IEEJ International Workshop on Sensing, Actuation, Motion Control, and Optimization

High-Bandwidth Current Control of PMSM
Based on Quasi Multirate Feedforward Contol
Shona Noguchi∗a)
Hiroshi Fujimoto∗Student Member,Masahiro Mae∗

Student Member

Senior Member

PI feedback control is commonly used as the current control of electric motors. However, its tracking performance
becomes worse when the reference value changes steeply. To drive motors at higher speeds, the reference tracking
performance needs to be guaranteed. We have proposed a current controller based on the perfect tracking control(PTC) before. It has improved the tracking performance of the current control for motors. However, there is still room
for improvement, so a new approach to current control of the permanent magnet synchronous motors (PMSMs) basedon the quasi multirate feedforward control is proposed. This method based on the quasi multirate control doubles thetracking point of the reference current waveform compared to the conventional method. In this paper, the theory of
the new control method is described at first. Next, the design of the controller is described. Finally, simulations andexperimental verifications were conducted to demonstrate the eﬀectiveness of the proposed method.
Keywords: permanent magnet synchronous motor (PMSM), current control, perfect tracking control, quasi multirate feedforward
control, tracking performance

1.

2.

Introduction

2.1 Perfect Tracking Control (PTC)
PTC is the control system that follows the reference value
without error at every sampling point [5]. There are two samplers for the reference r(t) and the output signal y(t), and one
holder on the input u(t), in a digital control system. Therefore, period parameters are notated as T r , T y and T u , which
represent the periods of r(t), y(t), and u(t), respectively.

In recent years, the electrification of vehicles such as hybrid vehicles (HVs) and electric vehicles (EVs) is trending
to prevent air pollution and global warming. Furthermore,some countries have declared the prohibition of selling gasoline and diesel engine cars [1]. Thus, it is expected that thenumber of electrified vehicles will increase in the future.
In general, the proportional integral (PI) feedback control isused as the current control of motors. However, the problem is that when the reference value is steeply changed, theoutput current cannot follow it. Therefore, it is necessary toguarantee the tracking performance of the reference value todrive motors at high speeds. Some papers reported experimental verifications of motors for real EVs. For example,
it is reported that the model predictive control has improved
the tracking performance of the torque reference better thanField Oriented Control (PI control) [2]. Another study analyzed the noise and the vibration of motors for EVs [3]. However, the tracking performance of the torque reference shownin the former study can be improved by the use of the perfect
tracking control (PTC) for the current control [4], and the latter one shows that the current control that has the tracking
capability up to the carrier frequency to remove the harmonics is necessary. To solve those issues, a new current control
method based on PTC is introduced in this paper.

2.2 PWM Hold model
The single phase inverter (Fig. 1) outputs either 0 V or ±E
V (Fig. 2). For the discretization of this system, a zero-orderhold (ZOH) is commonly used. The ZOH outputs V[k] at asampling point k as the discretized value. However, a moreexact plant model can be obtained by treating it as a hold thatthe width of the pulse is controlled instead of a zero-orderhold. To achieve this, a PWM hold is introduced [7]. Thecontrol system is discretized with the switching time ∆T [k] asa control input u[k]. The state space model with controllable
canonical form (1) and (2) can be described by the PWM
hold,x[k + 1] = A s x[k] + b s ∆T [k]y[k] = c s x[k],(1)(2)

where A s = e Ac Tu , b s = e Ac Tu /2 ∆T bE, c s = cc and if ∆T < 0,then the output voltage will be −E.

a) Correspondence to: noguchi.shona19@ae.k.u-tokyo.ac.jp∗The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba, 227-8561 Japan
Phone: +81-4-7136-3881
Fax: +81-4-7136-3881
c 2020 The Institute of Electrical Engineers of Japan.
⃝Proposed Control Method

2.3 Multirate Feedforward Control
To achieve PTC, a multirate feedforward control is presented[6]. The state equation is described up to n samples ahead by1

High-Bandwidth Current Control of PMSM Based on Quasi Multirate Feedforward Contol (Shona Noguchi et al.)V𝐸Δ𝑇[𝑘 + 1]

M

𝑘𝑇#𝑡2𝑡3

𝑘 + 1 𝑇#𝑘 + 2 𝑇#t

Δ𝑇[𝑘]−𝐸

Fig. 3.
Fig. 1.

