Model-Free Predictive Current Control of Five-Phase PMSM Drives

: Model predictive control is highly dependent on accurate models and the parameters of electric motor drives. Multiphase permanent magnet synchronous motors (PMSMs) contain nonlinear parameters and mutual cross-coupling dynamics, resulting in challenges in modeling and parameter acquisition. To lessen the parameter dependence of current predictions, a model-free predictive current control (MFPCC) strategy based on an ultra-local model and motor outputs is proposed for ﬁve-phase PMSM drives. The ultra-local model is constructed according to the differential equation of current. The inherent relation between the parameters in the predictive current model and the ultra-local model is analyzed in detail. The unknowns of the ultra-local model are estimated using the motor current and voltage at different time instants without requiring motor parameters or observers. Moreover, space vector modulation technology is employed to minimize the voltage tracking error. Finally, simulations and experiments are conducted to verify the effectiveness of the MFPCC with space vector modulation. The results conﬁrm that the proposed method can effectively eliminate the impact of motor parameters and improve steady-state performance. Moreover, this control strategy demonstrates good robustness against load variations


Introduction
With the developments of power electronics and control theories, multiphase motor drives have become a competitive alternative to traditional three-phase drives due to their inherent merits, including lower torque pulsations, a higher power density, enhanced fault-tolerant capability, and a higher number of control degrees of freedom [1][2][3].
Generally, the high-performance control strategies of multiphase motor drives are derived from traditional three-phase counterparts, e.g., field-oriented control [4] and direct torque control [5].Meanwhile, finite control set-model predictive control (MPC) has been widely investigated in conjunction with motor drives [6][7][8].Model predictive current control and model predictive torque control are two typical manners of MPC.The former concentrates on the control of currents, and the latter focuses on the regulation of torque and stator flux.In contrast to the field-oriented control and direct torque control algorithms, a better balance between steady-state performance and dynamic response can be implemented by MPC.Furthermore, it is easier to deal with nonlinear constraints, such as reducing switching frequency and suppressing harmonic current.Nevertheless, since the number of switching states and subspaces increases in multiphase drives, MPC suffers from unsatisfactory steady-state performance, huge computational burden, and parameter uncertainties.
To improve steady-state performance and mitigate the influences of voltage vectors in the harmonic subspace, a concept of virtual voltage vector is proposed by means of voltage vector synthesis, and the duty cycle modulation is accordingly designed to generate pulses for dual three-phase permanent magnet synchronous motors (PMSMs) [9,10].Later, the multivector MPC, combining two voltage vectors as outputs in each control cycle, is further developed to cope with the problem caused by the application of a single voltage vector [11].For computational burden reduction, a voltage preselection algorithm is proposed according to the location of the dead-beat voltage vector [12] or the reference stator flux vector [13].
To deal with the issues related to parameter dependence, model-free predictive control (MFPC) has become a promising alternative in the field of motor drives.Although the implementations and formulations are different, they share the common idea of making predictions free of a system model or at least less dependent on it.Existing methods are categorized into three main groups according to the extent to which they are model-free, namely, totally model-free, using an ultra-local model, and prediction correction [14].The first method is carried out by using the input and output information of systems without any model for prediction [15,16].The second method uses a model with one or more uncertain terms that should be estimated continually via the input and output data of systems [17][18][19][20][21][22][23].The last type is devoted to the methods with an ideal model of the plant.However, by using the input, output, and previous prediction data of the system, some correction factors are estimated to compensate for the predictions [24][25][26].The totally model-free scheme generally faces the stagnation of the data updating, and prediction correction needs the nominal value of motor parameters.Comparatively, MFPC with an ultra-local model has attracted more attention.
