Time-Optimal Model Predictive Control of Permanent Magnet Synchronous Motors Considering Current and Torque Constraints

In various permanent magnet synchronous motor (PMSM) drive applications the torque dynamics are an important performance criterion. Here, time-optimal control (TOC) methods can be utilized to achieve highest control dynamics. Applying the state-of-the-art TOC methods leads to unintended overcurrents and torque over- and undershoots during transient operation. To prevent these unintended control characteristics while still achieving TOC performance the time-optimal model predictive control (TO-MPC) is proposed in this work. The TO-MPC contains a reference prerotation (RPR) and a continuous control set model predictive flux control (CCS-MPFC). By applying Pontryagin's maximum principle, the TOC solution trajectories for states and inputs of the PMSM are determined neglecting current and torque limits. With the TOC solution, a flux linkage reference for the CCS-MPFC is calculated that corresponds to a prerotation of the operating point in the stator-fixed coordinate system. This prerotated flux linkage reference is reached in minimum time without overcurrents and torque over- as well as undershoots by incorporating current and torque limits as time-varying softened state constraints into the CCS-MPFC. Simulative and experimental investigations for linearly and non-LM-PMSMs in the whole speed and torque range show that, compared to state-of-the-art TOC methods, overcurrents and torque over- as well as undershoots are prevented by the proposed TO-MPC.


I. INTRODUCTION
D UE to the high torque and power densities of permanent magnet synchronous motors (PMSMs) with resulting low rotor inertia, PMSMs are well suited for highly dynamic applications. Since the torque control directly affects the jerk and acceleration of the drive, the torque response is an important performance criterion for characterizing the dynamics of the drive. Therefore, improving the torque response dynamics is an ongoing and important research topic in academia and industry [1], [2], [3], [4], [5], [6]. Manuscript

A. State-of-the-art Techniques
A desired torque can be achieved by various combinations of stator currents. Here, the operating point with the least losses is preferred. This loss-optimal reference operating point is usually calculated by an open-loop operating point controller (OPC) and fed to the underlying closed-loop controller. As control variables, currents or flux linkages in the stator-or rotor-fixed coordinate system can be used as well as the tupel consisting of torque and flux linkage amplitude.
To achieve time-optimal control (TOC) performance, the future transient state and input trajectories must be considered for optimization. Consequently, all controller types that do not optimize the whole future trajectory during transients, e.g., conventional proportional-integral field oriented control (PI-FOC) [7], deadbeat direct torque and flux control [8], and direct torque control [9] achieve time-suboptimal control performance. Particulary during larger torque variations, it may take many sampling periods to reach a new reference operating point. Accordingly, model predictive controllers (MPCs) [10], [11], [12], [13] may only achieve suboptimal control performance during transients because their prediction horizon is always restricted by computational capacity of the embedded controller device.
To increase the control dynamics to its maximum extent, control methods that solve the TOC problem online [2], [3], [4], [5], [6], must be applied. With the help of optimal control theory methods, e.g., Pontryagin's maximum principle, the TOC solution for input and state trajectories during transients for PMSMs can be derived. The resulting state trajectories reach the operating point in the shortest possible time. From previous publications in the field, two characteristics for the stator voltages u αβ = [u α u β ] (input trajectories) in the stator-fixed αβ-coordinate system of the TOC solution during transients can be observed [2], [3] as follows.
2) The stator voltages u αβ (t) are saturated by the input constraint (voltage hexagon or a circular approximation of the voltage hexagon). Constant u αβ (t) voltages lead to a linearly shaped trajectory of the PMSM's flux linkage ψ αβ (t) by neglecting the ohmic voltage drop, since the ordinary differential equation (ODE) of ψ αβ (t) (Faraday's law of induction) is equal to an integrator, and the intersection of the input constraints ensures fastest possible movements of ψ αβ (t) in the αβ flux linkage plane. To calculate this time-optimal stator voltage u αβ during transients, a nonlinear equation system must be solved iteratively in an online fashion. The publications [2], [3], [4] solve the TOC problem in every sampling instant. Here, the bisection method as numerical solver with a constant number of iterations per sampling instant can be applied [3]. For the rotating reference chase control (RRCC) method proposed in [6], one iteration to solve the TOC problem per sampling instant is executed based on the TOC solution of the previous sampling instant. Compared to the methods [2], [3], [4], [5], only the RRCC takes transient overcurrents as well as torque over-and undershoots into account and tries to prevent them heuristically. Nevertheless, these transient overcurrents as well as torque over-and undershoots can be reduced compared to the TOC methods [2], [3], [4], [5], but not prevented with the RRCC, which is shown in Section VI.

