Field-Weakening Control for SPMSM Based onFinal-State Control Considering Voltage Limit

Takayuki Miyajima1, Hiroshi Fujimoto1, and Masami Fujitsuna21 The University of Tokyo, 5-1-5 Kashiwanoha Kashiwa Chiba, Japan2 DENSO CORPORATION, 1-1 Syouwacho Kariya Aichi, Japan

Abstract— SPMSMs (Surface Permanent Magnet SynchronousMotors) are employed for many industrial applications. SPMSMdrive systems should achieve fast torque response and wideoperating range. In this paper, in order to fulfill fast response infield-weakening region, field-weakening control based on final-state control (FSC) is proposed. FSC settles state variables in afinal state from an initial state with feedforward input duringfinite time. Previously, in the single-input single-output system,FSC considering input limit with linear matrix inequality wasproposed. It is adapted to SPMSM which is the two-input two-output system in this paper. Finally, simulations and experimentsare performed to show that the proposed method can achieve fastfield-weakening control.

Index Terms—SPMSM, voltage limit, PWM hold model, final-state control, linear matrix inequality

I. INTRODUCTION

Permanent magnet synchronous motors (PMSMs) arewidely employed in many industries because of high effi-ciency and high power density. Especially, surface PMSMs(SPMSMs) are used as machine tool and electric powersteering. In these applications, SPMSM drive systems aredemanded high speed operation and fast torque responsebut output voltage is limited. Thus, in order to expand theoperating range, field-weakening control applies in the linearrange of inverter.

Under the voltage limit, the manipulated variable is volt-age phase only. However, conventional current vector controldetermines d-axis and q-axis voltage independently but itcannot operate voltage phase directly. Therefore, in field-weakening range, fast torque response is not achieved. Authorsproposed control methods for Interior PMSM (IPMSM) basedperfect tracking control [1] and PWM hold model [2] inovermodulation range [3][4]. These methods improve torqueresponse with feedforward (FF) controller but these methodsoperate d-axis and q-axis voltage independently too. Thus, ifoutput voltage is limited, it is not desired that FF controllersmend response.

Control method which can operate voltage phase directly isnecessary. Many methods for fast torque response under thevoltage limit had been studied in IPMSM. Methods which op-erate voltage phase with the voltage limiter [5][6][7], methodswhich control torque and modulation index by torque loop[8][9], and methods which operate voltage phase directly byfeedback (FB) controller [10][11] were proposed.

In this paper, for the fastest response under voltage limit,a field-weakening control based on final-state control (FSC)

[12] is proposed. In [13], FSC considering input limit withlinear matrix inequality (LMI) was proposed for single-inputsingle-output system. However, in the case of vector control,SPMSM is the two-input two-output system as dq-axis voltageand current. In this paper, field-weakening control is solvedin form of programming program whose evaluation functionand constraint functions are quadratic function and quadraticinequality, respectively. This proposed method determinesvoltage phase directly but the FF input is generated consideringthe voltage limit. Thus, the proposed method takes voltagephase in consideration. In addition, the proposed methodconsiders the voltage limit circle with transient term becauseit consists of PWM hold model.

II. MODEL AND DISCRETIZATION

A. dq Model of SPMSM

The voltage equation of SPMSM in dq-axis is representedin the form of state equation as y = x = [id iq]

T and u =[vd vq]

T by

x(t)=Ac(ωe)x(t)+Bc

u(t)−

[0 ωeKe

]T(1)

y(t)=Ccx(t) (2)[Ac(ωe) Bc

Cc 0

]:=

−RL ωe

−ωe −RL

1L 00 1

L

I 0

(3)

where vd, q are d-axis and q-axis voltage, R is stator windingresistance, L is inductance, ωe is electric angular velocity, id, qare d-axis and q-axis current, and Ke is back EMF constant.

The motor torque T is given by

T = Kmtiq (4)

where Kmt = PKe and P is the number of pole pairs. In thispaper, 2-phase/3-phase transform is 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 isnot suitable. When the pulse is allocated in the center ofcontrol period Tu, the plant in inverter drive system can bediscretized as follows [2].

