Control Method for IPMSM Based onPerfect Tracking Control and PWM Hold Model

in Overmodulation RangeTakayuki Miyajima

Yokohama National University79-5 Tokiwadai, Hodogaya-kuYokohama, 240-8501 Japan

miya@hfl.dnj.ynu.ac.jp

Hiroshi FujimotoThe University of Tokyo

5-1-5 Kashiwanoha,Kashiwa, Chiba, 277-0804 Japan

fujimoto@k.u-tokyo.ac.jp

Masami FujitsunaDENSO CORPORATION

1-1 Syouwacho,Kariya, Aichi, 448-8661 Japan

MASAMI FUJITSUNA@denso.co.jp

Abstract— Interior permanent magnet synchronous motors(IPMSMs) are employed for drive motors of electric and hybridvehicles. Therefore, the control method in overmodulation rangeof inverter is important for fast torque response and expandingoperation range of IPMSM. In this paper, the control methodsbased on perfect tracking control and PWM hold model areproposed in overmodulation range. The proposed methods do notneed to switch controllers in overmodulation range. One of theproposed methods simultaneously prevents windup phemonemonand voltage saturation which is caused by harmonic currentwith PWM hold model. Finally, simulations and experiments areperformed to show the advantages of the proposed methods.

Index Terms—IPMSM, overmodulation range, PWM holdmodel, perfect tracking control

I. INTRODUCTION

Permanent magnet synchronous motors (PMSMs) arewidely employed in many industries. Especially, IPMSMs areused as drive motors of electric and hybrid vehicles. So, itis necessary that IPMSM drive systems achieve fast torqueresponse and wide operation range.

For realization of these requests, high output voltage ofinverter is necessary. Therefore, the control in overmodulationrange of inverter is important in addition to linear range.One of the control methods of IPMSM in overmodulationrange is presented in [1] which is composed of current vectorcontrol and voltage phase control. However, this method needsto switch between two control methods. The switch makestracking performance worse, so it is desirable to control in alloperation ranges by only one controller.

When output voltage is saturated, flux-weakening and anti-windup controls are applied. In [2], a method which achievesflux-weakening and anti-windup controls simultaneously isproposed with feedback controllers and reference tables. Ingeneral, a control system consists of only feedback controllerlike [2]. However, feedforward controller is necessary for hightracking performance. [3] proposed a control system usingmodel predictive control (MPC). Though MPC controller hasno integrator, MPC needs high computational cost and onlyone voltage vector is output during one control period.

Moreover, harmonic current increases in overmodulationrange. [4] and [5] proposed the methods based on estimatedharmonic currents for suppressing the voltage saturation whichis caused by harmonic current. However, the discrete-timemodel was not discussed.

In this paper, control methods based on perfect trackingcontrol [6] and PWM hold model [7] are proposed withouta switch of controllers in overmodulation range. In addition,this method is based on PWM hold model for simultaneouslypreventing windup phenomenon and the voltage saturationwhich is caused by harmonic current is proposed. The PWMhold model is a discrete-time model considering inverter drivesystem. Finally, simulations and experiments are performed toshow the advantages of the proposed method.

II. MODEL AND DISCRETIZATION

A. dq Model of IPMSM

The voltage equation of IPMSM in dq axis is representedby [

vd

vq

]=

[R + sLd −ωeLq

ωeLd R + sLq

][idiq

]+

[0

ωeKe

], (1)

where vd, q are d-axis and q-axis voltage, R is stator windingresistance, Ld, q are d-axis and q-axis inductance (Ld �= Lq),ωe is electric angular velocity, id, q are d-axis and q-axiscurrent, and Ke is back EMF constant.

The motor torque T is given by

T = Kmtiq + Krtidiq, (2)

where Kmt = PKe, Krt = P (Ld − Lq), and P is thenumber of pole pairs. In this paper, 2-phase/3-phase transformis represented as absolute transformation.