The ultra-local model, derived from the mathematical model of motor drives, typically contains one or more uncertain terms that need to be identified.The estimation methods can be classified into two approaches.The first approach uses algebraic parameter identification techniques.The ultra-local model is usually studied in conjunction with model-free predictive current control (MFPCC).A nonlinear disturbance observer [17], a linear extended state observer [18], a sliding mode observer [19], and a high-gain disturbance observer [20] have been successively developed to estimate the known and unknown parts of the ultra-local model.Apart from the parameter-free predictive control, the steady-state performance enhancement is considered in [21]; a discrete-space vector modulation is developed for induction motor drives.Noticeably, the stability of the MFPCC with observers should be considered.The improper gain tuning of observers may result in instability issues.The second approach investigates the inherent nature of the control-variable ripples to estimate uncertain parts of the ultra-local model instead of adopting observers.In contrast, the research on the second method is relatively rare.In [22], the motor current time derivatives, which are expressed as functions of the phase angles of the basic voltage vectors, are predicted without motor parameters and are used for voltage vector optimization.The method lessens parameter dependence and eliminates the stagnation of data updating.In [23], a current update mechanism based on the two most recent current variations to reconstruct the PMSM model is developed.The nonparametric predictive current control can effectively reduce torque ripple and enhance current performance.
Very recently, MFPC has also been applied in multi-phase motor drives.An ultralocal-based MFPCC in the natural coordinate system has been proposed for five-phase induction motors [27], where the unknown terms of the ultra-local model are estimated by a linear extended state observer.Moreover, the MFPCC based on an ultra-local model and an extended state observer has been developed for an asymmetrical dual three-phase PMSM [28].The method was shown to provide superior current regulation when compared to the standard MPCC approach under uncertain parameter conditions.In [29], a MFPCC method with an optimal modulation pattern was proposed for dual three-phase PMSMs.A novel nonlinear disturbance observer was designed to overcome the limitation of an extended state observer.This strategy can effectively compensate for voltage distortion caused by inverter characteristics and improve steady-state performance.Additionally, the MFPCC strategy has been investigated in the harmonic current control of multiple three-phase permanent magnet synchronous generators [30].The ultra-local model-based predictive current controller was designed to regulate the current component in the fifthorder multi-dq coordinate.The uncertain parameter of the ultra-local model is estimated by the differential algebra method.However, the accuracy of estimation may be affected by the length of the gradient integral.
The abovementioned MFPC of multiphase drives mainly concentrates on ultra-localbased MFPC with observers, which would encounter the problems associated with gain tuning and system stability.In contrast, the ultra-local model using the output information of drives can avoid these drawbacks.To lessen the parameter dependence of current prediction, an ultra-local model-based MFPCC strategy is proposed for five-phase PMSM drives.The ultra-local model is constructed from the differential equation of current.The relation between the predictive current model and the ultra-local model is analyzed in detail.The unknowns of the ultra-local model are estimated by the current and voltage at different time instants without motor parameters.Moreover, space vector modulation (SVM) technology is employed to further reduce the voltage tracking error.The proposed methods not only theoretically eliminate the impact of machine parameters but also improve system performance.The contributions of this paper are summarized as follows: 1.
An MFPCC strategy based on an ultra-local model and drive information is proposed for five-phase PMSM drives; 2.
The unknown components of the ultra-local model are estimated with the current and voltage at different time instants without motor parameters or observers.
The organization of the rest of this paper is listed as follows.The conventional MPCC of a five-phase PMSM drive is presented in Section 2.Then, the MFPCC is detailed based on the ultra-local model and SVM in Section 3. In Section 4, simulated and experimental results are demonstrated to verify the effectiveness of the proposed control schemes.Finally, conclusions are drawn in Section 5.

Mathematical Model
A five-phase PMSM drive fed by a two-level voltage source inverter is presented in Figure 1.The inverter generates 32 basic voltage vectors in the fundamental and harmonic subspace, as shown in Figure 2.With Park transformation, the mathematic model of a five-phase PMSM in the rotating reference frame is expressed as where u dn , u qn , i dn , i qn , and L dn and L qn are the phase voltages, currents, and inductance components in the d n -q n reference frame, respectively, n = {1, 3}; ω e is the electrical angular velocity; R s is the stator resistance per phase; and ψ f is the fundamental magnitude of phase PM flux linkage.

