Selection of optimal weighting matrices

Inthe AMB control system,the physical limitationssuch as: maximum

bear-ing forces, expected disturbance forces, displacements, currents, and voltages

point at the formulationof thecontrol strategy asan LQ optimization

prob-lem to minimize the values of control variables with minimum control eort.

The LQ optimal control guarantees a desirablehigh-performance, stable

con-trol,andeasesthedesignofMIMOsystemsbyreducingthedegreesoffreedom

inthedesigniterations. However,itrequiresanaccurateplantmodel,accurate

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.2

−0.1 0 0.1 0.2

(A)

position [mm]

0 0.1 0.2 0.3 0.4 0.5 0.6

−7.5

−2.5 2.5 7.5

(B)

time [s]

current [A]

5 100 3000

0.3 1

5 (C) Tolerated disturbance

omega [rad/s]

force [kN]

(a) (b) (c) (d) (e)

Figure 4.12: Comparative simulationsof the axial suspension, with the

state-spacecontrollers structuredaccordingto points 1-4,are presented. In thetop

twoillustrations (A-B), the step responses of the systems with dierent

con-trollers,tothereferenceinputanddisturbanceforceareshown. Inthebottom

illustration(C),themaximaltolerabledisturbanceforcesoftheclosed-loop

sys-tems, as functions of frequency, are plotted. The plots (a-d) correspond to

theaxialsuspensionwiththecontrollers1-4, respectively. Theplots(e)

corre-spond tothereferencedisplacement,disturbanceforcescaled by theinverseof

thecurrentstiness, and physical force limit, in the top, middle, and bottom

rateplantmodel isaddressedin Chapter3and,fortunately, theKalmanlter

providesthe necessaryestimates and ltersnoisy measurements basedon the

speciederrorcharacteristicsofthesysteminputsandoutputs.

Regarding the plant, we assume that the pair

(A, B)

^is controllable and that the pair

(A, C)

^is observable. For LQR used for AMBs, some authors (e.g. Lösch,2002)pointadrawbackinthetrialanderrorofdesignprocedure,

namely, the selection of weighting matrices. In fact, there are at least two

practical design methods for LQ control, which address this issue: Bryson's

rulespresentedforinstancebyFranklinetal.(1998),LewisandSyrmos(1995),

and the method based on the asymptotic modal properties of LQR given by

Hiroe et al. (1993). Owing to the straightforward design and good results,

Bryson'srulesareselectedforutilization intheoptimalcontrollersynthesis.

TheprocedurefordesigningtheLQcontrollerisasfollows. IntheLQR,the

state-feedbackcontrollergainmatrices

[K I K]

^{minimizethequadraticintegral}

performanceindex

J q

^{, andthediagonalmatrix}

C si

^{selectswhich statesshould}

bekeptundercloseregulation,as

J q =

where

Q

^,

Q ¯

^,

R

^,and

N

^{arethescaledandtheunscaledstateweightingmatrix,}

control weighting matrix, and the weighting matrix responsible for the cross

eect between the stateand control, respectively. The weighting is arbitrary

buttheproblemdatamustsatisfycertainlimitingconditionswhensolvedwith

theMatlab(seeTheMath WorksInc. 1999).

Similarly,theoptimalestimatorgainmatrices

[L L dist ] ^T

^{aredeterminedby}

solvinganalgebraicRiccatiequationbasedonthenoiseintensitymatrix

R v

^of

thesensorsandtheprocessinputnoiseintensitymatrix

R w

^{. Thesecovariance}

matrices are formed basedon therms accuracyof the position measurements

andcontrolinputs. Forthestateestimator,theoptimalsolutionoftheKalman

lteris

˙¯

x = A¯ x + Bu + L (y − C¯ x − Du) .

^(4.16)

Theerrorcharacteristicsofthepositionmeasurementareinuencedbythe

ac-curacyofthesensors,theirsensitivitytoexternalelectromagneticinterferences,

the non-idealrotorsurface(runout), and presenceof unmodeled dynamics. If

R v

^{increases,thestabilityand peak} disturbanceforce rejectiondecrease. The errorcharacteristicsofthecontrolinputsdierconsiderably,dependingonhow

wellthesystem(andespeciallyactuator)islinearized andmodeled.

Letusconsidertheselectionoftheweightingmatricesfordesignoftheaxial

AMBcontroller. Now,assumingthelayoutvariationasinsubsection4.5.1,the

matrices areselected asfollows. Inthe LQR, thestates, which correspondto

theposition errorintegral,control current,position,andvelocityarescaledby

thesquareoftheirmaximumallowedvaluesas

Q ¯ = diag

1/x ² _max , 1/i ² _c,max , 1/x ² _max , 1/v _max ²

,

^(4.17)

and likewise thecontrol eortis scaled as

R = 1/i ² _c,max

^{. The}

N = 0

^{. In the}

similarmanner,intheLQestimator,theinitialcovariancematricesareselected

as

R v = (0.1 · x max ) ² , R w = (0.5 · i c,max ) ²

^{. Thesematricesareselectedinsuch}

5 100 3000

Figure4.13: Selecting weightofthepositionstate, e.g. for(A)

