized during steady-state and transient operations. This article proposes the incorporation of a harmonic reference generator (HRG) with a pulse clipping (PC) scheme into a model predictive control framework to achieve highest power output. The HRG calculates flux references in the stator-fixed coordinate system for an underlying continuous-control-set model-predictive flux control (CCS-MPFC). These reference trajectories contain the harmonic flux component induced by the voltage hexagon in the overmodulation range and specific switching states (e.g., six-step operation) of the inverter are ensured by the proposed PC scheme. This enables a seamless transition to the overmodulation range including six-step operation and increases the drive's power conversion to its maximum extent. Since the flux differential equation in the stator-fixed coordinate system, used as motor model for the HRG and CCS-MPFC, is able to represent PMSMs with linear and nonlinear magnetization, the proposed approach is well suited for the control of PMSMs with nonlinear magnetization. Extensive transient and steady-state experimental investigations on a highly utilized PMSM in the entire modulation and speed range prove the performance of the proposed approach.

. Linear and overmodulation range, cf. (12), as well as the definitions of the hexagon sectors, the elementary vectors s n , and the switching positions of the inverter. on the current limit. Above the nominal speed, the constantpower region is entered. Here, the voltage limit characterizes the maximum torque. Hence, it is crucial to fully utilize the dc-link voltage of the inverter in the constant-power region. By exploiting the overmodulation range up to six-step operation the dc-link voltage is fully utilized and the fundamental voltage can be increased by approx. 10 % compared to the linear modulation range (cf. Fig. 1). Hence, overmodulation and six-step operation are important to achieve maximum possible torque and power density.

A. State-of-the-Art Techniques
Current harmonics inevitably occur in the overmodulation range induced by the voltage constraints of the feeding inverter (hexagon) (cf. Fig. 1). The harmonics and the voltage constraint itself are limiting control strategies, e.g., proportional-integral field-oriented controllers (PI-FOC) in its standard formulation [1], [2], [3], [4], to the linear modulation range. However, control approaches exist that can exploit the overmodulation range.
In the context of PI-FOC, extensions can be applied to reduce the deterioration caused by the current harmonics to enable the utilization of the overmodulation range [5], [6], [7], [8], [9], [10]. In [5], low-pass filters in the feedback path are used to suppress current harmonics. Obviously, adding low-pass filters to the control loop will decrease its dynamic performance. Methods [6], [7], [8], [9], [10] are using model-based approaches to estimate the current harmonics and to compensate for them within the 0885-8993 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
control loop [6], [7], [8], [9], [10]. In [6], the harmonics in the overmodulation range are estimated in an open-loop fashion with the help of a harmonic motor model. Here, the estimator provides accurate observations during steady-state conditions but stability problems can occur during transient operation due to a difference of the system's eigenvalues [7]. To overcome this problem, a feedback path was added to the estimator in [7], [8], and [9]. Nevertheless, six-step operation cannot be realized. In contrast, six-step operation combined with PI-FOC can be achieved in [11] and [10]. In [11], a higher level open-loop torque controller enables the rotation of the reference voltage vector for the modulator outside of the voltage hexagon. By utilizing Bolognani's overmodulation scheme [12], six-step operation is ensured since the reference voltage vector for the modulator rotates that far outside the voltage hexagon that the voltage corrections of the PI current controller, which are induced by the current harmonics in the overmodulation range, do not affect the modulator. However, in between the linear modulation range and six-step operation, the PI controller's actuating voltage vector intersects the voltage hexagon. The resulting excitation of the PI-FOC induced by the current harmonics leads, therefore, to a deterioration and suboptimal control performance in this modulation range. In [10], a harmonic reference generator (HRG) is proposed that calculates the harmonic components of the current in the overmodulation range and adds them to the reference current by solving a computation time consuming boundary value problem with the help of Holtz's overmodulation scheme [13]. This enables high control performance during transient and steady-state operation in the entire overmodulation range. The dynamic control performance can be further increased compared to the PI-FOC by combining the HRG with a continuous-controlset model-predictive current controller (CCS-MPCC) [10]. Model predictive controllers (MPCs) are characterized by superior control dynamics and can handle state and input constraints inherently [14]. Therefore, MPC is well suited for the control of electrical drives, especially in the overmodulation range at the voltage limit [10], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. MPC methods can be divided into continuous-control-set (CCS) algorithms combined with a modulator [10], [21], [22], [23] and finite-control set (FCS) methods that directly determine the switching pulses for the inverter [15], [16], [17], [18], [19], [20], [24]. Despite the fact that the CCS-MPCC methods [21], [22], [23] and the FCS-MPCC methods [17], [18], [19], [20] increase the modulation index above the linear modulation range, they do not inherently achieve six-step operation. Nonetheless, the CCS-MPCC method [10] mentioned in the previous paragraph and the FCS-MPCC methods [15], [16], [24] are able to realize six-step operation. By sufficiently penalizing the inverter switching in the FCS-MPCC cost function, six-step operation can be enforced [15]. Method [16], in contrast, ensures six-step operation via a rectangular boundary area for the control error. With the help of a pulse clipping (PC) scheme, current control is enabled in the overmodulation range up to the six-step operation in [24].
A deadbeat flux control strategy is proposed in [25] that continuously transits from the linear to the overmodulation range including six-step operation. This is achieved by synthesizing a flux reference trajectory in the stator-fixed αβ coordinate system that is circular in the linear modulation range, hexagonal during six-step operation, and a combination of both in the overmodulation range.
Summarizing the literature in this field there is a need for closed-loop control schemes that are able to: 1) realize seamless transition from the linear modulation range to the overmodulation range including six-step operation; 2) handle highly utilized PMSMs with significant (cross-) saturation effects; 3) adding minimum computational overhead to the overall control framework.

