field-weakening control for spmsm

8/22/2019 Field-Weakening Control for SPMSM

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Field-Weakening Control for SPMSM Based on

Final-State Control Considering Voltage LimitTakayuki Miyajima1, Hiroshi Fujimoto1, and Masami Fujitsuna2

1 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 TermsSPMSM, voltage limit, PWM hold model, final-

state control, linear matrix inequality

I. INTRODUCTION

Permanent magnet synchronous motors (PMSMs) are

widely employed in many industries because of high effi-

ciency and high power density. Especially, surface PMSMs

(SPMSMs) are used as machine tool and electric power

steering. In these applications, SPMSM drive systems are

demanded high speed operation and fast torque response

but output voltage is limited. Thus, in order to expand the

operating range, field-weakening control applies in the linear

range of inverter.

Under the voltage limit, the manipulated variable is volt-

age phase only. However, conventional current vector control

determines d-axis and q-axis voltage independently but itcannot operate voltage phase directly. Therefore, in field-

weakening range, fast torque response is not achieved. Authors

proposed control methods for Interior PMSM (IPMSM) basedperfect tracking control [1] and PWM hold model [2] in

overmodulation range [3][4]. These methods improve torque

response with feedforward (FF) controller but these methods

operate d-axis and q-axis voltage independently too. Thus, ifoutput voltage is limited, it is not desired that FF controllers

mend response.

Control method which can operate voltage phase directly is

necessary. Many methods for fast torque response under the

voltage limit had been studied in IPMSM. Methods which op-

erate voltage phase with the voltage limiter [5][6][7], methods

which 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 with

linear matrix inequality (LMI) was proposed for single-input

single-output system. However, in the case of vector control,

SPMSM is the two-input two-output system as dq-axis voltage

and current. In this paper, field-weakening control is solvedin form of programming program whose evaluation function

and constraint functions are quadratic function and quadratic

inequality, respectively. This proposed method determines

voltage phase directly but the FF input is generated considering

the voltage limit. Thus, the proposed method takes voltage

phase in consideration. In addition, the proposed method

considers the voltage limit circle with transient term because

it 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(e)x(t)+ Bc

u(t)

[0 eKe

]T}(1)

y(t)= Ccx(t) (2)Ac(e) Bc

Cc 0

:=

RL ee RL

1L 00 1L

I 0

(3)

where vd, q are d-axis and q-axis voltage, R is stator windingresistance, L is inductance, e is electric angular velocity, id, q

are d-axis and q-axis current, and Ke is back EMF constant.The motor torque T is given by

T = Kmtiq (4)

where Kmt = P Ke 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 generally

applied. However, in the case of single-phase inverter, it

cannot output arbitrary voltage but only 0 or E (E is dc-bus voltage of single-phase inverter). Therefore, in order to

control instantaneous values precisely, the zero-order hold isnot suitable. When the pulse is allocated in the center of

control period Tu, the plant in inverter drive system can bediscretized as follows [2].

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i

d[k]

i

q[k]

Td[k]

Tq[k]

iu[k]

iw[k]

e[k]

uw

dq iq[k]

id[k]SVM

C[z]

C[z]

Decoupling Control

SPMSM+

INVT[k]

Tu

Vdc

Tu

Vdc

Current

Reference

Generator

+

+

+

+

+

+

e[k]+e[k]

Fig. 1. Block diagram of conventional method 1.

daxis

qaxis

T

Kmt

iqmax

iqmin

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] + BsT[k], y[k] = Csx[k], (6)

As := eAcTu, Bs := e

AcTu/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 assumed

that the speed variation during one control period can be

neglected. Hence, the back EMF eKe can be presumed as thezero-order hold. Therefore, the PWM hold model of SPMSM

is designed as E = Vdc in (5) as follows:

x[k + 1] = As(e)x[k] + Bs(e)T[k]

Bs2(e)[

0 eKe]T

, (8)

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

As(e) := eAc(e)Tu , Bs(e) := e

Ac(e)Tu2 BcVdc,

Bs2(e) := A1c (e)

(eAc(e)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(e), Bs(e), and Bs2(e) are functions of e. Thus,these functions are calculated again when rotor speed is

changed.