1

i∗

d[k]

i∗

q[k]

∆Td[k]

∆Tq[k]

iu[k]

iw[k]

θe[k]

uw

dq iq[k]

id[k]

SVM

C[z]

C[z]

−

Decoupling Control

SPMSM+

INVT

∗[k]

Tu

Vdc

Tu

Vdc

Current

Reference

Generator

−

+

+

+

+

+

+

θe[k]+∆θe[k]

Fig. 1. Block diagram of conventional method 1.

d−axis

q−axis

T ∗

Kmt

iq max

iq min

voltage limit circle

Fig. 2. Point at the intersection of voltagelimit circle with constant torque line.

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

x(t) = Acx(t) +Bcu(t), y(t) = Ccx(t). (5)

The precise discrete model in which the input is the ON time∆T [k] is obtained as

x[k + 1] = Asx[k] +Bs∆T [k], y[k] = Csx[k], (6)

As := eAcTu , Bs := eAcTu/2BcE, Cs := Cc. (7)

In derivation of (6) and (7), the voltage amplitude is consid-ered +E only. Therefore, if ∆T [k] is negative, the voltageamplitude is −E.

C. PWM Hold Model of SPMSM

(1) is discretized based on PWM hold. Here, it is assumedthat the speed variation during one control period can beneglected. Hence, the back EMF ωeKe can be presumed as thezero-order hold. Therefore, the PWM hold model of SPMSMis designed as E = Vdc in (5) as follows:

x[k + 1] = As(ωe)x[k] +Bs(ωe)∆T [k]

−Bs2(ωe)[0 ωeKe

]T, (8)

y[k] = Csx[k], (9)

As(ωe) := eAc(ωe)Tu , Bs(ωe) := eAc(ωe)Tu2 BcVdc,

Bs2(ωe) := A−1c (ωe)

(eAc(ωe)Tu − I

)Bc, (10)

where Vdc is dc-bus voltage of three-phase inverter, ∆T =[∆Td ∆Tq]

T , and ∆Td, q are d-axis and q-axis ON time.As(ωe), Bs(ωe), and Bs2(ωe) are functions of ωe. Thus,these functions are calculated again when rotor speed ischanged.

Here, in SPMSM drive systems with three-phase inverter,input of three-phase can be generated based on space vectormodulation (SVM) [15].

III. CONTROL SYSTEM DESIGN

A. Conventional Method 1

Fig. 1 shows the block diagram of conventional method 1.The decoupling control which is given by (11), (12) is applied.

vrefd [k] = v′dref

[k]− ωe[k]Lqiq[k] (11)

vrefq [k] = v′qref

[k] + ωe[k](Ldid[k] +Ke) (12)

The current PI controller is designed with pole-zero cancella-tion as follows:

C(s) =Ls+R

τs, τ = 10Tu (13)

By discretizing (13), C[z] is obtained by Tustin transformationwith control period Tu. Here, if Va :=

√v2d + v2q > Vmax (Va

is the voltage amplitude and Vmax :=√

32Vdc√

3is the maximum

value of voltage amplitude.), the integrator of C[z] is stopped.Current reference generator generates d-axis and q-axis

current references i∗d and i∗q from the point at the intersectionof voltage limit circle with constant torque line to make currentamplitude minimum as Fig. 2. Here, T ∗ is torque reference.The generation of current references is represented by

i∗d[k] =−ω2

eKeL

R2 + ω2eL

2+

√(iqmax − i∗q)(i

∗q−iqmin), (14)

i∗q [k] =

iqmax if T ∗[k] > Kmtiqmax

iqmin elseif T ∗[k] < KmtiqminT∗[k]Kmt

otherwise, (15)

iqmax := −ωeKeR− Va

√R2 + ω2

eL2

R2 + ω2eL

2, (16)

iqmin := −ωeKeR+ Va

√R2 + ω2

eL2

R2 + ω2eL

2. (17)