Model Predictive Current Control
Using the Euler forward formula to discretize the voltage equation above, the current predictive model can be derived as where Ts is the sampling time.
Considering the regulations of both fundamental and harmonic subspaces, the cost function is defined in terms of the absolute errors between the referenced and predicted currents in the d1-q1-d3-q3 axes as  =  −  ( + 1) +  −  ( + 1) +  −  ( + 1) +  −  ( + 1) where the superscript "ref" represents the reference value.Assessing all 32 voltage vectors will impose a huge computational burden on processors.A common manner is to use the virtual vector voltage, which is synthesized by large and medium vectors.Although this method can eliminate the harmonic voltage theoretically, harmonic currents are under open-loop conditions.Another practice is only evaluating the large voltage vectors and one zero voltage vector, and the voltage vector that minimizes the cost function will be applied in the next instant.By this, both the fundamental and harmonic subspaces can be regulated.

Model Predictive Current Control
Using the Euler forward formula to discretize the voltage equation above, the current predictive model can be derived as where T s is the sampling time.
Considering the regulations of both fundamental and harmonic subspaces, the cost function is defined in terms of the absolute errors between the referenced and predicted currents in the d 1 -q 1 -d 3 -q 3 axes as where the superscript "ref" represents the reference value.Assessing all 32 voltage vectors will impose a huge computational burden on processors.A common manner is to use the virtual vector voltage, which is synthesized by large and medium vectors.Although this method can eliminate the harmonic voltage theoretically, harmonic currents are under open-loop conditions.Another practice is only evaluating the large voltage vectors and one zero voltage vector, and the voltage vector that minimizes the cost function will be applied in the next instant.By this, both the fundamental and harmonic subspaces can be regulated.

Ultra-Local Model
To address the parameter issue related to the conventional MPCC, the ultra-local model is applied to replace the actual mathematical model [15].The modeling part is calculated with the input and output in real time, which is expressed as .y = F + αu (4) where u and y are the system input and output, α is the gain of the system input, and F represents the uncertain part of the system.
Electronics 2023, 12, 4848 5 of 16 where Discretizing the ultra-local above at the instants k and (k − 1), the current differences can be expressed as where ∆i dn (k) and ∆i qn (k) are the dn-axis and qn-axis current error between the k th and (k − 1) th sampling step.
Similarly, the current differences between the (k − 1) th and (k − 2) th sampling step is given as Subtracting ( 7) from ( 6) yields Considering the sampling time T s is short enough, T s R s /L dn 1 and T s R s /L qn 1.Thus, the parameter α can be derived as Subsequently, the uncertain component F can be calculated as Electronics 2023, 12, 4848 6 of 16 Substituting ( 9) and ( 10) into (2), the predictive current model is written as The best voltage vector can be finally obtained by optimizing the cost function consisting of the reference current and the predicted current.

SVM
It is noted that the parameters α and F are obtained using the current and voltage information at different sampling instants, being independent of motor parameters.Thus, the parametric influence is theoretically eliminated.Nevertheless, the control performance is highly associated with the sampling precision.Moreover, the inherent shortcoming of the conventional MPC that only one voltage vector is applied in each control period would result in large tracking errors in voltages and harmonic currents.
To solve the abovementioned problems, SVPWM technology is employed.Due to the high sampling frequency of the MPC, it can be assumed that the value of F is constant over a couple of consecutive sampling periods.The desirable voltage is obtained based on the dead-beat theory and ( 5), which is presented as The reference currents in the d 1 -q 1 -d 3 -q 3 axis are calculated using the maximum torque per ampere theory.When a PMSM without or with little saliency effect is studied, the references of i d1 ref , i d3 ref , and i q3 ref are set to zero.The reference of i q1 ref is equal to the output of the speed regulator.Figure 3 shows the flow diagram of the MFPCC with the SVM method.The control diagram of the proposed method for five-phase PMSM drives is illustrated in Figure 4.
The reference currents in the d1-q1-d3-q3 axis are calculated using the maximum torque per ampere theory.When a PMSM without or with little saliency effect is studied, the references of id1 ref , id3 ref , and iq3 ref are set to zero.The reference of iq1 ref is equal to the output of the speed regulator.Figure 3 shows the flow diagram of the MFPCC with the SVM method.The control diagram of the proposed method for five-phase PMSM drives is illustrated in Figure 4.