Here, in SPMSM drive systems with three-phase inverter,

input of three-phase can be generated based on space vector

modulation (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] = vd

ref[k] e[k]Lqiq[k] (11)

vrefq [k] = vq

ref[k] + e[k](Ldid[k] + Ke) (12)

The current PI controller is designed with pole-zero cancella-

tion as follows:

C(s) =Ls + R

s, = 10Tu (13)

By discretizing (13), C[z] is obtained by Tustin transformation

with control period Tu. Here, if Va :=

v2d + v2q > Vmax (Va

is the voltage amplitude and Vmax :=

32

Vdc3

is the maximum

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

current references id and iq from the point at the intersection

of voltage limit circle with constant torque line to make current

amplitude minimum as Fig. 2. Here, T is torque reference.The generation of current references is represented by

id[k] =2e KeL

R2 + 2e

L2+ (iq max iq )(iq iq min), (14)

iq [k] =

iq max if T[k] > Kmtiq max

iq min elseif T[k] < Kmtiq min

T[k]Kmt

otherwise, (15)

iq max := eKeR Va

R2 + 2e L

2

R2 + 2e L2

, (16)

iq min := eKeR + Va

R2 + 2e L

2

R2 + 2e L2

. (17)

If id[k] strengthens the field flux, id[k] = 0.

In SVM, e(= 0.5eTu) is added to e of the dq/2-phasetransform for sampling error compensation [16]. The ON time

limit in SVM is described by

T[k] =

{T[k]

|T[k]|Tmax if |T[k]| > Tmax

T[k] otherwise, (18)

where Tmax

(:=

32

Tu3

)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 in

Fig. 3. It consists of the 2-DOF control system.

The inverse system of (8) is obtained as

Tff[k] = B1s (e)As(e)x[k]+ B

1s (e)xd[k + 1]

+ B1s (e)Bs2(e)[

0 eKe]T

. (19)

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i[k]

C2[z]

+

+

+

i[k]e[k]

Tff[k]

x[k]

uw

dq S(Tu)

(SVM)HPWM

e[k]

Tu

VdcI +e[k]

e[k]Pn[z]

Current

Reference

Generator

T[k]B

1s

(e)As(e)x[k] + B1s

(e)i[k]

Tff[k]

SPMSM

INV+

T[k]C1[z]

+B1s

(e)Bs2(e) [0 eKe]T

Fig. 3. Block diagram of conventional method 2.

i[k]

C2[z]

+

+

+

i[k]e[k]

Tff[k]

x[k]

uw

dq S(Tu)

(SVM)HPWM

e[k]Tu

VdcI +e[k]

e[k]Pn[z]

Current

Reference

Generator

T[k]

Tff[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 is

not limited, C1[z] assures the perfect tracking at the sampletime. Here, x[k] := [id[k] iq [k]]T in Fig. 3 is the nominal

output 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 transient

state which does not cause the voltage limit, the FF controller

which is represented by (19) applies. In the other state, the FF

input is generated with FSC trajectory generator. The proposed

method switches two types of FF controller but the inverse

system is FSC as prescribed time interval is one. Therefore,

the stability is assured.

FSC transits an initial state to a final state by prescribed

time interval. Here, FSC considering the voltage limit is ex-

pressed as a programming program whose evaluation function

and constraint functions are quadratic function and quadratic

inequality in the form of LMI. The FF input called FSC

trajectory is generated by solving these LMIs.

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

i[k]

FSC Trajectory Generator

Tff[k]

C1[z]

B1

s (e)As(e)x[k] + B1

s (e)i[k]

Inverse system

+B1s

(e)Bs2(e) [0 eKe]T

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

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

Y = U (20)

Y := x[N] ANs x[0] 2UEM F (21)

:=

AN1

s Bs AN2

s Bs Bs

(22)

2 :=

AN1s Bs2 AN2s Bs2 Bs2

(23)

U :=

TT[0] TT[1] TT[N 1]T

(24)

UEM F := eKe[

0 1 0 1]T

(25)

Here, it is assumed that e is constant in this derivation.

The FF input U which satisfies (20) is not determined

uniquely. Thereupon, the square sum of the current error is

minimized. 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] =[

id iq

]T x[k]. (28)

From

A :=

As A2s A

Ns

T, (29)

B :=

Bs 0 0AsBs Bs 0

......

. . ....

AN1s Bs AN2s Bs Bs

, (30)

B2 :=

Bs2 0 0

AsBs2 Bs2 0...

.... . .

...

AN1s Bs2 AN2s Bs2 Bs2

, (31)

I :=

(i)T (i)T (i)TT

, (32)

E is given by

E = I Ax[0] BU B2UEM F. (33)

Moreover, (8) is controllable. Thus, is full row rank. There,

R2N(2N2) and R2N2 that fulfill = 0

and = I are defined. In addition, U is obtained by

U := [ ]U. (34)

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

U =[

Y q]

T , (35)

where q R(2N2)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] BY B2UEM F, (37)

R(q) := ZTQZ 2ZTQS(q), (38)

S(q) := Bq. (39)

With LMI, the condition which satisfies J < for provided

is given by (40) [13]. R(q) S(q)T

S(q) Q1

> 0. (40)

Next, the voltage limit is described with LMI. Voltage limits

at each sampling points are described as

TT[k]T[k] = T2d [k] + T2q [k]

T2max (k = 0, 1, , N 1). (41)

Here, g(i) R22N (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)} T2max. (43)

(43) is expressed with N LMIs byT2max U(q)

Tg(i)T

g(i)U(q) I

0, (44)

where i = 2k + 1, k = 0, 1, , N 1.

