Dual-Vector Predictive Torque Control of Permanent Magnet Synchronous Motors Based on a Candidate Vector Table

In order to reduce the torque ripple of permanent magnet synchronous motors (PMSMs), this paper proposes a dual-vector predictive torque control strategy based on a candidate vector table. The main feature of this strategy is that two vectors are acted in a control period to form a vector combination, and the vector combination can be either an effective-zero combination or an effective-effective combination. In the process of establishing the vector combinations, the switching frequency is also taken into account, therefore avoiding a high switching frequency, while effectively reducing the motor torque ripple. The candidate vector table is constructed offline, and three sets of candidate vectors and their duty cycles can be determined by looking up the table. Then the cost function is used to screen the action vectors from the three sets candidate vectors, so the two vectors acted in one control period and their duty cycles can be obtained simultaneously. Finally, the feasibility and effectiveness of the proposed method are verified on a 5.2 kW two-level inverter-fed PMSM drive system.


Introduction
Permanent magnet synchronous motors (PMSMs) have the advantages of simple structure, reliable operation, high power density, and have been widely used in many industrial fields, such as aerospace, mechanical manufacturing, rail traffic and other industrial fields [1][2][3].In recent years, PMSM research has gradually become a hot spot.Reference [4] adjusts the parameter of a PID controller by using a wavelet algorithm to control a PMSM.References [5][6][7] develop an observer to reduce the number of sensors in the PMSM control system.In order to obtain more accurate parameters in motor control, references [8,9] study PMSM parameter identification technologies.Predictive torque control (PTC) is applied to PMSM control systems in order to give comprehensive consideration to the dynamic and steady performance of the motor system, which is simple in principle and easy to implement online, and has received extensive attention in academia and industry [10,11].
In the voltage source inverter-permanent magnet synchronous motor (VSI-PMSM) drive system, finite control set predictive torque control (FCS-PTC) takes the discrete nature of inverter switching states into account.FCS-PTC substitutes switching states into the motor predictive model one by one, evaluates prediction results by the cost function, and chooses the switching state minimizes the

Dual-Vector Predictive Model of PMSM
The two-level voltage source inverter-fed PMSM system is shown in Figure 1.Switch variables are defined as Si, i ∈ {a, b, c}, representing the switch state of the inverter bridge arm switch tube: ON, OFF = 0, OFF, ON upper switch tube lower switch tube upper switch tube lower switch tube i According to the above definition, there are eight optional switch states in total, corresponding to 8 voltage vectors Vm (m ∈ {0, 1, 2, 3, 4, 5, 6, 7}.Among them, V1-V6 are effective vectors, V0 and V7 are zero vectors. Ri ω ψ t (1) where usd, usq, isd, isq, ψsd, and ψsq are d-axis and q-axis component of stator voltage, stator current and stator flux, respectively; ωr is the rotor electrical angular speed; Rs is the stator resistance.
The flux and torque equations are as: where ψf is the rotor flux; Ld, Lq are d-axis and q-axis inductance respectively; T e is the electromagnetic torque; p is pole pairs of motor.
According to the Equations ( 1) and ( 2), the stator current is selected as a state variable, and the state-space equation of motor can be obtained.Discretizing the state-space equation, then the stator current predictive model is obtained as: where Ts is the control period of the system; isd (k), isq (k), isd (k + 1) and isq (k + 1) are d-axis and q-axis components of stator current at kTs and (k + 1) Ts, respectively; u1sd (k), u1sq (k), u2sd (k), u2sq (k) are d-axis and q-axis stator voltage of the two voltage vectors at kTs respectively; t1 ∈ [0, Ts] is the action time of the first voltage vector.Combining ( 2) and (3), the stator flux predictive model and the torque predictive model are obtained as: According to the above definition, there are eight optional switch states in total, corresponding to 8 voltage vectors V m (m ∈ {0, 1, 2, 3, 4, 5, 6, 7}).Among them, V 1 -V 6 are effective vectors, V 0 and V 7 are zero vectors.
The voltage equations of the PMSM in dq frame are expressed as: where u sd , u sq , i sd , i sq , ψ sd , and ψ sq are d-axis and q-axis component of stator voltage, stator current and stator flux, respectively; ω r is the rotor electrical angular speed; R s is the stator resistance.
The flux and torque equations are as: where ψ f is the rotor flux; L d , L q are d-axis and q-axis inductance respectively; T e is the electromagnetic torque; p is pole pairs of motor.
According to the Equations ( 1) and ( 2), the stator current is selected as a state variable, and the state-space equation of motor can be obtained.Discretizing the state-space equation, then the stator current predictive model is obtained as: where T s is the control period of the system; i sd (k), i sq (k), i sd (k + 1) and i sq (k + 1) are d-axis and q-axis components of stator current at kT s and (k + 1) T s , respectively; u 1sd (k), u 1sq (k), u 2sd (k), u 2sq (k) are d-axis and q-axis stator voltage of the two voltage vectors at kT s respectively; t 1 ∈ [0, T s ] is the action time of the first voltage vector.Combining ( 2) and (3), the stator flux predictive model and the torque predictive model are obtained as: where ψ sd (k + 1), ψ sq (k + 1) are d-axis and q-axis stator flux at (k + 1) T s respectively; ψ s (k + 1), i s (k + 1), T e (k + 1) are stator flux, stator current, and electromagnetic torque at (k + 1)T s respectively.

