Continuous Control Set Model Predictive Torque Control With Minimum Current Magnitude Criterion for Synchronous Motor Drives

To achieve high efficiency and dynamics in electric drive applications, it is necessary to have accurate torque control. This is typically accomplished through a current regulator that is fed by references generated by various open-loop control strategies, in order to obtain the desired torque. As an alternative, this work presents a model predictive torque control. Starting from the torque reference, the algorithm generates optimal voltage references to the inverter-fed synchronous motor drive while working at maximum efficiency and considering the motor current limit. This feature is achieved by combining two different norms in the cost function. The electromagnetic torque is estimated from the measured currents and an accurate magnetic model of the motor. According to the paradigm of the more autonomous drives, an important feature is that the algorithm requires only knowledge of the motor model. This means that a tuning procedure for control weight is no longer required as analytically discussed in this work. Experimental validation of the proposed technique is performed on a test rig featuring an anisotropic permanent magnet motor in different dynamic operation, including flipping from the motor nominal working point to the generator one.

x k .
= x(k) Discrete-time x variable.x| y=z Stands for x calculated at y = z.
l 2 -norm of vector x.

I. INTRODUCTION
T HE pressing demand for energy-efficient and environ- mentally friendly electric drives across various civil and industrial sectors is driving research toward synchronous motors that utilize reduced or minimal amounts of rare-Earth permanent magnets.Anisotropic or pure reluctance motors, which fulfill this need, offer a degree of freedom in torque control but also bring inherent complexity due to their nonlinear magnetic model.
For efficient torque control, the standard approach is to obtain current references for torque demand through a maximum torque per current (MTPC) strategy, which can be implemented using either analytical expressions or online perturb-andobserve methods.An exhaustive and comprehensive example of current reference generation can be found in [1], but many other alternatives exist, see [2] and the related bibliography.
The torque control based on analytical methods requires an accurate model of the motor.Normally, this model is obtained through voltage and current measurements at various points within the operational range of the drive.The relationships between currents and flux linkages are then stored in look-up tables (LUTs) [3].A significant advancement in this area is the utilization of radial basis function (RBF) networks to derive precise and differentiable analytical expressions of flux linkages [4], which account for saturation and cross-coupling effects.However, a disadvantage of this approach is the necessity to train the neural networks, which adds complexity to the MTPC algorithm that may not be feasible for all microprocessors typically employed in the ac drives field.
In the perturb-and-observe methods, the MTPC is determined by adjusting the current vector angle of a common field-oriented control (FOC) algorithm.This adjustment is done by observing changes in the current-magnitude-reference signal generated by the speed proportional-integral (PI) controller [5].The advantage of this method is that it does not require motor equations or parameters.However, the perturbation signal used in this method can cause torque fluctuations, which may affect the speed, especially in the case of low inertia loads.Additionally, load torque disturbances can interfere with the MTPC algorithm, so a smooth steady-state condition is necessary for proper functioning.A good implementation example, together with an extensive discussion of pros and cons of different MTPC strategies, is described in [6].In case of magnetic saturation, PI current regulators need to adapt their gains to maintain a constant bandwidth [7].This is recommended to guarantee high performances and control loop stability [8], [9] but increases the complexity of the regulator's design.
Another approach to implementing torque control is through the use of a direct torque controller (DTC) [10].The fundamental principle of DTC involves directly selecting the voltage source inverter states while tracking the errors between the reference values and the estimated torque and stator flux linkage [11].In DTC methods, the relationship between torque and stator flux linkage is a critical aspect of achieving high efficiency.A notable example is presented in [12], which develops the MTPC control in a frame synchronous to the stator flux linkage.One advantage of this method is that it does not require the q-axis inductance, which is susceptible to magnetic saturation.However, due to the limited voltage combinations that can be applied during each control period, this approach involves a tradeoff between switching losses and torque ripple in steady-state conditions.
An interesting alternative is represented by model predictive controllers (MPCs).Over the past decade, they have collected significant interest in electric drive applications, primarily due to the availability of high computational power hardware at reduced costs.MPCs represent a promising class of controllers that possess inherent capabilities for online optimization with multivariable control, enabling the minimization of a cost function that may encompass tracking accuracy, fast dynamics, and high efficiency.In many respects, they are considered the ideal choice for the next generation of high-performance electric drives [13].A comprehensive and exhaustive presentation of MPCs can also be found in [14], [15], [16].
Traditionally, ac drives had limited computational power onboard, which necessitated early attempts at implementing MPC to rely on finite control set model predictive control (FCS-MPC).This approach allows the predictive algorithm to select a control voltage vector from the limited set of inverter states.An illustrative example can be found in [17].In contrast to conventional DTC, the stator flux is not explicitly controlled, and hysteresis bounds are not used.However, there are certain disadvantages, such as the need to determine three weighting coefficients.Additionally, FCS-MPC shares the ripple-related drawbacks of the DTC controller, as it applies only one full-amplitude state voltage vector at a time throughout the entire control period.
Recently, advancements have been made in improving the choice of weights by combining FCS-MPC with a neural network [18].However, this comes at the cost of increased complexity, which somewhat contradicts the requirement for low computational power.
It is worth saying that modern ac drives benefit from the availability of more powerful floating-point processors, which enable the utilization of continuous control set model predictive control (CCS-MPC) to a great advantage.By employing a space voltage modulator, the CCS-MPC algorithm can precisely select the required control action, resulting in a significant reduction in current and torque ripple while maintaining the same level of dynamic performance if no penalization on voltage variations is considered in the cost function.A comprehensive overview can be found in [19].When dynamics are the main concern, time-optimal MPC proves to be the best solution, as demonstrated in [20].However, when steady-state efficiency is the primary consideration, alternative solutions must be explored.This article addresses the latter topic.