Verifications 4.1. Simulations
Simulations in the environment of MATLAB/Simulink were conducted to validate the effectiveness of the proposed MFPCC with SVM of five-phase PMSM drives.For comparison, the conventional MPCC and the MFPCC were investigated simultaneously.The motor parameters are listed in Table 1. Figure 5 shows the simulated waveforms of phase currents, torque, and speed by three MPC methods under steady-state situations.The phase currents of the three MPC methods are balanced and sinusoidal, and the current component in the harmonic subspace is also well suppressed.Although the MFPCC can implement the parameter-independent control of five-phase PMSMs, large harmonics and ripples can be found in its performance when compared to the performance of the conventional MPCC and the MFPCC with SVM.
The detailed analyses of the waveforms are listed in Table 2.The total harmonic distortion (THD) values of the Phase A current are 8.68%, 9.32%, and 5.25% with respect to the conventional MPCC, MFPCC, and MPFCC with SVM.The torque ripples are 5.36%, 7.35%, and 3.34%, respectively.The lowest THD and torque ripple are found with the proposed MFPCC with SVM.It is also found that the steady-state performance of the MFPCC is slightly poorer than that of the conventional MPCC.The reasons are chiefly as follows.The current prediction in the MFPCC is implemented with two parameters, F and α, and the measured phase currents at the instant k without motor parameters.As shown in ( 9) and ( 10), the calculations of F and α require phase current variations ∆i(k) and ∆i(k − 1) and the voltage vector at the instants (k − 1) and (k − 2).The current variations can be obtained with the measured and restored values.However, the voltage vector cannot be measured directly, and it is equivalently regarded as the best vector outputted from the cost function optimization in the last control period.Meanwhile, the obtained best vector differs from the reference voltage vector due to the application of a single vector.These factors may lead to the deviation in the estimations of F and α and then the error in the current prediction.Consequently, the steady-state performance of the MFPCC slightly deteriorates when compared to the conventional MPCC.Fortunately, the SVM can guarantee the precision of the reference voltage vector and minimize the identification error of F and α.Thus, the behavior under steady-state conditions can be significantly improved.Figure 5 shows the simulated waveforms of phase currents, torque, and speed by three MPC methods under steady-state situations.The phase currents of the three MPC methods are balanced and sinusoidal, and the current component in the harmonic subspace is also well suppressed.Although the MFPCC can implement the parameter-independent control of five-phase PMSMs, large harmonics and ripples can be found in its performance when compared to the performance of the conventional MPCC and the MFPCC with SVM.The quantitative results demonstrate that the proposed MFPCC with SVM method not only eliminates the influences of motor parameters but also improves steady-state performance.

Experiments
Experiments were also performed to further verify the effectiveness of the proposed MFPCC scheme.Figure 6 shows the experimental setup.The parameters of the five-phase PMSM were similar to those in simulations.The control algorithms were implemented in a DSP-TMS320C28346-board-based controller.The reference speed was set as 200 rpm with a constant load of 15 Nm if not specially pointed out.The waveforms of phase currents, torque, speed, and harmonic current were outputted from the DO module and then displayed on an oscilloscope.
Experiments were also performed to further verify the effectiveness of the proposed MFPCC scheme.Figure 6 shows the experimental setup.The parameters of the five-phase PMSM were similar to those in simulations.The control algorithms were implemented in a DSP-TMS320C28346-board-based controller.The reference speed was set as 200 rpm with a constant load of 15 Nm if not specially pointed out.The waveforms of phase currents, torque, speed, and harmonic current were outputted from the DO module and then displayed on an oscilloscope.