B. Contribution
In this article, an HRG and a PC scheme is proposed. The model-based HRG calculates the reference flux that contains the harmonic component, which is induced by the voltage constraints of the feeding inverter (voltage hexagon) in the overmodulation range, and the PC scheme ensures the desired switching behavior of the inverter, e.g., six-step operation. This flux reference is calculated by solving an initial value problem (IVP) with the help of a motor model and voltage trajectories synthesized by Holtz's overmodulation scheme [13] for the desired modulation index. As a result of the proposed flux-based HRG and PC, the following advantageous properties are achieved.
1) The control scheme is able to seamlessly utilize the entire speed and modulation range including six-step operation without any control scheme reconfiguration. 2) High control dynamic is ensured even at the voltage limit.
3) The flux-based HRG approach is conceptually simpler, more accurate and reduces the computational load significantly compared to the current-based HRG proposed in [10].
2) Continuous-control-set model-predictive flux control (CCS-MPFC) that achieves highest tracking dynamics of the reference flux ψ * αβ supplied by the proposed HRG. 3) Flux observer that estimates the PMSM's fluxψ αβ in the entire speed range including standstill [37], [38]. 4) Space vector modulation (SVM) scheme that converts the voltages u αβ,ccs commanded by the CCS-MPFC into switching commands s abc,ccs [39]. In the following, these elements are explained in more detail.

A. Coordinate Systems
Transformations between the stator-fixed three-phase abc and the stator-fixed αβ coordinate system are calculated with the following matrices: Here, † denotes the Moore-Penrose pseudoinverse and bold symbols depict matrices or vectors. The transformations between the αβ and rotor-fixed dq coordinate system can be formulated as T dqαβ (ε(t)) = cos(ε(t)) sin(ε(t)) − sin(ε(t)) cos(ε(t)) wherein ε denotes the electrical angle of the PMSM. With (1) and (2), the transformation between the abc and dq coordinate system becomes

B. Discrete-Time PMSM Model
According to the Faraday's law of induction, the differential equation of the flux linkages ψ αβ = [ψ α ψ β ] T for a PMSM can be described as follows: Here, u αβ = [u α u β ] T represents the stator voltage, i αβ = [i α i β ] T represents the stator current, and R s represents the ohmic stator resistance. By applying the forward Euler method with a sampling time T s , the difference equation of (4) is given by

C. Inverter
For a three-phase two-level inverter, the stator voltages result as u αβ ∈ U αβ = u αβ ∈ R 2 |u αβ = u DC T αβabc d abc (6) where u DC is the dc-link voltage and d abc is the duty cycle vector of the inverter given by