A. Contribution
This article proposes continuous control set model predictive torque control (CCS-MPTC) for direct torque reference tracking while minimizing the magnitude of the current vector.
The contributions of this work can be summarized as follows.1) No current or flux regulators are implemented, eliminating the need for reference generation algorithms.2) Adoption of both l 1 -and l 2 -norms in the cost function to track the torque reference and minimize the magnitude of the current vector, respectively.3) Insights are provided on algorithm design.Specifically, the disadvantages of using the l 2 -norm for torque reference tracking are highlighted, and a criterion for choosing the weight coefficient is presented.The proposed minimization problem is formulated as a quadratic programming (QP) problem, allowing the utilization of generic QP solvers to find the optimal solution.Several high-efficiency solvers are available that can be implemented on real-time control platforms [21], [22], [23].
As is customary in MPC algorithms, precise knowledge of the motor parameters is required to avoid errors in torque reference tracking or suboptimal steady-state operating points.Although the description of motor parameter identification techniques is beyond the scope of this work, it is recommended to adopt the appropriate ones from the excellent options available in the literature [3], [24], [25].

B. Article Structure
The rest of this article is organized as follows.The discretetime synchronous motor model is described in Section II.The proposed CCS-MPTC and an insight into the cost function are both reported in Section III.To demonstrate the effectiveness of the proposed technique, the experimental results along with detailed discussions are presented in Section IV.Finally, Section V concludes this article.

II. DISCRETE-TIME SYNCHRONOUS MOTOR MODEL
Two discrete-time motor models are needed for the proposed algorithm, i.e., current and torque model.

A. Discrete-Time Motor Model
The discrete-time current dynamic model is obtained by applying Euler forward discretization to the voltage balance equation in the dq reference frame, as described in [26] and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.[27].This results in where where are the voltage, current, and flux vectors, respectively.R = R s I ∈ R 2×2 is the stator resistance matrix, with I is the identity matrix, ω k is the electrical speed and T s is the sampling and control time period.
The differential inductances matrix L dq (i dq ) is introduced as follows: where the cross-saturation inductances are equal, i.e., l dq (i dq ) = l qd (i dq ), [28].

B. Electromagnetic Torque Model
For the sake of simplicity, the torque model is directly written in the discrete-time domain without a lack of generality.The torque is given by where p is the pole pair number of the motor.In order to obtain a linear torque prediction at time k + 1, the torque relationship (3) has to be linearized with respect to the current i k dq .The first-order approximation is where The current difference Δi k+1 The final torque model by using ( 1), ( 4), and ( 6) is