B. Discretization based on PWM Hold

In order to discretize a plant, a zero-order hold is generallyapplied. However, in the case of single-phase inverter, itcannot output arbitrary voltage but only 0 or ±E (E is dc-bus voltage of single-phase inverter). Therefore, in order tocontrol instantaneous values precisely, the zero-order hold is

irefd [k]

irefq [k]

vrefd [k]

vrefq [k]

vrefu [k] iu[k]

iw[k]θe[k]

uw

iq[k]

id[k]vrefv [k]

vrefw [k]uvw

dqCd[z]

Cq[z]+

+

−

−

+

+

+

+

Decoupling Control

IPMSM+

INV dq

Fig. 1. Block diagram of current vector control.

not suitable. When the pulse is allocated in the center ofcontrol period Tu, the plant in inverter drive system can bediscretized as follows [7].

A continuous-time state equation of a plant is given by

x(t) = Acx(t) + Bcu(t), y(t) = Ccx(t). (3)

The precise discrete model in which the input is the switchingtime ΔT [k] is obtained as

x[k + 1] = Asx[k] + BsΔT [k], y[k] = Csx[k], (4)

As := eAcTu , Bs := eAcTu/2BcE, Cs := Cc. (5)

C. PWM Hold Model of IPMSM

The continuous-time state equation of (1) is described by

x(t)=Ac(ωe)x(t)+Bc

{u(t) − [

0 ωeKe

]T}

, (6)

y(t)=Ccx(t), (7)

[Ac(ωe) Bc

Cc 0

]:=

⎡⎢⎣

− RLd

ωeLq

Ld

−ωeLd

Lq− R

Lq

1Ld

00 1

Lq

I 0

⎤⎥⎦ ,(8)

where u = [vd vq]T and y = x = [id iq]T .(6) is discretized based on PWM hold. Here, it is assumed

that the speed variation during one control period can be dis-regarded. ωeKe can be considered to be a pulse of magnitudeωeKe and width Tu. Therefore, the PWM hold model of (6)is designed as E = Vdc in (5) as follows:

x[k + 1] = As(ωe)x[k]

+ Bs(ωe){

ΔT [k] −[0 ωeKeTu

Vdc

]T}

, (9)

y[k] = Csx[k], (10)

where Vdc is dc-bus voltage of three-phase inverter, ΔT =[ΔTd ΔTq]T , and ΔTd, q are d-axis and q-axis switchingtime. As(ωe) and Bs(ωe) are functions of ωe, so thesefunctions are calculated again when speed is changed.

III. CONTROL SYSTEM DESIGN

A. Conventional Method

The block diagram of current vector control is shown inFig. 1. The decoupling control which is shown by (11) and(12) is applied to the plant (1) as follows:

vrefd [k]= v′

dref [k] − ωe[k]Lqiq[k], (11)

vrefq [k]= v′

qref [k] + ωe[k](Ldid[k] + Ke). (12)

T ref [k]

iu[k]iw[k]θe[k]

IPMSM+

INV

vrefu [k]

vrefv [k]

vrefw [k]Calculator

Voltageδ[k]

uwTorque

Estimator iq[k]

id[k]T [k]

Cδ[z]+−

dq

Fig. 2. Block diagram of voltage phase control.

C2[z]

+

++

u0[k]

y0[k] y[k]

u[k]

e[k]S

(Tu)−

P c(s)y(t)

B−1(I − z−1A)S

D

z−1C

r(t)

r[k] = xd[k + 1]

C1[z]

+

+

(Tu)

PLANT

Fig. 3. Singlerate perfect tracking control system of multi-input multi-outputsystem.

In addition, the d-axis and q-axis current controllers Cd, q(s)are designed as

Cd, q(s) =Ld, qs + R

τs, (13)

where τ = 10Tu. By discretizing (13), Cd, q[z] are obtainedby Tustin transformation with control period Tu.

