PMSM Rotor Speed and Position Estimator.

UMPEDAC Postgraduate Srudents _{Renewable Energy Symposium (201}

l)

PMSM Rotor

Speed and

Position Estimator

Mohamed

Azmi

Said,

W.p.

Hew,

N.A.

Rahim

IJM Power Energy Dedicated Advanced Centre (UMpEDAC)

Level 4, Engineering Tower, University of Malaya 50603 Kuala Lumpur

azmisaid @siswa.um.edu.my

Abstract

-

Estimation techniques have been developed for flux observer,

rotor

position, and speed. The techniques have a common problem of the offset

in

voltage_sensor _and

current-sensor

outputs

increasing

during

integration,

causing

instability. _{This paper describes uo}

att"-pito

_{overcome the}

problem through estimations of rotor posiiion and speed of a permanent magnet synchronous

motor

(pMSM)

by

three methods:

with

cascaded low pass

filter

(LpF),

with

adaptive sensorless rotor flux _{observer, and with sliding mode observer}

(SMO). Simulation on Matlab/Simulink validaled the proposed

method.

The

advantages

and

drawbacks

of

_each

implementation are discussed.

Keywords - PMSM motor drives, sensorless, cascaded LpF, adaptive rotor _flux obsemer, SMO.

I.

INTRoDUCTION

Permanent

magnet

synchronous

motor (PMSM)

_is

popular for its high efficiency, _{high power density, very low} inertia, and high

reliability. In

_{field_oriented control (FOC)}

or

vector control schemes, continuous rotor information is

mandatory.

To

ensure

a

pMSM,s high

performance and

precise speed

control,

a

position encodei

is

required;

it

senses and _{continuously feeds back both position and speed.}

The position

encoder, _{though, imposes drawbacks}

on

_a system:

it

is

costly,

it

complicates

circuit,

is

sensitive to

noise,

and,

in

a

hostile

environment,

is

fragile [1].

_Its [image:1.588.46.265.541.650.2]

absence

is

thus desired. Figure 1

is

a

block dfagram

of

a sensorless PMSM _{vector control drive.}

Figure 1: Sensorless vector control pMSM drive

_

Direct

torque control

(DTC)

[2] is

less parameter_ dependent, has fast dynamic response, and does not need a

mechanical

rotor position

_sensor

for its

control

loop. Problems exist, though, namely,

drift in

stator_flux_linkage estimation, owing to measurement _{offset error and}

varyiig

stator resistance.

The

stator

flux

linkage

in

DTC

scheme

is

estimated by integrating the difference between input voltage and

voltagl

drop across stator resistance:

)"r=vrt-R,

Rotor

flux

linkage

1o&)

aad

)r(k)at

the

tth

sampting are calculated from stator voltage and current:

)o(k)=1o,0_r*(V,u-r-R"ir)2"

e)

1o&)=

Xsrr,-t*(Verr,t-R,lr)4

(3)

(l)

fi,at

+

l,tt=o

).,1k1

=

(4)

From (5), rotor position

₀

_{canbe calculated as:}

o=tan,(+9)

(5)

\xD(k)

)

Rotor

position

angle

0

depends

on

estimated stator flux, which

is

determined

by

integrating the difference between input voltage and voltage drop across resistance, as Equation

(l)

shows. There is an offset error produced by current and

voltage sensors. When stator

flux is

indirectly

integrated, any _{dc offset error is also integrated. This eventuallyiauses} a large

drift in

stator

flux

_{estimation. The offset error is} non_

periodical

and

unidirectional.

It

can

not be

rectified

by calculation alone.

It

must be overcome

by

some means

of

compensation (see Figure 7 for offset of

flux

locus).

A.

Flw

estimator with a programmable cascaded low_pass

filter

Figure

2 is a

block

diagram

of

the

three_stage

programmable cascaded

LPF

[3].