Synthesized Vector Set
After combining basic voltage vectors to generate synthesized vectors, the action vector combination in one control period can be either an effective-zero combination or an effective-effective combination.The two basic voltage vectors in the combination are expressed as (V m , V n ), m, n ∈ {0, 1, 2, 3, 4, 5, 6, 7}.There are 28 combinations without considering the sequence of the two vectors.It is worth noting that, given the principle of the minimum action times of the inverter bridge arm, some of these combinations are unreasonable.For example, (V 1 , V 7 ) can be replaced by (V 1 , V 0 ) with the same effect but fewer action times of inverter bridge arm, and two opposite voltage vectors (V 1 , V 4 ) can be replaced by one effective vector and one zero vector.There are 18 reasonable combinations of voltage vectors.Provided different duty cycles, these combinations can generate an infinite number of voltage vectors, which are denoted as synthesized vectors v = dV m + (1 − d) V n , where the duty cycle d ∈ [0,1].The collection of terminals of these synthesized vectors constitutes 18 lines as shown in Figure 2.These lines include six sides of the regular hexagon 'abcdef', three sides of the triangle 'ace', three sides of the triangle 'bdf', and six line segments 'ao', 'bo', 'co', 'do', 'eo', 'fo'.

Synthesized Vector Set
After combining basic voltage vectors to generate synthesized vectors, the action vector combination in one control period can be either an effective-zero combination or an effective-effective combination.The two basic voltage vectors in the combination are expressed as (Vm, Vn), m,n ∈ {0, 1, 2, 3, 4, 5, 6, 7}.There are 28 combinations without considering the sequence of the two vectors.It is worth noting that, given the principle of the minimum action times of the inverter bridge arm, some of these combinations are unreasonable.For example, (V1, V7) can be replaced by (V1, V0) with the same effect but fewer action times of inverter bridge arm, and two opposite voltage vectors (V1, V4) can be replaced by one effective vector and one zero vector.There are 18 reasonable combinations of voltage vectors.Provided different duty cycles, these combinations can generate an infinite number of voltage vectors, which are denoted as synthesized vectors v = dVm + (1 − d) Vn, where the duty cycle d ∈ [0,1].The collection of terminals of these synthesized vectors constitutes 18 lines as shown in Figure 2.These lines include six sides of the regular hexagon 'abcdef', three sides of the triangle 'ace', three sides of the triangle 'bdf', and six line segments 'ao', 'bo', 'co', 'do', 'eo', 'fo'.(1) If the synthesized vector terminal is on the regular hexagon 'abcdef', the two basic voltage vectors (Vm, Vn) of this synthesized vector satisfy the following relationship: For example, in Figure 2, the terminal of the synthesized vector va = daV4 + (1 − da) V5 is on the 'de' of the regular hexagon.
