IEEJ Journal of Industry Applications
Vol.10 No.4 pp.428–435
DOI: 10.1541/ieejjia.20004635

Paper

Quasi Multirate Feedforward Current Control
toward Nyquist Frequency of PWM for SPMSM
Shona Noguchi∗ Student Member, Masahiro Mae∗
Hiroshi Fujimoto∗a) Senior Member

Student Member(Manuscript received May 12, 2020, revised Feb. 24, 2021)
J-STAGE Advance published date : April 23, 2021

A high-bandwidth current controller is necessary to suppress torque ripple. To drive motors at high speeds, thereference-value-tracking performance must be guaranteed. To ensure the reference-value-tracking performance aroundthe Nyquist frequency, the proposed method samples reference values twice more than the conventional multirate
feedforward control for a second-order system. We have previously proposed a current controller based on the perfect
tracking control. However, there is still room to improve the reference value tracking performance; therefore, a new
approach based on the quasi multirate feedforward current control has been proposed and applied to permanent magnet
synchronous motors. The proposed method doubles the tracking points of the reference current waveform comparedwith the conventional methods. The enhancement of the proposed approach has been demonstrateted using simulations
and experimental verifications.
Keywords: permanent magnet synchronous motor (PMSM), current control, perfect tracking control, quasi multirate feedforward
control, tracking performance

1.

The torque ripple reduction eﬀect is applied through the experiment with motors for actual EVs at a speed of up to
300 rpm. Another study reported that the torque ripple is suppressed by injecting the compensation current determined by
the gradient descent optimization (GDO) (7) . The compensation signal determined by the GDO can suppress the torqueripple at minimal current. Another study is reported thatthe feedforward controller based on the perfect tracking control (PTC) can improve the reference value tracking performance (8) and suppress the torque ripple at a higher speed (9) (10) .
However, they have not discussed the torque ripple reductioneﬀect at a high-speed range of more than 1000 rpm and thetracking performance of the current control.
Though an experiment on motors used for actual EVs,
it was reported that the low current control bandwidth isa problem causing the noise and vibration (11) . Thus, it isnecessary for an application to reduce the noise and vibration within a wide bandwidth through a control approach tobroaden the bandwidth of the current control. In general, theproportional-integral (PI) feedback control is used as the current control of a motor. However, as a problem of the PI control when the reference current is steeply changed, the output
current cannot follow it owing to the limitation of the highbandwidth control from the modeling errors. Therefore, theproposed method is designed with a feedforward controller
for guaranteeing the tracking performance of the reference
value to the drive motors.
The proposed feedforward controller is designed using the
PTC. Furthermore, a quasi multirate deadbeat control is proposed as a new digital control method (12) . This method applies a new feedforward control approach using multirate
feedforward control, which is improved by increasing theIntroduction

The development of electrified vehicles such as hybrid vehicles (HVs) and electric vehicles (EVs) for the prevention of
air pollution and global warming is increasing. Furthermore,some countries have declared the prohibition of selling gasoline and diesel engine cars (1) . It is therefore expected that the
number of electrified vehicles will increase in the future (2) .
In the use of electric motors, the motor noise and vibration
are well-known problems. Such problems are tackled by the
car manufacturer, and it has been reported that they are suppressed by reducing the excitation force with an optimized
magnetic circuit (3) . Torque ripple, one of the causes of suchproblems, is generated from the spatial harmonics included in
the permanent magnet flux distribution, such as the distorted
back electromotive force (EMF) (4) .
There are two approaches used to suppress the torque ripple. One is a mechanical approach, of which skew is thebest-known method (5) . Although this is a simple way to suppress the torque ripple, there are disadvantages in the processing cost and the reduction of the torque output. By contrast,
the control approach has no such disadvantages because additional processing is unnecessary. Therefore, we have taken
a control approach.
Several papers have reported the suppression of the torqueripple. For example, it was reported that the torque ripple issuppressed with a Kalman filter to assume the actual electricangle to transform between the dq-axis and 3-phase model (6) .
a) Correspondence to: Hiroshi Fujimoto. E-mail: fujimoto@k.utokyo.ac.jp∗
The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba 227-8561, Japanc 2021 The Institute of Electrical Engineers of Japan.
⃝