The

discrete transfer characteristic of an LPF is:

1T

=-

1-,-t

,'C(7-Z-t)

1+r' '

's'

T,

where 7

is

the sampling

time

and ? is

the

time

constant. The corresponding difference equation is:

y(kT,)

=

{;tr"*(kT")

+

ly(kT,

-7"))

(1)

(6)

Y

X

IIMPEDAC Postgraduate Students Renewable Energy Symposium (2011)

X

and

G

arefunctions offrequency or the rotor speed' at steady state, and are calculated as follows:

(8)

(e)

where

n

is the number of filters cascaded' When the signal

is

dc,

C

arrd

G

become

infinite'

thus the

filter

cannot

oerform the

integration. Implementation

of

the

cascaded

[Fi

"ii"*t

,t e

flrix

locus to

rimain

centered on the

origin'

The integration process however, requires knowledge of the

iJ,iJ

rtir"t

flui

position'

If

the initial-position.information

i,

tt

"

.orrt otter

ii

inaccurate' the motor may

initially

rotate

i,

tfr"

*ro.rg

direction. Cascaded LPF depends heavily too or,

-rno,ot

i"qu"r"y. At

zero and

very

low

speeds' its implementation draws a disadvantage'

B.

Adaptive sensorless

rotorflw

observer

Adaptive

sensorless

rotor

flux

observer

can

be

used to

"uf"of*"

rotor

position and

speed

[4]'

From Equation (1)'

,h"

;;;;"t

oi

a PMSM motor

is

given

by

the following statoivoltage and flux linkage equation:

where

0

is

the

rotor

angle'

Electrical

rotor

speed is expressed from mechanical rotor speed:

y

(kT,) -+ ).oor).,

Figure 2: Block diagram of a thrce-stage programmable LPF

[image:2.569.69.548.50.841.2] [image:2.569.320.547.82.329.2]

Figure 3: Signal flow graph of the rotor angle observer

Figure 4: SPeed estimator

Figure

3

shows how this

is

obtained

by

a feedback

of

the estimated error .2,,

,,",

- n*.

For a known magnitr'rde

of

the

permanent

magnet,

lr,uf

=

X* is

used'

The

observer may then be seen as a rotor

field

angle estimator'

At

zero and very low speeds, BEMF gives no information

of

,tL

n"iA"rgf".

i,t"*

principle is introduced to the situation

by

impresslng

a

current

in

the

direction

of

the

default

"rti*ui"a

anlle.

rhe

current

then

forces

the

permanent

;rg*

t"

ai"g,

to that angle; this way, the estimated angle

U""L"t

conlct

only

witli

the error caused

by friction

and load.

Fizure

4

shows the signal

flow

graph

for

the

rotor

speed

esimator.

The

*gGtt)

uro"rt

solves

the

modulus

problem. The algorithm for the estimator then becomes:

+

=v"

-

R,i" +

rrbr,,"r

-

l.rLtu

(16)

i,

=

i,

-

L,i,

(17)

+=v"

-

R,i,

clt

)',=L,ir+1,

^

^

ie

A, =

Lu€'

49=r=p@*""h

(10)

( 11)

(t2)

(13)

Estimation

of 2,

requires

an

open

integration

of

the

voltage equation. The offset contained

in

the inputs hence

makes

the

output

drift.

The offset

is

modeled

as

)

',

'

Estimator for the rotor

flux

is:

oi

= v,

-

R,i,

+

r)ou

(14)

dt

l,

=

lr"it

(15)

where fiouhas to be designed

in

a way that leads

to

a

flux

estimate with constant amplitude

l2.l'

x(kT,) '+ (v* =uo'ioR)o'lva

=u,-

inR)

AmplindeComPensation Gyr(kT")

(so')

tanl

-

I

\,J

a

IIMPEDAC Postgraduate Students Renewable Energy Symposium (2011)

0)=

d0,

;=

^

(20)

C.

Sliding mode observer

In

sliding

mode,

the

observers

are

insensitive

to parameoor variations and disturbance. Sliding mode observer has been presented for robust estimation

ofthe

state variable

of

controlled objects

[5-7].

Compared

with

other methods,

the

observer

is

more robust

to

operating conditions and parameter uncertainties [8].

To

estimate

rotor

position and angular

speed

of

a

PMSM, the motor

is

generally

modeled

in

stationary reference frame. The angular speed and position information are ready to be extracted in this frame.

