Speed and Position Estimation for PM Speed and Position Estimation for PM nc

en

ce

CE

CE Speed and Position Estimation for PM Speed and Position Estimation for PM

PF-014893

Speed and Position Estimation for PM Speed and Position Estimation for PM Synchronous MotorSynchronous Motor usingusing

S lfS lf C t d B kC t d B k EMF ObEMF Ob Co

nfe

ren

Co

nfe

ren

is, F

RA

NC

is, F

RA

NC Speed and Position Estimation for PM Speed and Position Estimation for PM

Synchronous MotorSynchronous Motor usingusingS lfS lf C t d B kC t d B k EMF ObEMF ObSelfSelf--Compensated BackCompensated Back--EMF ObserversEMF Observers

ec

tro

nic

s e

ctr

on

ics

2006

20

06 ––

Par

iP

ari

Marco TURSINI Marco TURSINI Roberto PETRELLARoberto PETRELLA Alessia SCAFATI Alessia SCAFATI

SelfSelf--Compensated BackCompensated Back--EMF ObserversEMF Observers

ust

ria

l Ele

ust

ria

l Ele

ber

6b

er 6

--10,

210

, 2 Marco TURSINI, Marco TURSINI, Roberto PETRELLARoberto PETRELLA, Alessia SCAFATI, Alessia SCAFATI

University of L’AquilaUniversity of L’Aquila

EE

E In

du

EE

E In

du

Nov

emb

Nov

emb University of L AquilaUniversity of L Aquila

Dept. of Electrical and Information Engineering (DEIE)Dept. of Electrical and Information Engineering (DEIE)

University of UdineUniversity of Udine

32

32

nd

nd

II

SS16SS16 1 1 Ad d t l d b ti f AC d i

University of UdineUniversity of UdineDept. of Electrical, Managmt and Mechanical Eng. (DIEGM)Dept. of Electrical, Managmt and Mechanical Eng. (DIEGM)

Thursday, November 9Thursday, November 9thth, 2006, 8:00 AM, 2006, 8:00 AM

SS16SS16--1 1 Advanced control and observation of AC drives

PF-014893

Aim of the paperAim of the paperAim of the paperAim of the paper

- provide full analytical description of previously-proposed speed andposition observers for PM synchronous motors, based on back-EMFestimation

- calculate the (steady-state) rotor position estimation errorss

- propose a real-time compensation strategy for rotor position estimationerror in a transducer-less PMSM drive

- validate analytical results by means of simulation analysis and comparey y y pthem with experiments

PF-014893

TransducerTransducer--less drive schemeless drive scheme*αv

*qv

*qi

*rω

SVRiqRvel dq*dv

*βv PWM

Rid αβ

0=*di

dv βv

dq αβqi αiib

iaˆ

ˆ

Luenberger/Sliding Mode

αv

v

αi

iαβ 3βidi

ibic

ˆ

rsinθθ

αˆiv βˆivrθ *

gObserverβv βi

Observerrω

* * i i PMSM

rcosθ

KalmanFilter

)(θ 1rsin

)(θ 1rcos

αv βv αi βi PMSMFilter)(ω 1r

⎟⎠⎞⎜

⎝⎛ += 22

βαβ ˆˆˆθ iiir vvvarccos

PF-014893

Permanent Magnet Synchronous MotorPermanent Magnet Synchronous Motor

ωr dβs

iβ

Hypothesis:Hypothesis:

-- symmetricalsymmetrical--sinusoidal machinesinusoidal machine

θr

qβ vβ

yy

-- non salient rotornon salient rotor

-- represented by tworepresented by two--phase phase

αsαN

equivalent statorequivalent stator--fixed fixed αβαβwindingswindings

vαiαS

[ ] [ ] [ ]vBvBiAi i −+=&

Dynamical model:Dynamical model:

[ ] [ ] [ ]vBvBiAi i+

Tiii ],[ βα=T

[ ] [ ]IAssLR−=

Tvvv ],[ βα=

Tiii vvv ],[ βα= s

[ ] [ ]IBsL1−=

PF-014893

BackBack--EMF and PMSM Extended ModelEMF and PMSM Extended Model

The backThe back--EMF components contain the information on the rotor position:EMF components contain the information on the rotor position:

rreri

rreri

kvkv

θωθθωθ

β

α

cos)(sin)(

=−=

In order to arrange backIn order to arrange back--EMF observer, backEMF observer, back--EMF components are added EMF components are added to the state through a couple of fictitious dynamic equations:to the state through a couple of fictitious dynamic equations:to the state through a couple of fictitious dynamic equations:to the state through a couple of fictitious dynamic equations:

[ ] [ ] [ ]0

−+= iv

vBvBiAi&

&

0=iv

Tii ][ Tii vviix ],,,[ βαβα= Extended stateExtended state

PF-014893

State and disturbance observersState and disturbance observers

[ ] [ ] [ ] zvBvBiAi i +−+=& ˆˆˆ

[ ]zLvi =&

[ ] )( iiK ˆ [ ] )( iiK

Luenberger observer (LO)Luenberger observer (LO) Sliding mode observer (SMO)Sliding mode observer (SMO)

[ ] )(1 iiKz −⋅= [ ] )( iisgnKz −⋅= 1

)(][][][ ~ˆ~ˆ iiKvBxAx)& −++= )(][][][ ˆ~ˆ~ˆ iisgnKvBxAx −++=&)(][][][ )(][][][ g

[ ] [ ] [ ][ ]TIklIkK Gain matrixGain matrix[ ] [ ] [ ][ ]TIklIkK =

k,k, ll Gain coefficientsGain coefficients

PF-014893

Why and how to develop a compensation law ?Why and how to develop a compensation law ?Rotor position estimation error (difference between estimated and actual Rotor position estimation error (difference between estimated and actual position) dynamics: position) dynamics:

[ ] [ ][ ] [ ]iie

ieii

eklveekeBeAe

−−=

−+=&&

&If known, they it can be onIf known, they it can be on--line compensated !line compensated !

Solution of the problem is difficult in the timeSolution of the problem is difficult in the time--domain.domain.

Let us consider the equivalent sLet us consider the equivalent s--domain transfer functions:domain transfer functions:

( ) [ ] ( ) [ ] ( ) ( )sEksEBsEAesEs +

Af l l iAf l l i

( ) [ ] ( ) [ ] ( ) ( )( ) ( ) ( )sEklsVesEs

sEksEBsEAesEs

iiee

ieiii

−−=−

−+=−&

0

0

After some calculations:After some calculations:

( ) ( )( ) ( )sVssss

LsE i

si

&

21

1−−

= ( ) ( )( ) ( )sVssss

hssE ie&

21

2−−

+−=( )( )ssss 21 ( )( )ssss 21

sLlm −=jhhs Δ±−=Δ±−=21, ( ) ( ) mkLRkLRkh ssss −+=Δ+= 241

21 ;

PF-014893

BackBack--EMF estimation errorEMF estimation errorThe asymptotic stability is a sufficient condition for the existence of the The asymptotic stability is a sufficient condition for the existence of the sinusoidal steadysinusoidal steady--state response to sinusoidal excitation: state response to sinusoidal excitation:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

+=

= )φω(ω)φω(ω

ω ωωv

vr Frre

Frreve tsink

tcoskjFte 2

2

⎦⎣ v

( )⎪⎧ += )φω(α Frve tcosAte ( )( )⎪⎩

⎪⎨⎧

+=

+

)φω()φω(

β

α

v

v

Frve

Frve

tsinAtetcosAte

( )( )+Δ+Δ−−

−−Δ+Δ+=

= ωωω

ωω)(ω ωω )])(([

)()(222

222222

2

32

hh

hhhjF

rrr

rrv r ( ) 2ωω rervv kjFA =

Actual rotor speedActual rotor speed

⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ+−−Δ

== )(

)ω(ωφωω 2

22

23

hhharctang rr

Frv

rω

PF-014893

Calculation of the rotor position estimation errorCalculation of the rotor position estimation error

αβαβ components of the estimated backcomponents of the estimated back--EMF:EMF:

)t(cosAtsink)t(evv ii φωωω ++−=+=

)t(sinAtcosk)t(evv

)t(cosAtsink)t(evv

v

v

Frvrreeii

Frvrreeii

φωωω

φωωω

βββ

ααα

++=+=

++−=+=

⎟⎞

⎜⎛ Fv φcosAˆ

Rotor position estimation errors:Rotor position estimation errors:

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−=−

v

v

Fvre

Fvrr φsinAk

φcosAarctg

ωθθ Luenberger observerLuenberger observer

( )lLarctg srrr ωθθ =−

F i t f t t th ( t dF i t f t t th ( t d t t ) t iti t t ) t iti

Sliding mode observerSliding mode observer

For a given set of motor parameters, the (steadyFor a given set of motor parameters, the (steady--state) rotor position state) rotor position estimation error depends on rotor speed, but it is independent on feeding estimation error depends on rotor speed, but it is independent on feeding current (i.e. load torque)current (i.e. load torque)