428

Quasi Multirate Feedforward Current Control（Shona Noguchi et al.）sampling period of the double reference value (13) . Although
the eﬀectiveness of the proposed current control method has
been verified through a simulation and experiment, in this paper the eﬀectiveness while the motor is rotated is discussed
based on the frequency response of the tracking error whencompared to the PI and the multirate feedforward control. In
this study, the authors improved the tracking performance ofthe reference value by adopting the quasi multirate control.
This paper is organized as follows: First, the control theory
is described in Section 2. The conditions of the simulationand its result are discussed in Section 3. The experimental
results are provided in Section 4. Finally, some concluding
remarks are presented in Section 5.

(1) and (2) can be described using the PWM hold,x[k + 1] = A s x[k] + b s ∆T [k], · · · · · · · · · · · · · · · · · · (1)y[k] = c s x[k], · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (2)2.1 PWM Hold Model
Fig. 1 shows the circuit diagram of the single-phase inverter. It outputs either 0 V or
±E V as shown in Fig. 2. Fig. 3 shows the hold models. A
zero-order hold (ZOH) is commonly used for the discretization of a system. The ZOH outputs V[k] at a sampling point k
as the discretized value. However, a more exact plant modelcompared to the ZOH can be obtained by treating the pulse
width as the control input and discretizing it. To achieve this,a pulse width modulation (PWM) hold is introduced (14) . The
control system is discretized with the on-time ∆T [k] as a control input u[k].
The state-space model with the controllable canonical form

where A s = e Ac Tu , b s = e Ac Tu /2 bE, c s = c c , and if ∆T < 0,then the output voltage will be −E.
2.2 Multirate Feedforward Control
The multirate
feedforward control based on the perfect tracking control(PTC) is described to design the proposed method. The PTC
is the control system that tracks reference values without
errors at every sampling point (15) . Fig. 5 shows the block
diagram of PTC. It was reported that this phenomenon isachieved by the multirate feedforward controller based on
the PTC (16) . This control system is the two degrees of thefreedom composed by the feedforward and the feedback controller.
If there are disturbances or modeling errors, the feedback
controller CPI suppresses them. There are two samplers forthe reference r and the output y, and one holder for the input u, in a digital control system. Therefore, the sampling
parameters are notated as T r , T y , and T u , which represent the
sampling periods of r, y, and u, respectively.
Fig. 4 shows the sampling period in a multirate system.
Here, l is defined as the index of the reference sampling pointof a reference value. The state equation is described up to n
samples ahead by lifting. Here, n is the plant order. The controllable canonical forms (1) and (2) are discretized with thesampling period T u . The discretized A, B, C, and D matricesof the control system are⎛⎞
A s n−1 b s · · · A s b s b s ⎟⎟⎟⎜⎜⎜ A s n
⎜
⎟
cs
0
···
0
0 ⎟⎟⎟⎟!
" ⎜⎜⎜⎜
⎜⎜⎜
⎟
A B
cs bs
···
0
0 ⎟⎟⎟⎟ .
= ⎜⎜⎜⎜ c s A s
C D
⎜⎜⎜
..
..
..
.. ⎟⎟⎟⎟⎟
..
⎜⎜⎜
.
.
.
.
. ⎟⎟⎟
⎝
⎠
n−2
c s A n−1
c
A
b
·
·
·
c
b
0
s s
s
s s
s
· · · · · · · · · · · · · · · · · · · · (3)Fig. 1. Circuit diagram of single-phase inverter

The controllability matrix is a full rank because the system iscontrollable, and thus B matrix is a non-singular matrix. The
control input uo [l] is described as follows from (3) to achieve
PTC.

2.