B. Contribution
To prevent transient overcurrents as well as torque over-and undershoots in a TOC framework, the method of time-optimal model predictive control (TO-MPC) 1 is proposed in this article. The TO-MPC contains a continuous control set model predictive flux control (CCS-MPFC) and a reference pre-rotation (RPR) that manipulates the flux linkage reference of the CCS-MPFC and ensures TOC behavior, see Fig. 1. Reason for the application of the CCS-MPFC is that compared to other control methods, e.g., PI-FOC, an MPC provides the possibility to naturally take into account current and torque limits that can be formulated as time-varying softened state constraints.
Moreover, the propsed TO-MPC is incorporated in a torque control scheme, cf., Fig. 1, with the following additional elements.
3) Space vector modulation (SVM) scheme that converts the voltages u αβ commanded by the CCS-MPFC into switching commands s abc [18]. The proposed RPR solves the set of nonlinear equations of TOC numerically in every sampling instant without considering torque and current limits. With the TOC solution, the reference flux linkage ψ * αβ for the CCS-MPFC is calculated. This flux linkage reference corresponds to a prerotation of the flux linkage reference ψ dq (i * dq ) in the dq-coordinate system transformed to the αβ-coordinate system with the momentary electrical rotor angle ε. To steer the PMSM's flux linkage to the flux linkage reference of the RPR, the CCS-MPFC including torque and current limits is applied. As a result of the proposed TO-MPC, the following advantageous properties for the overall control scheme can be accomplished.
1) The TO-MPC is able to achieve minimum settling times during transient operation in the entire speed and torque range thanks to the reference flux linkage manipulation for the CCS-MPFC by the RPR. 2) Compared to TOC methods proposed in [2], [3], [4], [5], [6], overcurrents as well as torque over-and undershoots during transient operation are prevented by time-varying torque and current limits implemented as linear state constraints for the quadratic program (QP) of the CCS-MPFC.

C. Article Structure
The rest of this article is organized as follows. Section II describes the general control framework. The discrete-time prediction models for flux linkage, current, and torque are derived in Section III. In Section IV the RPR is presented. Section V focuses in detail on the CCS-MPFC. Extensive simulative and experimental investigations are discussed in Sections VI and VII. Finally, Section VIII concludes this article.

II. GENERAL CONTROL FRAMEWORK
In the following, the overall control scheme shown in Fig. 1 is explained in more detail.

B. Inverter
For a three-phase, two-level inverter, the stator voltages of a motor with star-connected windings are given by where, u DC is the dc-link voltage and d abc ∈ [0, 1] 3 the duty cycle vector of the inverter.

C. Operating Point Control
To minimize ohmic losses the operating point control (higher level open-loop torque controller) proposed in [14] is utilized, which selects the operating point i * dq based on the MTPC and MTPV strategies. Here, the nonlinear magnetization with significant (cross-)saturation effects of highly utilized PMSMs are linearized online and optimal operating points i * dq are calculated analytically. The nonlinear magnetization is taken into account iteratively with a successive linearization and analytical calculation of the subsequent optimal operating point. To reduce computation time only one iteration to calculate the optimal operating point per controller cycle (sampling instant) is executed by the OPC in this article.

D. 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 [16], [17]. Here, the flux linkage estimate of a current model via a current-to-flux linkage look-up table (LUT), see Fig. 10, is combined with the flux linkage estimate of a voltage model (6).

III. DISCRETE-TIME PMSM MODEL
The proposed TO-MPC requires motor model variants for predicting the flux linkage in the αβ-coordinate system, the current in the dq-coordinate system [19], [20], [21], as well as the airgap torque. These models are derived in the following.

A. Flux Linkage Model in the αβ-Coordinate System
According to Faraday's law of induction, the differential equation of the flux linkages ψ αβ = [ψ α ψ β ] for a PMSM can be described as follows: where, i αβ = [i α i β ] represents the stator current, and R s the ohmic stator resistance. By applying the forward Euler method with a sampling time T s , the discrete-time flux linkage difference equation in the αβ-coordinate system is given by with: where, I is the identity matrix.