D. Operating Point Control
To minimize ohmic copper losses, the OPC (higher level open-loop torque controller) proposed in [34] that selects the mean current operating point i * dq based on the MTPC and MTPV strategies is utilized. Here, the nonlinear magnetization with significant (cross-)saturation effects of highly utilized PMSMs is linearized online and optimal operating points i * dq (i.e., the harmonic-free fundamental current) are calculated analytically. The nonlinear magnetization is taken into account iteratively by a successive linearization and analytical calculation of the subsequent optimal operating point. In this article, only one iteration to calculate the optimal operating point per controller cycle (sampling instant) is executed by the OPC in order to reduce computation time.

E. Continuous-Control-Set Model Predictive Flux Control
The cost function for the CCS-MPFC with a prediction horizon of N = 1 is designed to penalize the weighted squared Euclidean distance from the predicted fluxψ αβ [k + 1] to the reference flux at the next time step ψ * αβ [k + 1]. The optimization problem can, thus, be defined as To solve the linearly constrained quadratic program (8), any standard quadratic programming (QP) solver can be utilized. For the practical implementation of this contribution, the embedded solver of the MATLAB MPC toolbox was chosen [40], [41]. To Fig. 3. Exemplary voltage trajectory u α of the applied SVM scheme [39] for a mean voltage u αβ,ccs that is located in sector 2 of the voltage hexagon with the pulse durations t n for the corresponding switching vector s n (cf. Fig. 1) and equally shared zero vector pulse duration (t 1 = t 8 ).
compensate for the control delay due to the digital implementation a one-step state prediction is applied before the QP solver is called [42].

F. Overmodulation Scheme
As mentioned in Section I-B, the proposed HRG uses voltage trajectories and a motor model (5) in the αβ coordinate system to calculate the reference flux linkage trajectories in the overmodulation range by solving an IVP. With a given fundamental voltage vector in the dq coordinate system and the corresponding modulation index, an overmodulation scheme is able to synthesize these voltage trajectories for a sixth of the fundamental wave (one sector of the voltage hexagon). To utilize the entire modulation range, an overmodulation method must be applied, which also covers the six-step operation. Here, a few of these can be mentioned [12], [13], [27], [28]. In this article, the Holtz's method [13] was utilized because of its low-harmonic distortion [29]. It is recommended to familiarize first with the overmodulation techniques [12], [13], [27], [28] before implementing the proposed HRG.

G. Gopinath-Style Flux Observer
The CCS-MPFC requires the knowledge of the PMSM's momentary flux linkage ψ αβ [k]. Hence, the flux linkage is estimated with the help of a Gopinath-style flux observer [37], [38]. Here, the flux linkage estimate of a current model via a current to flux linkage look-up table (LUT), see Fig. 10, is fused with the flux linkage estimate of a voltage model (5). The flux linkage estimates of both models are fused as a function of the speed ω. Thus, the estimate of the current model dominates for low speeds and the estimate of the voltage model dominates for high speeds. The speed at which both models contribute equally to the observer's flux linkage estimateψ αβ is called crossover frequency ω 0 . This frequency is a tuning parameter of the observer and is usually set to values smaller than the rated frequency of the motor.

H. Space Vector Modulation
To generate the pulses s abc,ccs that represent a mean voltage u αβ,ccs during one sampling period T s (cf. Fig. 2), the SVM method proposed in [39] is applied. An exemplary voltage trajectory synthesized by the applied SVM scheme is depicted in Fig. 3. Fig. 4. Simplified HRG conceptual scheme applied to a general closed-loop control system with the mean reference r, harmonic content of the referencer, reference r, system output y, and control error e.