Control System Designu o[l] = B−1 (I − z−1 A)x[l + 1]"!
I0x[l + 1], · · · · · · · · · · · · · · (4)=−B−1 A B−1

Fig. 2. Output of single-phase inverter
Fig. 4. Multirate sampling periods

Fig. 3. DC voltage waveform discretized using PWM
hold

Fig. 5. Block diagram of PTC
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Quasi Multirate Feedforward Current Control（Shona Noguchi et al.）

To sample the reference value n times as fast as the normal
multirate control beforehand, the number of switches can beincreased. Therefore, the tracking performance of a reference
value can be improved by designing the multirate controller
based on the PTC and oversampling reference values withoutchanging a carrier frequency.

Simulation

3.

3.1 Plant ModelingIn this study, SPMSM is the target plant, and its dq coordinate model is shown in Fig. 7.
Now, if the d-axis current id = 0 is maintained by the current controller and the decoupling control, the plant modelon the q-axis can be assumed to be a DC motor, as shownin Fig. 8. The transfer function of the controlled system canbe expressed in the same way as the DC motor. The transfer
function is from the voltage input to the current output in thecontinuous-time. Because the transfer function from the voltage input to the current output must include the feedback loop
owing to the eﬀect of the back EMF, the plant model would
be the second-order plant model, which is given as follows:

Fig. 6. Principle of quasi multirate control

yo [l] = z−1 Cx[l + 1] + Duo [l], · · · · · · · · · · · · · · · · · · (5)where z = e sTr . The feedforward control input is generated by(4) with one-sample ahead desired state trajectory x[l + 1].
This guarantees the perfect tracking of the nominal plant at
every period T r .
2.3 Quasi Multirate Feedforward Control
The proposed quasi multirate feedforward control is described in thissubsection. Fig. 6 shows the principle of the quasi multirate control. The on-time is generated between the sampling
points of index l corresponding to the control period. In themultirate control system, u o[l] of (4) consists ofi
Js + B=.· · · · · · · · · · (9)2v JLs + (JR + LB)s + (BR + Ke Kt )The controllable canonical form of this model is describedas follows with the state variable x(t) = [x ẋ]T .

Tu o[l] = [u1 u2 . . . un ]= [u[k] u[k + 1] . . . u[k + n − 1]]T , · · · · · · · · · (6)ẋ(t) = A c x(t) + b c u(t), · · · · · · · · · · · · · · · · · · · · · · · · · (10)y(t) = c c x(t), · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (11)where each control input u is⎛⎞n)⎜⎜⎜ −1⎟⎟−1−1⎜u[k] = ⎜⎝ B11 − z
B1m Am1 ⎟⎟⎟⎠ x1 [l + 1],

where
Ac =

*

0

1

+

, · · · · · · · · · · · · · · · · · (12)e Kt
− BR+K− JR+LB
JL
JL
*
+T
1
bc = 0, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (13)JLc c = [B J]. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (14)

m=1

⎞
⎛
n
)
⎟⎟
⎜⎜⎜ −1−1−1
B2m Am2 ⎟⎟⎟⎠ x2 [l + 1],
u[k + 1] = ⎜⎜⎝ B22 − z
m=1

..
.

⎛
⎞
n
)
⎜⎜⎜ −1⎟⎟−1−1
u[k + n − 1] = ⎜⎜⎝ Bnn − z
Bnm Amn ⎟⎟⎟⎠ xn [l + 1].

As described in the equations above, the simulation is conducted by applying multirate feedforward control as a conventional method and the quasi multirate feedforward control
as a proposed method to the SPMSM plant model, which isdefined through (9).

m=1

· · · · · · · · · · · · · · · · · · · · (7)Here, Amn is the element of the matrix A at the mth row andnth column and B−1
nm is the element of the inverse matrix of
B at the nth row and mth column. From (7), the equation ata sampling point of the reference value lT r is described asfollows:
u′o [l] = u[k] + u[k + 1] + · · · + u[k + n − 1].· · · · · · · (8)Thus, PWM pulses are combined into one pulse, and u′o [l] described by combining PWM pulses is a new control input (17) .
A single pulse can be obtained as shown in Fig. 6. As a result, the number of output pulses of the quasi multirate feedforward control is smaller than that of the multirate feedforward control under the condition in which the reference sampling period is the same. This is because the average output
voltage between kT u and (k + n)T u is the same as before combining the pulses. However, the response between sample
points of a reference trajectory can deteriorate because noneof the added output pulses are generated accurately when
considering the state at that time.

