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Mixed-Weights Least-Squares Stable Predictive Control Algorithm with Soft and Hard Constraints

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Mixed-Weights Least-Squares Stable Predictive Control Algorithm with Soft and Hard Constraints

    Mixed-Weights Least-Squares Stable Predictive Control Algorithm with Soft and

    Hard Constraints

ChineseJ.Chem.Eng.,11(5)565570(2003)

    ;AlgorithmwithSoftandHardConstraints

    ;ZHOULifang(~立芳)andSHAOZhijiang(g之江)

    ;InstituteofSystemsEngineering,ControlScienceandEngineeringDepartment,ZhejiangUniversity,Hangzhou

    ;310027.China

    ;AbstractMixedweightleastsquares(MwLS)predictivecontrolalgorithm,comparedwithquadraticprogram

    ;ming(QP)method,hastheadvantagesofreducingthecomputerburden,quickcalculationspeedanddealingwith

    ;thecaseinwhichtheoptimizationisinfeasible.Butitcanonlydea1withsoftconstraints.Inordertodea1with

    ;hardconstraintsandguaranteefeasibility,animprovedalgorithmisproposedbyrecalculatingthesetpointaccord

    ;ingtothehardconstraintsbeforecalculatingthemanipulatedvariableandMWLSalgorithmisusedtosatisfythe

    ;requirementofsoftconstraintsforthesystemwiththeinputconstraintsandoutputconstraints.Thealgorithmcan

    ;notonlyguaranteestabilityofthesystemandzerosteadystateerror,butalsosatisfythehardconstraintsofinput

    ;andoutputvariables.Thesimulationresultsshowtheimprovedalgorithmisfeasibleandeffective. ;Keywordsmixedweightleastsquares,predictivecontro1,softconstraints,hardconstraints,feasibility

    ;1INTRoDUCTIoN

    ;Modelpredictivecontrol(MPC)hasbeenwidely

    ;andsuccessfullyappliedinprocessindustries[1-4].

    ;Thereareseveralreasons:MPCisacontrolmethod

    ;ologythatdealseasilywithcomplexdynamics,inter

    ;actionsandmeasurabledisturbance;itiseasytoan

    ;derstandbyplantoperators.Butthemainreasonfor

    ;itsacceptanceisprobablytheMPCtakesintoaccount

    ;processandoperatingconstraintsexplicitly.Butbe

    ;causeanoptimizationproblem,usuallyaquadratic

    ;program,issolvedonlineateachsamplingtime,the

    ;onlinecomputationaldemandishigh.whichlimits ;itsapplicationinlarge-scaleandcomplexcontrolsys-

    ;tems.HighcomputationaldemandofMPCisalso ;amajorobstacleinapplyingittosystemswithshort ;samplingtimeintervalsuchasflightcontrol,idlespeed ;control,etc.Inordertoreduceonlinecomputational

    ;demands.manymethodshavebeenproposed[5‟bJ:only

    ;thefirstcontrolmoveisimplementedalthoughmcon

    ;trolmovesarecalculatedateachsamplingtimes,by ;consideringthatu[(1)/k],…,u[(+m1)/]are

    ;unconstrainedandonlythefirstcontrolactionufk/k) ;isconstrained;undertheconditionoftheoptimization ;problembeingfeasibleatthefirstsamplingtime,the ;currentcontrolactionisconstrainedandtheotherfu

    ;turecontrolactionsareassumedtobeunconstrained: ;orconverttheconstraintpredictivecontrolintoop

    ;timallinearapproximation.Theabovemethodsare ;donesurroundingthequadraticprogramming(QP) ;methodandhavetheirshortcoming.Fortheexist

    ;ingconstraints,thestabilityofcloseloopcontrolsys

    ;temcannotbeguaranteed.Baseonstablegeneralized ;predictivecontrol,RossiterandKouvaritakis[71mixed ;weightsleastsquares(MWLS)algorithmwithguaran

    ;teedstability,whichmodifiedtheLawsonalgorithmIsJ, ;insteadofusingQPtosolvetheoptimizationproblem. ;MWLSalgorithmhassomesignificantadvantagesover ;OP:itcannotonlyreducetheonlinecomputational

    ;demandsbutalsocopewiththecaseofanemptyfea

    ;sibleset.