If i∗d[k] strengthens the field flux, i∗d[k] = 0.In SVM, ∆θe(= 0.5ωeTu) is added to θe of the dq/2-phase

transform for sampling error compensation [16]. The ON timelimit in SVM is described by

∆T [k] =

∆T [k]

|∆T [k]|∆Tmax if |∆T [k]| > ∆Tmax

∆T [k] otherwise, (18)

where ∆Tmax

(:=

√32Tu√3

)is the maximum value of dq-axis

ON time vector amplitude and ∆T [k] is the limiter output. Inthis paper, all limiters calculate in the same way as (18).

B. Conventional Method 2

The block diagram of conventional method 2 is shown inFig. 3. It consists of the 2-DOF control system.

The inverse system of (8) is obtained as

∆T ff [k] =−B−1s (ωe)As(ωe)x[k]+B−1

s (ωe)xd[k + 1]

+B−1s (ωe)Bs2(ωe)

[0 ωeKe

]T. (19)

2

i∗[k]

C2[z]

+

+

+

i[k]e[k]

−

∆T ff [k]

x[k]

uw

dq S

(Tu)

(SVM)HPWM

θe[k]Tu

Vdc

I +∆θe[k]

θe[k]Pn[z]

Current

Reference

Generator

T ∗[k]−B

−1

s(ωe)As(ωe)x[k] + B

−1

s(ωe)i

∗[k]

∆T ff [k]

SPMSM

INV+

∆T [k]C1[z]

+B−1

s(ωe)Bs2(ωe) [0 ωeKe]

T

Fig. 3. Block diagram of conventional method 2.

i∗[k]

C2[z]

+

+

+

i[k]e[k]−

∆T ff [k]

x[k]

uw

dq S

(Tu)

(SVM)HPWM

θe[k]Tu

Vdc

I +∆θe[k]

θe[k]Pn[z]

Current

Reference

Generator

T ∗[k]

∆T ff [k]

SPMSM

INV+

∆T [k]C1[z]

Fig. 4. Block diagram of proposed method.

From this stable inverse system, the FF controller C1[z] isdesigned. Therefore, if the plant is nominal and the input isnot limited, C1[z] assures the perfect tracking at the sampletime. Here, x[k] := [id[k] iq[k]]

T in Fig. 3 is the nominaloutput which takes the input limit in consideration. If the inputis limited, x[k] do not equal to the current reference delayedone sample i∗[k − 1].

The FB controller C2[z] suppresses the error e[k] betweenx[k] and i[k]. Here, C2[z] is same as the FB controller ofconventional method 1. However, in conventional method 2,anti-windup control is not necessary because x[k] considersthe input limit.

C. Proposed Method

Fig. 4 and 5 show the block diagrams of the proposedmethod and FF controller C1[z], respectively. The proposedmethod has two FF controllers. In steady state and transientstate which does not cause the voltage limit, the FF controllerwhich is represented by (19) applies. In the other state, the FFinput is generated with FSC trajectory generator. The proposedmethod switches two types of FF controller but the inversesystem is FSC as prescribed time interval is one. Therefore,the stability is assured.

FSC transits an initial state to a final state by prescribedtime interval. Here, FSC considering the voltage limit is ex-pressed as a programming program whose evaluation functionand constraint functions are quadratic function and quadraticinequality in the form of LMI. The FF input called FSCtrajectory is generated by solving these LMIs.