Fig. 7. Block diagram of SPMSM at dq-axis

Fig. 8. Block diagram of DC motor
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Quasi Multirate Feedforward Current Control（Shona Noguchi et al.）Fig. 9. Current control based on PTC
Table 1. Motor parameters
Parameter

Value

Resistance R[Ω]Inductance 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]DC input voltage of inverter E [V]Reference current input Iqref [A]

0.15673.65
9.084 × 10−4
4.00 × 10−440.17270.172710250
1

3.2 Feedforward Controller Design
Fig. 9 shows
the block diagram used in the simulation. The simulation
model is designed as a 3-phase model. Table 1 shows theparameters of the simulation. The plant model is a secondorder model owing to a feedback loop by the back EMF, andtherefore the transfer function from the voltage input to thecurrent output is described as (9).
The A, B matrices of the multirate feedforward controllers
and P[z] are designed by the dq-axis model of the SPMSM.
They are discretized using the PWM hold. The controllable
canonical form is described through the transfer function of
the plant (9) and is then substituted for (1) and (2) to determine the parameters under the discrete-time. The parameters
are substituted for the equation (3)–(5) to determine the A, B
matrices of the multirate feedforward controllers.
Fig. 10 shows a pulse pattern comparison between a multirate and the quasi multirate feedforward control. Fig. 10
assumes that the plant order is 2, for example. The sampling
period of the reference value through the proposed method T r′
can be reduced compared to a conventional multirate feedforward control. In Fig. 10, T r′ is defined as T r /2, and the feedforward calculation is held every T r′ /2, whereas the pulses are
generated at T r′ , which equals to the carrier frequency. Thus,
the calculation of the control input of the proposed method
considered with an oversampling is increased twofold compared with a conventional multirate feedforward control.
3.3 Feedback Controller Design
The current controller CPI is designed as a PI feedback controller. The gainsof the PI feedback controller are described as follows.

Fig. 10. Comparison of PWM pulse pattern between
quasi multirate and multirate control

ω = 2π f, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (17)1
ζ = √ . · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (18)2
The cut-oﬀ frequency of the closed-loop system f is designed
at 400 Hz. Thus, the transfer function of this feedback controller is as follows:1
CPI = K p + Ki .· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (19)s
For the experiment, the transfer function is discretized by theTustin transform when considering the ZOH.
3.4 Simulation Results
The frequencies of the reference current fref are 10, 100, 1000, and 2500 Hz in thesimulation. The sampling periods of the multirate feedforward control are T r = 200 µs and T u = 100 µs, respectively.
By contrast, T r′ of the quasi multirate feedforward control is
100 µs without changing the carrier frequency fc . This simulation assumes that the load motor is rotating at a speed inwhich the frequency of the 6th harmonics is as same as thatof the input current. The input rotation speed is calculated by
the following:
60 fref.· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (20)6P
The simulation results are given in Fig. 11, which shows
the current waveform of the PI control, the multirate feedforward control, and the quasi multirate feedforward control. The conventional methods apply PI control and multirate feedforward control, and the proposed method applies
the quasi multirate feedforward control described in this paper. Although the waveform changes depending on phases,
ω=K p = 2ζωL − R, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (15)Ki = ω2 L. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (16)Here, the poles of the PI feedback controller are placed as aclosed system, which becomes a Butterworth filter (18) ,431

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Quasi Multirate Feedforward Current Control（Shona Noguchi et al.）(a) fref = 10 Hz, ω = 25 rpm(b) fref = 100 Hz, ω = 250 rpm(c) fref = 1000 Hz, ω = 2500 rpm(d) fref = 2500 Hz, ω = 6250 rpm

Fig. 11. Simulation q-axis waveform of 6th current control(a) fref = 1000 Hz, ω = 2500 rpm(b) fref = 2500 Hz, ω = 6250 rpm