B. Current Model in the dq-Coordinate System
To derive the difference equation of the current i dq , the motor model (6) 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 with the electrical angular velocity ω [19], [20], [21]. To link the variation in flux linkage ψ dq with the variation in current i dq , the differential inductance matrix must be applied. By approximating (9) and inserting (10) in (7), the discrete-time current difference equation of a PMSM considering (cross-)saturation effects in the dq-coordinate system evaluates to with:

C. Torque Model
The airgap torque of the PMSM is given by By linearizing (12) with respect to the current, the torque-tocurrent relation can be approximated by . (13) Here, the partial derivative is calculated from (12) and (9). The current difference Δi dq in (13) can be expressed as using the current model (11). Inserting (15) in (13) leads to the following torque model: with:

IV. REFERENCE PREROTATION
The task of the RPR is to calculate a reference flux linkage ψ * αβ [k + 1] without respect to overcurrents as well as torque over-and undershoots for the CCS-MPFC such that the required time to reach the operating point i * dq during transient operation is minimal. The RPR, therefore, inherently solves the TOC problem.

A. Steady-State Control Conditions
, the reference flux linkage must be equal to The flux linkage reference (17) is derived with a current-to-flux linkage LUT, see Fig. 10, for the operating point i * dq , which is calculated by the OPC, to take (cross-)saturation effects into account. For a constant operating point i * dq and speed ω, in the linear modulation range, the operating point in the αβ flux linkage coordinate system ψ αβ (i * dq , ε) rotates with a constant magnitude and angular velocity ω, cf., (17) and Fig. 3(b).