    ;Ifafeasibleregionexists.thenMWLScanonly ;convergetotheconstrainedoptimum,butwhenthe ;constraintsoptimizationproblemisinfeasible.MWLS ;willconvergetothepointthatminimizesthe”maxi—

    ;mumconstraintviolation”.So.itcanonlydealwith

    ;thesoftconstraintsinsteadofhardconstraints.al

    ;thoughtheyoftenoccur.Whenthevariablesen

    ;counterthephysicalconstraintsfwhichcanneverbe ;surpassed1,theinfeasibilityinshorttermwilldegrade ;theperformanceofthesystem.ormakeitunstable. ;Itissignificanttostudythestabilitywhentheinfea

    ;sibilityappearsinshortterm.Thereasonsthatresult ;intheinfeasibilityaredisturbanceandchangeofset

    ;point;.Itisditticulttostudytheeffectofdisturbance ;andthereportisscarce.Fortheeffectofsetpointon

    ;feasibilitywhenhardconstraintsexist,thefeasibility ;ofgeneralizedpredictivecontrolalgorithmwithinput ;constraintswasstudied[9J.

    ;butonlyinputhardcon

    ;straintsareconsideredinthestudiedsystem.Soin ;thispaper.basedontheadvantagethattheMWLS ;algorithmcandcalwithsoftconstraintsandreduce ;theonlinecomputationaldemands,theinput/output ;hardconstraintsandsoftconstraintsareconsidered ;inMWLSalgorithmanditsfeasibilityisresearched. ;Received20021120,accepted20030828.

    ;SupportedbytheNationalKeyBasicResearchandDevelopment(No.2002CB312200)

    ;Towhomcorrespondenceshouldbeaddressed.Email:lfzhou@iipc.zju.edu.cn ;

    ;566ChineseJ.Ch.E.(Vo1.11,No.5)

    ;TheimprovedMWLSalgorithmisproposedinwhich ;thesetpointisrecalculatedwhenthehardconstraints ;areviolatedtoguaranteefeasibility.Thesimulation ;resultsshowtheimprovedalgorithmiseffectiveand ;feasible.

    ;2STABLEMIX.WEIGHTLEAST?SQUARES

    ;PREDICTIVEC0NTR0LALGoRITHM

    ;Consideringthesingleinputsingleoutput(SISO) ;system

    ;y(k)=G()u()(1)

    ;G?=

    ;

    ;[b(0)+b(1)z+-??+b(nb)z]

    ;n(0)+n()??‟+.(n.)z.(2)

    ;whereu(k)expressesthemanipulatedvariableand ;y(k)expressesthecontrolledvariable,a(z)denotes ;thetransferfunctionoftheSISOsystem.n?andnb

    ;meantheorderofdenominatorandnumeratorofc(z) ;respectively.

    ;2.1Predictivemodel

    ;Formastabilizingfeedbackloopthatthesystem ;inputu,outputYandcommandinginputCarerelated ;by

    ;u()=A(z)c(k)(3)

    ;y(k+1)=B()c()

    ;SothatthesystemofEqs.(3)and(4)isequivalent ;tothatofEqs.(1)and(2).Multiplyingbothsidesof ;Eq.(3)by?,thefuturecontrolincrementAu(k+J) ;canbeexpressedas

    ;Au(k+J)=A(z)c(k+j),J=1,…,P1(5) ;where

    ;

    ;A(z)=AA(z)=(1)A(z)

    ;d(o)+(1)+?..+(+1)z(.+) ;Pisthepredictivehorizon.Toguaranteethestability

    ;ofsuchacloseloopsystem,afterthecontrolhorizon

    ;N,commandinginputCissettothesteadystatevalue

    ;C~o

    ;.