Fig. 12. Simulation q-axis current error between input and output

multirate feedforward control, and the cross mark is the sampling point of the quasi multirate feedforward control. The
proposed method can track the reference trajectory because
its waveform is closer to the reference waveform.
Moreover, the error between the input and output currentof the multirate feedforward control and the quasi multirate
feedforward control is shown in Fig. 12. From Fig. 12, thequasi multirate feedforward control can cross nearly 0 at every carrier period compared to the multirate feedforward control. Theoretically, the tracking error can be 0 at every sampling point T r using the conventional method and T r′ by thethe simulation results are as shown in Fig. 11 when the phase
diﬀerence is 0. The PI control cannot track the reference
value as it changes more steeply from Fig. 11. However, themultirate feedforward control and the quasi multirate feedforward control can track the reference value under 1000 Hz.
Therefore, they can track the reference value under highbandwidth. Moreover, Fig. 11(d) shows a comparison between the multirate feedforward control as the conventional
method and the quasi multirate feedforward control as the
proposed method at the Nyquist frequency of the multirate
control. The circle mark indicates the sampling point of the432

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Quasi Multirate Feedforward Current Control（Shona Noguchi et al.）

, 1 T
T

|ER (jωref )| = , 1 TT0

e2 (t)dt(i (t))2 dt
0 qref.· · · · · · · · · · · · · · · · (21)outputoutputHere, T = 1/ fref and e(t) = iqref (t) − iq (t), where iq (t)is the output currents controlled by the PI control, the multirate feedforward control, and the quasi multirate feedforward
control. From Fig. 13, the RMS value ratio (21) between thereference and the error of the quasi multirate feedforward
control is smaller than the multirate feedforward control as
the frequency of the reference value steeply changes. Thus,
the proposed method can track the reference value more accurately. However, each max error value in Fig. 12 is larger
than the multirate feedforward control. It is considered thatthe quasi multirate feedforward control contains the modeling error because the control inputs of the quasi multirate

Fig. 13. Simulation frequency response of inter-sample
tracking error

proposed method, respectively. However, this phenomenon
was not observed in this simulation because the motor isrotating at high-speed, and thus the coordinate transformation error can appear because the electric angle θe steeplychanges (19) .
Furthermore, Fig. 13 shows the ratio of the reference andthe error value by the root mean square (RMS). The ratio of
the RMS is calculated as
Fig. 14. Experimental setup (SPMSM)(a) fref = 10 Hz, ω = 8.33 rpm(b) fref = 100 Hz, ω = 83.3 rpm(c) fref = 1000 Hz, ω = 833 rpm(d) fref = 2500 Hz, ω = 2083 rpm

Fig. 15. Experimental q-axis waveform of 18th current control
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Quasi Multirate Feedforward Current Control（Shona Noguchi et al.）(a) fref = 1000 Hz, ω = 833 rpm(b) fref = 2500 Hz, ω = 2083 rpm

Fig. 16. Experimental q-axis current error between input and output

feedforward control are the sum of two pulses.
From these simulation results, the proposed method thatsamples twice more than the conventional method can track
the reference trajectory more accurately without changing thecarrier frequency. These results verify the eﬀectiveness of the
proposed method.
4.

ExperimentFig. 14 shows the motor bench used in the current controlof the experimental validation. Although the parameters are
the same as the simulation, the rotation speed of the test motor has been set at a speed of 1/3 of the rotation speed obtained by (20) assuming the 18th harmonic in this experiment. The current waves are obtained as follows: First, they
are observed using an oscilloscope. Next, the electric angle
θe is obtained on the sampling points. Finally, the coordinatetransformation is processed with the observed current waves
and θe , which is interpolated linearly. The PI control and themultirate feedforward control are demonstrated as the conventional methods, and the quasi multirate feedforward control is the proposed method.
The experimental results of the current waveform areshown in Fig. 15. From Fig. 15, the current controllers basedon the multirate feedforward control improved the tracking
performance better than the PI control, similar to the simulation results. Moreover, the proposed method successfully
demonstrated a better reference tracking performance thanthe conventional methods at a high-bandwidth, as indicatedin Fig. 15(d).
Furthermore, the tracking error between the input and output current of the multirate feedforward control and the quasi
multirate feedforward control is shown in Fig. 16. The proposed method can track a reference trajectory more accurately for less tracking error compared to the conventional
method, similar to the simulation shown in Fig. 16. Furthermore, Fig. 17 shows the frequency response of the RMS ratio
between the reference and the error values from the results in
Fig. 15. The RMS ratio is calculated by (21) as T = 3/ fref .
The RMS ratio of the proposed method is less than that of
the conventional methods, and therefore the proposed method
can improve the reference tracking performance at a highbandwidth without changing the carrier frequency. Thus, theFig. 17. Experimental frequency response of intersample tracking error