(2) If the synthesized vector terminal is on the triangle 'ace' or 'bdf', the two basic voltage vectors (Vm, Vn) of this synthesized vector satisfy the following relationship: (1) If the synthesized vector terminal is on the regular hexagon 'abcdef', the two basic voltage vectors (V m , V n ) of this synthesized vector satisfy the following relationship: For example, in Figure 2, the terminal of the synthesized vector v a = d a V 4 + (1 − d a ) V 5 is on the 'de' of the regular hexagon.
(2) If the synthesized vector terminal is on the triangle 'ace' or 'bdf', the two basic voltage vectors (V m , V n ) of this synthesized vector satisfy the following relationship: For example, the terminal of synthesized vector Energies 2019, 12, 163 5 of 15 (3) If the synthesized vector terminal is on any of the line segments 'ao', 'bo', 'co', 'do', 'eo', 'fo', the two basic voltage vectors (V m , V n ) of this synthesized vector satisfy the following relationship: For example, the terminal of synthesized vector After the abovementioned vector synthesis process, voltage vectors in the synthesized vector set are increased to an infinite number.The following is to construct a candidate voltage vector table containing duty cycle information.The voltage vector acted on the inverter is determined by a two-step screening method.

Desired Voltage Vector
According to the principle of torque and flux deadbeat, the desired voltage vector can be obtained and denoted as D-VV (desired-voltage vector).The process of solving the desired voltage vector is as follows [35,36].
According to the mathematical model of the motor, the change rate of torque can be expressed as: Discretize Equation (10), and let T e (k + 1) be equal to the torque reference value T ref e , the torque deadbeat line equation is obtained as: where: B = ( The voltage model Equation ( 1) is discretized by Euler method.By neglecting the resistance term and decoupling the cross coupling term, the stator flux can be approximately discretized as [36]: Let |ψ s (k + 1)| be equal to the flux reference value |ψ s ref |, then the flux deadbeat circle equation can be obtained as: Combining Equations ( 11) and ( 15), the voltage vector that meets the inverter voltage limit can be obtained as: where: Energies 2019, 12, 163 Translating ( 16) from synchronous dq coordinate to stationary αβ coordinate, there is: where u sα (k) and u sβ (k) are the D-VV inαβ coordinate; θ is the rotor position angle.
The space position angle of the DB-VV is:

Construct Candidate Vector Table Offline
The vectors in the synthesized vector set are screened according to the amplitude and position angle of the desired voltage vector.First, the space region of the synthesized vector set is divided into 12 large sectors, numbered I-XII.The region division rule is shown in Figure 3.The 18 line segments formed by the synthesized vector terminals in the set divide each large sector into three small right triangle regions with equal areas, as shown by 1 2 3 in the sector II.Thus, the region of the synthesized vector set can be divided into 36 small right triangle regions with equal areas. where: Translating ( 16) from synchronous dq coordinate to stationary αβ coordinate, there is: where usα (k) and usβ (k) are the D-VV inαβ coordinate; θ is the rotor position angle.
The space position angle of the DB-VV is:

