model-based design of voltage phase controller for...

Model-based Design of Voltage Phase Controller for SPMSMin Field-weakening Region

Takayuki Miyajima and Hiroshi FujimotoThe University of TokyoKashiwa, Chiba, Japan

[email protected], [email protected]

Masami FujitsunaDENSO CORPORATION

Kariya, Aichi, JapanMASAMI [email protected]

Abstract— This paper investigates the nonlinear characteris-tic between voltage phase and torque and proposes a model-based design of voltage phase controller. PMSM drive systemsshould realize a quick torque response and have a wideoperating range. However, in high-speed region, the controlinput is voltage phase only and the plant is nonlinear dueto the inverter output voltage amplitude saturation. Becauseof this nonlinear characteristic between current and voltagephase, model-based design have not yet been carried out. Asbasic study, this paper proposes a model-based voltage phasecontroller for SPMSM. Simulation results and experimentalresults show the effectiveness of the model-based design method.In addition, the analysis of the derived plant model shows therelationship between q-axis current and voltage phase is a non-minimum phase system.

I. INTRODUCTION

PMSMs (Permanent Magnet Synchronous Motors) arewidely employed in many industrial applications becauseof high efficiency and high power density. Interior PMSMs(IPMSMs) are used in driven systems of electric vehicles,hybrid electric vehicles. Surface PMSMs (SPMSMs) are usedfor electric power steering systems. In these applications,the inverter output voltage is limited by the volume ofthe installed battery pack. Therefore, in order to achievequick torque response under the voltage saturation, highperformance PMSM control scheme is required.

Feedforward control methods based on model predictivecontrol [1]–[3] have been proposed. The model-predictivecontrol determines control input which optimizes the definedcost function such as torque tracking error. It can easilyconsider the voltage limit. The authors proposed feedforwardfield-weakening control method based on the final-state con-trol [4]. However, feedforward control does not guaranteerobustness to parameter variation. Thus, robust feedbackcontrol method is necessary. The voltage limiter methods[5], [6] operate current controller output so as to quicken itsresponse. During this operation, the current feedback loopbecomes an open-loop. The modulation feedback methods[7]–[9] cannot achieve quick torque response because it hastwo feedback loops.

During field-weakening control, the voltage amplitudeis fixed and the control input is the voltage phase only.The voltage phase control [10], [11] operates voltage phasedirectly to compensate torque tracking error. However, thecontroller gains are determined by trial and error because of

its nonlinear characteristics. This nonlinearcharacteristic hasnot been discussed precisely. In induction motors, transferfunction between voltage phase and q-axis current had beenanalyzed [12], [13]. However it focused only on resonancepeak of linearized transfer function in order to design thetorque reference filter of modulation feedback system.

In this paper, a precise model-based design of voltagephase controller is proposed. This paper considers SPMSMfor basic study. The precise plant model is derived based onlinearization to use in the model-based design. An analysisof the proposed plant model shows that the plant is a non-minimum phase system during motor mode. A non-minimumphase system is the system whose zero is unstable [14].Feedback control systems of a non-minimum phase systemhave the trade-off between undershoot and settling time.This trade-off is an achievable performance limitation ofvoltage phase control. The proposed model-based designmethod selects a PID controller as voltage phase controllerin order to place all closed-loop poles to arbitrary valuesbut these poles are slower than a unstable zero of the plant.Finally, simulations and experiments are performed to showthe advantages of the proposed methods.

II. MODEL AND LINEARIZATIONA. dq Model of SPMSM

The voltage equations of SPMSM are represented by

x(t) =

[ −RL ωe

−ωe −RL

]x(t) +

[vd(t)L

vq(t)−ωeKe

L

],

= f(x, u) (1)vd(t) = −Va(t) sin δ(t), vq(t) = Va(t) cos δ(t), (2)x(t) := [id(t) iq(t)], u(t) := [Va(t) δ(t)], (3)

where vd, vq, id, and iq are the d- and q-axis voltagesand currents, , L is the inductance, R is the stator windingresistance, ωe is the electric angular velocity, Ke is the backEMF constant, Va is the voltage amplitude, and δ is thevoltage phase.

The torque T is described as

T (t) = Kmtiq(t), (4)

where Kmt := PKe and P is the number of pole pairs.In this paper, the 2-phase/3-phase transform is absolutetransformation.

