第十一章多变量系统的耦合性分析

Chapter11InteractionAnalysisforMIMOSystemsLiShaoyuanE-mail:[e-mail]ONSInteractionMeasureofControlLoops2x2system(s)(s)mg(s)(s)mg(s)y(s)(s)mg(s)(s)mg(s)y22212122121111Onemanipulatedvariableaffectsmorethanonecontrolledvariabletransferfunctionmatrix-non-diagonalstructuretransmissioninteraction:existswhenchangeinonecontrollersetpointaffectsoutputthroughtheactionsofothercontroller(s)Experiment1Unitstepchangeinm1withallloopsopenBothy1andy2change,Atsteadystate,thechangeiny1asaresultofm1bey1m;y1m=K11Asnootherinputvariablechange,andallthecontrolloopsareopened,thereisnofeedbackcontrolinvolved.Experiment2Unitstepchangeinm1withLoop2closedDuetothechangeinm1:1.y1changesbecauseofg11,andiy2changebecauseg21,2.Loop2manipulatingm2untily2isrestoredtoitsinitialstate.3.Thechangesinm2returntoaffecty1viatheg12.Changesiny1fromtwosourcesDirectinfluenceofm1ony1;y1mTheretaliatoryactionfromLoop2counterm1ony2say,y1r.RelativeInteraction111*mryyy*1221111111122*1KKyKKKK*1111yymrmmyyy11111Thenetchangeiny1,y1*givenAtsteadystate,ItisgivenbyAninteractionmeasureSteady-StateProcessGainMatrixusesteady-stategainmatrixtoevaluateprocessbehaviour-interactionusefulgaininfofordeterminingcontrollooppairingsDistillationExample-GainMatrixKG()....000747006670117301253Relativemagnitudes,signsGsesesesesssss().......007471210066715101173117101253102132332RGARelativeGainArrayijijukjijykiyuyugainwithotherloopsopengainwithotherloopsclosedkkconstantconstant,,relativegain-ratioofgainwithotherloopsopentootherloopsclosedmathematicaldefinition:matrixofrelativegains-describescompletesteady-stateinteractionbehaviour11122122nnnnnn.....................212222111211canbecalculateddirectlyfromgainmatrixKintermsof“Hadamardproduct”-element-wisematrixmultiplication.*TKKPropertiesoftheRGAscale-independent-fromdefinitionroworcolumnelementssumto1Maybesensitivetoerrorsinthegains-becarefulusingempiricaltransferfunctionsClosed-loopsteady-stategainandopen-loopgainijijijKK1*NegativerelativegainforI-j:Changeinyiinresponsetomjoppositeinsignforotherloopsareopenandclosed.Theinput/outputpairingsarepotentiallyunstableandbeavoided.CALCULATINGRGA’SFROMFIRSTPRINCIPLESWeillustratetheprocedurebya2x2system.Forthesteady-state,themodelis:Foropen-loopconditionsKeepy2=0inchangesinm1andyield22212122121111mKmKymKmKy1111Kmyopenloopsall122212mKKm12221121111mKKKmKyTHEMATRIXMETHODFORCALCULATINGRGA’SLetLetTheelementsoftheRGAcanbeobtainedfromtheelementsofthesetwomatrices:KsGs)(lim0T-)(KR1ijijijKrLOOPPAIRINGUSINGRGAInterpretationoftheRGA--gainamplification-loopgainincreasedifotherloopsareclosed0ij0ij10ijLookatbothsignandmagnitude--gainreversaloccurswhenotherloopsareclosed-undesirable--openloopgainiszero-controlactiondependsonothercontrolloopsbeingclosed-undesirable–instabilityduetoincorrectcontrolleraction--lossofcontrolwhenotherloopsareclosed-gaingoestozero–undesirable1ij--notransmissioninteraction–open-loopgainunaffectedbyotherloops–desirable-one-wayinteractionstillpossible1ij--attenuationofopen-loopgainwhenotherloopsareclosedijijijijKK1*BASICLOOPPAIRINGRULESRule1:PairinputandoutputvariablesthathavepositiveRGAelementsthatareclosestto1.0.wheredenotesthepairedRGAelementscorrespondingtothekthalternativemin1.0kijkInterpretationoftheRGA-DistillationExamplerelativegainarray-09.609.509.509.6Gsesesesesssss().......007471210066715101173117101253102132332pairing-useu1-y1andu2-y2loops–avoidnegativerelativegainsADDITIONALRULESTheNiederlinskiCriteriontestforinstabilitybasedonsteady-stateinformationarrangetransferfunctionmatrixsothatpairingsareondiagonalloopscontainintegralactionsteadystate,neededforanalysisusingsteady-stategainsstable-processelementsmustbeopen-loopstablerationaltransferfunctionstransferfunctionsconsistofratiosofpolynomialsin“s”i.e.,nodeadtime,whichintroducesexponentialspropertransferfunctions-orderofdenominatorgreaterthanorderofnumeratorindividualcontrolloopsareclosed-loopstableNiederlinskiCriterionclosed-loopmulti-loopsystemisunstableifsufficientconditiontoidentifyinstability–ifconditionisnotsatisfied,wehavefailedtodetectinstability(asopposedtoconfirmingstability)--thisdoesNOTimplystabilityuseasascreeningtool-identifyproblemcases1(0)det((0))0..,0productof(0)diagonalelementsniiiGGNIiegExample130050.0118128.0135046.015117.0)(sssssGpEvaporatorprocessforproducingalumina:05.0128.0046.0117.0Krelativegainis0.5foreachpairing-nopreferenceindicatedre-arrangementofgainmatrixisn’tnecessarygiven1-1,2-2pairing(pairedtransferfunctionsareonmaindiagonal)•Note-notimedelays,assumingintegralcontrol,propertransferfunctions,open-loopstable1det((0))0.0117200.0059niiiGgNiederlinskitest:-instabilityisNOTdetectedExampleLooppairingfora3x3system311113111135)0(GK15.45.45.415.45.45.410ByRGA,the1-1/2-2/3-