In the rectangular wave operation, the fundamental com-ponent of output voltage is constant. Therefore, the motortorque T is controlled by the voltage phase δ. The blockdiagram of voltage phase control [1] is shown in Fig. 2. ThePI controller Cδ[z] calculates δ which reduces torque error.Next, the rectangular wave voltage is generated based on δ.However, the relationship between T and δ is nonlinear, soCδ[z] cannot be designed by model-based design. In this paper,the proportional gain and the integral gain of Cδ(s) were setto be 1.05 and 175 by try and error. By discretizing Cδ(s),Cδ[z] is obtained by Tustin transformation with Tu.

When the absolute value of dq axis voltage is√

32

Vdc

2 ×1.28 or more, the control structure is changed from the currentvector control to the voltage phase control. Here, coefficient√

3/2 is the coefficient to transform two-phase into three-phase. Then, the initial state variable of Cδ is compensatedwith the voltage phase at this point.

On the other hand, when the operation range is assumedthat the absolute value of dq axis voltage in the case of current

vector control is√

32

Vdc

2 × 1.1 or less, the control structure ischanged to the current vector control.

B. Perfect Tracking Control

Perfect tracking control (PTC) system which consists of the2-DOF control system as show in Fig. 3. The feedforwardcontroller C1[z] is a stable inverse system of the plant asthe reference is state variable xd[k+1]. Therefore, the perfecttracking is assured at the sample time. Moreover, the feedbackcontroller C2[z] suppresses the error between the output y[k]

B−1(ωe)(I − z−1A(ωe))

z−1C

iref [k] =

C1[z]

C2[z]

+

++

i[k]

e[k]−

ΔT 0[k]

Tu

VdcI

ΔT [k]

i0[k]

[0 ωeKeTu/Vdc]T

xd[k + 1]

+

+

uw

dq S(Tu)

IPMSM

INV+

θe + Δθe

θe

SVM

Fig. 4. Block diagram of proposed method 1.

α

β

V1

V2V3

V4

V6V5

V0

ΔTβ

ΔTαΔT1

ΔT2

θ

U VW

(1, 0, 0)

(1, 1, 0)(0, 1, 0)

(0, 1, 1)

(0, 0, 1) (1, 0, 1)

(0, 0, 0)

I

VIVI

V

II

III

Fig. 5. Space Vector Modulation(SVM).

Vdc

Vuv

Vj

√3/2(ΔTi + ΔTj)

−Vdc

Vvw

√3/2ΔTj

Vwu

√√√√√√3

2

ΔTi

2

V0 Vi

kTu (k + 1)Tu

Vi V0

√√√√√√3

2

ΔTi

2

−Vdc

Fig. 6. Pulse of line voltage.

and the nominal output y0[k] to assure robustness only whendisturbances or plant variations exist.

Here, a discrete-time state equation is given by

x[k + 1] = Ax[k] + Bu[k], y[k] = Cx[k]. (14)

Therefore, a stable inverse system of the plant is obtained as(15) and the nominal output is obtained as (16).

u0[k]=B−1(I − z−1A)xd[k + 1] (15)

y0[k]= z−1Cxd[k + 1] + Du0[k] (16)

C. Proposed Method 1

The inverse system of (9) is given by

ΔT 0[k] = B−1(ωe)(I − z−1A(ωe))xd[k + 1]

+[0 ωeKeTu

Vdc

]T

. (17)

Therefore the proposed method 1 is designed as Fig. 4. Thefeedback controller C2[z] is the same Cd, q[z] as the currentvector control of the conventional method. In this paper,anti-windup control is not applied for comparison of PTCstructures.

1) Input Generation: In the case of the proposed method,the control input is the switching time ΔTd and ΔTq becausethe control system is designed by dq model. However, in orderto apply the three-phase inverter, the switching time of three-phase have to be generated. Then, precise method in [8] isapplied as input generation of three-phase inverter.