PF-014893

Simulation resultsSimulation resultsSteadySteady state estimation errorsstate estimation errorsLuenberger SM O

Comments:Comments:

LO t iti ti ti LO t iti ti ti

SteadySteady--state estimation errorsstate estimation errorsg800 1600 16000 ∞ k←

-7.41 -6.22 -5.19 -5.08 1000 rpm  

-16.97 -13.22 -10.37 -10.08 2000 rpm -- LO rotor position estimation LO rotor position estimation

errors tend to SMO ones for high errors tend to SMO ones for high values of gain values of gain kk

-32.63 -21.99 -15.52 -14.93 3000 rpm

Position error )( θθ rr − [degrees]

-- for the same value of gain for the same value of gain ll, , SMO has smaller estimation SMO has smaller estimation error with respected to LOerror with respected to LO-5

0

degr

ees]

-5.078

-- estimation error reaches nonestimation error reaches non--negligible values at high speednegligible values at high speed

-- compensation is possible by compensation is possible by

-10

mat

ion

erro

r [d

-14.928 -10.079-6.22

LO p p yp p yconsidering the analytical error considering the analytical error as a first attemptas a first attempt

-- actual implementation could actual implementation could -20

-15

Posi

tion

estim l = −28.28; k = 1600 -13.22

SMO

LO

LO

l = −28.28

l = −56.56; k = 1600-- actual implementation could actual implementation could

lead to different results due to lead to different results due to unun--modelled effectsmodelled effects0 500 1000 1500 2000 2500 3000 -25

Rotor speed [RPM]

P

-21.99 SMO l = −56.56

PF-014893

Simulation resultsSimulation resultsTransient estimation errorsTransient estimation errors0

1 Lu e n be rge r O bse rve r

Conditions:Conditions:

Transient estimation errorsTransient estimation errors

-3 -2 -1 0

on e

rror [

degr

ees]

0 .5e-3

P os ition es tim a tion e rro r C om pen sa ted pos itio n es tim a tio n e rro r

Conditions:Conditions:

-- offoff--line operation of the line operation of the observerobserver-6

-5 -4

Posi

tion

estim

atio

l 28 28lo ad in sertion

-- nono--load speed transient from 0 load speed transient from 0 to 1000 RPMto 1000 RPM

-- 1.2 Nm (60% rated) load torque 1.2 Nm (60% rated) load torque

0 .0 5 0 .1 0 .1 5 0 .2 0 .2 5 0 .3 0 .3 5 0 .4 0 .4 5 0 .5 -8 -7

T im e [s ]

l = −28 .28 k = 1 600

-6 .2 2 lo ad in sertion

1 S lid in g M o de O b serve r1.2 Nm (60% rated) load torque 1.2 Nm (60% rated) load torque insertion at 0.35sinsertion at 0.35s

-- position estimation error is position estimation error is compensated by means of compensated by means of -2

-1 0

ror [

degr

ees]

-5 .5e -3

P iti ti ti compensated by means of compensated by means of analytical valuesanalytical values

-5 -4 -3

ition

est

imat

ion

err P o s itio n e s tim a tio n e rro r

C o m p e n sa te d p o s itio n e s tim a tio n e rro r

0 0 .0 5 0 .1 0 .1 5 0 .2 0 .2 5 0 .3 0 .3 5 0 .4 0 .4 5 0 .5 -7 -6

T im e [s ]

Pos

i

lo ad in sertion-5 .08

l = −2 8 .2 8

PF-014893

Experimental resultsExperimental resultsThe drive systemThe drive systemThe drive systemThe drive system

Braking test bench

IEEIEE 488 t ll d h t i b k488 t ll d h t i b k

PMSM under test

IEEIEE--488 controlled hysteresis brake488 controlled hysteresis brake(Magtrol HD(Magtrol HD--710710--8NA8NA--0040)0040)

PMSM under test

rated power /current 630 W - 2.5 A rms (*) rated speed/torque 3000 rpm (*) - 2.0 Nm (*) pole pairs 4 no load back EMF @ rated speed 82 72 Vrms (*)no-load back-EMF @ rated speed 82.72 Vrms (*)stator resistance 1.9 Ω stator inductance 6 mH rotor inertia 22⋅10-5 kgm2 inverter DC voltage 300 V

Control & power electronics

1616--bit fixedbit fixed--point DSP microcontrollerpoint DSP microcontrollerIGBT intelligent power moduleIGBT intelligent power module