0 20 40 60−1

−0.5

0

0.5

1

Real axis [krad/s]

Imag

inar

y ax

is [k

rad/

s]

ωe: large

ωe: large

ωe: large

(a) Speed varies

−5 0 5 10 15 20−1

−0.5

0

0.5

1

Real axis [krad/s]

Imag

inar

y ax

is [k

rad/

s]

T : large

(b) Torque varies

−1 0 1 2−1

−0.5

0

0.5

1

Real axis [krad/s]

Imag

inar

y ax

is [k

rad/

s]

(c) Torque varies (zoom)

Fig. 1. A zero and poles of ΔP22(s) in field-weakening region

B. Linearization

In field-weakening region, the manipulated variable is thevoltage phase only. Therefore, the control system designis difficult due to the nonlinear characteristic between thevoltage phase and the q-axis current (torque) as shown in(1). Thus, the voltage equation of SPMSM is linearized formodel-based design.

Consider the equilibrium point (xo, uo) which satisfiesf(xo, uo) = 0, where uo = [Vao δo]

T and xo =[ido iqo]

T . (1) can be linearized around this equilibriumpoint by using first-order Taylor series as follows:

d

dt

[ΔidΔiq

]= A

[ΔidΔiq

]+B

[ΔVa

Δδ

], (5)

A :=∂f

∂x

∣∣∣∣(xo,uo)

=

[ −RL ωe

−ωe −RL

], (6)

B :=∂f

∂u

∣∣∣∣(xo,uo)

=

[ − 1L sin δo −Vao

L cos δo1L cos δo −Vao

L sin δo

], (7)

Δid := id − ido, Δiq := iq − iqo,

ΔVa := Va − Vao, Δδ := δ − δo.

The transfer functions from the voltage amplitude and thevoltage phase to the d- and q-axis currents are obtained by[

ΔidΔiq

]=

[ΔP11(s) ΔP12(s)ΔP21(s) ΔP22(s)

] [ΔVa

Δδ

], (8)

ΔP11(s) =− 1

L sin δo{s+ R

L − ωe tan(π2 − δo

)}s2 + 2R

L s+R2

L2 + ω2e

, (9)

ΔP12(s) =−Vao

L cos δo(s+ R

L + ωe tan δo)

s2 + 2RL s+

R2

L2 + ω2e

, (10)

ΔP21(s) =1L cos δo

(s+ R

L + ωe tan δo)

s2 + 2RL s+

R2

L2 + ω2e

, (11)

ΔP22(s) =−Vao

L sin δo{s+ R

L − ωe tan(π2 − δo

)}s2 + 2R

L s+R2

L2 + ω2e

,

(12)

The proposed model-based design method for voltage phasecontroller uses ΔP22(s) which is the linear transfer functionbetween q-axis current and voltage phase.

TABLE INOMINAL PARAMETERS UNDER THE TEST

R 33.7 [mΩ]

L 0.185 [mH]

Ke 11.60 [mV/(rad/s)]

P 7

dc-bus voltage Vdc 12.0 [V]

maximum modulation index Mmax 1.0

C. Analysis of linearized plant modelFrom the viewpoint of zeros, the characteristics of the

transfer functions from voltage phase to d- and q-axis cur-rents. The zeros of ΔP12(s) and ΔP22(s) are representedby (13) and (14), respectively.

z12 = −R

L− ωe tan δo. (13)

z22 = −R

L+ ωe tan

(π2− δo

). (14)

The poles are described by

p1, p2 = −R

L± jωe (15)

Namely, the poles and the zeros are functions of the operatingpoint. In the field-weakening region, R

L � ωe. Therefore,during motoring mode (0 < δo < π), ΔP12(s) and ΔP22(s)have a stable zero and a unstable pole, respectively. On theother hand, under regeneration (−π < δo < 0), ΔP12(s)and ΔP22(s) have an unstable zero and a stable pole,respectively. The imaginary parts of poles is depend on theoperating frequency.

Fig. 1(a) reports the values of a zero and poles when thespeed varies from 900 [rpm] to 1400 [rpm] under 0 [Nm].Here, Table I illustrates the nominal parameters of the test.The zero is unstable. By increasing motor speed, the zeroslows because the voltage phase increases in field-weakeningregion.