Construct Candidate Vector Table Offline
The vectors in the synthesized vector set are screened according to the amplitude and position angle of the desired voltage vector.First, the space region of the synthesized vector set is divided into 12 large sectors, numbered I-XII.The region division rule is shown in Figure 3.The 18 line segments formed by the synthesized vector terminals in the set divide each large sector into three small right triangle regions with equal areas, as shown by ①②③ in the sector II.Thus, the region of the synthesized vector set can be divided into 36 small right triangle regions with equal areas.The duty cycle of the voltage vector corresponding to the vertex of each small right triangle region can be directly obtained by geometric relationship.The voltage vectors corresponding to the three vertexes are taken as the candidate voltage vectors.Taking Figure 3 as an example, if Equation ( 21) yields a desired voltage vector located in the sector II, the specific position of the desired voltage vector is identified by the amplitude information of desired voltage vector as follows.
When the desired voltage vector satisfies: Then the desired voltage vector is located in region 1 ; When the desired voltage vector satisfies: Energies 2019, 12, 163 7 of Then the desired voltage vector is located in region 2 ; When the desired voltage vector satisfies: Then the desired voltage vector is located in region 3 ; When the desired voltage vector is located in other regions, the calculation method is similar.Merely the region in the first quadrant needs calculating, for the other quadrants have a symmetric relationship with the first one.Without loss of generality, when the voltage vector is located in the small region 2 in the sector II, V 2 can be determined, and the three sets candidate voltage vectors are denoted as v(1), v(2), v(3) respectively, with duty cycles directly obtained according to the geometric relationship:

Realization of Control Algorithm
The control block diagram of the proposed strategy is shown in Figure 4.It mainly includes following parts: delay compensation [37,38], torque and flux calculation, judgment of the region where the D-VV is located on, candidate voltage vector table, predictive model and cost function.The implementation process of algorithm is as follows: (1) The stator three-phase current, DC bus voltage and motor rotor position at the current time are obtained by sampling.(2) The i sd (k + 1) and i sq (k + 1) are calculated by using the sampled current values i sd (k) and i sq (k) according to Equation ( 4).(3) The ψ s (k + 1) and T e (k + 1) are calculated by using the compensated current value i sd (k + 1) and i sq (k + 1).Output the variable at (k + 1) T s to the predictive model part as the predicted starting value; (4) The ψ s (k + 1) and T e (k + 1) are also taken as the input of the D-VV calculation part.The D-VV is obtained by the deadbeat principle of torque and flux, and then the region is judged according to its position angle and amplitude information.(5) Three sets of candidate voltage vectors and their duty cycles can be obtained by looking up the table.Finally, the candidate voltage vectors and their duty cycles are substituted into the predictive model of PMSM, and the voltage vector acted on the inverter is screened by cost function.The cost function is defined as: where Te (k + 2) and ψs (k + 2) are the predictive values of the electromagnetic torque and stator flux at (k + 2) Ts; λΨ is weight coefficient, its value is tuned empirically based on the branch and bound principle [39], and it is set to 600.

Experimental Platform
The proposed control strategy is experimentally tested on a 5.2 kW two-level voltage source inverter-fed PMSM system, and the experimental platform is shown in Figure 5.The control system is mainly composed of a control circuit and a power circuit.The cores of the control circuit are TI's floating-point digital signal processor (DSP) chip TMS320F28335 and field programmable gate array (FPGA) chip EP1C6.The DSP chip is mainly used to sample DC bus voltage, three-phase stator current and realize the control algorithm of PMSM.The FPGA chip is an auxiliary control unit, which is mainly used to realize the adjustment of the encoder interface circuit and the output signal.In addition, the control circuit also includes some peripheral circuits, such as AD conversion, Butterworth filter, dead band, sampling circuit, drive protection circuit and so on.The power circuit The cost function is defined as: where T e (k + 2) and ψ s (k + 2) are the predictive values of the electromagnetic torque and stator flux at (k + 2) T s ; λ Ψ is weight coefficient, its value is tuned empirically based on the branch and bound principle [39], and it is set to 600.