In PWM hold model of SPMSM, a final state x[N ] is

i∗[k]

FSC Trajectory Generator

∆T ff [k]

C1[z]

−B−1

s(ωe)As(ωe)x[k] + B

−1

s(ωe)i

∗[k]

Inverse system

+B−1

s(ωe)Bs2(ωe) [0 ωeKe]

T

Fig. 5. Feedforward controller C1[z] of proposed method.

represented with an initial state x[0] as follows:

Y = ΣU (20)

Y := x[N ]−ANs x[0]−Σ2UEMF (21)

Σ :=[AN−1

s Bs AN−2s Bs · · · Bs

](22)

Σ2 :=[AN−1

s Bs2 AN−2s Bs2 · · · Bs2

](23)

U :=[∆T T [0] ∆T T [1] · · ·∆T T [N − 1]

]T(24)

UEMF := −ωeKe

[0 1 · · · 0 1

]T(25)

Here, it is assumed that ωe is constant in this derivation.The FF input U which satisfies (20) is not determined

uniquely. Thereupon, the square sum of the current error isminimized. The evaluation function J is represented by

J = ETQE, Q > 0, (26)

E := [eT [1] eT [2] · · · eT [N ]]T , (27)

e[k] := i∗ − x[k] =[i∗d i∗q

]T − x[k]. (28)

From

A :=[As A2

s · · ·ANs

]T, (29)

B :=

Bs 0 · · · 0

AsBs Bs · · · 0...

.... . .

...AN−1

s Bs AN−2s Bs · · · Bs

, (30)

B2 :=

Bs2 0 · · · 0

AsBs2 Bs2 · · · 0...

.... . .

...AN−1

s Bs2 AN−2s Bs2 · · · Bs2

, (31)

I∗ :=[(i∗)

T(i∗)

T · · · (i∗)T]T

, (32)

E is given by

E = I∗ −Ax[0]−BU −B2UEMF . (33)

Moreover, (8) is controllable. Thus, Σ is full row rank. There,Σ⊥ ∈ R2N×(2N−2) and Σ† ∈ R2N×2 that fulfill ΣΣ⊥ = 0and ΣΣ† = I are defined. In addition, U is obtained by

U := [Σ† Σ⊥]U . (34)

3

By assigning (34) to (20), Y = [I 0]U is given. Therefore,U is represented by

U =[Y q

]T, (35)

where q ∈ R(2N−2)×1 is a free parameter.By substituting (33), (34), and (35) to (26), the evaluation

function J is transformed as follows:

J = R(q) + ST (q)QS(q), (36)

Z := I∗ −Ax[0]−BΣ†Y −B2UEMF , (37)

R(q) := ZTQZ − 2ZTQS(q), (38)

S(q) := BΣ⊥q. (39)

With LMI, the condition which satisfies J < γ for providedγ is given by (40) [13].[

γ −R(q) S(q)T

S(q) Q−1

]> 0. (40)

Next, the voltage limit is described with LMI. Voltage limitsat each sampling points are described as

∆T T [k]∆T [k] = ∆T 2d [k] + ∆T 2

q [k]

≤ ∆T 2max (k = 0, 1, · · · , N − 1). (41)

Here, g(i) ∈ R2×2N (i := 2k + 1) of which (1, i) th and (2,i+ 1) th entries are 1 and other entries are 0 is defined. g(i)separates input vector ∆T [k] as follows:

∆T [k] = g(i)U(q). (42)

With g(i), (41) is transformed as

g(i)U(q)T g(i)U(q) ≤ ∆T 2max. (43)

(43) is expressed with N LMIs by[∆T 2

max U(q)Tg(i)T

g(i)U(q) I

]≥ 0, (44)

where i = 2k + 1, k = 0, 1, · · · , N − 1.By minimizing γ under (40) and (44), the FF input which

drives an initial state x[0] to a final state x[N ] within theprescribed time interval N for PWM hold model of SPMSMand minimizes square sum of current error to fulfill voltagelimit is obtained. Moreover, the proposed method takes volt-age limit circle with transient term in consideration exactlybecause PWM hold model treats transient term as difference.

IV. SIMULATION

The parameters of SPMSM 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].