𝑇%

Circuit Diagram of Single Phase Inverter

𝐸𝐸𝑢'() = V[k]𝑇&𝑢[𝑘]

𝑘𝑇#

Fig. 2.
Hold

Output of Single Phase Inverter

𝑢+,- = Δ𝑇[𝑘]𝑘 + 1 𝑇#+𝐸

lifting. Here, n is the plant order. The controllable canonicalstate equation (1)–(2) is discretized with the sampling period
T u of the multirate holder. The discretized A, B, C, D matrices of the control system are
An−1· · · A s b s b s  Anss bscs0···00 ) (A Bcs bs···00  .
=  c s A sC D

..
..
..
.. ..
.
.
.
.
. 
n−2c s An−1cAb···cb0s sss ss
(3)

(4)

yo [l] = z−1 Cx[l + 1] + Du[l],(6)𝑢[𝑘]+𝑢[𝑘 + 1]

𝑘+2
𝑘+1𝑘𝑙𝑙+1Fig. 4. Principle of Quasi Multirate ControlFrom (7) and (8), the equation at the next discretization pointof the reference value (l + 1)T r
u[k + 2] = u[k + 1] + u[k].

(9)Thus, a PWM pulse can be made into the one pulse by adding
two pulse widths, and the newly derived u[k + 2] is outputtedto the plant [9]. The one pulse can be obtained as shown inFig. 4. As a result, the number of output pulses of the quasi
multirate PTC is smaller than that of the multirate PTC, sothat the switching loss can be reduced while guaranteeing thereference value tracking at every sampling point. This is because the average output voltage between kT u and (k + 2)T u
is as same as before combining the pulses. For this reason,
the number of the switching can be doubled. This method
is shown in Fig. 5. The new sampling period of the reference signal T r′ is defined as T r /2 and feedforward calculation is held every T u while the pulses are enhanced at T r′ that
equals to the carrier frequency. Therefore, with keeping zerotracking error at every T r′ , the proposed multirate feedforward
control can be held.

The controllability matrix is full rank since the system is controllable, so B matrix is a holomorphic matrix, then (3) canbe rewritten asuo [k] = B−1 (I − z−1 A)x[k + 1]()0I
=x[l + 1],−B−1 A B−1+(a) Centered pulse

DC Voltage Waveform Discretized by PWM(5)where, z = e .
Therefore, the nominal output that is a stable inverse model ofthe plant is described. The inverse system that uses the statevariable of one sample ahead x[l + 1] as the input variable
can be calculated by treating with the multirate conversion.
This guarantees the perfect tracking of the nominal plant at
every period T r .
sT r

3.

Simulation

3.1 Plant ModelingIn this study, SPMSM is a plant, and its dq coordinate modelis shown in Fig. 6. Now, if the d-axis current id = 0 is maintained by the current controller and non-interference control,
the plant model on the q-axis can be assumed to be a DC motor as shown in Fig. 7 This is expressed in the same way as
a transfer function from the voltage input to current output
in the continuous time in this model is obtained. Since thetransfer function from the voltage input to the current output
must include the feedback loop due to the eﬀect of the reverse

2.4 Quasi Multirate Control

The quasi multirate PTC is proposed in this subsection. Fig.
4 shows the principle of the quasi multirate control. The
pulse width is generated between the sampling points using
the state variable k corresponding to the control period. In
the second order multirate control system, the first and second lines of the (4) areu[k + 1] = A s x[k] + b s u[k],u[k + 2] = A s x[k + 1] + b s u[k + 1].