Normal Operations
Figure 7 demonstrates the measured waveforms of the currents, speed, and torque of the three MPC methods under normal operations.The phase currents are almost balanced, and the q 3 -axis currents of the three methods are well regulated, fluctuating around zero.Similar to the simulated results, large harmonics and ripples can be found with the current and the torque by the MFPCC.Table 3 lists the detailed analyses of the performance of different control schemes.The harmonic spectra of the Phase E current considering up to the 40th order are shown in Figure 8. Considering the magnitude percentage of the fundamental component is 100%, it is suppressed to better demonstrate the results of high-order contents.The THD values of the conventional MPCC, MFPCC, and MFPCC with SVM are 24.08%,26.37%, and 8.04%, respectively.The torque ripples are 11.15%, 16.53%, and 8.14%, respectively.In Figure 8a,b, the harmonic components ranging from 2nd to 20th dominate the spectra.Comparatively, higher-order contents are well reduced by the MFPCC with SVM, where their magnitude percentages are less than 5%.The quantitative results demonstrate that the behavior of the MFPCC method is slightly poorer than that of the other two MPC methods.The reason may lie in the fact that the parameter-free prediction is highly associated with the precision of the current and the voltage.The error of the sampling would affect the calculation of the reference voltage vector.In contrast, the most desirable performance is found in the performance of the MFPCC with SVM, where the phase currents are smooth, and the torque ripple is minimized.Owing to the modulation technology, the impact of measurement errors has been suppressed remarkably.
The results indicate that the proposed MFPCC with SVM can achieve the parameterindependent control of five-phase PMSM drives and significantly improve steady-state performance.
the 40th order are shown in Figure 8. Considering the magnitude percentage of the fundamental component is 100%, it is suppressed to better demonstrate the results of highorder contents.The THD values of the conventional MPCC, MFPCC, and MFPCC with SVM are 24.08%,26.37%, and 8.04%, respectively.The torque ripples are 11.15%, 16.53%, and 8.14%, respectively.In Figure 8a,b, the harmonic components ranging from 2nd to 20th dominate the spectra.Comparatively, higher-order contents are well reduced by the MFPCC with SVM, where their magnitude percentages are less than 5%.
Electronics 2023, 12, x FOR PEER REVIEW 11 of 17    The quantitative results demonstrate that the behavior of the MFPCC method is slightly poorer than that of the other two MPC methods.The reason may lie in the fact that the parameter-free prediction is highly associated with the precision of the current and the voltage.The error of the sampling would affect the calculation of the reference voltage vector.In contrast, the most desirable performance is found in the performance of the MFPCC with SVM, where the phase currents are smooth, and the torque ripple is minimized.Owing to the modulation technology, the impact of measurement errors has been suppressed remarkably.