B. Transient Control Conditions
During transient control conditions the RPR must solve the TOC problem. For a general nonlinear dynamical system the TOC problem is given by where, the timet (18a) is minimized that is needed to steer a dynamical system (18b) from an initial state x 0 (18c) to its reference x * (18d) with respect to state (18e) and input constraints (18f). In this work the differential equation characterizing the dynamical system (18b) corresponds to Faraday's law of induction (5), the state x corresponds du the flux linkage ψ αβ , the input u to the voltages u αβ , the state constraints to current and torque constraints, and the input constraints to the voltage hexagon.
By applying Pontryagin's maximum principle to find the timeoptimal stator voltages u αβ (t), i.e., solving the TOC problem, during transients to steer the PMSM's flux ψ αβ to its reference ψ * αβ for neglected state constraints, e.g., overcurrents and torque over-and undershoots, the following two characteristics are valid [2], [3].
2) The stator voltages u αβ (t) are saturated by the input constraint. This results in linearly shaped trajectories of the flux linkage ψ αβ (t) of the PMSM when neglecting the ohmic voltage drop.
To calculate this time-optimal stator voltage, a set of nonlinear equations for the timet, that is needed to steer the PMSM's flux ψ αβ to its reference ψ * αβ , and the stator voltage u αβ must be solved [2], [3] where, the voltage drop of the ohmic stator resistance R s is neglected and u max represents a circular approximation of the voltage hexagon. This approximation is set to which corresponds to the fundamental voltage of six-step operation [22]. The left-hand side of (19a) represents the prerotation of the flux reference ψ * αβ as a function of the rotation angle ω[k]t and the right-hand side represents the linear evolution of ψ αβ as a function oft and u αβ . After a time duration oft, the predicted flux ψ αβ must coincide with the reference flux ψ * αβ , whereby a constant voltage u αβ with maximum amplitude u max is applied.
To solve the system of nonlinear equations (19) in every sampling instant, numerical methods must be applied. A method to iteratively solve (19) fort is given in pseudocode Algorithm 1 from line 1 to 9 and a graphical representation is depicted in Fig. 2. Here, N iteration steps are performed from line 6 to 9 to approximate the solution of (19) witht ≈t N and ψ * αβ (kT s +t) ≈ ψ * αβ,N . The iteration step from line 6 to 9 is similar to the iteration step that is only performed once per sampling instant, which is proposed for the RRCC in [4]. In line 10-14 of Algorithm 1, a distinction between transient and steady-state control operation is made. If the timet to steer ψ αβ to its reference ψ * αβ is smaller than a chosen threshold t thresh , steady-state control operation is present and ψ * αβ [k + 1] is set according to (17). Otherwise, the controller is in a transient operation state and the reference ψ * αβ is prerotated by the angle ωt, cf., (19a). The threshold t thresh is a tuning parameter and must be set to t thresh ≥ T s . Here, t thresh = T s would be the obvious choice. Since the voltage hexagon is approximated circularly, slightly increased values for t thresh are recommended, e.g., t thresh = 1.1T s . . . 1.5T s .
Applying the deadbeat flux control law with the calculated flux linkage reference ψ * αβ [k + 1] of the RPR results in time-optimal transient and accurate steady-state operation. To saturate the voltages (21) to the voltage hexagon, the minimum phase error dynamic overmodulation scheme [23] must be applied.
However, TOC performance is achieved by applying the deadbeat flux control (21) with the RPR, neither unintended violations of the current limit nor over-and undershoots of the torque can be prevented. This is shown by a simulation, cf., Fig. 3, with steps in the reference torque T * and the corresponding reference currents i * dq calculated by the OPC. For the sake of clarity, only the discrete-time samples with T s = 62.5 µs that are synchronized with the SVM are shown. Therefore, the current, torque, and flux ripples induced by the switching of the inverter are not visible in the following figures. For this simulation a linearly magnetized PMSM (LM-PMSM), with the parameters given in Table I, is used.
In Fig. 3(a), the flux linkage reference ψ * αβ , calculated by the RPR, and the resulting u αβ , calculated with the deadbeat control law (21), are depicted. Here, nearly constant flux linkage   references ψ * αβ and voltages u αβ during transients can be seen. The slight deviations of constant voltages u αβ during transients are caused by RPR with the circular approximation of the voltage hexagon, the neglected ohmic voltage drop, and the finite number of iterations. However, these deviations can be considered as minor, which results in the characteristic time-optimal linear evolution of the flux linkage ψ αβ during transients in the αβ flux linkage plane, cf., Fig. 3(b). Furthermore, the time-optimal trajectories can be visualized in the dq current coordinate system, see Fig. 3(c). Here, the current change caused by the induced voltage, characterized by E i (i dq ), is optimally exploited by the RPR to achieve TOC performance. Nevertheless, current limits are violated and overand undershoots of the torque cannot be avoided, see Fig. 3(a) and (c).
To investigate the influence of the number of RPR iterations N on the control performance, the transients 1 and 2 of the scenario depicted in Fig. 3 are shown in Fig. 4 for N = {0; 1; 2; 5; 100}. For N = 0 no prerotation of the flux linkage reference is conducted and N = 100 represents an approximation of N = ∞. It can be seen that the deadbeat controller without prerotation of the reference (N = 0) is not able to reach its reference within the given time. For N ≥ 5 the time-optimal characteristic linearly shaped flux linkage trajectories during transient operation are achieved due to the fast convergence of the RPR. Increasing N to more than 5 would not improve the  results significantly but increase the computational load, which is not desirable.

V. CONTINUOUS CONTROL SET MODEL PREDICTIVE FLUX CONTROL
To overcome the unintended control characteristic of timeoptimal controllers that do not consider current and torque limits during transient operation, e.g., [2], [3], [5] or the deadbeat controller (21), a CCS-MPFC with state and input constraints and the reference flux linkage ψ * αβ calculated by the RPR, is proposed in the following. With the help of state constraints for the CCS-MPFC, current limits are taken into account and overas well as undershoots of the torque are prevented.

A. Current Constraints
Although the OPC selects operating points i * dq within the current limit I max , this limit can be violated during transient processes, see Fig. 3(c). This can lead to thermal overload or increased thermal cycling of the inverter semiconductors due to their small thermal time constants. To prevent this, a dynamic current limit I max,dyn is introduced. The dynamic current limit is a tuning parameter and must satisfy I max,dyn ≥ I max .
Incorporating a circular current constraint for the predicted current i dq [k + 1] into the CCS-MPFC would lead to an optimization problem with quadratic inequality constraints. Since these would increase the computational burden further than linear inequality constraints, the circular dynamic current constraint I max,dyn is approximated with a linear time-varying current constraint, see Fig. 5. This current constraint can be formulated as a linear inequality constraint for the predicted Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
To prevent nonmonotonic torque dynamics due to positive i d currents, as it can be seen in the operating point change 2 in Fig. 3(c), an additional linear current constraint with the tuning parameter i d,max ≥ 0 A can be added, see Fig. 5. Both current constraints (22) and (23) can be rewritten in matrixvector notation To solve the optimization problem of the CCS-MPFC (32), the inequality constraint (24)