ΔTd and ΔTq are transformed into ΔTα and ΔTβ by thedq/2-phase transform. Here, Δθe(= 0.5ωeTu) is added to θe

of the dq/2-phase transform for sampling error compensation[9]. Δθe is a pole position variation between the samplingpoint and the center of the PWM pattern output point.

0.7 0.72 0.74 0.76 0.781

1.2

1.4

1.6

1.8

|Δ T|/Tu

Wei

ght f

acto

r k

Fig. 7. Weight factor k.

V1

V2

V0

ΔT ij

ΔT ′ij

√√√√3

2ΔT1

√3/2ΔT ′

2√3/2ΔT2

√√√√3

2ΔT ′

1

Fig. 8. Output vector in overmodula-tion range.

The area is decided by ΔTα and ΔTβ in Fig. 5. The orderand the switching times

√3/2ΔTi, j of the output vectors Vi, j

in each areas are decided. Here,√

3/2 is the coefficient totransform two-phase into three-phase. For example, the pulsesof Fig. 6 are output in the area VI.

2) Input Generation in Overmodulation Range: In this sec-tion, a method for applying input generation to overmodulationrange is explained.

A technique of SVM in overmodulation range is proposedin [10]. This technique has two methods and has to switchbetween two methods according to modulation index. In thispaper, SVM in overmodulation range is achieved withoutswitching methods.

The range of switching time in overmodulation range isgiven by

ΔTlim1 < |ΔT | :=√

ΔT 2d + ΔT 2

q ≤ ΔTlim2, (18)

where ΔTlim1 =√

32

Tu√3

and ΔTlim2 =√

32

2π Tu. When the

switching time reference ΔT is in this range, ΔTi and ΔTj

are changed as follows. First, the longer switching time (ΔTi

or ΔTj) is multiplied by the weight factor k in Fig. 7. If thelength of ΔTi and ΔTj is same, the leading switching time ismultiplied by k. The factor k is decided as the fundamentalcomponent of the actual input is equal to the constant referenceduring 1/6 periods of θe. Finally, the other switching vector isoutput during the rest of Tu.

In the case of Fig. 8, ΔT2 is longer than ΔT1, so ΔT2 ismultiplied by k and changed into ΔT ′

2. Then, ΔT1 is changedinto ΔT ′

1 if√

3/2(ΔT ′2 + ΔT1) > Tu. The actual input

is changed from ΔT ij into ΔT ′ij by the above-mentioned

process.

D. Proposed Method 2

In the proposed method 1, the nominal output i0 is thetarget trajectory delayed one sample. When the fundamentalcomponent ΔT is saturated (|ΔT | > ΔTlim2), the fundamen-tal component of the current is decreased. However, becausei0 is not considered the saturation, the windup phenomenonis caused. In addition, the harmonic current which is causedby overmodulation range is not considered. The outputs offeedback controllers in dq axis oscillate in order to suppressharmonic current because i0 contains no harmonic current.The oscillated outputs make the tracking performance worse.Therefore, the proposed method 1 cannot archieve high track-ing performance. In the proposed method 2, nominal output i0

B−1(ωe)(I − z−1A(ωe))

z−1C

iref [k] =

C1[z]

C2[z]

+

++

i[k]

e[k]−

ΔT 0[k]

i∗0[k]

ΔT [k]

i0[k]

[0 ωeKeTu/Vdc]T

xd[k + 1]

+

+

i0[k]

uw

dq S(Tu)

IPMSM

INV+SVM

θe + Δθe

θe

NOE

Tu

VdcI

ΔT ′[k]

Fig. 9. Block diagram of proposed method 2.

ΔT [k]

ΔT ′[k]+

−ΔT h[k] i0[k + 1]

ΔT o[k]

+

+

ΔT 0[k]

PWMHoldModel

Fig. 10. Block diagram of nominal output estimator.

which considers the saturation of the fundamental componentand the harmonic current is estimated based on PWM holdmodel.

