Unbalance force rejection control

4.6 MIMO case position control

4.6.2 Unbalance force rejection control

Inthemagneticallybornerotor,whichspinsaboutaxedaxis,therearealways

somesinusoidaldisturbanceforcesactingontherotor. These forcesarecaused

bytheunbalance,whichcanbedescribedasadiscrepancybetweenthexedaxis

ofrotation(usually,theassumedaxisofgeometryoftherotor)andtheprincipal

inertiaaxis. Apartfromanunbalance,allotherdiscrepanciesandimperfections,

which make the system non-axisymmetrical, may cause additional harmonic

disturbances. The resultingvibrational bearing forces are proportional to the

squareoftherotationalspeed.

The unbalance compensation in the AMB applications is referredto asan

andthe possibilityof changingthestiness anddampingprovide better

capa-bilities to deal with the unbalance than in the traditional bearings. Control

theory, rotordynamics, and the literature on AMBs provide dierent UFRC

methods. Basically, acompensationmechanismissynchronizedwith the

rota-tional speed. It injects compensating harmonicsignals, of proper amplitudes

andphases,tothecontrolsystem. Ingeneral,itispossibleeither tocancelthe

position vibrations(i.e. cancel the eect of the unbalance forces onthe rotor

position)ortocancelthemagneticforcevibrations(i.e. canceltheeectofthe

unbalanceforcesonthecontrolcurrents).

Asanexample,anadaptivefeedforwardcompensationanddiscreteFourier

transform (DFT) are applied by Bleuler et al. (1994). Grochmal and Lynch

(2006)andtheysugestthereduced-orderdisturbanceobserverwiththeobserver

gainscomputedanalytically(based onthedesiredlocationofeigenvalues)and

scheduled accordingto the rotationalspeed. The interestingapproach is

pre-sentedbyLum etal.(1996),wheretheobserver-basedimbalance compensator

performsanon-lineidenticationofthephysicalcharacteristicsoftheimbalance

andusestheresultto tunethecompensator. This lastcompensatortechnique

workesundervaryingrotorspeed. Manyother compensationmethodsarealso

available. Ashort reviewofdierentcompensationmethods andreferenceson

themaregivenbyBleuleretal.(1994).

Inthiswork,theobserver-basedUFRCisconsideredintwovariants. Inthe

rstone,thepositionvibrationsarecanceledand thesystemcantoleratehigh

sinusoidalunbalanceforceswhileitmaintainsaccuraterotorposition.Itresults

in low measured vibration amplitudes. In the second variant, the magnetic

force vibrations are canceled and the rotor (considered as rigid) spins about

its principal inertia axis withouttouching thesafety bearings (assuming that

theclearances are largeenoughto accommodatethe unbalance). This results

inminimal controlcurrents. Both casesaredesignedasaperiodicdisturbance

estimatorthat iscombinedwiththeconstantdisturbanceestimator.

Inthecaseof positionvibrationcancellation,twodierentdesignmethods

areemployedandcomparedwithrespecttotheirrobuststability,forthe

equiva-lentperformance. Forthebettercomparison,boththedesignmethodsareused

in such a waythat theyresult in the sameclosed-loopeigenvalues. The rst

designgeneralizes the LQ designof the full stateestimator with theconstant

disturbanceestimate,asin (4.11), totheonethatincludes asinusoidal

distur-bance. Thecomputationoftheproportionalgainmatrix

L

^and^thedisturbance estimatorgainmatrix

L dist

^is^based^on^the^augmented^plant^model

x ˙

Similarlytothedesignoftheconstantdisturbancemodel,alsoherethe

distur-Thenegativedamping

ζ = − 0.05

^of^a^sinusoidaldisturbancemodelisassumed.

This perturbation is introduced in order to move the roots of the augmented

plantmodelfrom theimaginaryaxis. Inaddition,

ζ

^aects^the^resulting^speed

oftheintegration,forthesinusoidaldisturbanceestimate. Now,theaugmented

modelusedin theimplementationappliesthedisturbancestatematrixsuchas

A ⁰ _dist =

The second design is based on the pole placement. It applies the closed-loop

roots, which result from theKalmanlter designcomputed forthe state

esti-mateswithoutthedisturbances,plusthearbitrarilyaddedintegralpoles.