Fig. 6 shows simulation results when step torque referencewas given and the rotor velocity was 1000 [rpm]. Here, thevoltage amplitude of current reference generator was set to be0.98Vmax. The dq-axis ON time vector amplitude ∆Ta and the

TABLE IPARAMETERS OF SPMSM

Inductance L 1.80 [mH]Resistance R 0.157 [Ω]

Pairs of poles P 4Back EMF constant Ke 74.6 [mV/(rad/s)]

dq-axis ON time vector phase (voltage phase) δ are representedby

∆Ta =√∆T 2

d +∆T 2q , (45)

δ = tan−1 −∆Td

∆Tq. (46)

Dotted lines in Fig. 6(c), 6(g), and 6(k) are described ∆Tmax.Conventional method 1 cannot operate the voltage phase

because it consists of current vector control. Therefore, thevoltage phase variation was slow and fast response was notachieved. In addition, when the voltage amplitude was not lim-ited, current response still is slow because anti-windup controlmade closed-loop discontinuous and conventional method 1 isonly composed with FB controllers. Conventional method 2uses a FF controller and does not apply anti-windup controlbecause nominal output takes voltage limit in consideration.Thus, it controlled faster than conventional method 1 but itcannot operate voltage phase. Therefore, by operating voltagephase, settling time should be able to be shortened.

In the proposed method, Q = I and the FSC trajectory wasmade to be the shortest trajectory. The prescribed time intervalwas N = 40. During the beginning of response, variation of d-axis current is larger by decreasing q-axis current. This maded-axis current lager than d-axis current reference quickly.Furthermore, excess d-axis current made variation of q-axiscurrent large. Finally, the proposed method achieved the fastestresponse and the proposed method shortened the settlingtime a half as short as conventional method 2. In general,this operation is executed on experiential grounds or by FBcontroller but the proposed method optimally operates withfinal-state control.

V. EXPERIMENT

Experimental results are shown in Fig. 7. Experimentswere performed in the same condition as simulations. Theparameters of SPMSM and the control period were same too.From a point of view of velocity control by load motor, Vdc

was set to be 80 [V]. In the controller calculation, the nominalthree-phase inverter dc-bus voltage was Vdcn = 36 [V] and theinput was multiplied by Vdcn/Vdc to compensate the differencebetween Vdc and Vdcn in SVM. ∆θ in experiments is 0.9ωeTu

to consider the computational time 0.4Tu.Variations of d-axis reference in all experimental results are

due to speed variation. In conventional method 1, the voltagephase variation was slow and anti-windup control deterioratescurrent response. Conventional method 2 improved the settel-ing time a little by FF controller.

4

0 10 20 30 40−25

−20

−15

−10

−5

Time [ms]

Cur

rent

[A]

id

id*

(a) id (Conventional 1)

0 10 20 30 400

2

4

6

8

10

12

14

Time [ms]

Cur

rent

[A]

iq

iq*

(b) iq (Conventional 1)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

Am

plitu

de o

f inp

ut [m

s]

(c) ∆Ta (Conventional 1)

0 10 20 30 40−0.4

0

0.4

0.8

1.2

Time [ms]

Ang

le o

f Inp

ut [r

ad]

(d) δ (Conventional1)

0 10 20 30 40−25

−20

−15

−10

−5

Time [ms]

Cur

rent

[A]

id

id*

(e) id (Conventional 2)

0 10 20 30 400

2

4

6

8

10

12

14

Time [ms]

Cur

rent

[A]

iq

iq*

(f) iq (Conventional 2)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

Am

plitu

de o

f inp

ut [m

s]

(g) ∆Ta (Conventional 2)

0 10 20 30 40−0.4

0

0.4

0.8

1.2

Time [ms]

Ang

le o

f Inp

ut [r

ad]

(h) δ (Conventional 2)

0 10 20 30 40−25

−20

−15

−10

−5

Time [ms]

Cur

rent

[A]

id

id*

(i) id (Proposed)

0 10 20 30 400

2

4

6

8

10

12

14

Time [ms]

Cur

rent

[A]

iq

iq*

(j) iq (Proposed)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

Am

plitu

de o

f inp

ut [m

s]

(k) ∆Ta (Proposed)

0 10 20 30 40−0.4

0

0.4

0.8

1.2

Time [ms]A

ngle

of I

nput

[rad

]

(l) δ (Proposed)

Fig. 6. Simulation results.