Fig. 9 shows the block diagram of the proposed method2. “NOE” in Fig. 9 is the Nominal Output Estimator. Theestimated nominal output i0 is used in overmodulation rangeand i0 is used in linear range. The reason to use two kindsof nominal outputs is to attenuate the harmonic current by thefeedback controller.

1) Nominal Output Estimator: Fig. 10 shows the blockdiagram of the nominal output estimator.

The fundamental switching time ΔT o for estimation is thefeedforward controller output ΔT 0. If ΔT 0 is

|ΔT 0| :=√

ΔT 2d0 + ΔT 2

q0 > ΔTlim2, (19)

the fundamental component is saturated. At this point, ΔT 0

is obtained as,

ΔT o =1

|ΔT 0|[ΔTd0

ΔTq0

]ΔTlim2. (20)

The harmonic switching time ΔT h is calculated from theinput to match the phase of the harmonic component. ΔT ′ isΔT ′

ij which is transformed into dq coordinate. ΔT ′ containsthe fundamental and harmonic component, so ΔT h is calcu-lated from the difference between ΔT ′ and the fundamentalcomponent ΔT . Here, if |ΔT | > ΔTlim2, ΔT is limited asfollow:

ΔT =1

|ΔT |[ΔTd

ΔTq

]ΔTlim2. (21)

i0 is estimated by the PWM hold model of IPMSM and theswitching time by the above-mentioned process.

IV. SIMULATION

The parameters of IPMSM for the simulation are shown inTable I. The dc-bus voltage of the three-phase inverter Vdc is36 [V]. The control period Tu is 0.1 [ms].

TABLE I

PARAMETERS OF IPMSM.

d-axis inductance Ld 0.920 [mH]q-axis inductance Lq 2.114 [mH]

Resistance R 154.9 [mΩ]Number of pole pairs P 3

EMF constant Ke 47.21 [mV/(rad/s)]

A. Simulation I

In the simulation I, the responses are compared in therectangular wave operation in the case of the conventionalmethod on steady state. It is assumed that the motor speedis controlled at ωe=490 [rad/s] by the load motor. Simulationresults are shown in Fig. 11.

The “flag” in the conventional method is a signal ofswitching controllers. When the flag is “high”, the voltagephase control is applied. On the other hand, the current vectorcontrol is applied when the flag is “low”. T , T0, and T ∗

0 arethe torque estimated from i, i0, and i∗

0 by (2) respectively. Theovermodulation range of the proposed method is represented asthe range between two dotted lines like Fig. 11(d) (0.0707 ∼0.0780 [ms]).

In the conventional method, the switch of controllers makesthe tracking performance worse when the control structureswitches to the current vector control. In addition, the controlsystem is composed by only feedback controllers, so thetracking performance is worse.

In the proposed methods, there is no switch of controllers.However, in the proposed method 1, the input |ΔT | is oscil-lated in Fig. 11(d), so it cannot track the reference. On theother hand, the proposed method 2 can track the referencebecause of the estimated nominal output which contains theharmonic current. The nominal output estimator is composedby the PWM hold model, so there is little error between T ∗

0

and T .Here, the very low frequency torque ripple is caused in the

proposed method 2. This reason is the voltage phase variationwhich is caused by asynchronous PWM.

B. Simulatuion II

In the simulation II, the tracking performance is comparedby sinusoidal torque reference of 50 [Hz]. It is assumed thatthe motor speed is controlled at ωe=530 [rad/s] by the loadmotor. Simulation results are shown in Fig. 12.

The conventional method is not satisfied the internal modelprinciple, so it cannot track the reference. In the proposedmethods, |ΔT 0| is excessive value, so the fundamental com-ponent is saturated. Therefore, the windup phenomenon iscaused in the proposed method 1. On the other hand, theproposed method 2 can achieve the perfect tracking in linearrange because the nominal output is considered the saturationof the fundamental component.