B. Torque Constraints
To prevent over-as well as undershoots and to ensure monotonic torque trajectories during transients, the predicted torque T [k + 1] must satisfy Both conditions (26) and (27) can be considered with and Furthermore, (28) and (29) can be rewritten in vector notation . (30) Similarly to the current constraint (24), the torque constraint (30) must be mapped to the optimization variable u αβ [k] of the CCS-MPFC (32). This is achieved by inserting the torque prediction model (16)

C. Quadratic Optimization Problem
The cost function of the CCS-MPFC is designed to penalize the Euclidean distance from the predicted fluxψ αβ [k + 1] to the reference flux linkage ψ * αβ [k + 1] of the RPR, cf., Section IV. Furthermore, input constraints (voltage hexagon) and state constraints (torque and current, cf., Sections V-A and V-B) must be fulfilled. Thus, the optimization problem can be defined as For the practical implementation of this contribution the state constraints (32c) were softened with the help of a slack variable formulation to ensure feasibility of the optimization problem [24], [25]. To solve the linearly constrained quadratic program (32), any standard QP solver can be utilized. In this work, the embedded solver of the MATLAB MPC toolbox was chosen [26], [27]. To compensate for the control delay due to the digital implementation, a one-step state prediction is applied before the QP solver is called [28].

VI. SIMULATIVE INVESTIGATION
On the basis of a simulation, the TO-MPC (RPR combined with the CCS-MPFC of Section V) is investigated. The LM-PMSM motor model characterized by Table I is applied for this study. All following simulative investigations were conducted with the software Simulink from MathWorks. Here, the inverter and motor are modeled in a quasi-time continuous environment based on their equations reported in Sections II and III. The TO-MPC is simulated in a discrete-time subsystem with the controller sampling time T s = 62.5 µs. The resulting ODE of the Simulink model is solved with the adaptive Runge-Kutta method ode45 [29]. The parameters of the TO-MPC and the simulation settings are listed in Table II. Since the TO-MPC is synchronized with the SVM, current, torque, and flux ripples induced by the switching of the inverter are not visible in the following figures.
Based on torque step responses, the transient control performance of the proposed method is analyzed for different initial rotation angles ε(t = 0 s) and different constant speeds. To illustrate the working principle of the TO-MPC an exemplary video animation is available in [30] for the same scenario as depicted in Fig. 3.

A. Initial Rotor Angle Investigation
Due to the voltage constraint (32d) and the cost function (32a) of the TO-MPC, the corners (elementary vectors) of the voltage hexagon are preferred as input variables u αβ during transients, since these often reduce the cost function the most as long as torque and current constraints (32c) are not active. For this reason, not only the flux linkage trajectory in the αβ-coordinate system during transients depends on the initial rotor angle, but also the current trajectories in the dq-coordinate system and the torque, see Fig. 6. Nevertheless, similar settling times for the torque without over-and undershootings of the torque and without violating the dynamic current limit I max,dyn are achieved, see Fig. 6(a) and (c). In Fig. 6(b), the flux linkage trajectories for a step response to the rated torque are depicted for equidistant distributed initial rotor angles ε 0 = ε(t = 0 s) from 0 to π/3. Here, the flux linkage trajectory for ε 0 = 0 is equal to the trajectory for ε 0 = π/3 rotated by an angle of −π/3 due to the symmetry of the voltage hexagon and, therefore, results in identical dq current and torque trajectories.

B. Speed Dependency Investigation
Since the prerotation of the flux linkage reference depends on the angular velocity ω, the transient trajectories of torque and dq current differ during transient operation for different motor speeds n me even for an identical initial rotor angle ε 0 and identical torque reference trajectories, see Fig. 7. Here, step responses to maximum and minimum torque are commanded. For speeds of n me = {0, 2750} min −1 the maximum and minimum motor torque can be realized. However, for a speed of n me = 13000 min −1 , the OPC selects the intersections of the voltage limit and the MTPV trajectory as operating points i * dq which results in reduced torque magnitudes (flux weakening operation) compared to the rated operating point, cf., Fig. 7(b). Nevertheless, maximum and minimum possible torques are achieved.
Although the initial dq current i dq (t = 0 s) = 0 A is outside the voltage limit for n me = 13000 min −1 and, thus, a torque undershoot is inevitable, the optimization problem (32) of the  III  REQUIRED SAMPLING INSTANTS TO CONDUCT THE OPERATING POINT  CHANGES DEFINED IN FIG. 3 FOR TOC WITH AND WITHOUT STATE  CONSTRAINTS AS WELL AS THE TO-MPC  TO-MPC remains feasible due to its softened state constraints (32c) and a solution is found that limits the torque undershoot.