Fig. 1(b) illustrates the characteristic between the zero andthe torque when the torque is changed from 0 [Nm] to 3 [Nm]at 1000 [rpm]. Increment of torque makes unstable zero slow.In high-torque region, the zero is extremely slow as shownFig. 1(c). This slow unstable zero makes a trade-off betweenquick torque response and small undershoot.

i∗d[k]

i∗q[k]

vd[k]

vq[k]

iu[k]iw[k]θe[k]

uw

dq iq[k]

id[k]PWM

C[z]−

DecouplingSPMSM

+INV.

T ∗[k]−

+

++

+

++

θe[k]+Δθe[k]

1

Kmt

Control

Abs

√2

3

2

Vdc

−

CMI [z]

C[z]

+M∗

i [k]

Mi[k]

Fig. 2. The block diagram of the conventional method 1.

III. CONTROL SYSTEM DESIGN

The proposed method is compared with two conventionalmethods. The conventional method 1 is the modulationfeedback control. The conventional method 2 consists of thecurrent vector control, the voltage phase control which isdesigned by trial and error, and the control switch structure.On the other hand, the proposed method designs voltagephase controller by model-based design. Briefly, the differ-ence between the conventional method 2 and the proposedmethod is the design of voltage phase controller only.

A. Conventional method 1: modulation feedback control

Fig. 2 shows the block diagram of the conventional method1. The conventional method 1 has two feedback loops:current loop and modulation index loop.

In the current loop, the coupling terms in (1) are rejectedby the decoupling controls which are represented by

vd[k] = v′d[k]− ωe[k]Liq[k], (16)vq[k] = v′q[k] + ωe[k](Lid[k] +Ke), (17)

where v′d and v′q denote the d- and q- axis current feedbackcontroller outputs, respectively.

The d- and q-axis current feedback controllers are de-signed to be a plant-pole cancellation feedback control asfollow:

C(s) =Ls+R

τs, τ = 1.0 [ms], (18)

where τ is the selected bandwidth of the current loop. Thediscretized controller C[z] by the Tustin transform is appliedto the control system.

In the modulation index loop, the modulation index con-troller CMI [z] which is PI controller obtains the d-axiscurrent reference i∗d[k] to put the current references withinthe voltage limit circle. In this paper, the gains of CMI(s) aredesigned as the proportional gain and the integral gain are10 and 100, respectively, by trial and error. By discretizingCMI(s), CMI [z] is obtained by Tustin transformation withTu.

B. Conventional method 2: voltage phase controller is de-signed by trial and error

If the voltage amplitude is not saturated, the conventionalmethod 2 applies current vector control to the control system.On the other hand, at the operating point on the voltage

limit circle, the torque is controlled with the voltage phasecontroller.

1) Current vector control: The block diagram of currentfeedback control is described in Fig. 3. The current feedbackcontroller C[z] and the decoupling control are the same withthe conventional method 1.

When the voltage reference is saturated before switchingto the voltage phase control, it is limited as

V a[k] =

{V a[k]

|V a[k]|Vamax (|V a[k]| > Vamax)

V a[k] (otherwise)(19)

where V a = [vd vq]T , V a is the limited voltage reference,

and Vamax(:=√3/2MmaxVdc) is the maximum voltage

amplitude. Here,√3/2 is the coefficient to transform two-

phase into three-phase. Under voltage amplitude saturation,an anti-windup control in [15] is applied.

2) Voltage phase control [10]: The conventional voltagephase controller reported in Fig. 4(a) uses a PI controllerwhich is expressed as

CδPI =KP s+KI

s, (20)

where KI and KP are the integral and proportional gains.These gains are determined by trial and error so that thetransient response becomes small damped oscillation. Thecontrol gains are KP = 0.0001 and KI = 2. By discretizingwith Tustin transform, CδPI [z] is obtained.

During the voltage phase control, constant voltage ampli-tude Vamax is given.

3) Control switch structure: When the voltage amplitudeis limited in transient response, d-axis current and q-axiscurrent cannot track the reference. Therefore, The switchingcondition from the current vector control to the voltage phasecontrol is based on d-axis current error. If the absolutevalue of d-axis current error which is represented by (21)exceeds the baseline X1, the mode of controller is changedfrom the current vector control to the voltage phase control.At this point, state variables of voltage phase controller iscompensated with the current feedback controller outputs.