In the proposed method, previously, the FSC trajectorywhose the prescribed time interval is N = 40 was calculatedoff-line. When torque reference was obtained, the FSC trajec-tory was output as FF input. If the response is not settled,FF input was generated from (19). When torque referencewas obtained, q-axis current was decreased. Therefore, duringthe beginning of response, q-axis current raised the slowestresponse. However, this made larger variation of d-axis cur-rent. Therefore, q-axis response was fast by excess d-axiscurrent. d-axis and q-axis current were not settled within theprescribed time interval because of speed variation but theproposed method achieved the fastest response.

VI. CONCLUSION

Field-weakening control for SPMSM based final-state con-trol considering voltage limit was proposed in this paper.The FF controller of proposed method is composed withPWM hold model. Therefore, the proposed method can takevoltage limit circle with transient term in consideration strictly.Simulations and experiments were performed to show thatcurrent responses remarkably are deteriorated on voltage limitin conventional methods. On the other hand, the proposed

method can achieve fast response on voltage limit becausefeedforward input operates voltage phase quickly.

In our future paper, current response of PMSM on voltagelimit will be analyzed. In this paper, prescribed time interval Nwas provided previously and FF input was calculated off-line.In our future work, online calculation will be realized.

REFERENCES

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[2] 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.

[3] T. Miyajima, H. Fujimoto, and M. Fujitsuna: “Control Method forIPMSM Based on Perfect Tracking Control and PWM Hold Modelin Overmodulation Range”, The 2010 International Power ElectronicsConference, pp.593-598, 2010.

[4] T. Miyajima, H. Fujimoto, and M. Fujitsuna: “Control Method forIPMSM Based on PTC and PWM Hold Model in OvermodulationRange -Study on Robustness and Comparison with Anti-WindupControl-”, The 2010 IEEE Energy Conversion Congress and Exposi-tion, pp.2844–2850, 2010.

[5] J.-K. Seok, J.-S. Kim, and S.-K. Sul: “Overmodulation Strategy forHigh-Performance Torque Control”, IEEE Trans. Power Electronics,Vol.13, No.4, pp.786–792, 1998.

5

0 10 20 30 40−25

−20

−15

−10

−5

Time [ms]

Cur

rent

[A]

id

id*

(a) id (Conventional 1)

0 10 20 30 400

5

10

15

Time [ms]

Cur

rent

[A]

iq

iq*

(b) iq (Conventional 1)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

Am

plitu

de o

f inp

ut [m

s]

(c) ∆Ta (Conventional 1)

0 10 20 30 40−0.4

0

0.4

0.8

1.2

Time [ms]

Pha

se o

f inp

ut [r

ad]

(d) δ (Conventional 1)

0 10 20 30 40−25

−20

−15

−10

−5

Time [ms]

Cur

rent

[A]

id

id*

(e) id (Conventional 2)

0 10 20 30 400

5

10

15

Time [ms]

Cur

rent

[A]

iq

iq*

(f) iq (Conventional 2)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

Am

plitu

de o

f inp

ut [m

s]

(g) ∆Ta (Conventional 2)

0 10 20 30 40−0.4

0

0.4

0.8

1.2

Time [ms]

Pha

se o

f inp

ut [r

ad]

(h) δ (Conventional 2)

0 10 20 30 40−25

−20

−15

−10

−5

Time [ms]

Cur

rent

[A]

id

id*

(i) id (Proposed)

0 10 20 30 400

5

10

15

Time [ms]

Cur

rent

[A]

iq

iq*

(j) iq (Proposed)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

Am

plitu

de o

f inp

ut [m

s]

(k) ∆Ta (Proposed)

0 10 20 30 40−0.4

0

0.4

0.8

1.2

Time [ms]P

hase

of i

nput

[rad

]

(l) δ (Proposed)

Fig. 7. Experimental results.

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