1/(0.1 · x max ) ²

thestinessisathighest

awaythattherearecertainsafetymarginsbetweentheactualerror

character-istics(the rmsaccuracyofthepositionmeasurementsandthermsaccuracyof

thecontrol inputs),andthe assumedstochastic properties(of theprocessand

measurement noise), otherwise they are used as the design parameters. F

ur-thermore,theintegralstate(ifpresent),controlcurrent,androtorpositionare

keptunder thecloseregulation(

C si = [1 1 1 0]

^{). Leavingthecurrentcontrolto}

theinner control loop (

C si = [1 0 1 0]

^{) would increasethe system}performance in terms of the tolerated disturbance force, but on the slight expense of the

stability. The weighting of the state related to the displacement contributes

mostly to the bearing stiness (roughly 2-3 times as much as to the bearing

damping), and the weighting of the state related to the velocity contributes

mostly to the bearing damping(Fig 4.13 and 4.14). The smallerthe allowed

maximumpositionandvelocity,thebiggertheresultingstinessanddamping.

The assumed maximum velocity, in the weight, equals to the velocity of the

rotor that is accelerated from its central position to

x max

^{when applying the}

maximumbearingforce. Increasing theavailable controleortin

R

^{, increases}

both thebearing stiness anddamping(4.15). Inthe caseofthe unbiased

es-timator, thegaincomputation is basedonthe augmented plant model, where

thedisturbancemodelis

A dist = 1/τ I

^{. If}

τ I

^{decreases,the} peak-tolerated-force increasesconsiderably(as atrade-obetweenlow-andhigh-frequencyforces),

butinexpenseofthestabilitymargins. Theeectoftheintegraltimeconstant

iscomparedinFig 4.16.

Thepolesoftheestimatorbecomefasterwhentheratioof

R w /R v

^increases.

Themost straightforwardway to increasethestabilitymargins ofthe LQ

de-sign is to increase the noise rms values in the process covariance matrix

R w

(Fig 4.17). This way, theestimator poles arefaster than the controllerpoles,

butthesystembecomesmorenoisesensitive.

There exist a trade-o between low and high frequencies when selecting

theweightingmatricesin regardto theperformanceand tolerateddisturbance

forces. Inmostcasestheweightsthatresultin acontrolsystemthat tolerates

the disturbances at low frequencies at the same time decrease the tolerated

disturbancesathighfrequencies. Lowtolerateddisturbancesathighfrequencies

indicatehighopen-loopgainsforfrequenciesabovethecrossoverfrequencyand

5 100 3000

Figure4.14: Selectingweightof thevelocitystate,e.g. for(A)

1/(0.05 · v max ) ²

thedampingisat highest

5 100 3000

Figure 4.15: Selecting weight ofthe control eort, e.g. for (A)

R = 1/(0.25 · i c,max ) ²

^{thestinessandthedampingareat lowest}

5 100 3000

Figure4.16: Selectingdisturbancemodel,e.g. for(A)

A dist = 1/(0.05)

5 100 3000

Figure4.17: Selectingthecovariancematrix,i.e. for(A)

R w = (0.01 · i c,max ) ²

Table4.2: Parametersinthedesignoftheaxial LQcontroller

Q ¯ = diag

1/i ² _c,max , 1/x ² _max , 0

,

R = 1/i ² _c,max , C si = [1 1 0] , R w = (0.5 · i c,max ) ²

^,

R v = (0.1 · x max ) ² ,

constantdisturbanceestimator

A dist = 1/(0.3) ^(A)

(A)

The

A dist = 1/(0.05)

^{inFig.4.12,4.17,and4.15.}

Finally,aftersomeiterations,theselectedparametersoftheaxialcontroller

arepresentedinTable4.2. Thecontrolsystemtoleratesagaindropandincrease

bythefactorof0.29(at37rad/s)and2.4(at915rad/s),respectively. Thephase

marginis

32 ^◦

^{,at 383rad/s.}

In the case of the LQ control and considerable actuator delay, the

actua-tor model with a approximated delay should be included in the plant model

(increasesthe numberof statesin anestimator), otherwiseit mayreduce the

stabilitymarginsand systemperformance. Thereducedbiascurrentadds not

onlytheplantuncertaintiesowingtoits changeabledynamics thatare

depen-dentonthesignalamplitudeand frequency, but alsoinuences theachievable

maximumtolerated disturbance force, asshown in Fig. 4.18A.In addition, in

Fig.4.18B,thestabilityof theaxialsuspensionwith thederivedcontroller,as

a function of

k x

^and

k i

^{, is examined. Figure 4.18B is obtained by iterating}

overthevalues of

k x

^and

k i

^{, and verifyingthe stability of thesystemin each}

iteration.

In document Design and implementation of FPGA-based LQ control of active magnetic bearings (sivua 98-103)