Y1[k] =

{i∗d − id + Y1[k − 1] (Va ≥ Vamax)0 (otherwise)

(21)

On the other hand, the switching condition from thevoltage phase control to the current vector control is based onthe value of d-axis current. If id > 0, the torque reference

i∗q[k]

vd[k]

vq[k]

iu[k]

iw[k]

θe[k]

uw

dq iq[k]

id[k]

PWMC[z]

C[z]

−

Decoupling Control

SPMSM+

INV−

++

+++

T ∗[k]1

Kmt

i∗d[k] = 0 +

Fig. 3. Block diagram of the current vector control (conventional method 2 and proposed method).

can be achieved by the current vector control. Therefore,the control system is switched to the current vector control.(22) expresses the sum of the d-axis current under smallerq-axis current tracking error than X2. If Y2 is smaller thanthe baseline X3, the voltage phase controller is applied tothe control system. At this point, state variables of C[z] iscompensated with the d-axis and q-axis voltages.

Y2[k] =

{id[k] + Y2[k − 1] (|i∗q − iq| ≤ X2)0 (otherwise)

(22)

X1, X2, and X3 are determined by trial and error. In thispaper, they are selected as X1 = 100, X2 = 1, and X3 = 40.

C. Proposed method: Model-based voltage phase control

The current vector control and control the switch structureof the proposed method are the same as the conventionalmethod 2 but the design method of voltage controller isdifferent. The proposed method designs voltage phase con-troller with model-based design. Fig. 4(b) represents theblock diagram of model-based voltage phase control.

The voltage phase controller CδPID[z] is designed withthe linearized plant model ΔP22(s). However, this modelis linearized around an equilibrium point. Here, the voltagephase controller is used during the voltage amplitude satu-ration. Thus, the equilibrium point is an intersection of thevoltage limit circle and the torque constant line. The voltagephase on this intersection can be determined from the q-axiscurrent reference i∗q uniquely as follows:

δo[k] = sin−1

{(R2 + ω2

eL2)i∗q [k] + ωeKeR√

R2 + ω2eL

2Vamax

}

− tan−1 R

ωeL. (23)

In this paper, in order to place all close-loop poles atarbitrary values, the authors selected a PID controller as thevoltage phase controller. By using the linearized plant modelΔP22(s) on the equilibrium point (Vamax, δo), CδPID(s) isdesigned by pole placement. However, in high-speed region,the plant has fast complex conjugate poles. All closed-loop poles should be faster than the plant poles. However,fast closed-loop poles cause undershoot because the plantis a non-minimum phase system. Therefore, at the high-speed region, the proposed method places all closed-looppoles on a circle which goes through the plant poles andhas a center at the origin. In this paper, the real parts ofall close-loop poles are placed at -750 [rad/s]. Cδ[z] is

+

iq[k]

−

uw

dq S(Tu)

(PWM)HPWM

θe[k]

θe[k]

SPMSM

INV.+

Va max

δ[k]i∗q [k]

id[k]

CδPI [z]T ∗[k]1

Kmt

(a) conventional method 2

+

iq[k]

−

uw

dq S(Tu)

(PWM)HPWM

θe[k]

θe[k]

SPMSM

INV.+

Va max

δ[k]i∗q [k]

id[k]

f (xo, uo) = 0

CδPID[z]

uo = [Va max δo]T

T ∗[k]1

Kmt

(b) proposed method

Fig. 4. Block diagram of the voltage phase control.

obtained by Tustin transformation with control period Tu.Here, the parameters of ΔP22(s) vary depending on theoperating point, namely, the proposed model-based voltagephase controller is a variable gain controller.

IV. SIMULATION

The proposed method is evaluated firstly by simulationresults. The parameters under simulations are the same asTable I. The sampling period Tu is 0.1 [ms].

Fig. 5, Fig. 6, and Fig. 7 show step torque responses at800 [rpm]. Here, the “SW” represents the mode of controller.When the SW is “high”, the voltage phase control is applied.On the other hand, the current vector control is applied whenthe SW is “low”. The conventional method 1 has the currentloop and the modulation index loop. The modulation loopis the outer of the current feedback loop which has lowbandwidth due to voltage saturation. Therefore, it cannotachieve quick current response. On the other hand, afterswitching to voltage phase control, The conventional method2 and the proposed method realize quicker current responsein comparison with the conventional method 1. However,the gains of the conventional method 2 are small for smalldamped oscillation. As a result, current response is slow.The proposed method designs high-bandwidth voltage phasecontroller with precise plant model. It achieves quickestcurrent response.

