F orce-eld linearization

AsalreadymentionedinChapter1,themajornonlinearityintheAMBcontrol

systemistheforce-eldnon-linearrelation. A popularandconvenientmethod

forcompensatingactuatorsisthatofinversenonlinearities(INL).

4.3.1 Inverted nonlinear force eld

IntheINLmethod,tocompensatetheactuatornonlinearities,weintroducethe

0 500 1000 1500 2000 2500 3000 3500 4000 4500 0.2

0.4 0.6 0.8 1

frequency [rad/s]

magnitude [pu]

0 500 1000 1500 2000 2500 3000 3500 4000 4500

−80

−60

−40

−20 0

frequency [rad/s]

phase [degrees]

(A) (B) (C) (D)

positive force negative force

Figure 4.5: Frequency responses of theradial AMB model with the nonlinear

controller(compensated

L dyn

^{andtheforceeld): (A)}simulation,(B) approxi-mationbasedon(3.49),(C)approximationbasedon(3.54),(D)approximation

basedon(3.53).

−150

−100

−50 0

Bode Diagram

Magnitude (dB)

10 ¹ 10 ² 10 ³ 10 ⁴ 10 ⁵

−225

−150

−75 0

Phase (deg)

(A) (B) (C)

frequency [rad/s]

Figure4.6: Bodediagramsof theclosed-looptransferfunction ofthe actuator

fordierentapproximationsofthePWMdelay(basedonanalyticalmodel): (A)

second-orderdelay,(B)rst-orderdelay,(C)nodelay. Themaximaldelay,which

is twice aslongas theaveragemodulationdelay, wasused in the simulations.

For the selected carrier frequency, the inuence of the delay (in the control

bandwidth)isinsignicant.

L i n e a r

Figure 4.7: Principle of theINL compensation method (A), and theprinciple

ofthemodelreferencemethod(B)

−2000

Figure 4.8: Inverted relation (A) of the force-current-displacement

character-istics from Fig. 3.10 and the compensating surface (B), are presented. The

surfaceswerebuiltfortheradialAMBasin TableB.1.

ableinterpolation. TheINL compensation principleispresentedin Fig.4.7A.

Letus assumethat wewantto compensate theforce generated by thepair of

the opposite electromagnets. First, the inverted relation of the

force-current-displacementcharacteristics(Fig.4.8A)isdeterminedandstoredinthelook-up

table. Thevalueofthecontrolcurrentcanbeobtainedfromthetableofstored

valueswith force andposition asthe table entries. The closestentry orother

various types of interpolations can be used. When the compensation of the

nonlinearactuatorisdirectlyincorporatedintothecontroller,thecontrolinput

becomesforceinsteadofcurrent,andthepositionstinessiscanceledoutfrom

theclosed-loopsystemdynamics.

Itisalsopossibletoapplythecompensationoftheactuatorasasingle

look-up table with theentries (

x, i c

^{), such that there are no changes in the upper}

control. Thisisrealizedbyrstcomputingtheforcereferencefrom(2.14),and

secondbycombiningthereferenceforcewiththeinvertednonlinearityintoone

compensationsurfaceshowninFig.4.8B.

In mostcases, it is moreconvenientto compensatethe force generated by

bythesingle electromagnet. Albeit,bothapproachesare equivalent. It is

im-portanttonoticethatforce-eldcompensationisaccurateforthestaticsignals.

Asanexample,itcannotfullycompensatetheactuatornonlinearitiesfor

high-frequencysignalsand high signalamplitudes,when only one electromagnetis

active(Fig.4.5),to saynothingofthesaturation.

4.3.2 Model reference method

Analternativesolutionfortheactuatorlinearization,whichcandirectlyutilize

theforcecharacteristics(nonlinearitydonothastobeinverted)anditisapplied

parallel withthe existingcontroller,is theapproachnamed hereasthemodel

referencemethod. The intrinsicidea of thissolution is acomparison between

the linear and nonlinear reference approximations as shown in Fig. 4.7. In

the studied case, the control current and rotor displacement are utilized in

computationoftheelectromagneticforces. Theforcesarecalculatedusingboth

thelinearizedandinterpolation-basednonlinearbearingmodels. Thedierence

between these two forces is employed in a feedforward control manner as a

compensationcurrent. Astotheforce-eldcompensationinAMB,thepractical

realizationcanbereducedto asinglelook-uptableaswiththeINLmethod.

The presented method canbeshownto be equivalentto the INL method.

In the application of the INL method to an AMB actuator, when using the

conventional controllerwith thecontrol outputin theform of thecontrol

cur-rent, the compensated control current

i cc

^{equals the control current plus the}

compensationcurrent,thatis

i cc = i c + i cp

^{. The}compensationcurrentcanbe expressedas

i cp = k i − k ^∗ _i (x, i cc )

k _i ^∗ (x, i cc ) i c + k x − k ^∗ _x (x, i cc )

k ^∗ _i (x, i cc ) x,

^(4.4)

where

k ^∗ _i

^and

k _x ^∗

^{arethecurrentstinessandpositionstinessdependentonthe}

positionandcompensatedcontrolcurrent(thevaluesofthecoecientswithout

the star are equal to the nominal values). When

k _i ^∗

^{in the} denominator of Eq.(4.4)isreplacedbytheconstantnominalcurrentstiness,thecompensation

current becomes equalto the compensation current from the model reference

methodwiththefeedforwardgainequalto

1/k i

^.

Inthecaseofthecompensationbasedonacontrolcurrent(compensationof

theforcegeneratedbythepairoftheelectromagnets),theINLismoreaccurate

thanthemodelreferencemethod. However,themodelreferencemethodismore

genericandeasiertoapply.

The benets andtrade-os of both compensation methods alongwith

sys-temresponses withand withoutcompensationcanbefoundin PublicationV.

Figure 4.9 shows the position responses for the impulse external disturbance

of the systemwith and without theINL compensation. The disturbance was

equivalenttothepointmassofabout3kgfalling ontherotorfromtheheight

of12cm. Theadvantageofthecompensatedcontroloverthenon-compensated

one is apparent in smaller position deviation from the reference point but in

−50 0 50 100 150 200 250

−100

−50 0 50 100 150 200

x [µm]

y [ µ m]

compensated non−compensated

Figure4.9: Positionresponsesfortheimpulseexternalforceofthecompensated

andnon-compensatedcontrolsystems

In document Design and implementation of FPGA-based LQ control of active magnetic bearings (sivua 90-94)