Experimental Platform
The proposed control strategy is experimentally tested on a 5.2 kW two-level voltage source inverter-fed PMSM system, and the experimental platform is shown in Figure 5.The control system is mainly composed of a control circuit and a power circuit.The cores of the control circuit are TI's floating-point digital signal processor (DSP) chip TMS320F28335 and field programmable gate array (FPGA) chip EP1C6.The DSP chip is mainly used to sample DC bus voltage, three-phase stator current and realize the control algorithm of PMSM.The FPGA chip is an auxiliary control unit, which is mainly used to realize the adjustment of the encoder interface circuit and the output signal.In addition, the control circuit also includes some peripheral circuits, such as AD conversion, Butterworth filter, dead band, sampling circuit, drive protection circuit and so on.The power circuit mainly includes: voltage stabilizing capacitor, uncontrollable rectifying circuit, and IPM driving circuit of inverters.The input terminal of the control system is connected with an uncontrollable rectifier bridge by the grid and the output terminal is connected with the PMSM by an intelligent power module (IPM) power module.In the experiment, the control of the PMSM is realized by controlling the action of the switching tube in the IPM power module.The load consists of a generator and a resistance box.In the experiment, the load torque is adjusted by adjusting the resistance value of resistance box.PMSM parameters are listed in Table 2.
To prove the proposed method effective, torque and flux amplitude tracking experiments are carried out first, then comparison experiments concerning steady-state and dynamic performance are carried out, with traditional FCS-PTC and Duty-PTC as references.In general, a higher average switching frequency leads to better control performance.Therefore, the switching frequencies of the proposed strategy and FCS-PTC have to be roughly consistent for the experimental results to be tenable.In this regard, the control period of the traditional FCS-PTC is set as 100 µs, and the control period of the Duty-PTC and the proposed control strategy are set as 150 µs.
circuit of inverters.The input terminal of the control system is connected with an uncontrollable rectifier bridge by the grid and the output terminal is connected with the PMSM by an intelligent power module (IPM) power module.In the experiment, the control of the PMSM is realized by controlling the action of the switching tube in the IPM power module.The load consists of a generator and a resistance box.In the experiment, the load torque is adjusted by adjusting the resistance value of resistance box.PMSM parameters are listed in Table 2.  To prove the proposed method effective, torque and flux amplitude tracking experiments are carried out first, then comparison experiments concerning steady-state and dynamic performance are carried out, with traditional FCS-PTC and Duty-PTC as references.In general, a higher average switching frequency leads to better control performance.Therefore, the switching frequencies of the proposed strategy and FCS-PTC have to be roughly consistent for the experimental results to be tenable.In this regard, the control period of the traditional FCS-PTC is set as 100 μs, and the control period of the Duty-PTC and the proposed control strategy are set as 150 μs.

Torque and Flux Amplitude Tracking Experiment
Torque and flux amplitude tracking performance of the proposed strategy is verified without speed controller as follows.Figure 6 shows the waveforms of torque and stator flux amplitude, where the torque reference value is a step signal from 400 Nm to −200 Nm, and the flux amplitude reference value is also a step signal from 1.6 Wb to 1.2 Wb and then to 1.6 Wb.According to Figure 6, under the proposed strategy, the actual torque and flux are able to track their references rapidly and accurately, when the references change abruptly.

Torque and Flux Amplitude Tracking Experiment
Torque and flux amplitude tracking performance of the proposed strategy is verified without speed controller as follows.Figure 6 shows the waveforms of torque and stator flux amplitude, where the torque reference value is a step signal from 400 Nm to −200 Nm, and the flux amplitude reference value is also a step signal from 1.6 Wb to 1.2 Wb and then to 1.6 Wb.According to Figure 6, under the proposed strategy, the actual torque and flux are able to track their references rapidly and accurately, when the references change abruptly.

Steady Performance Comparison
The steady-state performances of the three control strategies are compared under three working conditions.Working condition 1: The motor operates at low speed with light load.The speed and torque are set as 10 r/min and 100 Nm respectively.Working condition 2: The motor operates at medium speed with half load.The speed and torque are set as 25 r/min and 500 Nm respectively.Working condition 3: The motor operates at high speed with full load.The speed and

Steady Performance Comparison
The steady-state performances of the three control strategies are compared under three working conditions.Working condition 1: The motor operates at low speed with light load.The speed and torque are set as 10 r/min and 100 Nm respectively.Working condition 2: The motor operates at medium speed with half load.The speed and torque are set as 25 r/min and 500 Nm respectively.Working condition 3: The motor operates at high speed with full load.The speed and torque are set as 40 r/min and 1000 Nm.
Figures 7-9 show the the experimental waveforms of torque, flux amplitude, and phase A stator current under three working conditions of three strategies respectively.According to the waveforms, the proposed strategy causes less torque and flux ripple than the other two.Under the proposed strategy, phase A stator current is closer to an ideal sinusoidal wave.In conclusion, the proposed strategy improves steady-state control performance in contrast to traditional FCS-PTC and Duty-PTC, under low-speed light-load, medium-speed half-load, or high-speed full-load working conditions.