Now, the state and disturbanceestimate feedback matrix is

[K, K dist ] = [K, C dist ]

Theunbalanceresponsesofthesimulatedcontrol systemwith andwithout

the UFRC are presented in Fig.4.24 and 4.25. The applied step disturbance

forceis

f = 0.2 · i c,max k i ≈ 402 N

^and^the^applied ^couple^unbalance¹^is^equal^to

500 g · mm

^thatcorrespondstothedisturbanceforceamplitudeequalto

137 N

ateachradialbearing,for

Ω = 5000 rpm

Theresultingclosed-looppolesarethesameforbothaforementioneddesign

methods, when theplant model without the residualmodes is applied.

How-ever, the closed-loop pole patterns for the designed controllers and the plant

that included theresidual modes(Fig. 4.25)aredierent for both controllers;

namelyforthepoleplacementdesignofintegrators,theresidualmodesareless

damped(aboutsixtimes)thanfortheoptimaldesign. Thisdierencecanalso

be noticed when comparing the singularvalue plots of the output sensitivity

transfer functions for both controllers (4.26). The singular values plot of the

outputsensitivitytransferfunction correspondstothedisturbanceattenuation

performanceinthemeasuredsignal,anditisequivalenttothesensitivityofthe

SISOcase,whichindicates therobuststability.

In thecaseof magneticforce cancellation,thecomputation of

L

^and

L dist

and the feedback are the same as for the disturbance force cancellation, but

the augmented plantmodels used in the design and implementation have the

output equation such as

y = [C, 0, I, 0] x

^and

y ¯ = [C, 0, I, 0] ¯ x

^, respectively.

Thiseliminatestheoscillatingcomponentfrom theerrorsignal.

The experimental evaluations ofUFRCfor thepositionvibration

cancella-tion and forthe magnetic force cancellation are presented in Fig. 4.27and in

Fig.4.28, respectively. Infact,the rotorwasnotrotating; however,theeect

ofcoupleunbalancewasgeneratedthroughthecurrents.

4.6.3 Comparison of LQ and PID controllers

Aslongasarigidrotor(forslowrotation)isconsidered,itispossibleto apply

the decentralized control (four SISOcontrollers) to support the rotorradially

in x and y directions. The low-order PID controllersmay be easily designed

manually in asimilar fashion asthe one for the axial suspension (see section

coupleunbalanceassumingnostaticunbalance,theremainingforcesineachplane(in

oppositesidesofthecenterofrotormass)actinoppositedirectionsandareequalinmagnitude

0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 4.24: System withoutUFRCand thesystem with UFRC that cancels

theunbalanceforcesarepresentedunderthecoupleunbalance. The(A)stands

forthemeasuredposition

x A

^and^control^current

i c,x,A

^;^the^(B)^is^the^reference

position andthescaleddisturbanceforce

f dist /k i

0 0.1 0.2 0.3 0.4 0.5 0.6

controlled flexible mode roots residual mode roots

integrator roots

Figure4.25: Intheleft-handillustrations,thesystemresponseswithUFRCfor

cancelingthemagneticforcevibrations(thereforeminimizingcontrolcurrents),

arepresented. Themeasuredposition

x A

^and^control^current

i c,x,A

^are^indicated

by(A). Thereference position

x ref,A

^and^the ^scaleddisturbance force

f dist /k i

are indicated by (B). In the right-hand illustration, the closed-loop roots of

thesystemusedin thecontroldesign(B)andthecontrolsystemthatincludes

non-controlledexiblemodes(A)in theplant,areshown.

10 200 1000 3000 10000 10 ⁻²

10 ⁻¹ 10 ⁰ 10 ¹

Sensitivity

frequency [rad/s]

gain [dB]

Figure4.26: Singularvaluesplotoftheoutputsensitivityfunction forthe

sys-temswithUFRC,arepresented. Theblueplotcorrespondstothesystemdesign

basedontheoptimalgainselectionandtheredplotcorrespondstothesystem

designbasedonthepole placement.