Parameter Variations
Since the structure of the motor cannot be changed, the parameter values in the predictive model are modified to imitate the parameter uncertainties that occur during real implementation.The MFPCC and the MFPCC with SVM are both parameter-independent.Therefore, the parameter-mismatch test is conducted for the conventional MPCC.As mentioned previously, the load torque is set as 15 Nm; the reference of the q 1 -axis current corresponds to 4.76 A. The experimental waveforms under 200% of the nominal inductance and 150% of the nominal flux-linkage magnitude are demonstrated in Figure 9b.The experimental waveforms under 50% of the nominal inductance and 75% of the nominal fluxlinkage magnitude are illustrated in Figure 9c. Figure 10 shows the current performance by the MFPCC and the MFPCC with SVM.Electronics 2023, 12, x FOR PEER REVIEW 13 of 17  Electronics 2023, 12, x FOR PEER REVIEW 13 of 17  Slight differences can be seen among the waveforms under the nominal and mismatched parameters.According to the detailed analyses in Table 4, drive performance varies with parameter variations in the predictive models.The largest tracking error of the q 1 -axis current and the highest THD value are with the results under 200% of the nominal inductance and 150% of the nominal flux-linkage magnitude, indicating the performance deteriorates more significantly when the values of the parameters in algorithms are higher than the actual ones.Moreover, the tracking error of the q 1 -axis current of the MFPCC is relatively high.However, the tracking error and the current harmonic are both reduced in the MFPCC with SVM.The results under parameter variations verify that the MFPCC with SVM can suppress the current tracking error and ensure desirable performance.The test with varying loads was performed to validate the robustness of the MFPCC with SVM against load disturbances, as depicted in Figure 11.Since the load is provided by a magnetic brake, the electromagnetic torque cannot be changed ideally; thus, the time of the transient process is relatively long.
The load decreases from 15 Nm to 5 Nm while the speed reference remains unchanged.Similar responses toward the load variation are found with the speed, torque, phase current, and q 3 -axis current of the three MPC methods.During the unloading process, the q 3 -axis current hardly changes, while the measured speed increases and recovers to its reference shortly.Under steady-state situations, the speed-tracking error is evaluated in terms of RMES, as listed in Table 5.Similar to the current performance assessment above, the largest speed tracking error is found with the MFPCC, and the least is found with the MFPCC with SVM.The settling time of the speed of the MPCC, MFPCC, and MFPCC with SVM is 0.90 s, 0.93 s, and 0.92 s, respectively.Although large ripples are seen with the torque of the MFPCC with SVM under dynamic conditions, its torque pulsation in steady states is relatively low.The results verify that the MFPCC with SVM is robust to load disturbances.Table 6 presents a brief comparison of the three MPC methods.All the methods do not require gain tuning.The conventional MPCC is model-based, while the other two methods are model-free, eliminating the impacts of parameter uncertainties.The previous current and voltage information are required in the MFPCC to identify the unknown parts of an ultra-local model.The information uncertainty would cause errors in current prediction.However, this influence is alleviated in the MFPCC with SVM since the voltage tracking error is minimized.Moreover, since the MPCC and the MFPCC need to evaluate basic voltage vectors to determine the optimal one, their computational burden is relatively high.
The MFPCC with SVM calculates the reference voltage vector based on the ultra-local model instead of assessing switching states; thus, the computation cost is reduced.The test with varying loads was performed to validate the robustness of the MFPCC with SVM against load disturbances, as depicted in Figure 11.Since the load is provided by a magnetic brake, the electromagnetic torque cannot be changed ideally; thus, the time of the transient process is relatively long.

Conclusions
This article proposed a model-free control algorithm based on an ultra-local model for a five-phase PMSM drive system that does not require motor parameters.The ultra-local model for the fundamental subspace and harmonic subspace was derived based on the current differential equation.The inherent relation between the mathematical model and the ultra-local model was theoretically investigated.For the estimation of the unknown terms in the ultra-local model, the current and voltage data were employed instead of

Figure 4 .
Figure 4. Control diagram of the MFPCC with SVM for five-phase PMSM drives.Figure 4. Control diagram of the MFPCC with SVM for five-phase PMSM drives.

Figure 9 .Figure 10 .
Figure 9. Experimental waveforms of the currents of the conventional MPCC with parameter mismatch: (a) nominal parameters; (b) 200% of the nominal inductance and 150% of the nominal fluxlinkage magnitude; (c) 50% of the nominal inductance and 75% of the nominal flux-linkage magnitude.

Figure 9 .
Figure 9. Experimental waveforms of the currents of the conventional MPCC with parameter mismatch: (a) nominal parameters; (b) 200% of the nominal inductance and 150% of the nominal flux-linkage magnitude; (c) 50% of the nominal inductance and 75% of the nominal flux-linkage magnitude.

Figure 9 .Figure 10 .
Figure 9. Experimental waveforms of the currents of the conventional MPCC with parameter mismatch: (a) nominal parameters; (b) 200% of the nominal inductance and 150% of the nominal fluxlinkage magnitude; (c) 50% of the nominal inductance and 75% of the nominal flux-linkage magnitude.

Table 2 .
Evaluations of simulated current and torque performance of the three MPC methods.

Table 3 .
Evaluations of experimental current and torque performance of the three MPC methods.

Table 3 .
Evaluations of experimental current and torque performance of the three MPC methods.

Table 4 .
Experimental performance of the three MPC methods under parameter mismatch.

Table 5 .
Evaluations of the dynamic performance of the three MPC methods.

Table 6 .
Comparison of the three MPC methods.