0 0.1 0.2 0.3−25

−20

−15

−10

−5

0

5

10

15

Time[s]

Cur

rent

[A]

idi∗d

(a) d-axis current id

0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗q

(b) q-axis current iq

0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

Time[s]

Mod

ulat

ion

inde

x

(c) Modulation index

0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

pha

se[r

ad]

(d) Voltage phase δ

Fig. 5. Simulation result (800 [rpm], conventional method 1).

0 0.1 0.2 0.3−25

−20

−15

−10

−5

0

5

10

15

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

Time[s]

Mod

ulat

ion

inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

pha

se[r

ad]



0 0.1 0.2 0.3−25

−20

−15

−10

−5

0

5

10

15

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

Time[s]

Mod

ulat

ion

inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

pha

se[r

ad]


Fig. 7. Simulation result (800 [rpm], proposed method).

Fig. 8, Fig. 9, and Fig. 10 describe step torque responses at1000 [rpm]. According to this simulation, torque is controlledby voltage phase controller only.

Both conventional methods cannot achieve quick currentresponse. In contrast, the proposed method shortens thesetting time.

V. EXPERIMENT

Experiments were conducted under the same conditionas simulations. In the conventional method 1, d-aix currentreference is limited within -80 [A] to avoid the divergenceof the modulation index loop.

Fig. 11, Fig. 12, and Fig. 13 reports step torque responsesat 800 [rpm]. Current responses in the conventional method1 is very slow because the modulation index loop hasa cascaded feedback systems. The conventional method 2cannot quick response because it is designed by trial anderror. On the other hand, the proposed method achieves quickcurrent response owing to the precise model-based design.

The experimental results at 1000 [rpm] are shown in Fig.14, Fig. 15 and Fig. 16. The proposed method controls q-axiscurrent quickly. The proposed design method realizes highperformance.

VI. CONCLUSIONIn order to achieve high bandwidth control under voltage

saturation, this paper proposes a precise model-based designof voltage phase controller for SPMSM. For the model-based design of voltage phase controller, the voltage equationof SPMSM is linearized. Detailed analysis of the proposedprecise plant model shows that the relationship between q-axis current and voltage phase is a non-minimum phasesystem depending on the operating point. The simulationresults and experimental results verified the effectiveness ofthe proposed design method.

In our future works, the authors will apply this proposedtheory to IPMSMs.

ACKNOWLEDGMENTThis research was partly supported by the Ministry of

Education, Culture, Sports, Science and Technology grantNo. 22246057.

REFERENCES

[1] P. Vaclavek and P. Blaha: “Model based High-performance PMSMDrive Control”, SICE Annual Conference 2010, pp.227–232, 2010.

[2] T. Geyer: “Computationally Efficient Model Predictive Direct TorqueControl”, IEEE Trans. Power Electronics, Vol. 26, No. 10, pp. 2804–2816, 2011.

[3] T. Maeda and S. Doki: “Improvement of Torque Control System ofPMSM based on Model Predictive Control”, 37th Annual Conferenceon IEEE Industrial Electronics Society, pp. 1891–1896, 2011.

0 0.1 0.2 0.3−35

−30

−25

−20

−15

−10

−5

0

5

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−5

0

5

10

15

20

25

30

35

Time[s]

Cur

rent

[A]

iqi∗q


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

Time[s]

Mod

ulat

ion

inde

x


0 0.1 0.2 0.3−1.5

−1

−0.5

0

0.5

1

1.5

Time[s]

Vol

tage

pha

se[r

ad]



0 0.1 0.2 0.3−35

−30

−25

−20

−15

−10

−5

0

5

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−5

0

5

10

15

20

25

30

35

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

Time[s]

Mod

ulat

ion

inde

x


0 0.1 0.2 0.3−1.5

−1

−0.5

0

0.5

1

1.5

Time[s]

Vol

tage

pha

se[r

ad]



0 0.1 0.2 0.3−35

−30

−25

−20

−15

−10

−5

0

5

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−5

0

5

10

15

20

25

30

35

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

Time[s]

Mod

ulat

ion

inde

x


0 0.1 0.2 0.3−1.5

−1

−0.5

0

0.5

1

1.5

Time[s]

Vol

tage

pha

se[r

ad]


Fig. 10. Simulation result (1000 [rpm], proposed method).

[4] T. Miyajima, H. Fujimoto, and M. Fujitsuna: “Feedforward controlfor SPMSM with Final State Control Based on Voltage Limit Circlewith Transient Term”, The 2011 IEEE Energy Conversion Congressand Exposition, pp.3913–3919, 2011.