eﬀectiveness of the proposed method is also validated, similar to the simulation results.
5.

Conclusion

A new current control method based on the quasi multirate feedforward control is proposed in this paper. Althoughprevious research shows that the multirate feedforward controller can improve the output tracking performance compared with the PI control, the proposed method has an improved tracking performance with an increase in the sampling
period of reference trajectory double that of the conventional
methods. The eﬀectiveness of the proposed method has been
demonstrated in simulations and experiments.
The torque ripple suppression eﬀect by the PTC based on areference value oversampling similar to quasi multirate feedforward control is confirmed in (19) . However, the experiment
was conducted using a motor bench, and thus a future study
will be conducted to demonstrate the eﬀectiveness of the proposed method for electric vehicle motors.

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Systems, The University of Tokyo (2021) (in Japanese) (to be published)

Shona Noguchi (Student Member) received the B.E. degree fromWaseda University in 2019. He is currently working
towards the M.E. degree in the Department of Electrical Engineering and Information Systems, Graduate
School of Engineering, the University of Tokyo. His
research interests are in motor drive, motion control,
and power electronics. He is a student member of The
Institute of Electrical Engineers of Japan.

Masahiro Mae (Student Member) received the B.E. and M.S. degrees
from The University of Tokyo in 2018 and 2020, respectively. He is currently working towards the Ph.D.
degree in the Department of Electrical Engineering
and Information Systems, Graduate School of Engineering, The University of Tokyo. His research interests are in control engineering, precision motion control, multirate control, and multi-input multi-output
systems. He is a student member of the Institute ofElectrical and Electronics Engineers and the Societyof Instrumental and Control Engineers.
Hiroshi Fujimoto (Senior Member) received the Ph.D. degree in the
Department of Electrical Engineering from the University of Tokyo in 2001. In 2001, he joined the Department of Electrical Engineering, Nagaoka University of Technology, Niigata, Japan, as a research associate. From 2002 to 2003, he was a visiting scholar in
the School of Mechanical Engineering, Purdue University, U.S.A. In 2004, he joined the Department ofElectrical and Computer Engineering, Yokohama National University, Yokohama, Japan, as a lecturer andhe became an associate professor in 2005. He had been an associate professor of the University of Tokyo from 2010 to 2020 and became a professor
from year 2021. He received the Best Paper Awards from the IEEE Transactions on Industrial Electronics in 2001 and 2013, Isao Takahashi PowerElectronics Award in 2010, Best Author Prize of SICE in 2010, the Nagamori Grand Award in 2016, and First Prize Paper Award IEEE Transactions
on Power Electonics in 2016. His interests are in control engineering, motion control, nano-scale servo systems, electric vehicle control, motor drive,
visual servoing, and wireless motors. He is a senior member of IEE of Japanand IEEE. He is also a member of the Society of Instrument and Control
Engineers, the Robotics Society of Japan, and the Society of Automotive
Engineers of Japan. He is an associate editor of IEEE/ASME Transactions
on Mechatronics from 2010 to 2014, IEEE Industrial Electronics Magazine from 2006, IEE of Japan Transactions on Industrial Application from2013, and Transactions on SICE from 2013 to 2016. He is a chairperson of
JSAE vehicle electrification committee from 2014 and a past chairperson ofIEEE/IES Technical Committee on Motion Control from 2012 to 2013.

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IEEJ Journal IA, Vol.10, No.4, 2021