    ;ThenthefutureinputincrementvectorZxU(k)

    ;canbeexpressas

    ;ZxU(k)=F.c(k)+.c(k1)+M.Coo ;zxu(k)=[Au(k)Au(k+1)? ;c(k)=[c(k)c(k+1)…

    ;c(k1)=[c(k1)c(k)

    ;

    ;Au(k+P1)

    ;c(k+N1)]T

    ;vector.For

    ;is7‟.c..can

    ;aSISOsystem,ifthesetpointofoutput

    ;becalculatedby

    ;Coo=r/B(1)

    ;andF,H,Mare

    ;F.(,J)=

    ;:

    ;l,P,j:,…,N

    ;.c,=+

    ;.J1)+‟.+2)

    ;,J=l,…,(n.+2)

    ;0i?N

    ;liN

    ;M.()={?(1)0<({一?)?(n.+2)

    ;=1

    ;(i

    ;i:1,…,P

    ;else

    ;Usingthesamemethod,thefutureoutputvector

    ;Y(k+11canbepredictedas ;r(k+1)=rbc(k)+HbC(k1)+Mbc..(9) ;r(k+1):[y(k+1)y(k+2)…(+P)]T

    ;whererb}Hb}Mbare

    ;

;Hb(i,J

    ;Mb(i)=

    ;i=1,…,P;J=1,…,?

    ;Mb(i1)

    ;i=1,-,P

    ;else

    ;b+1)

    ;(n6+1)

    ;ThelasttwotermsofEqs.(6)and(9)areknown,SO

    ;they”canberewrittenas

    ;.

    ;c(kn.1)]Twhere ;wherec(k)denotesthefuturecommandinginputyea

    ;tor,andc(k1)denotesthepastcommandinginput

    ;October,2003 ;AU(k):F.c(k)+Au! ;r(k+1)=Fbc(k)+, ;?u,=.c(k1)+M.c.. ;yf=Hbc(k1)+Mbc.. ;(11)

    ;(12)

    ;,,,

    ;1

    ;+

    ;0

    ;

    ;,

    ;?

    ;d

    ;

    ;?

    ;O

    ;,,,

    ;.,

    ;0

    ;

    ;Mixed?-WeightsLeast?-SquaresStablePredictiveControlAlgorithmwithSoftandHardConstraints5

    67

    ;assumingtheperformanceindexas

    ;J=lIY(k+1)R(k+1)11+allau(k)ll

    ;=

    ;[c(k)c0()]Ts[c()c0()]+ ;where

    ;s=ffb+rf.

;c0()=(s)[f[R(+1)yY]~FTAuI]

    ;R(k+1)isthesetpointvectorofoutput(n(k+1)= ;[11…1]Tr=Er),isweightofcontrolincrement,

    ;isconstant.TheperformanceindexEq.(13)can ;alsobeexpressedas

    ;J=lIs[(c(k)c0()

    ;2.2constraints

    ;Inmostpracticalcontrolengineeringapplications ;thereexistconstraintsonsysteminputsandoutputs. ;Forexample,anover/undershootofmorethanacer. ;tainpercentageintheoutputrendersaparticular ;controllerunacceptable.Alternatively.theremavbe ;somedesignobjectivesconcerningspeedofresponse ;andrisetime.Aconvenientwavofencapsulatinga ;hostofsuchrequirementsistostipulateanenvelope ;withinwhichthesystemstepresponsemustlie;this ;canbedonebyspecifyinglowerandupperboundsfor ;theinstantaneousvaluesoftheoutputs.Ontheother ;hand.1owerandupperboundsoftenexistfortheab ;soluteandincrementalvaluesoftheinputs.because ;ofinherentnonlinearitiessuchassaturationandslow ;ratelimitsonactuators.OPmethodisoffenused ;tocopewiththeseconstraints.Butitnotonlyhas ;largeonlinecomputationaldemandsbutalsocandeal ;withthecaseonlywhenitisfeamble.Inorderto ;overcomethesedisadvantages,RossiterandKouvari

    ;tamsproposedMWLSpredictivecontrolalgorithm ;basedontheLawsonalgorithm.MWLSderiveda ;unifiedrepresentationfortheconstraints. ;1f(+J1)”.nt.1??di

    ;I(+J)IltI?Yradl.