Steady Performance Comparison
The steady-state performances of the three control strategies are compared under three working conditions.Working condition 1: The motor operates at low speed with light load.The speed and torque are set as 10 r/min and 100 Nm respectively.Working condition 2: The motor operates at medium speed with half load.The speed and torque are set as 25 r/min and 500 Nm respectively.Working condition 3: The motor operates at high speed with full load.The speed and torque are set as 40 r/min and 1000 Nm.
Figures 7-9 show the the experimental waveforms of torque, flux amplitude, and phase A stator current under three working conditions of three strategies respectively.According to the waveforms, the proposed strategy causes less torque and flux ripple than the other two.Under the proposed strategy, phase A stator current is closer to an ideal sinusoidal wave.In conclusion, the proposed strategy improves steady-state control performance in contrast to traditional FCS-PTC and Duty-PTC, under low-speed light-load, medium-speed half-load, or high-speed full-load working conditions.

Dynamic Performance Comparison
Figure 10 shows the experimental waveforms of torque, speed and phase A stator current of three control strategies with speed at 20 r/min and load torque stepping from 200 to 500 Nm.It can be seen that three control strategies can follow the change of load torque rapidly, and the settling time is almost equal.In addition, the proposed strategy has better torque control performance and smoother sinusoidal current waveform, indicating that the proposed control strategy achieves better steady-state control performance and retains the advantages of rapid dynamic response.
Quantitative comparison of three control strategies about the average switching frequencies (favsw), torque ripple standard deviation (δT), flux amplitude ripple standard deviation (δψ) and current harmonic distortion rate (ITHD) are shown in Table 3 (I-III).Table 4 shows the comprehensive comparison of the three control strategies It can be seen that the proposed control strategy can improve the steady-state control performance of the motor with the same average switching frequency as the traditional FCS-PTC's.Compared with Duty-PTC, the proposed control strategy can obtain higher steady-state control accuracy with lower average switching frequency.The dynamic response time of three control strategies is almost equal, indicating that the proposed strategy preserves the advantage of rapid dynamic response while improving the steady-state control performance of motor.Figure 11 shows histograms of the standard deviation of torque ripple and flux amplitude ripple of three control strategies.According to Figure 11, under working condition 1, compared with the traditional FCS-PTC and Duty-PTC, the torque ripple is reduced by 30.65% and 16.87%, and the flux amplitude ripple is reduced by 46.87% and 33.76% in the proposed

Dynamic Performance Comparison
Figure 10 shows the experimental waveforms of torque, speed and phase A stator current of three control strategies with speed at 20 r/min and load torque stepping from 200 to 500 Nm.It can be seen that three control strategies can follow the change of load torque rapidly, and the settling time is almost equal.In addition, the proposed strategy has better torque control performance and smoother sinusoidal current waveform, indicating that the proposed control strategy achieves better steady-state control performance and retains the advantages of rapid dynamic response.Quantitative comparison of three control strategies about the average switching frequencies (f avsw ), torque ripple standard deviation (δ T ), flux amplitude ripple standard deviation (δ ψ ) and current harmonic distortion rate (I THD ) are shown in Table 3 (I-III).Table 4 shows the comprehensive comparison of the three control strategies It can be seen that the proposed control strategy can improve the steady-state control performance of the motor with the same average switching frequency as the traditional FCS-PTC's.Compared with Duty-PTC, the proposed control strategy can obtain higher steady-state control accuracy with lower average switching frequency.The dynamic response time of three control strategies is almost equal, indicating that the proposed strategy preserves the advantage of rapid dynamic response while improving the steady-state control performance of motor.with the traditional FCS-PTC and Duty-PTC, torque ripple is reduced by 33.09% and 15.13%, and the flux amplitude ripple is reduced by 45.74% and 28.17% in the proposed control strategy.Under working condition 3, compared with the traditional FCS-PTC and Duty-PTC, the torque ripple is reduced by 31.30% and 18.74%, and the flux amplitude ripple is reduced by 41.24% and 17.39% in the proposed control strategy.(III) Quantitative comparison three control strategies with Figure 9.