V. EXPERIMENT

In order to reduce computational cost, As(ωe) and Bs(ωe)of the PWM hold model are calculated by 4th order Taylor

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

Time[s]

Tor

que[

Nm

]

T^

Tref

flag

(a) Torque (Conventional)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

Time[s]

Tor

que[

Nm

]

T^

T0

(b) Torque (Proposed 1)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

Time[s]

Tor

que[

Nm

]

T^

T0*

T0

(c) Torque (Proposed 2)

0 0.02 0.04 0.06 0.08 0.10.055

0.06

0.065

0.07

0.075

0.08

0.085

Time[s]

Sw

itchi

ng T

ime[

ms]

(d) |ΔT | (Proposed 1)

0 0.02 0.04 0.06 0.08 0.10.055

0.06

0.065

0.07

0.075

0.08

0.085

Time[s]

Sw

itchi

ng T

ime[

ms]

(e) |ΔT | (Proposed 2)

Fig. 11. Simulation results for comparison in rectangular wave operation of conventional method.

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Time[ms]

Tor

que[

Nm

]

T^

Tref

flag

(a) Torque (Conventional)

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Time[ms]

Tor

que[

Nm

]

T^

T0

(b) Torque (Proposed 1)

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Time[ms]

Tor

que[

Nm

]

T^

T0*

T0

(c) Torque (Proposed 2)

0 10 20 30 400.06

0.07

0.08

0.09

0.1

0.11

Time[ms]

Sw

itchi

ng T

ime[

ms]

(d) |ΔT | (Proposed 1)

0 10 20 30 400.06

0.07

0.08

0.09

0.1

0.11

Time[ms]

Sw

itchi

ng T

ime[

ms]

(e) |ΔT | (Proposed 2)

Fig. 12. Simulation results of torque tracking characteristic.

series. The encoder resolution is 1000 [pulse/r]. In order toreduce oscillation of input which is caused by ωe, S method[11] is applied. Δθ in experiments is 0.9ωeTu to consider alsothe computational time 0.4Tu.

A. Experiment I

The speed is controlled at ωe=460 [rad/s] by the load motor.Experimental results are shown in Fig. 13.

In the proposed method 2, the error between T ∗0 and T is

existed because of modeling error, so the input oscillation isnot suppressed completely. However, the proposed method 2can achieve the best tracking control.

B. Experiment II

The speed is controlled at ωe=510 [rad/s] by the load motor.Experimental results are shown in Fig. 14.

In the proposed method 2, windup phenomenon is notcaused because the nominal output estimator considers the

saturation of the fundamental component. In addition, perfecttracking is achieved in the linear range.

VI. CONCLUSION

In this paper, the control methods based on PTC and PWMhold model in overmodulation range are proposed. Simulationsand experiments are performed to show that the proposedmethod can control in overmodulation range without switchof controllers.

Moreover, we show that the proposed method 1 cannotachieve high tracking performance because the nominal outputis not considered the input saturation of the fundamentalcomponent and the harmonic current. In the proposed method2, a method based on estimated nominal output with PWMhold model is proposed for simultaneously preventing windupphenomenon and the voltage saturation which is caused byharmonic current. Simulations and experiments are performed

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

Time[s]

Tor

que[

Nm

]

T^

Tref

flag

(a) Torque (Conventional)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

Time[s]

Tor

que[

Nm

]

T^

T0

(b) Torque (Proposed 1)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

Time[s]

Tor

que[

Nm

]

T^

T0*

T0

(c) Torque (Proposed 2)

0 0.02 0.04 0.06 0.08 0.10.06

0.07

0.08

0.09

0.1

Time[s]

Sw

itchi

ng T

ime[

ms]

(d) |ΔT | (Proposed 1)

0 0.02 0.04 0.06 0.08 0.10.06

0.07

0.08

0.09

0.1

Time[s]

Sw

itchi

ng T

ime[

ms]

(e) |ΔT | (Proposed 2)

Fig. 13. Experimental results for comparison in rectangular wave operation of conventional method.