    ;J=1.?.P

    ;Au(k+J1)1?oL

    ;“centre,oLandYcentredenotethecenterofsoftcon—

    ;straintsofinput,inputincrementandoutputrespec

    ;tively.”radiLls,Yradiusdenotetheradiusofsoftcon

    ;straintsofinputandoutputvariablerespectively.Ap

    ;plyingEqs.(11)and(12),theaboveconstraintscanbe ;rewrittenasthefollowingform

    ;mc()1?1

    ;(1)Forinputconstraints,theparametersin ;Eq.(16)are

    ;m=

    ;u

;E3ro

    ;“radius

    ;EAu$+?(1)——Ucentre

    ;Uradius

    ;whereEjdenotesthesumofthefirstJrowvectorsof

    ;theidentitymatrix.

    ;(2)Forinputincrementconstraints,theparame

    ;tersinEq.(16)are

    ;m:

    ;ejI

    ;

    ;

    ;a

    ;,i:

    ;JQ

    ;eiAuI

    ;(18)

    ;ejdenotesthejthcolumnvectoroftheidentityma-

    ;trix.

    ;(3)Foroutputconstraints,theparametersin

    ;Eq.(16)are

    ;m=

    ;ej

    ;Yradius,

    ;ejyfY

    ;centre

    ;Yradius

    ;(19)

    ;Combiningtheaboveconstraintstogethermeansthe

    ;feasibleregionofstableconstraintpredictivecontrol

    ;Eq.(14)suchas

    ;MC(k)l,ll..?1

    ;whereMandt,consistofmandofEqs ;(19)respectively(M?R.P,,?R.P).

    ;2.3Controllaw

    ;(20)

    ;(17)

    ;Forcalculatingthecontrolaction,weshould

    ;minimizethecostofEq.f141undertheconstraints

    ;Eq.(20).ThefollowingextensionofLawson‟salgo—

    ;rithmprovidesasensiblewaytocombinetheobjec

    ;tirewiththerequirementthattheinput/outputcon

    ;straintsbesatisfied

    ;M

;i+

    ;w

    ;l

    ;Ls

    ;+]/z[c()c0()]]l[(?][M()l,]f2

    ;withi+1)beingapositiverealscalarweight.and ;()beingadiagonalmatrixofpositivereal ;weightsdefinedby

    ;=

    ;1le

    ;?1?j1

    ;j=1j=1

    ;(22)

    ;withe)=MC?(k)l,,C?(k)representsthevec

    ;torc(k)whichminimizes%wLS,denotesthe

    ;/thdiagonalelementofWl.Theweightscanbe

    ;initializedas(.)=1and.):m,withmdenot

    ;ingthenumberofrowsinM.Then,withincreasing ;i,thealgorithmcanonlyconvergetothevectorC(), ;whichminimizesthecostofEq.(21)underthecon

    ;straintsEq.(20).