C. Time-Optimality Investigation
As a result of the state constraints (32c) the space of possible state trajectories during transient operation is restricted. To empirically investigate whether the TO-MPC leads to time-optimal trajectories even with active state constraints, the TO-MPC's control performance is compared to a TOC with the same torque and current constraints. This TOC with state constraints takes the whole future state and input trajectory during transient operation into account to minimize the time to reach the reference operating point i * dq , i.e., its prediction horizon is practically infinitely long. The resulting nonlinear constrained optimization problem was solved with a sequential quadratic programming algorithm of the MATLAB optimization toolbox. This is feasible for an offline simulation comparison, but of course the computational burden of the TOC solution is by far higher than the one step prediction required to solve the TO-MPC problem.
The sampling instants that are required to conduct the operating point changes of Fig. 3 for the TO-MPC and the TOC with state constraints are listed in Table III. Here, the TO-MPC achieves the same transient control performance as the TOC with state constraints. Thus, empirical evidence was provided that the TO-MPC solution requires the same transient performance as a TOC approach at significantly reduced computational cost (one-step versus unlimited prediction steps).
Furthermore, the required sampling instants for the TOC without state constraints are listed in Table III. This TOC corresponds to the deadbeat flux control law (21). It can be seen that only for the operating point changes 2 and 5 the required sampling steps are reduced compared to the TOC with state constraints and the TO-MPC for the price of unintended transient current and torque trajectories, cf., Fig. 3.

D. Comparison to State-of-the-art Methods
In this section the control performance of the TO-MPC is compared to the state-of-the-art continuous-control-set methods of PI-FOC, RRCC [6], and a CCS-MPFC without state constraints and without RPR. Thus, the optimization problem of this simple, standard CCS-MPFC evaluates to The parameters of the decoupled PI-FOC current controllers are listed in Table II. Compared to the PI-FOC, both the RRCC and CCS-MPFC (33) does not contain any tuning parameters. All controllers are sampled with T s = 62.5 µs and synchronized with the SVM in the same way as the TO-MPC.
For this comparison, the same reference torque trajectory, as in Fig. 7(a) consisting of steps to rated motor and generator operation at rated speed (n me = 2750 min −1 ) with an initial rotor angle ε(t = 0 s) = 0 is selected. The resulting torque and current trajectories for the TO-MPC, PI-FOC, RRCC, and CCS-MPFC are depicted in Fig. 8. Since the torque and current constraints are not taken into account by the PI-FOC and CCS-MPFC, overcurrents and torque over-and undershoots  cannot be prevented. Furthermore, increased settling times of the PI-FOC and CCS-MPFC especially for the transients starting at t = 0 ms and t = 8 ms can be observed compared to the TO-MPC and RRCC. The RRCC and the TO-MPC achieve similarly fast settling times since both use a prerotation of the reference flux linkage. However, the RRCC takes current and torque constraints only into account heuristically and, therefore, prevention of overcurrents and torque over-and undershoots cannot be guaranteed.