150 150 μμs control and modulation cycless control and modulation cycles

PF-014893

Experimental resultsExperimental resultsSMO operationSMO operationrω

uncompensated uncompensated

500 RPM500 RPM SMO operationSMO operation

)( rmonrˆ θθ , −

Conditions:Conditions:

-- nono--loadload99°°

500 RPM500 RPM

-- 500 to 2000 RPM500 to 2000 RPM

-- 500 RPM slow varying ramp 500 RPM slow varying ramp incrementsincrements

theoretical compensationtheoretical compensation

99°°

incrementsincrements

Comments:Comments:

-- noise is related to residual noise is related to residual noise is related to residual noise is related to residual “chattering” on the back“chattering” on the back--EMF EMF estimatesestimates

theoretical compensation is not theoretical compensation is not empirical compensationempirical compensation -- theoretical compensation is not theoretical compensation is not the exact one, but provides good the exact one, but provides good resultsresults

i i l ti i i i l ti i

p pp p

-- empirical compensation is empirical compensation is obtained by varying the slope of obtained by varying the slope of the compensating anglethe compensating angle

PF-014893

Experimental resultsExperimental resultsLO operationLO operationrω

uncompensated uncompensated

500 RPM500 RPM LO operationLO operation

)( rmonrˆ θθ , −

Conditions:Conditions:

-- nono--loadload99°°

500 RPM500 RPM

-- 500 to 2000 RPM500 to 2000 RPM

-- 500 RPM slow varying ramp 500 RPM slow varying ramp incrementsincrements

99°°

incrementsincrements

-- same gains as SMOsame gains as SMO

Comments:Comments:Comments:Comments:

-- position error has different slopeposition error has different slope

-- noise is lowernoise is lowerempirical compensationempirical compensationp pp p

PF-014893

Experimental resultsExperimental resultsSMO torque/speed analysisSMO torque/speed analysisSMO torque/speed analysisSMO torque/speed analysis

Comments:Comments:

-- load torque affects position load torque affects position estimation error (differently estimation error (differently from theoretical results)from theoretical results)))

-- the effect of the load torque the effect of the load torque is quite small (less than 2 is quite small (less than 2 degrees in every condition) degrees in every condition) degrees in every condition) degrees in every condition)

-- overover--compensation is compensation is obtained if theoretical laws obtained if theoretical laws are adopted (dotted lines)are adopted (dotted lines)are adopted (dotted lines)are adopted (dotted lines)

-- compensation with compensation with theoretical values leads to a theoretical values leads to a maximum error of 5 degreesmaximum error of 5 degreesmaximum error of 5 degreesmaximum error of 5 degrees

PF-014893

ConclusionsConclusions

TheThe possibilitypossibility toto compensatecompensate thethe intrinsicintrinsic rotorrotor positionposition estimationestimation errorerror inin aatransducertransducer--lessless controlcontrol schemescheme forfor PMPM synchronoussynchronous motorsmotors basedbased onon backback--emfemf observersobservers hashas beenbeen analysedanalysed::

-- itit isis possiblepossible toto evaluateevaluate aa theoreticaltheoretical compensationcompensation lawlaw forfor thethe steadysteady--statestate operation,operation, bothboth withwith LuenbergerLuenberger andand SlidingSliding ModeMode observersobservers

emfemf observersobservers hashas beenbeen analysedanalysed::

pp gg gg

-- suchsuch lawslaws yieldyield satisfactorysatisfactory compensationcompensation alsoalso duringduring fastfast transientstransients

-- itit hashas beenbeen provenproven thatthat thethe presencepresence ofof aa KalmanKalman filterfilter inin cascadecascade toto thethebackback--EMFEMF observerobserver doesdoes notnot introduceintroduce additionaladditional positionposition errorerror atat steadysteady--state,state, andand negligiblenegligible oneone duringduring transientstransients ifif slowslow rampramp--varyingvarying speedspeedstate,state, andand negligiblenegligible oneone duringduring transientstransients ifif slowslow rampramp varyingvarying speedspeedreferencereference isis usedused

-- thethe unun--modelledmodelled effectseffects inin thethe actualactual drive,drive, motormotor andand realreal--timetime systemsystemthethe unun modelledmodelled effectseffects inin thethe actualactual drive,drive, motormotor andand realreal timetime systemsystemintroduceintroduce somesome discrepanciesdiscrepancies fromfrom theorytheory:: exactexact compensationcompensation requiresrequiresaa properproper adjustmentadjustment byby experimentalexperimental teststests