III. HARMONIC REFERENCE GENERATOR
To prevent a performance deterioration of the CCS-MPFC during steady-state operation due to the inevitable current and flux harmonics in the overmodulation range, these flux harmonics are calculated and added to the mean reference flux ψ * dq (i * dq ) by the HRG with the help of a motor model and voltage trajectories by solving a numerical IVP. These calculated flux harmonics are added to the mean reference flux ψ * dq (i * dq ). Therefore, under the assumption of an accurate motor model to calculate the flux harmonics, the harmonic content of the flux during steady-state operation in the overmodulation range is contained in the reference flux as well as in the actual flux of the PMSM. This results in a vanishing steady-state control error without a harmonic component. Hence, the controller is not deteriorated in the overmodulation range anymore by applying the HRG. A simplified block diagram of an HRG applied to a general closed-loop control system for steady-control conditions to illustrate the concept is depicted in Fig. 4. The calculation steps and methodology of the HRG are described in more detail in the following. The concept is explained for highly utilized PMSMs with significant (cross-)saturation. A flowchart of the HRG algorithm is shown in Fig. 5.

A. Modulation Index
To calculate the required modulation index m for the current reference i * dq that is provided by the OPC (cf. Fig. 2), the steadystate fundamental voltage vector u dq that is needed to reach i * dq must be calculated. With the help of current to flux linkage LUTs (cf. Fig. 10), the mean flux linkage reference ψ * dq (i * dq ) is given. To derive a functional relationship that links ψ * dq , i * dq , and u dq , the motor model (5) must be transformed to the dq coordinate system with the identity considers the rotation of the dq against the αβ coordinate system during one sampling period [43], [44]. By rearranging (9) under the assumption of steady-state conditions the fundamental voltage u dq evaluates to Based on u dq and u DC , the modulation index is defined as When the modulation index is in the linear modulation range |u dq | ≤ u DC / √ 3, no further calculations have to be performed by the HRG and the flux linkage reference for the CCS-MPFC is set to

B. Voltage Trajectories
In the following, operation within the overmodulation range is considered. With the help of an overmodulation scheme [13], the constrained voltages u αβ as a function of the modulation index m and the angle of its fundamental component, denoted by ∠u αβ,f , are given. The fundamental voltage u αβ,f with length u αβ,f = 2 π u DC m rotates with a speed of ω in the αβ coordinate system.
With the modulation index m, the voltage trajectories over a sixth of the fundamental wave are given with the help of Holtz's overmodulation scheme [13] in the αβ coordinate system. Due to the symmetry of the voltage hexagon, it is sufficient to calculate the voltage trajectories for one sector (one sixth of the fundamental wave). In Fig. 6, an exemplary continuous voltage trajectory u αβ (ε) in sector 2 of the voltage hexagon (cf. Fig. 1) is depicted. Fig. 7 contains the continuous voltage trajectories as a function of the modulation index m and the angle of the fundamental voltage, denoted by ∠u αβ,f , and can, therefore, be seen as an extension of Fig. 6. The angle of the fundamental voltage is given by In Fig. 6, sector 2 of the voltage hexagon was chosen because the symmetry of the voltages with respect to the αβ coordinate system leads to a simpler interpretation of the voltage trajectories than, e.g., sector 1. Even if the values of Fig. 7 are valid only for sector 2, the characteristic of the applied overmodulation scheme [13] will be defined uniquely with Fig. 7 since the voltage values for any other sector can be calculated by a simple rotary transformation for the desired sector of the voltage hexagon. Furthermore, the dual-mode property of the applied overmodulation scheme [13] can be seen in Fig. 7, because the voltage u β is constant for a modulation index m ≥ 0.952, which means that the voltage u αβ always intersects the boundary of the hexagon.