VII. EXPERIMENTAL INVESTIGATION
All following experimental results have been obtained on a laboratory test bench, see Fig. 9. The electrical drive system under test is a highly utilized interior permament magnet synchronous motor (Brusa: HSM1-6.17.12-C01) for automotive applications and a two-level IGBT inverter (Semikron: 3×SKiiP 1242GB120-4D). The datasheet parameters, flux linkages and differential inductances can be seen in Table IV, Figs. 10 and 11. 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 IV.   TABLE IV  PMSM, DC-LINK, INVERTER, CONTROL, AND TEST BENCH PARAMETERS OF  THE EXPERIMENTAL TEST SETUP   TABLE V  TURNAROUND TIMES OF THE CONTROL STRATEGY The turnaround times of the OPC, GFO, RPR, CCS-MPFC, auxiliary functions and the overall control strategy are listed in Table V. Here, the low computational load of the RPR with an iteration number of N = 5 can be seen. The turnaround time reported for the auxiliary functions cover, e.g., SVM, coordinate transformations, analog-digital conversion, as well as processor and host computer communication that must be executed in addition to the OPC, GFO, RPR, and CCS-MPFC, are summarized. Compared to the other parts of the overall control scheme, the CCS-MPFC demands a variable number of calculation steps per controller sample. This is due to the varying required number of iteration steps for the utilized embedded active-set solver, cf., [26], [27], to find the global optimum of the QP (32). To reduce the number of iterations of the active-set solver the solution of the previous sampling instant was used as initial solution guess (hot start). Furthermore, in simulations and experiments it has not been observed that more than 11 iterations are required to find the optimum. Therefore, the required turnaround time of   Table V.
In order to prove the effectiveness and performance of the TO-MPC, several representative experiments in the torque and speed range were carried out and 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, depicted in the following figures is not measured directly with the help of a torque sensor since 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. Instead, the torque is estimated viâ

A. Initial Angle Investigation
In Fig. 12, the trajectories for a step response to the rated torque are depicted for equidistant distributed initial rotor angles ε 0 = ε(t = 0 s) from 0 to π/3 at rated speed. Here, torque overand undershoots as well as overcurrents are prevented. Due to nonideal effects, e.g., parameter and model inaccuracies as well as measurement noise, prediction errors can lead to slight violations of the torque and current constraints. Nevertheless, these violations can be considered minor. Similar to the simulative investigation in Section VI-A for the linear-magnetized PMSM, the current trajectories differ for different initial rotor angles for the highly utilized PMSM with significant cross saturation, cf., Fig. 12(a). However, the resulting torque trajectories are similar for different initial rotor angles, see Fig. 12(c).

B. Speed Dependency Investigation
The torque and current trajectories for torque reference steps to rated motor and generator operation for different speeds in the whole speed range from standstill to maximum speed are depicted in Fig. 13. Due to the voltage limit for n me = {6000, 11000} min −1 , the rated torque can no longer be achieved, see Fig. 13(a). However, the OPC ensures that the maximum and minimum possible torques are achieved for the corresponding speed. Because of the nonlinear magnetization, no sectionwise linear current trajectories occur at standstill compared to the current trajectories of the PMSM with linear magnetization, cf., Fig. 7(b).

C. Dynamic Current Limit Investigation
The state constraints (32c) ensure that any selected voltages u αβ respect the dynamic current limit I max,dyn . Therefore, the dynamic current limit I max,dyn restricts the set of feasible voltages u αβ during transient operation at the current limit the more, the smaller I max,dyn is chosen. To investigate the influence of the dynamic current limit, the torque, current, and flux linkage trajectories are depicted in Fig. 14 for torque reference steps to rated motor and generator operation at rated speed n me = 4750 min −1 . Here, slightly reduced settling times of the torque for increased dynamic current limits I max,dyn can be observed since the trajectories of the currents reach the isotorque loci earlier, see Fig. 14(a). However, the flux linkage reference operating points are reached with similar settling times, cf., Fig. 14(c) and (d). During the transient from the rated generator to motor operating point at approx. 4 ms, there are current samples near the d-axis which are slightly below and above the dynamic current limit I max,dyn . This deviation of the current trajectory from the dynamic current limit can be explained by prediction errors of the current prediction model (11) due to the rapidly varying differential inductance L qq , cf., Fig. 11.

VIII. CONCLUSION AND OUTLOOK
In this article, the concept of TO-MPC for PMSM was proposed and investigated in the whole speed and torque range via simulations and experiments. Here, TOC performance is achieved with the help of a prerotation of the flux linkage reference for a CCS-MPFC. Thanks to the incorporation of linearized time-varying current and torque limits as softened state constraints of the TO-MPC, overcurrents as well as torque over-and undershoots can be effectively prevented compared to the state-of-the-art TOC methods for PMSMs.
In the future, the extension of the TO-MPC to the overmodulation range up to the six-step operation will be investigated. This should be achieved by incorporating the harmonic reference generator approach presented in [32] with the prerotation of the flux linkage reference to enable TO-MPC in the whole modulation range.