III. PROPOSED CCS-MPTC ALGORITHM
The proposed CCS-MPTC control algorithm, sketched in Fig. 1, aims to determine the optimal voltages by solving an optimization problem.The cost of the optimization problem to be minimized considers the torque reference tracking τ * − τ k+1 and the current magnitude i k+1 dq .Only the torque reference tracking term is weighted by a parameter c τ .As an original contribution of this article, the choice of norms and the analytical design of weight c τ is discussed.As a distinctive feature of the proposed algorithm, the controller is able to track the reference torque and minimize the current vector amplitude at a steady state.
The minimization problem of the proposed CCS-MPTC control algorithm at time k is defined as min where (M , m) and (G i , g i ) are related to voltage and current constraints, respectively, defined in ( 10) and ( 14).The current constraint (G i , g i ) is already mapped to the input space with the current prediction model (1).The choice of norms for the current and the torque control error in the cost function ( 8) is discussed in Section III-C.It is worth recalling that the goal of the proposed CCS-MPTC is able to track the torque reference without offset and minimize the current vector magnitude during stationary operations.This can be obtained only with an appropriate design of c τ weight which is discussed in Section III-B.Despite the presence of an l 1 -norm, the proposed minimization problem can be rewritten into a standard QP problem: This is a crucial aspect for a real-time implementation using generic QP-solvers.All mathematical steps are reported in Appendix A. In this work, the embedded active-set algorithm of the Matlab MPC toolbox was chosen [29].The one-step delay introduced by the digital implementation of the controller is adequately compensated as suggested in [30].

A. Voltage and Current Constraints
A three-phase voltage inverter is considered.In order to obtain a feasible voltage vector u k+1 dq , the hexagonal constraint in the dq current plane is implemented as linear inequality constraints [26], [31], [32].Matrices M and m in (8) are defined as where u bus is the measured bus voltage and T dqαβ (ϑ k ) (where ϑ is the electrical position) is a matrix transformation from the stator-fixed αβ frame to the rotor-fixed dq frame.
In order to keep the current vector magnitude lower than the nominal current, avoiding overcurrents during transient operations, a linear current constraint has to be defined due to the linearly-constrained QP problem implementation.Therefore, the circular current constraint has to be linearized as proposed in [33].The linear time-varying current constraint, which is a function of the instantaneous current i k dq , implemented in this article is An additional constraint is necessary to avoid undesired current trajectories in the right-half dq current plane, that is where i d,pos is a positive d-current quantity.For this article, i d,pos is set at 0.1 I n .Allowing a small positive i d current during transients enables faster dynamics [33].However, this does not affect the efficiency at a steady state, where the proposed torque control ensures MTPC operation.The current constraints (11) and (12) are sketched in Fig. 2, where the forbidden regions are shadowed in red.Both current constraints (11) and ( 12) are combined in a matrix-vector notation Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Since the current constraints in (13) have to be written as a function of the voltages for being implemented in (8), the current prediction model ( 1) is used in ( 13) to obtain From a practical implementation point of view, the current constraints are softened with the help of a slack variable to ensure problem feasibility [34], [35].

B. Selection of the Weight c τ : Case of Isotropic Motor
Initially, to ease the design of the proposed cost function in (8), the case of an isotropic permanent magnet synchronous motor (PMSM) is considered.Here, operation in the constant-torque region is assumed.This results in an MTPC trajectory that lies in the q-axis of the dq current frame.The cost function ( 8) is rewritten as where the torque is τ = 3 2 pΛ pm i q with Λ pm the permanent magnet flux.Two portions of the cost function (15), namely J 1 and J 2 , are used to highlight the different contributions of current magnitude and torque reference tracking error.Since the torque τ does not depend on the d-axis current, i d = 0 will always result during steady-state operations to ensure the minimization of the cost function (15).For the sake of simplicity, the cost function J is considered as a function of the q-axis current only.
To ensure that the motor torque can reach the reference at the steady-state condition with zero-offset following a transient, a c τ value greater than a minimum value, namely c lim τ , must be used.In particular, for a given torque reference τ * , the current that assures J 2 = 0 in (15) and, therefore, leads to zero steadystate torque control error, is denoted as i * q .The value of c lim τ is calculated by imposing the derivative of (15) equal to zero at i * This means that the absolute value of the slope of parabola J 1 at i q = i * q has to be equal to the absolute value of the slope of J 2 .It is worth noting that ( 15) is a continuous function, but its derivative is not due to the l 1 -norm.It is possible to demonstrate that the minimum of the cost function lies in i * q only if c τ ≥ c lim τ .Hence, the choice of c τ ≥ c lim τ guarantees the offset-free capability at a steady state.Furthermore, c τ affects the torque control dynamics, which is investigated in Section IV.
A geometrical interpretation of the effects of different c τ values can be drawn from the cases in Fig. 3(a)-(c).If c τ < c lim τ , the cost function has a minimum in i min q < i * q , see Fig. 3(a).Thus, the reference torque cannot be reached.Imposing c τ ≥ c lim τ in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.(15) leads to J 2 = 0, i.e., zero steady-state control error, and the minimum of J is guaranteed for i q = i * q .As desired, this allows zero steady-state torque control error, see Fig. 3(b) and (c).
The value of c τ that leads to offset-free torque control depends on the operating point, cf.(17).The nominal operating point represents the worst case and must be selected to calculate c lim τ for attaining offset-free torque control in the whole operating region.