C. Solving the IVP
The calculation of the flux linkage references in the αβ coordinate system is performed for sector 2 of the voltage hexagon by solving an IVP with the help of the overmodulation scheme's voltage trajectories (see Figs. 6 and 7). Solutions for other sectors are calculated by a rotary transformation of the solution of sector 2. The approach to solve the IVP and to calculate the flux linkage reference ψ * αβ [k + 1] is summarized in Fig. 8. In a first step, a biased flux linkage reference ψ * αβ,b [k + 1] is calculated by applying the difference equation of the flux linkage (5) and neglecting the ohmic voltage drop of the stator resistance with the initial condition ψ * αβ,b (∠u αβ,f = π 3 ) = 0. Here, the angle π 3 as lower integration limit is needed since the voltage trajectories are given for sector 2 of the voltage hexagon. Instead of calculating the integral term in (15) with the help of the overmodulation scheme [13] and its nonlinear equations online, two-dimensional LUTs representing the integral terms of the  voltage trajectories are utilized for online operation (see Fig. 9). Here, each LUT must be evaluated only once to calculate the integral term of (15). It is worth to note that the precalculation of the integral of the modulation index dependent voltage trajectories depicted in Fig. 9 is fully independent of the specific motor and inverter parameters and, therefore, does not limit the scope of the proposed approach.
Due to the initial condition ψ * αβ,b (∠u αβ,f = π 3 ) = 0, the calculated flux linkage reference ψ * αβ,b [k + 1] is biased. Since the flux linkage reference trajectory in the αβ coordinate system for one electrical revolution must be zero-mean, the bias of ψ * αβ,b [k + 1] has to be eliminated. This bias corresponds to the mean of ψ * αβ,b (cf. Fig. 8) for one electrical revolution and evaluates to Here, the biased flux linkage reference ψ * αβ,b (∠u αβ,f = 2π 3 ) corresponds to the end of voltage hexagon sector 2 and is calculated similar as (15) By incorporating the ohmic voltage drop of the stator resistance and the bias ψ * αβ,b , the flux linkage reference ψ * αβ [k + 1] is given by is performed in (18). Since the harmonic current induced by the voltage constraints in the overmodulation range is not known, only its fundamental i * dq is considered for the ohmic voltage drop. Furthermore, the approach of scaling and rotating the flux linkage reference to incorporate the ohmic voltage drop in (18) is only valid for sinusoidal quantities (cf. Appendix). Due to the fact that the flux linkages, voltages, and currents in the overmodulation range in the αβ coordinate system are not sinusoidal and the ohmic voltage drop of the harmonic content of the stator current is not taken into account, a systematic error is inevitable. However, the resulting error in the calculation of ψ * αβ [k + 1] can be considered small, cf. Sections III-E and V, compared to rated quantities and nonideal effects in the application, e.g., measurement noise and slot-harmonics due to the winding scheme of the PMSM.

D. Reference Voltage
The reference voltage u * αβ that is needed for the PC strategy corresponds to the mean voltage that would be actuated by the overmodulation scheme [13] during the next sampling instant. This reference voltage is defined by and is calculated by utilizing the two-dimensional LUTs depicted in Fig. 9. Here, each LUT must be evaluated only twice to calculate u * αβ .

E. Simulative Accuracy and Computation Time Investigation
To investigate the accuracy of the calculated reference trajectories and the computational load of the proposed HRG, simulative investigations are carried out for a highly utilized PMSM and are shown in the following. The harmonic content of the flux reference can be visualized more clearly in the dq coordinate system since its fundamental is constant. For this reason, the reference trajectory ψ * αβ is transformed to the dq coordinate system for the remaining figures of this article. It is worth to note that this transformation is only performed for the visualization and not for the actual implementation of the controller. The parameters and nonlinear magnetization of the employed PMSM are given Table I and Fig. 10. Furthermore, the results are compared to the HRG presented in [10]. To prevent confusion, the HRG proposed in this work is denoted as ψ αβ -HRG in this section and the already published HRG of [10] is denoted as i dq -HRG. In [10], different variants of the i dq -HRG were presented. For the following comparisons, the most accurate i dq -HRG, with the variable supporting point method and exact discretization, is chosen. Compared to the ψ αβ -HRG, an additional tuning parameter N BP must be selected for the i dq -HRG. This tuning parameter corresponds to the number of supporting or grid points that are used to solve the boundary value problem of the i dq -HRG.
To investigate accuracy of the ψ αβ -HRG, the mean error is depicted in Fig. 11. Here, the error e dq is defined as the difference of the current trajectory calculated by the corresponding HRG and the trajectory obtained by a time-domain simulation (TDS) (see Fig. 12). For this simulative investigation, the current trajectory of the ψ αβ -HRG is determined with the help of flux linkage to current LUTs. The TDS solution is obtained by applying Holtz's overmodulation scheme [13] in an open-loop manner with a sufficiently long simulation time such that the natural damping of the PMSM leads to steady-state harmonic trajectories. Here, the resulting ordinary differential equation of Fig. 11. Mean error of the current trajectories for six-step operation calculated by the proposed ψ αβ -HRG and the i dq -HRG [10] for the rated operating point (cf. Fig. 12). the quasi-continuous TDS was solved with an adaptive Runge-Kutta method [45]. Although the mean error e dq of the i dq -HRG decreases for an increasing number of supporting points N BP [10], a significant mean error remains during six-step operation (m = 1) for the applied highly utilized motor since the i dq -HRG linearizes the motor's magnetization around the operating point i dq , which leads to a systematic error. The flux linkage difference (5) used by the ψ αβ -HRG is equally valid for linearly and nonlinearly magnetized PMSMs, which leads to a reduced mean error for six-step operation compared to the i dq -HRG. Due to the omission of the harmonic component of the ohmic voltage drop in the flux linkage difference equation (15), caused by the voltage constraint in the overmodulation range, a systematic error is also introduced by the ψ αβ -HRG, which is, however, smaller than with the i dq -HRG. Moreover, the error of the ψ αβ -HRG can be considered small compared to the rated current (cf. Table I) and nonideal effects in the actual application, e.g., measurement noise. For these reasons, the ψ αβ -HRG should be preferred for highly utilized PMSM in terms of accuracy. In Fig. 13, the required execution time of the ψ αβ -HRG and the i dq -HRG, implemented in C running on a Lenovo T14 s with an AMD Ryzen 7 1.9 GHz processor and 32 GB of RAM, is depicted. Here, it is worth to note that only one processor core was used. Since a square matrix with a dimension proportional to N BP must be inverted for the i dq -HRG, the required execution time increases quadratically with the number of supporting points. The mean execution time for the ψ αβ -HRG with 1.9 μs is significantly smaller than for the i dq -HRG (cf. Fig. 13). For this reasons the ψ αβ -HRG has to be preferred in terms of computational load.