𝑢[𝑘 + 1]

(7)
(8)
2

High-Bandwidth Current Control of PMSM Based on Quasi Multirate Feedforward Contol (Shona Noguchi et al.)𝑇3 ′𝑇3 ′(=(a)1/𝑓Centeredpulse*)𝐸𝑢[𝑘 + 2]+𝑢[𝑘 + 3]𝑢[𝑘]+𝑢[𝑘 + 1]

Prop.

Table 1.

𝑙+1𝑙

𝑙+2

𝑇%
𝑇% (= (a)1/𝑓*Centered)pulse𝐸𝑢[𝑘]Conv.

𝑢[𝑘 + 1]𝑘+2𝑘+1𝑘Fig. 5. Comparison of PWM Pulse Pattern between
Quasi Multirate and Multirate Control
𝒗𝒅𝑖345𝒊𝒅

1
𝐿𝑠 + 𝑅++Parameters of the motor for the simulation
Parameter

Value

Time Constant τ [ms]Resistance R[Ω]Inductans L [mH]Inertia J [kg・m2 ]Viscous Resistance D [N・m・s/rad]Pole Pairs P
Induced Electromotive Voltage Ke [V・s/rad]
Torque Constant Kt [N・m/A]
Carrier Frequency fc [kHz]Control Input Period T u [s]DC Input Voltage of Inverter E [V]Reference Current Input Iqref [A]Reference Current Frequency fref [kHz]1.00.1567
2.28
9.084 × 10−4
4.00 × 10−440.1727
0.1727101.00 × 10−4250
12.5𝑢

𝐵"# (𝐼 − 𝑧 "# 𝐴)𝐷ω* 𝐿1
𝐿𝑠 + 𝑅𝒊𝒒𝑻𝐾'

1
𝐽𝑠 + 𝐷ω𝒎1𝑠Fig. 8.

θ𝒎-𝑖𝐶,-𝒊1
𝐿𝑠 + 𝑅𝑻𝐾'

1
𝐽𝑠 + 𝐷ω𝒎1
𝑠i
Js + B.
=2v JLs + (JR + LB)s + (BR + Ke Kt )Current Control based on PTC

ẋ(t) = A c x(t) + b c u(t),y(t) = C c x(t)0e Kt− BR+KJL
1 Tb c = [0] ,JLc c = [B J].

1− JR+LB
JL(10)1.
τs + 1(17)3.3 Simulation
Fig.8 shows the block diagram of the proposed method. Table. 1 shows the parameters of the simulation. The frequencyof the reference current is 2.5 kHz (= fc /4) in the simulation.
The simulation results in Fig. 9 shows the current waveformof the multirate PTC (a) and quasi multirate PTC (b) respectively. From Fig. 9, PTC can follow the reference value andthe proposed method is more accurate than the multirate PTC
in Fig. 9(a). Moreover, the error between the input and output current is shown in Fig. 10. From Fig. 9(b), the proposed
method is a more accurate waveform compared with the multirate PTC. Furthermore, Fig. 10(b) shows that the proposed
method guarantees the performance of reference value tracking twice as much as the multirate PTC. Therefore, the proposed method can follow the reference waveform more accurately.

(11)
(12)]
,

=Furthermore, to design the feedforward controller, the transfer function from the voltage input to the current output isdescribed as (10). This is discretized by the PWM hold.
The controllable canonical state equation is described by thetransfer function of the plant (10) and then is substituted for(1)–(2) to determine the parameters in the discrete-time. Theparameters are substituted for the equation (3)–(6) to determine A, B, C, D matrices of the multirate feedforward controller.

The controllable canonical form of this plant is described asfollows, with the state variable x(t) as the current and the differential of that [i i̇]T

Ac =+

iref

voltage, the plant model would be second order plant model.

[(ZOH)-iBlock Diagram of DC Motorwhere,+θ𝒎𝐾*Fig. 7.