0 0.02 0.04 0.06 0.08 0.1

−2 0 2

control current [A]

without UFRC

0 0.02 0.04 0.06 0.08 0.1

−50 0 50

time [s]

position [ µ m]

i _cx,B i cy,B

x B y _B

0 0.02 0.04 0.06 0.08 0.1

−2 0 2

control current [A]

with UFRC

0 0.02 0.04 0.06 0.08 0.1

−50 0 50

time [s]

position [ µ m]

i cx,B i _cy,B

x _B y B

Figure4.27: MeasuredresponsesofthecontrolsystemwithoutandwithUFRC

for theunbalance force cancellation,when the coupleunbalance is

500 g · mm

for

Ω = 5000 rpm

0 0.02 0.04 0.06 0.08 0.1

Figure4.28: MeasuredresponsesofthecontrolsystemwithoutandwithUFRC

forthemagneticforcecancellation,whenthecoupleunbalanceis

250 g · mm

^for

Ω = 5000 rpm

4.4). However,fortherotormodelthatincludestheexiblemodes,theobtained

PIDcontrollersareunstable,and theyhavetobeimproved.

Let us consider a leadcompensatorin Eq. (4.5). We include the bending

modes in the design. Now, we include the rst bending mode into the SISO

plantmodel. Weassumethat it doesnotaecttheend-B oftherotor(as the

nodeof thebending modeis veryclose to theactuatornode). Next, webuild

theapproximatedplantmodelfortheend-Aas

G(s) = k i

modalmassoftherstbendingmode,modaldamping,modalstiness,andthe

rst modal inuence factor at the end-A (Lantto, 1999), respectively. Inthis

case,theinuencefactormayberoughlyapproximatedbythedistancebetween

theactuator node and thenode of therst bending mode. Other parameters

areobtainedfrom theFEM model. Now,wecanuse theapproximatedmodel

inthecontrollerdesign.

Wemodifytheoldcontrolbyadjustingthefrequencyofthemaximumphase

inordertoobtainthesucientstabilitymargins,foreachloopatatime. After

someiterationwiththecontrollerparameters(decreasingtheproportionalgains,

slightlyincreasing thecurrentcontrol bandwidth,andselecting the

ω ld = 0.7 · 2π · 260 rad/s

^), ^when^taking ^the^values ^according^to^Table^4.3, ^as^the ^starting

point, weobtain thestable decentralizedcontroller. In further iterations, the

resultingmultiple SISO,overallcontrolleris testedfor stabilitywith theplant

modelthat includestherstthreebending modes.

Thestepresponsesoftheclosed-loopsystem(withaleadcompensator),and

thecomparisonoftheunbalanceresponsesofthedecentralizedPIDcontrolwith

the centralized LQ position control,for dierent rotational speeds, are shown

inFig.4.29.

Forthe PIDcontrol oftheradialsuspension, thestepresponses arenotas

goodasfortheLQcontrol,eventhoughthePID-basedcontrolassumesa40%

0 0.1 0.2 0.3 0.4 0.5 0.6

1000 2000 3000 4000 5000 6000

0.1

Figure 4.29: Onthe left, the stepposition and disturbanceresponses for PID

radialsuspension. Ontheright,thecomparisonof thecentralizedLQposition

control with high stiness (A), low stiness (B), and the decentralized PID

control(C), fordierentrotationalspeeds,insimulations,ispresented.

responses,thebehaviorforlowerfrequenciesisalsonottoogoodbecauseofthe

lowstiness(requiredforstability).

FortheLQMIMO control,which isableto providehigher gains(and

bet-ter damping) without loosing the stability, the gyroscopic eect is taken into

account. Inconsequence,theoptimalcontrolsolutionvarywith therotational

speed. TheLQcontrolcanbetuned,andremainstable,tohavegoodunbalance

rejectionbothatloworhighfrequencies. Thelowstiness controldesignmay

be obtained by, for example, increasing the available maximum displacement

in the weightingmatrix

Q ¯

^. ^It ^occurs ^that ^the ^eect ^of ^gyroscopic ^coupling ^is

relativelyweak,whencomparingthecentralizedLQcontrollerwithUFRCand

thedecentralizedonewithUFRC,theunbalanceresponsesarethesameforthe

under-criticaloperation.