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Time[ms]

Tor

que[

Nm

]

T^

Tref

flag

(a) Torque (Conventional)

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Time[ms]

Tor

que[

Nm

]

T^

T0

(b) Torque (Proposed 1)

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Time[ms]

Tor

que[

Nm

]

T^

T0*

T0

(c) Torque (Proposed 2)

0 10 20 30 400.06

0.07

0.08

0.09

0.1

0.11

Time[ms]

Sw

itchi

ng T

ime[

ms]

(d) |ΔT | (Proposed 1)

0 10 20 30 400.06

0.07

0.08

0.09

0.1

0.11

Time[ms]

Sw

itchi

ng T

ime[

ms]

(e) |ΔT | (Proposed 2)

Fig. 14. Experimental results of torque tracking characteristic.

to show the fast torque response of the proposed method 2.

REFERENCES

[1] H. Nakai, H. Ohtani, E. Satoh, and Y. Inaguma: “Development andTesting of the Torque Control for the Permanent-Magnet SynchronousMotor”, IEEE Trans. Ind. Electron., Vol. 52, No. 3, pp. 800–806, 2005

[2] T.-S. Kwon, G.-Y. Choi, M.-S. Kwak, and S.-K Sul : “Novel Flux-Weakening Control of an IPMSM for Quasi-Six-Step Operation”, IEEETrans. Ind. Appl., Vol.44, NO.6, pp. 1722–1723, 2008

[3] H. Kobayashi, H. Kitagawa, S. Doki, and S. Okuma : “Realization of aFast Current Control System of PMSM based on Model Predictive Con-trol”, The 34th Annual Conference of the IEEE Industrial ElectronicsSociety, pp. 1343–1348, 2008

[4] A. M. Khambadkone and J.Holtz: “Compensated Synchronous PI Cur-rent Controller in Overmodulation Range and Six-Step Operation ofSpace-Vector-Modulation-Based Vector-Controlled Drives”, IEEE Trans.Ind. Electron., Vol. 49, No. 3, pp. 574–580, 2002

[5] S. Lerdudomsak, S. Doki, and S. Okuma: “Harmonic Currents Estima-tion and Compensation for Current Control System of PMSM in Over-modulation Range -Analysis for Robustness to Parameter Variations-”,The 34th Annual Conference of the IEEE Industrial Electronics Society,pp. 1216–1221, 2008

[6] H. Fujimoto, Y. Hori, and A. Kawamura: “Perfect Tracking Controlbased on Multirate Feedforward Control with Generalized SamplingPeriods”, IEEE Trans. Ind. Eletron., Vol. 48, No. 3, pp. 636–644, 2001

[7] K. P. Gokhale, A. Kawamura, and R. G. Hoft: “Deat beat microprocessorcontrol of PWM inverter for sinusoidal output waveform synthesis”,IEEE Trans. Ind. Appl., Vol. 23, No. 3, pp. 901–910, 1987

[8] K. Sakata and H. Fujimoto: “Perfect Tracking Control of Servo MotorBased on Precise Model with PWM Hold and Current Loop”, The ForthPower Conversion Conference, pp. 1612–1617, 2007

[9] J. Kudo, T. Noguchi, M. Kawakami, and K. Sano: “Mathematical ModelErrors and Their Compensations of IPM Motor Control System”, IEEof Japan Technical Meeting Record, IEE Japan, SPC-08-25, pp. 25–31,2008 (in Japanese)

[10] J. Holtz, W. Lotzkat, and A. M. Khambadkone: “On Continuous Controlof PWM Inverters in Overmodulation Range Including the Six-StepMode”, IEEE Trans. Power Electronics, Vol. 8, No. 4, pp. 546–553,1993

[11] T. Tsuji, M. Mizuochi, H. Nishi, and K. Ohnishi: “A Velocity Measure-ment Method for Acceleration Control”, The 31st Annual Conferenceof the IEEE Industrial Electronics Society, pp. 1943–1948, 2005