IV. PULSE CLIPPING
As seen in the previous section, even with exact knowledge of the motor parameters and flux linkages, errors in the solution of the IVP are present. In addition to these systematic errors, the solution of the IVP would also deviate from the PMSM's flux linkage and current trajectories due to inaccurate motor parameters based on various influences, such as temperature, aging, and further nonmodeled parasitic effects, such as slot harmonics or nonideal inverter switching behavior.
When the flux linkage reference trajectory of the HRG in steady state differs from the flux linkage trajectory of the PMSM, the CCS-MPFC tries to follow the reference trajectory of the HRG with voltage commands that would deviate from the reference voltage u * αβ [k] of the overmodulation scheme (cf. Section III-D). Due to these voltage commands six-step operation cannot be ensured since additional switching commands are introduced. Additionally, the desired switching behavior that only nonzero switching vectors should be active if the reference voltage u * αβ [k] of the HRG lies on the hexagon boundary, cannot be realized. To achieve the desired switching behavior in the overmodulation range including six-step operation despite deviations between reference and actual flux linkage trajectory, a PC method is proposed and described in the following.
The pulses s abc,ccs of the SVM calculated on the basis of u αβ,ccs (cf. Fig. 2) are clipped if the lengths of the pulses fall under certain thresholds that depend on the reference voltage u * αβ [k] of the HRG. Here, the following three different scenarios, cf. Fig. 14, defined subsequently must be distinguished (sector 2 of the voltage hexagon is assumed).   Fig. 15. Simulative step response for the highly utilized PMSM characterized by Table I and Fig. 10 to rated operation at n me = 5350 min −1 for T s = 62.5 μs with and without activated PC scheme.
The reference voltage u * αβ [k] is located on the boundaries without being in a corner of the voltage hexagon. Therefore, only nonzero active switching vectors are desired. For this reason, the threshold for clipping nonzero active switching vectors remains the interlocking time T i . For the zero vectors an additional threshold T c is introduced as tuning parameter to prevent undesired switching behavior. 3) u * αβ = 2u DC 3 : The reference voltage u * αβ [k] is located in the corner of the voltage hexagon. Here, no switching is desired during the sampling period. For this reason, the threshold T c is active for zero and nonzero switching vectors. The threshold T c should be selected as small as possible, but large enough that six-step operation without additional switching pulses can be ensured. The tuning of T c is done on a test bench at the rated operating point. Here, T c is increased starting from T i = T c until six-step operation without additional switching pulses is achieved.
In Fig. 15, a simulative step response for the highly utilized PMSM with the parameters defined in Table I and the flux linkages shown in Fig. 10 to the rated operating point at n me = 5350 min −1 is depicted. Here, for t < 0 s, a modulation index within the linear modulation range is required and, therefore, the switching behavior of the inverter with and without PC is the same. By applying the step response to rated operation the full dc-link voltage is required (m = 1). Due to model and numerical inaccuracies of the CCS-MPFC and the HRG, short switching pulses are inserted by the inverter without the PC, which results to slightly reduced fundamental voltages and currents. In contrast, if the PC scheme is active, these short pulses can be suppressed and the desired switching behavior (six-step operation) and maximum power conversion of the drive system can be ensured.