In

stationary

condition,

aB

reference

frame can

be written as:

with

/, >

**

(r,;,lrrl).

The sliding hyper plane on the stator current

errors

S

=

,

=Vo ;rf

is aefinea according to the

sliding mode observer theory. The system behavior can be examined

by

applying equivalent control method, when the

sliding mode

occurs

after

a

finite time

interval

io=0and

lp=0.

From Equations (21) and(22),

Setting

=

0

glves:

",(,-i)*

ei,-or)

(re)

t-+

o.l

t-trL)

A=l

:

*l''=l-',',i)

lu

-r)

(30)

i*=0,

ip

di_ R. I

I

;=

T'"

T""*T'"

dio

=-R,.-l r.*!r.

dt

LU

LP

LP

eo

= -2oo)"

sin

(4

)

e _o

=

),0a" cos

(4

₎

io

=

o)"eO;

ip

=

A"eo

The sliding

mode

observer uses

only

motor equation. Its main equations are:

di,__R" _1

I

/i\

,=-r,"

Tr"*-stgnud)

dfu R. I

1

/r\

i=-

,,0

Trr*-sign\iB)

*=-lr"-+",+l,ignQ.)

;=

-i,',

--1,"0

*1"*('o)

To

extract

eoand

e,

from the

corresponding equivalent

control values, a

low

pass

filter

is used

with

Zo,

zO

as

filter outputs:

u "n

o

=

(lrsign

(i,))"n

=

" o

u"qo

:

(lrrign

(i,)),,

=

,,

zo(t)

= eoQ) +

A,

(/)

zo1)=eoU)+L"(t)

(2t)

(22) (23) (24)

(2s)

elechical

(26) (27)

(28)

(2e)

(3 1)

(32)

(33)

(34) The motor speed changes slowly,

implying

that

A =

0

,

the model for the induced back EMF is:

where

l,

is

constant observer

gain,

and io

=

io

-

io

,

ip

=

tf

-

lp

arc observer mismatches. Assuming that the motor parameters are identical with those

in

the model, the mismatch dynamics is:

Zp

car. not be used directly. Equations (33) and

(34)

are used for better filtering and estimate the rotor angular speed and position. The observer undertaking the filtering task is:

bo

=

-fo"eo

-

L(A,

- z,)

(35)

bo

=

fo"eo

-

L(ap

-

,)

(36)

ir"

= (0,

-

z,hp

-

@o

-

zpb*

e7)

where

/,

is

a

constant observer gain. Mismatches

in

the back EMF equations are:

where Vo=0o-eo

Ap=Ap-ep

d"

=

6"

-

0)"are observer errors. According to relations

in

(21) and(22),

to

=

-6"0p

+

a"ep

-lr(0r- e")

(38)

io

=

d)"eo

-

@"€o

-

L(oB

-

"r)

(3e)

ir"

=

(ao

-

e,Yp

-(ao

-

"pb-

(40)

The dynamics are distributed

by

the unknown induced

EMF components. However, the back EMF components are

bounded, so they may be suppressed

by

continuous input,

oo

=

-loo)"ri,

(4)

UMPEDAC Postgraduate Students Renewable Energy Symposium (2011)

ao

=

-)"oo)"sin

(4

)

,0"

sin

(4)

uoa

cos(4)

"""beobtainedas:

.ir(ti)=

+?

As Figure

9

shows, adaptive sensorless rotor

flux

observer "un

UJop"t","d

from zero speed. The switch from open-loop

""rt

"f

tL

start up to closed loop can be

full of

noise and

give extreme speed transients. Adaptive sensorless operates

Lo*,".o

speed without changing the control structure'

SMO simulation results show accurate speed and position

"rGuti*.

Figures

10

and

1l

show

the

actual

and

the

"rti*ut"a

anglis. The position angle error

as shown in Figure

l2

is under 0.015p.u, below 5 degrees of electrical angle' (43)

"o'(4)=

+?