C. Disadvantage of l 2 -Norm for Torque Reference Tracking
The mathematical proof for avoiding l 2 -norm in torque reference tracking problems is provided in the following.In case of an l 2 -norm, (15) can be rewritten as In the same fashion as to calculate c lim τ in ( 16), the minimum of (17) for a given reference torque, which corresponds to a unique value of i * q , is calculated as : The term (τ * − 3 2 pΛ pm i * q ) at the denominator of ( 19) is actually zero, since τ * = 3 2 pΛ pm i * q .In other words, the only way to achieve offset-free torque reference tracking is to set c lim τ → ∞, which is not realizable.

D. Selection of Weight c τ : Case of Anisotropic Motor
A general formulation for the computation of the weight c τ in ( 8) is carried out in case of an anisotropic synchronous motor.Since the cost in ( 8) is a function of both d and q currents, the gradient is calculated in the desired working point i * dq .The weights c τ d and c τ q that guarantee that the minimum lies in i * dq are calculated as follows: It is worth noting that ( 16) is a particular case of (20).By choosing the maximum of c τ,d and c τ,q , i.e., c lim τ = max(c lim τ,d , c lim τ,q ) for the scalar weight c lim τ , offset-free torque reference tracking can be also achieved for anisotropic synchronous motors, e.g., synchronous reluctance motors (SynRM)s or internal PMSM (IPMSM).
The worst case i * dq point regarding the selection of c lim τ is the one that produces the nominal torque τ n and it lies on the MTPC trajectory.This corresponds to the intersection between the circular current constraint and the MTPC curve.It is worth highlighting that the calculation of c lim τ can be computed offline once the motor model is known.

IV. EXPERIMENTAL RESULTS
The experiments to test the feasibility and performance of the proposed CCS-MPTC were conducted on an IPMSM, whose parameters, as well as the inverter and system data, are listed in Table I.In the following, it will be referred to as motor under test (MUT).The proposed control algorithm is synthesized in Matlab-Simulink and thanks to the Matlab code generator it can be directly downloaded and used with the dSpace MicroLabBox control platform.The current sampling synchronous with the space vector modulation allows ripple-free measurements.The nonlinear current-to-flux linkage and the differential inductances maps are stored as LUTs and reported in Fig. 4. Rotor angle dependence is omitted in the stored LUTs, so that any angledependent effect, e.g., cogging torque, is not considered in this work.
All experimental outcomes are reported in per unit (p.u.), i.e. they are normalized with respect to nominal values.The MUT is connected to the prime mover, which is a speed-controlled synchronous motor.The torque reported in the following figures is estimated using (3).

A. Investigation on c τ Value
As a first test, the weight c τ is linearly increased from 10 % to 200 % of c lim τ while a constant reference torque τ * = τ n is applied.The results are reported in Fig. 5.
The purpose of the test with results in Fig. 5 was to confirm the theoretical analysis in Section III-D.It is worth recalling that if c τ < c lim τ , the reference torque cannot be reached.The current trajectory in the dq plane follows the MTPC curve during the transient until the nominal current value is reached.The nominal torque is obtained when c τ ≥ c lim τ .Torque reference steps were applied to the MUT with different c τ values.The results are reported in Fig. 6.
Here, the influence of c τ on torque reference tracking dynamic can be seen.On the one hand, when c τ = c lim τ the reference torque can be reached, see c τ = 0.95 c lim τ versus c τ = c lim τ in Fig. 6.On the other hand, when c τ > c lim τ , the performance in terms of torque reference tracking are boosted, see current trajectories for c τ = 1.5 c lim τ and c τ = 3 c lim τ in Fig. 6.As a conclusive remark, once c lim τ is calculated, c τ can be used as a tuning parameters to change the behavior of the controller to improve the performances during transients.It is worth recalling that the MTPC condition is always guaranteed during steady-state operation.During transients, the proposed controller selects alternative paths accordingly to the c τ value.