A. Test Setup
All following experimental results have been obtained on a laboratory test bench, which is depicted in Fig. 16. The electrical drive system under test is a highly utilized interior PMSM (Brusa: HSM1-6.17.12-C01) for automotive applications and a two-level insulated-gate bipolar transistor (IGBT) inverter (Semikron: 3×SKiiP 1242GB120-4D). The datasheet parameters and flux linkages can be seen in Table I and Fig. 10. As load motor, a speed-controlled induction machine (Schorch: LU8250M-AZ83Z-Z) is mechanically coupled with the test motor.
The test bench is further equipped with a dSPACE DS1006MC rapid-control-prototyping system. All measurements have been obtained by the dSPACE analog-digital-converters, which have been synchronized with the control task. The most important inverter, test bench, and control parameters are listed in Table II. The turnaround times of the OPC, GFO, HRG, CCS-MPFC, auxiliary functions, and the overall control strategy are listed in Table III. Here, the low computational load of the HRG can be seen. Under the item auxiliary functions, the turnaround times of all calculations, e.g., PC, SVM, coordinate transformations, analog-digital conversion, as well as processor and host computer communication that must be executed in addition to the OPC, GFO, HRG, and CCS-MPFC, are summarized. In order to prove the effectiveness and performance of the control strategy with the proposed HRG and PC, several representative experiments in the constant-torque and constant-power region were carried out that are shown in the following. Since the discrete-time measurement samples are synchronized with the SVM, the current ripple induced by the switching of the inverter is not visible in the following figures. Furthermore, the torque ,   TABLE II  DC-LINK, INVERTER, CONTROL, AND TEST BENCH PARAMETERS OF THE  EXPERIMENTAL TEST SETUP   TABLE III  TURNAROUND TIMES OF THE CONTROL STRATEGY depicted in the following figures is not measured directly with the help of a torque sensor. Instead, the torque was estimated with LUTs of the flux linkages ψ dq (i dq ) using the following relationship: The reliance on the estimation (21) rather than the direct shaft torque measurement can be legitimated for the following two reasons. 1) Highly dynamical experiments are investigated and, therefore, the moment of inertia of the rotor shaft as well as the limited bandwidth of the torque sensor distort the measurement. 2) Due to the finite stiffness of the connected motor shafts, mechanical resonance frequencies occur, which can be excited by the torque harmonics in the overmodulation range and, thus, further distort the torque measurement results.