@)

The signal

flow

diagram

for

sliding mode observer can be

implemented as shown in Figure 5'

Figure 5: Block Dagram of SMO

TABLEIV.

DATA OF PMSM

Number of Pole

Pairs

P 8

Armature

resistance

R

t.t

o

Magnet flux

linkage

)' 0.28 Wb

d-axis

inductance

La 8.65 mH

o-axis

inductance

Lo 8.65 mH

Inertia

J

0.0051kg.m'

Friction

factor

F

0.0050 N.m.s

II.

RESULTS

The

observer behaviors

were

tested

in

Matlab/Simulink

rofi**"

simulation

tool.

Parameters

of

the

PMSM

model

are as tabulated in Table I.

-0.25

[image:4.540.283.502.129.622.2] [image:4.540.305.495.180.379.2] [image:4.540.291.506.378.616.2]

0.25

LD'wb

Figure 6: Flux locus for ideal measurement of cunent and voltage without dc offset.

-0.25

0.25

lD,w

Figure 7: Flux locus with stator cunent measurement considering & offset'

In

sensorless drive operation, the estimated stator

flux

is used to calculate torque,

flux,

and rotor position' DTC dt'ive

implementation

requires

all

three

observer outputs

as

feJJbacts

to

closed-loop

control.

Meanwhile,

FOC

drive

need position

feedback

to

transform

the

stator

current measuied

from

stationary

reference

aB

frame

to

rotating reference frame

f,q

.

Figure

6

shows the

flux

locus calculated

from

an ideal

inpuritator

current

without

the

-p-resence

of

dc

offset'

In

aitual

implementation,

with

an offset measurement present' there is

u'd.ift io

stator flux locus, caused by the offset error in flux linkage estimation.

Cascaded

LPF

will

compensate

the-offset.error;

see

Figure 8.

A filter

resets the integrator used in estimating the

*tit*

nu*

fnkage. The

flux

locus is seen to remain centered

oo

th"

o.igin.

It-clearly

starts

from

the origin, requiring an

initial

rotoi

position. The observer also depends

on

motor

frequency 6a, requiring the motor to run at some speed flrst before the cascaded LFF can be implemented'

)'o,wb

Cumrt Obcarus

Eq. (24,(28)

LPF

Back EMF

Obsfls

Eq. (36),(37),(38) (41),(42),(43),(44)

rslrn0.)- Z5

)'o,wb

--t!-

R\

Ur

/r'

t/

\

\\

/#

\

-*=

--#

{

UMPEDAC _{Postgraduate Students Renewable Energy Symposium (2011)}

0.5

0.25

0

-0.25

-

0.5

Figure 8: Flux locus calculated by using flux astimator with programrnable cascaded low-pass fi lter

)"a,wb

-0.2

-0.2s

0

0.25

0.5

)'D,Wb

Figure 9: Flux locus calculated based on active rotor flux observer. However, at zero and low speeds, adaptive sensorless requires a

non-,oo

i

"a

reference value.

[image:5.573.18.513.43.836.2]

0.04 0.06 0.08 0.1 0.J2 0..t4 0.16 0.18 0.2 0.22 Tme (s) Figure l0: Actual rotor position angle

0.u 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 cn

Tire (s)

Figure 1 1: SMO estimated position

Figure 12: SMO position estimation error

M.

CoNCTSIoNs

Offset error due to the presence of dc in stator current and voltage measurement

in

PMSM motor drive causes drifts

in

stator-flux calculation, in turn causing inaccurate estimation of rotor angle position. Cascaded l,PF is able to diminish the

drift

but the calculation

is

assumed running at steady state. Adaptive sensorless rotor

flux

observer may be used at zero or low speeds but the current reference must be set to a non-zero value

initially.

SMO can calculate rotor position angle accurately, with low position-error.

ACKNoWLEDGMEI.IT

The

support

of

UM

Power Energy Dedicated Advanced

Centre (UMPEDAC)

is

grarefully

acknowledged. M.A.

Said's

research

study

is

funded

by

Universiti

Teknikal

Malaysia Melaka.

(Text edited by wirani-umpedac@yahoo.com) RepgneNcEs

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