Moreover,forbothcontrolsystems(PID-and LQ-based),thevariationsof

rotationalspeedresultinrelativelyinsignicantchangesoftheclosed-looppoles

andzeros(Fig4.30). InFig4.30,onlythepolesandzeros,whichhavepositive

orzeroimaginaryparts,andwhichareclosetotheimaginaryaxis,areshown.

Figure 4.31presentsthe start-up of the rotorwhen a centralized LQ

con-trollerwith theactivelycontrolled rstexible mode andthe constant

distur-banceestimatorwasemployed. Sincetheestimatorhasnoknowledgeaboutthe

initial states, it isstarted together with themeasurementsbefore theintegral

action isenabled to work asalter withzero control currents. It is sucient

toenableonlytheestimationoftherigidbodymodesduringthisinitialphase.

InFig.4.32,thestart-upoftherotorwhenPIDcontrollerisapplied,isshown.

The measured responses of the LQ controller and PID controller to the

stepdisturbanceforce introducedthrough theAMB at theend-Bof therotor

(

f dist = k i i c,y,B

^,^where

i c,y,B = 2A

⁾^are^presentedⁱⁿ^Fig.^4.33^and ^4.34,

respec-tively. Figure 4.35 shows the control currentsof the LQ controller when the

stepdisturbanceisapplied.

Tosumup,in thestudied rotortherelativepolarinertia isverysmalland

thereforethegyroscopiceecthassmallinuence ontheresponsesofthe

mul-tiple SISOandMIMO control. The dierencesin theunbalance responses for

−250 −200 −150 −100 −50 0

Figure4.30: Intheleft-handillustration,thepole-zeromap oftheclosed-loop

systemwiththePIDcontroller,andintheright-handillustration,thepole-zero

mapoftheclosed-loopsystemwiththeLQcontrollerarepresented. Thepoles

areplottedas x'sandthezerosareplottedaso's.

stiness solution). It isnot possibleto obtain optimalsolutionfor allthe

fre-quencyranges,andhencethecontroldesignis acompromisebetweenlowand

highfrequencies.

In document Design and implementation of FPGA-based LQ control of active magnetic bearings (sivua 108-115)

4.6 MIMO case position control

4.6.2 Unbalance force rejection control

L

L dist

x ˙

ζ = − 0.05

ζ

A 0 dist =

[K, K dist ] = [K, C dist ]

f = 0.2 · i c,max k i ≈ 402 N

500 g · mm

137 N

Ω = 5000 rpm

L

L dist

y = [C, 0, I, 0] x

y ¯ = [C, 0, I, 0] ¯ x

0 0.1 0.2 0.3 0.4 0.5 0.6

x A

i c,x,A

f dist /k i

0 0.1 0.2 0.3 0.4 0.5 0.6

controlled flexible mode roots residual mode roots

integrator roots

x A

i c,x,A

x ref,A

f dist /k i

10 200 1000 3000 10000 10 −2

10 −1 10 0 10 1

Sensitivity

frequency [rad/s]

gain [dB]

0 0.02 0.04 0.06 0.08 0.1

−2 0 2

control current [A]

without UFRC

0 0.02 0.04 0.06 0.08 0.1

−50 0 50

time [s]

position [ µ m]

i cx,B i cy,B

x B y B

0 0.02 0.04 0.06 0.08 0.1

−2 0 2

control current [A]

with UFRC

0 0.02 0.04 0.06 0.08 0.1

−50 0 50

time [s]

position [ µ m]

i cx,B i cy,B

x B y B

500 g · mm

Ω = 5000 rpm

0 0.02 0.04 0.06 0.08 0.1

250 g · mm

Ω = 5000 rpm

G(s) = k i

ω ld = 0.7 · 2π · 260 rad/s

0 0.1 0.2 0.3 0.4 0.5 0.6

1000 2000 3000 4000 5000 6000

0.1

Q ¯

f dist = k i i c,y,B

i c,y,B = 2A

−250 −200 −150 −100 −50 0

A ⁰ _dist =

10 200 1000 3000 10000 10 ⁻²

10 ⁻¹ 10 ⁰ 10 ¹

i _cx,B i cy,B

x B y _B

i cx,B i _cy,B

x _B y B