By minimizing under (40) and (44), the FF input whichdrives 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 voltage

limit is obtained. Moreover, the proposed method takes volt-

age limit circle with transient term in consideration exactly

because PWM hold model treats transient term as difference.

IV. SIMULATION

The parameters of SPMSM for the simulation are shown in

Table 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, the

voltage amplitude of current reference generator was set to be

0.98Vmax. The dq-axis ON time vector amplitude Ta and the

TABLE I

PARAMETERS OF SPMSM

Inductance L 1.80 [mH]

ResistanceR

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

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

Ta =

T2d + T2

q , (45)

= tan1TdTq

. (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, the

voltage phase variation was slow and fast response was not

achieved. In addition, when the voltage amplitude was not lim-

ited, current response still is slow because anti-windup control

made closed-loop discontinuous and conventional method 1 is

only composed with FB controllers. Conventional method 2

uses a FF controller and does not apply anti-windup control

because nominal output takes voltage limit in consideration.

Thus, it controlled faster than conventional method 1 but it

cannot operate voltage phase. Therefore, by operating voltage

phase, 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 interval

was 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 fastest

response and the proposed method shortened the settling

time a half as short as conventional method 2. In general,

this operation is executed on experiential grounds or by FB

controller but the proposed method optimally operates with

final-state control.

V. EXPERIMENT

Experimental results are shown in Fig. 7. Experiments

were performed in the same condition as simulations. The

parameters of SPMSM and the control period were same too.

From a point of view of velocity control by load motor, Vdcwas set to be 80 [V]. In the controller calculation, the nominal

three-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. in experiments is 0.9eTuto 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 deteriorates

current response. Conventional method 2 improved the settel-

ing time a little by FF controller.

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0 10 20 30 4025

20

15

10

5

Time [ms]

Current[A]

id

id

*

(a) id (Conventional 1)

0 10 20 30 400

2

4

6

8

10

12

14

Time [ms]

Current[A]

iq

iq

*

(b) iq (Conventional 1)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

Amplitudeofinput[ms]

(c) Ta (Conventional 1)

0 10 20 30 400.4

0

0.4

0.8

1.2

Time [ms]

Angle

ofInp

ut[rad]

(d) (Conventional1)

0 10 20 30 4025

20

15

10

5

Time [ms]

Current[A]

id

id

*

(e) id (Conventional 2)

0 10 20 30 400

2

4

6

8

10

12

14

Time [ms]

Current[A]

iqiq

*

(f) iq (Conventional 2)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]

A

mplitudeofinput[ms]

(g) Ta (Conventional 2)

0 10 20 30 400.4

0

0.4

0.8

1.2

Time [ms]

AngleofInput[rad]

(h) (Conventional 2)

0 10 20 30 4025

20

15

10

5

Time [ms]

Current[A]

id

id

*

(i) id (Proposed)

0 10 20 30 400

2

4

6

8

10

12

14

Time [ms]

Current[A]

iq

iq

*

(j) iq (Proposed)

0 10 20 30 400.06

0.065

0.07

0.075

0.08

Time [ms]


(k) Ta (Proposed)

0 10 20 30 400.4

0

0.4

0.8

1.2

Time [ms]

AngleofInput[rad]

(l) (Proposed)

Fig. 6. Simulation results.

In the proposed method, previously, the FSC trajectory

whose 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 reference

was 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 the

proposed 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 with

PWM hold model. Therefore, the proposed method can take

voltage limit circle with transient term in consideration strictly.Simulations and experiments were performed to show that

current responses remarkably are deteriorated on voltage limit

in conventional methods. On the other hand, the proposed

method can achieve fast response on voltage limit because

feedforward input operates voltage phase quickly.

In our future paper, current response of PMSM on voltage

limit 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.

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0 10 20 30 4025

20

15

10

5

Time [ms]

Current[A]

id

id

*

(a) id (Conventional 1)

0 10 20 30 400

5

10

15

Time [ms]

Current[A]

iq

iq

*

(b) iq (Conventional 1)

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(c) Ta (Conventional 1)

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(e) id (Conventional 2)

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Time [ms]

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(f) iq (Conventional 2)

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0.08

Time [ms]

A

mplitudeofinput[ms]

(g) Ta (Conventional 2)

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0.4

0.8

1.2

Time [ms]

Phase

ofinput[rad]

(h) (Conventional 2)

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Time [ms]

Current[A]

id

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Time [ms]

Current[A]

iq

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(j) iq (Proposed)

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Time [ms]


(k) Ta (Proposed)

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0

0.4

0.8

1.2

Time [ms]

Phase

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(l) (Proposed)

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field-weakening control for spmsm

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