+Ls + R.
(16)
τsThus, the transfer function from iref to i is seen as the firstorder system as follows.

Fig. 6. Block Diagram of SPMSM at dq-axis with Decoupling Control+Plant(SPMSM)

CPI (s) =𝐾*𝒗

Inverter

S𝑍 "# 𝐶-𝐻,/0+

ω* 𝐿

𝒗𝒒 ++(13)
(14)
(15)

As described in the equations above, the simulation is performed by applying the multirate PTC and the quasi multirate
PTC to the plant model (10).
3.2 Controller Design
The plant model has a feedback loop due to the reverse voltage, so the multirate controllers are applied. A, B, C, D matrices of the PTC controller are determined by the dq-axis
model of the SPMSM. The current controller CPI is designed
as a pole-zero canceled PI feedback controller as follows.

4.

ExperimentFig. 11 shows the motor bench that is used in the current
control. The results of the current wave are shown in Fig.
13 and 12. The PI control and the multirate PTC are demonstrated as the conventional methods, and the quasi multirate
3

High-Bandwidth Current Control of PMSM Based on Quasi Multirate Feedforward Contol (Shona Noguchi et al.)1.51.5
1.5

110.5

0.5

Current [A]

011

0.5
0.5

0-0.5

-0.5

-1-1-1.5

Current [A]Current [A]

1.5

-1.5
00.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

00.05

0.1

0.15

Time [ms]

0.2

0.25

0.3

0.35

00

-0.5
-0.5

0.4

Time [ms](a) Multirate PTC(b) Quasi Multirate PTC-1-1

Fig. 9.

Current Waveform
-1.5
-1.5
30

0.8

Error
SamplingPoint

0.6

0.4

0.2

0.2

0-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

Fig. 13.

-0.8
00.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

0.05

0.1

0.15

Time [ms]

0.2

0.25

0.3

0.35

0.4

Time [ms](a) Multirate PTC

Fig. 10.

0.3
3.3

0.4
3.4

0.5
3.5

0.6
3.6

0.7
3.7

0.8
3.8

0.9
3.9

41
10-3-3
10

Current Waveform (Proposed Method)previously reported that the multirate PTC controller can improve the output tracking performance compared with PI control, the proposed method improved the tracking performance
twice as much as the multirate PTC. The eﬀectiveness of theproposed method has been demonstrated in simulations andexperiments. The experiment has been conducted by the motor bench, so it is a future work to demonstrate the eﬀectiveness of the proposed method for electric vehicle motors.

0-0.2

0.2
3.2

Time [ms]
[ms]
Time
Error
SamplingPoint

0.6

0.4

Current [A]Current [A]

0.1
3.1

0.8(b) Quasi Multirate PTC

Error between Current Input and Output

References
(1)

(2)

(3)

Motor Bench (SPMSM)

1.5

1.5

11

0.5

0.5

Current [A]Current [A]Fig. 11.

0-0.5(4)

(5)

(6)

0-0.5

-1-1-1.5

-1.5
30

0.1
3.1

0.2
3.2

0.3
3.3

0.4
3.4

0.5
3.5

0.6
3.6

Time [ms](a) PI Control

Fig. 12.

0.7
3.7

0.8
3.8

0.9
3.9

41
10-3

(7)
30

0.1
3.1

0.2
3.2

0.3
3.3

0.4
3.4

0.5
3.5

0.6
3.6

0.7
3.7

0.8
3.8

Time [ms]

0.9
3.9

41
10-3

(8)

(b) Multirate PTC

Current Waveform (Conventional Methods)(9)

PTC is the proposed method. From the results, the proposed
method successfully demonstrated two times better reference
tracking performance than the conventional methods, as thecurrent output matches roughly every 0.1 ms (twice more redcircles than the proposed method). It guarantees to follow thereference twice better than the conventional methods.
5.

Conclusion

A new current control method based on the quasi multirate PTC is proposed in this study. Though our group had
4

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