Conclusions
In this paper, the two-level inverter-fed PMSM drive system is taken as the research object.By analyzing the characteristics of the synthesized vector set, a dual vector predictive torque control strategy based on the candidate vector table is proposed.

Figure 3 .
Figure 3. Space region division of synthesized vector set distribution.

Figure 3 .
Figure 3. Space region division of synthesized vector set distribution.
obtained by the deadbeat principle of torque and flux, and then the region is judged according to its position angle and amplitude information.(5)Three sets of candidate voltage vectors and their duty cycles can be obtained by looking up the table.Finally, the candidate voltage vectors and their duty cycles are substituted into the predictive model of PMSM, and the voltage vector acted on the inverter is screened by cost function.

Figure 4 .
Figure 4. Block diagram of the dual-vector predictive torque control based on candidate vector table.

Figure 4 .
Figure 4. Block diagram of the dual-vector predictive torque control based on candidate vector table.

Figure 6 .
Figure 6.Experimental waveforms of torque and flux amplitude tracking.

Figure 6 .
Figure 6.Experimental waveforms of torque and flux amplitude tracking.
Energies 2018, 11, x FOR PEER REVIEW 12 of 15 control strategy.Under working condition 2, compared with the traditional FCS-PTC and Duty-PTC, torque ripple is reduced by 33.09% and 15.13%, and the flux amplitude ripple is reduced by 45.74% and 28.17% in the proposed control strategy.Under working condition 3, compared with the traditional FCS-PTC and Duty-PTC, the torque ripple is reduced by 31.30% and 18.74%, and the flux amplitude ripple is reduced by 41.24% and 17.39% in the proposed control strategy.

Figure 10 .
Figure 10.Dynamic experimental waveforms of load toque step change from 200 to 500 Nm (a) Traditional FCS-PTC (b) Duty-PTC (c) Proposed strategy.

Figure 10 .
Figure 10.Dynamic experimental waveforms of load toque step change from 200 to 500 Nm (a) Traditional FCS-PTC (b) Duty-PTC (c) Proposed strategy.
Figure 11 shows histograms of the standard deviation of torque ripple and flux amplitude ripple of three control strategies.According to Figure 11, under working condition 1, compared with the traditional FCS-PTC and Duty-PTC, the torque ripple is reduced by 30.65% and 16.87%, and the flux amplitude ripple is reduced by 46.87% and 33.76% in the proposed control strategy.Under working condition 2, compared Energies 2019, 12, 163 of 15

Table 1 .
Then, 3 sets candidate voltage vectors are substituted into the predictive model.Finally the voltage vectors minimize the cost function are selected.Table 1 is a candidate voltage vector table which contains the duty cycle information when the desired voltage vector is in regions I, II, III of the first quadrant.Candidate voltage vector table.

Table 2 .
Parameters of the PMSM.

Table 2 .
Parameters of the PMSM.
(I) Quantitative comparison of three control strategies with Figure7.(III)Quantitativecomparison of three control strategies with Figure9.ItemTraditional FCS-PTC Duty-PTC Proposed Strategy

Table 3 .
Quantitative comparison.Quantitative comparison of three control strategies with Figure 7. Quantitative comparison of three control strategies with Figure 8.

Table 4 .
The comprehensive comparison of three control strategies.