[5] B.-H. Bae and S.-K. Sul: “A Novel Dynamic Overmodulation Strategyfor Fast Torque Control of High-Saliency-Ratio AC Motor”, IEEETrans. Ind. Appl., Vol.41, No.4, pp.1013–1019, 2005.

[6] S. Lerdudomsak, S. Doki, and S. Okuma: “Voltage Limiter CalculationMethod for Fast Torque Response of IPMSM in OvermodulationRange”, The 35th Annual Conference of the IEEE Industrial Elec-tronics Society, pp. 1385–1390, 2009.

[7] K. Kondo, K. Matsuoka, Y. Nakazawa, and H. Shimizu: “Torque feed-back control for salient pole permanent magnet synchronous motor atweakening flux control range”, T.IEEJapan, Vol. 119-D, No. 10, pp.1155–1164, 1999 (in Japanese).

[8] 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”,IEEE Trans. Ind. Appl., Vol. 44, NO. 6, pp. 1722–1723, 2008.

[9] H. Liu, Z. Q. Zhu, E. Mohamed, Y. Fu, and X. Qi: “Flux-WeakeningControl of Nonsalient Pole PMSM Having Large Winding Inductance,Accounting for Resistive Voltage Drop and Inverter Nonlinearities”,IEEE Trans. Power Electronics, Vol. 27, No. 2, pp. 942–952, 2012.

[10] 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.

[11] W. Hatsuse, Y. Notohara, K. Ohi, K. Tobari, K. Tamura, C. Unoko,and Y. Iwaji: “A Stable Field-Weakening Control Using Voltage PhaseOperations in the High-Power Region”, The 2010 International PowerElectronics Conference, pp.599–604, 2010.

[12] Y. Nakazawa, S. Toda I. Yasuoka, and H. Naito: “One-Pulse PWMMode Vector Control for Traction Drives”, IEEE Power Electronics inTransportation, pp. 135–141, 1996.

[13] Y. Nakazawa, S. Toda, and I. Yasuoka: “A New Vector Control for

Induction. Motor Drives in Full Block Mode of Inverters”, T.IEEJapan,Vol. 119-D, No. 9, pp. 1071–1080, 1998 (in Japanese).

[14] J. B. Hag and D. S. Bernstein: “Nonminimum-phase zeros - much todo about nothing - classical control - revisited part II”, IEEE Controlsystem magazine, Vol. 27, No. 3, pp. 45–57, 2006.

[15] K. Ohishi, E. Hayasaka, T. Nagano, and H. Masaya: “High-performance speed servo system considering Voltage saturation of avector-controlled induction motor”, IEEE Trans. Ind. Electron., Vol.5., NO. 3, pp. 795–802, 2006.

0 0.1 0.2 0.3−100

−80

−60

−40

−20

0

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗q


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

Time[s]

Mod

ulat

ion

Inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

Pha

se[r

ad]


Fig. 11. Experimental result (800 [rpm], conventional method 1).

0 0.1 0.2 0.3−100

−80

−60

−40

−20

0

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

Time[s]

Mod

ulat

ion

Inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

Pha

se[r

ad]



0 0.1 0.2 0.3−100

−80

−60

−40

−20

0

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

Time[s]

Mod

ulat

ion

Inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

Pha

se[r

ad]


Fig. 13. Experimental result (800 [rpm], proposed method).

0 0.1 0.2 0.3−100

−80

−60

−40

−20

0

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗q


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

Time[s]

Mod

ulat

ion

Inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

Pha

se[r

ad]


Fig. 14. Experimental result (1000 [rpm], the conventional method).

0 0.1 0.2 0.3−100

−80

−60

−40

−20

0

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

Time[s]

Mod

ulat

ion

Inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

Pha

se[r

ad]



0 0.1 0.2 0.3−100

−80

−60

−40

−20

0

Time[s]

Cur

rent

[A]

idi∗d


0 0.1 0.2 0.3−10

0

10

20

30

40

Time[s]

Cur

rent

[A]

iqi∗qSW


0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

Time[s]

Mod

ulat

ion

Inde

x


0 0.1 0.2 0.3−2

−1

0

1

2

Time[s]

Vol

tage

Pha

se[r

ad]


Fig. 16. Experimental result (1000 [rpm], proposed method).

model-based design of voltage phase controller for...

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