    ;ChineseJ.Ch.E.11(5)565(2003)

    ;

    ;568ChineseJ.Ch.E.(Vo1.11,No.5)

    ;3THEIMPRoVEDMWLSALGORITHM

    ;Therearetwokindsofconstraints.Oneishard ;constraintsorphysicalconstraints.Thoselimitscan ;neverbesurpassedandaredeterminedbythephys

    ;icallyfunctioningsystem.Forinstance,avalvecan- ;notbeopenedmorethan100%.Theotherissoft ;constraintsoroperatingconstraints.These1imitsare ;fixedbytheplantoperator,andintheformofband ;withinwhichthevariablesareexpectedtoLbe.MWLS ;canonlyconvergetotheconstrainedoptimumifa ;feasibleregionexists.Iftheconstrainedoptimization ;problemisfeasible,MWLSwil1convergetothepoint ;thatminimizesIIMC(k)ll..andthusminimizes

    ;the”maximumconstraintviolation”.So.MWLScan

    ;onlydea1withsoftconstraints.Whenthehardcon

    ;straintsareviolated.MWLSwil1beinfeasibleinshort ;term.Inordertodea1withhardconstraints,anim- ;provedMWLSpredictivecontro1algorithmispro

    ;posedinthispapertoguaranteethefeasibility. ;Thehardconstraintsofinputsandoutputsareex

;pressedasfollows

    ;lu(t+J1)Ucent.-hI?Uradius—h

    ;IAu(k+J1)I?Q—h

    ;Iy(k+J).t.hI?radius—h

    ;J=1.?--,P(23)

    ;whereUcentre_}l,a

    ;t1andYcentre

    ;hdenotethecenter

    ;ofconstraintsofinput,inputincrementandoutput

    ;respectively.radius_

    ;h,Yradius_hdenotetheradiusot ;constraintsofinputandoutputvariablesrespectively.

    ;Firstly.assumetheoutputsetpointrcanbereached

    ;whenthesystemisclosetosteadystate,whichmeans

    ;thesetpointcanbefollowedunderthehardcon

    ;straintscondition.Itcanbeexpressedas

    ;n

    ;[‰,‰…]…

    ;m‰,

    ;x]

    ;Umin=radiushUcentreh

    ;II1ax=radius_h+Ucentreh

    ;Thisassumptioncanbesatisfiedinpracticalsystems.

    ;WeknowthatC0andcontaintheoutputsetpoint

    ;r.So.rewriteCoandas

    ;C0():(s.)[r[R(+1)y1]AFTAu,]

    ;=

    ;s1rS2(24)

    ;where

    ;mc()/21,j/22,j?

    ;s=(S2E)…]

    ;October,2003

    ;(25)

    ;S2:(S.)[r~HbC(k

    ;forinputconstraints

    ;1)XFTH.c(k1)]

    ;EjH.c(k1)+u(k1)Ucentreh

    ;Uradiush

    ;EjM0

    ;B(1)dius_h——_————————r

    ;forinputincrements

    ;//1.

    ;J

    ;ejH.c(k1)

;a_h

    ;ejMa

    ;B(1)a_h

    ;andforoutputconstraints ;//1,

    ;j

    ;//2,j

    ;ejHbC(k1)Ycentreh

    ;Yradiush

    ;ejMb

    ;B(1)ydih

    ;SubstitutingtheunconstrainedoptimizationCointo

    ;theconstraintIIMC(k)Vl2II..?1,thenwe ;canobtain

    ;II(MSl2)r(MS2+v1)ll~?1—

    ;IIWlr—w2ll..?1

    ;ri?r?r(26)

    ;where

    ;x

    ;(min(,

    ;rmax

    ;wl=[,1,2

    ;2:[,l,2

    ;i?[1,3×

    ;i?[1,3×

    ;P

    ;,

    ;3×P]T

    ;,

    ;3×P]T

    ;So,theoutputsetpointrischeckedfirstlyateachsam ;piingtime.Calculatingtheminimumrandmaximum

    ;r(rminandrmax),ifr?[rminrmax],thenuseMWLS

    ;tocalculatec(k),whichmustsatisfytherequirement

    ;ofhardconstraintscompletely.Ifrrmin7”maxj,it ;meansthatwecannotfindoutc(k)thatsatisfiesthe

    ;demandsofhardconstraintswhenthesetpoint1sr?

    ;/,??

    ;,/

    ;\,?/

    ;P

    ;

    ;1

    ;一?1

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