B. Torque Reference Tracking at Different Speeds
The result of a torque reference step from motor to generator mode and vice versa is reported in Fig. 7.The test is conducted at different operating points, including a full nominal torque reversal, while the motor is dragged by a prime mover.Two different speeds were considered, which are half and nominal speed of the MUT.The weight c τ was set at the value c lim τ for both tests.The proposed control algorithm guarantees both torque reference tracking and MTPC criteria in every steady-state condition.
The test at 0.5 ω n and τ * = τ n reported in Fig. 7(a) is discussed in more detail in the following.A rated torque reference step was applied at point O , and the transient ended at point A .After a short steady-state period, the torque reference sign was suddenly changed [time 8 ms in Fig. 7(a)].The current trajectory moved from point A to point C passing through points B and O .This is due to the effects of the back-electromotive-force, that is present in E i (1), as shown in [33,Fig. 3].The behavior of the dq currents during the test of Fig. 7(a) is reported in Fig. 8.The current ripple is very limited and comparable with that of conventional modulator-based schemes, e.g., PI-FOC.
In general, all considerations made for the test at 0.5 ω n and τ * = τ n reported in Fig. 7(a) hold for all the remaining tests of Fig. 7.It is worth highlighting that the transient behavior of the torque can be improved by adjusting the c τ weight base on necessity.

V. CONCLUSION
The CCS-MPTC that was proposed in this article is capable of effectively tracking the reference torque without steady-state control error satisfying the MTPC criteria by solving a single optimization problem.It is a remarkable achievement within the field of electric drives, which has never been proposed yet.The main contribution of the proposed work can be summarized as follows: 1) implementation of a singular controller that is able to determine the optimal voltages based on a given torque reference; 2) the only control weight can be calculated with an analytical formulation; 3) the optimization problem is written in a QP formulation that is suitable for real-time application.The only requirement of the proposed algorithm is the knowledge of the motor model.It is worth highlighting that no offline computation of the MTPC trajectory nor online signal-injectionbased techniques are needed in this work.The current limitation has been always guaranteed due to the presence of current constraints in the problem formulation.

APPENDIX A QP Formulation in Case of l 1 -Norm in the Cost Function
The minimization problem of a variables vector x where an l 1 -norm appears in a standard quadratic cost function can be written as min In problem ( 21) matrix Q must be positive definite and vector c T > 0 for each element.If these hypotheses are satisfied, standard qp formulation of problem ( 21) can be derived in following two steps [36]: 1) substitution of the absolute value with an additional optimization variable y = x 1 ; 2) incorporation of the relation y = x 1 with linear inequality constraints.The resulting QP problem can be written as min Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

δ i x 2 i + δ i y 2 i= x T Qx ( 23 ) since x 2 i
Qx − x T δIx + y T δIy = x T Qx − = |x i | 2 = y 2 i .Parameter δ has to be set in order to guarantee Q − δI > 0. The dimension of the initial QP problem (21) is increased by one when the l 1 -norm is incorporated into a standard QP problem.Ismaele Diego De Martin received the B.S. and M.S. degrees in mechatronics engineering from the University of Padova, Vicenza, Italy, in 2018 and 2021, respectively.He is currently working toward the Ph.D. degree in mechatronics engineering with the Department of Management and Engineering, University of Padova, Vicenza.His main research interests include predictive control techniques for ac motors.Anian Brosch received the bachelor's and master's degrees in mechanical engineering from the Munich University of Applied Sciences, Munich, Germany, in 2016 and 2018, respectively.Since then, he has been a Research Associate with the Department of Power Electronics and Electrical Drives, Paderborn University, Paderborn, Germany.His research interests include identification and control of electrical drives, in particular model predictive control of highly utilized permanent magnet synchronous motors.

TABLE I SYSTEM
AND MUT PARAMETERS