B. Constant-Torque Region
In the constant-torque region, an operating point on the MTPC trajectory is selected to minimize copper losses. Hereby, the maximum torque is only constrained by the current limit.
To investigate the transient torque control performance, a step response from zero reference torque to rated torque (maximum torque) is shown in Fig. 17 at nominal speed. Here, the CCS-MPFC reaches the flux linkage reference trajectory calculated by the HRG within in approx. 3 μs and switches seamlessly to six-step operation thanks to the PC see Fig. 17(b). The elementary vectors s n = {1, 8} are defined as in Fig. 1. Moreover, in Fig. 17(a), the robustness of the strategy is demonstrated as it overcomes external disturbances.
1) The rapidly increasing current amplitude leads to a shorttime reduction of the dc-link voltage due to the limited capacity of the dc-link capacitor and the finite settling time of the voltage controller of the dc-link circuit.
2) The rapidly increasing torque leads to an acceleration of the rotor shaft and, therefore, to a short-time increase of the rotor speed due to the limited moment of inertia of the coupled rotor shaft (load and test motor) and the finite settling time of the load machine's speed controller. Both effects lead to a short-time contraction of the voltage limit. Therefore, an operating point i * dq with less flux linkage amplitude and less torque is selected by OPC. Hence, in this particular scenario, the constant-power region is entered transiently for this short-time with decreased voltage limit instead of the constant-torque region. Nevertheless, the controller is able to maintain six-step operation and provides maximum torque in this challenging scenario. Operation feasibility at transiently varying dc-link voltages and speeds is an important feature of the presented methodology, as this problem is encountered in many industrial applications (e.g., automation and traction drives).
To analyze the control performance during transient speed changes, the speed was increased from 4700 min −1 to 5350 min −1 by the speed-controlled load machine while maximum torque operation was attained by the test machine (see Fig. 18). The operating point chosen by the OPC corresponds to the intersection of MTPC trajectory and current limit (see Fig. 12). Due to increasing speed, the voltage limit decreases, whereby the entire overmodulation range is covered. To reach the mean current operating points i * dq specified by the OPC, the HRG adds a harmonic component to the mean flux linkage reference in the overmodulation range. This harmonic component increases with the modulation index (see Fig. 18). However, slight control deviations can be observed. This can be explained by remaining systematic model mismatches (cf. Section IV). In the linear modulation range, as expected, the flux reference contains no additional harmonic components added by the HRG and no additional torque, current, and flux linkage harmonics exist due to the voltage limit.
In the overmodulation range additional torque, current, and flux linkage harmonics are created (see Fig. 19). From a modulation index of m = 0.96, the actuated mean voltage vectors are located solely on the boundary of the voltage hexagon, which means that the zero voltage vectors are no longer selected (cf. Fig. 19). By reaching the nominal speed (n me = 5350 min −1 ), six-step operation has to be applied, which leads to high torque, current, and flux linkage harmonics, see Fig. 20(a).

C. Constant-Power Region
Above the nominal speed, the constant-power region is entered. Since the short-circuit point of the PMSM considered in this article lies on the current limit, only the intersection of voltage and current limits is chosen as operating point for maximum torque operation. For PMSMs with a short-circuit point within the current limit the intersection of voltage limit and MTPV trajectory has to be selected as operating point for maximum torque operation above a certain speed.
To investigate the control performance in the constant-power region, the speed was increased by the load machine from nominal to maximum speed while maximum torque operation was achieved by the test machine (see Figs. 20 and 22). Here, only minor deviations between estimatedψ αβ and reference flux ψ * αβ exist. Since six-step operation is used continuously during the speed transient and, therefore, the dc-link voltage is utilized to its maximum extend, maximum torque generation is ensured. To further analyze the PMSM's torque, current, and flux trajectories as well as the switching pattern enlarged snapshots from Fig. 20 are shown in Fig. 21. Here, decreasing phase current harmonics for increasing motor speed exist. The following two reasons are responsible for that: 1) the six-step operation's voltage harmonics have a decreasing impact on the currents and fluxes for increasing frequencies due to the low-pass characteristic of the stator winding system; 2) the degree of saturation of the highly utilized PMSM is lower for operating points at high speeds, which leads to larger differential inductance values, see Fig. 10, resulting in lower current harmonics. Despite the slight control deviation caused by the discussed systematic modeling errors (cf. Section IV), overmodulation including six-step operation can be ensured thanks to the proposed HRG and PC, which successfully enables the maximum power conversion of the drive system.

VI. CONCLUSION
In this article, an HRG combined with a PC method for CCS-MPFC of highly utilized PMSM was presented and experimentally investigated. Here, a harmonic component is added to the flux reference by the HRG and undesired switching pulses are suppressed by the PC in the overmodulation range. Thanks to the HRG and PC, the CCS-MPFC can seamlessly transit to the overmodulation range up to and including six-step operation without transitioning between different control strategies. This enables maximum power conversion of the drive system during transients and steady state since the dc-link voltage of the inverter is utilized to its maximum extend. Due to the formulation of the HRG and PC, only minor additional computational time and resources are required, which improves the applicability on low-cost hardware. In the future, the applicability of the proposed methods to other motor types, e.g., synchronous reluctance and induction motors, will be investigated.