PrinciplesofAutomaticControlPrinciplesofAutomaticControl自動(dòng)控制原理Topic5(Chapter6intext顧申申Shenshen(KevinPh.D.(CUHK),AssociateProfessorDepartmentofAutomationShanghaiReviewforthepreviousThreeobjectivesReviewforthepreviousThreeobjectivesindesigningacontrolsystem(Topic–––Transientresponse(Topic3);Stability(Thistopic);Steadystateerror(NextStabilityisthemostimportantsystemspecification.Ifasystemisunstable,transientresponseandsteady-stateerrorsaremootInthistopic,wewillstudyhowtodeterminewhetherasystemisstableornot.2NewterminologiesinthisColumnNewterminologiesinthisColumnQuadrantalsymmetrical象限對(duì)稱(chēng)不穩(wěn)Lefthalf-plane(lhpRighthalf-plane(rhpRouthtableRow3LearningOutcomesforTopicLearningOutcomesforTopicAftercompletingthistopic,youwillbeableMakeandinterpretabasicRouthtabletodeterminethestabilityofasystem;MakeandinterpretaRouthtablewhereeitherthefirstelementofarowiszerooranentirerowiszero.4BriefRouth-HurwitzRouth-HurwitzBriefRouth-HurwitzRouth-HurwitzCriterion:SpecialZeroOnlyintheFirstEntireRowisRouth-HurwitzCriterion:Additional5BriefThreeobjectivesindesigningacontrolsystemBriefThreeobjectivesindesigningacontrolsystem(Topic–––Transientresponse(Topic3);Stability(Thistopic);Steadystateerror(NextStabilityisthemostimportantsystemspecification.Ifasystemisunstable,transientresponseandsteady-stateerrorsaremootIfanengineermakesamistakeinhisstabilityanalysis,andwhathethinkisastablesystemisactuallyunstable:–––Unexpectedunboundedsystemresponse;Damagetoproperty;Injuryordeathtopeopleinthevicinity6WhatisTherearemanyWhatisTherearemanydefinitionsforstability,dependinguponthekindofsystemorthepointofview.––StabilitydefinitionforlinearsystemsfromtheviewpointofnaturalStabilitydefinitionforlinearsystemsfromtheviewpointoftotal7StabilitydefinitionforlinearsystemsfromtheviewpointofnaturalresponseTotalStabilitydefinitionforlinearsystemsfromtheviewpointofnaturalresponseTotalresponse=Naturalresponse+Forced–––Stable:Naturalresponsedecaystozeroastimeapproachesinfinity;Unstable:Naturalresponseincreaseswithoutbound;Marginallystable:NaturalresponseneitherdecaynorgrowwithoutboundbutThesedefinitionsrelyonadescriptionofthenaturalItmaybedifficulttoseparatethenaturalresponsefromtheforcedresponse.8StabilitydefinitionforlinearsystemsfromStabilitydefinitionforlinearsystemsfromtheviewpointoftotalresponseIftheinputisboundedandthetotalresponseisnotapproachinginfinityastimeapproachesinfinity,thenthenaturalresponseisobviouslynotapproachinginfinity.––Stable:IfeveryboundedinputyieldsaboundedoutputUnstable:Ifanyboundedinputyieldsanunbounded9HowtodeterminewhetherasystemisHowtodeterminewhetherasystemisstableorFocusonthenaturalresponsedefinitionsofThepolesofthetransferfunctiongeneratetheformofnaturalresponse.(Topic&p2=Exponentialdecay&p3c=Decayingoscillation&p4c=Pureoscillation=Exponentialp6&=IncreasingStablesystemshaveclosed-loopStablesystemshaveclosed-looptransferfunctionswithpolesONLYinthelefthalf-plane;Unstablesystemshaveclosed-looptransferfunctionswithatleastonepoleintherighthalf-planeand/orpolesofmultiplicitygreaterthan1ontheimaginaryaxis.Marginallystablesystemshaveclosed-looptransferfunctionswithonlyimaginaryaxispolesofmultiplicity1andpolesintheleftExampleofStableExampleofStableExampleofUnstableExampleofUnstableItisnotalwaysItisnotalwaysasimplemattertodetermineifafeedbackcontrolsystemisstable.Weknowthepolesoftheforwardtransferfunctioninthefollowingsystem,butwedonotknowthelocationofthepolesoftheequivalentclosed-loopsystem.WhatdoyoudowhentheWhatdoyoudowhentheclosedlooptransferfunctionpolynomialishorrendous?+54.32s233++0.0032s+Oneoption–plugintoacalculatororMatlabtosolveforGoodforcheckingaspecificsystemconfiguration,butwon’tgiveyouarangeofallowablesystemparameters.ExamplewhereyouhaveanunknownSpringconstantKExamplewhereyouhaveanunknownSpringconstantKinthedenominatorpolynomial:s234+++0.0032s+UsetrialanderrorvaluesofKtofindstablesystemconfigurations,butcouldbepainful!Solution–theRouth-HurwitzcriterionforThisisamethodforfindingouthowmanyclosed-loopsystempolesareinthelefthalfplane,righthalfplane,andontheimaginaryaxis.Doesn’ttelluswherethepolesarelocated,butthisdoesn’tmatterforsimplyworkingoutwhetherasystemisstableRouth,E.J.DynamicsofRouth,E.J.DynamicsofaSystemofRigidBodies,6thed.Macmillam,London,1905ThismethodrequirestwoGenerateadatatablecalledaRouthInterprettheRouthtabletotellhowmanyclosed-loopsystempolesareinthelefthalf-plane,therighthalf-plane,andonthejω-axis.Thepowerofthemethodliesindesignratherthananalysis.EdwardJohnRouth(1831-GeneratingaBasicRouthBeginbylabelingtheGeneratingaBasicRouthBeginbylabelingtherowswithpowersofsfromthehighestpowerofthedemoninatoroftheclosed-looptransferfunctiontos0.Nextstartwiththecoefficientofthehighestpowerofsinthedenominatorandlist,horizontallyinthefirstrow,everyotherInthesecondrow,listhorizontally,startingwiththenexthighestpowerofs,everycoefficientthatwasskippedinthefirstrow.Theremainingentriesarefilledinas–EachTheremainingentriesarefilledinas–Eachentryisanegativedeterminantofentriesintheprevioustworowsdividedbytheentryinthefirstcolumndirectlyabovethecalculatedrow.Theleft-handcolumnofthedeterminantisalwaysthefirstcolumnoftheprevioustworows,andtheright-handcolumnistheelementsofthecolumnaboveandtotheThetableiscompletewhenalloftherowsarecompleteddownto––MaketheRouthtablefortheMaketheRouthtableforthesystemshowninthefollowingForconvenience,anyrowoftheRouthtablecanbemultipliedbyaForconvenience,anyrowoftheRouthtablecanbemultipliedbyapositiveconstantwithoutchangingthevaluesoftherowsbelow.InterpretingtheBasicRouthTheRouth-Hurwitzcriteriondeclaresthatthenumberofrootsofthepolynomialthatareintherighthalf-planeisequalInterpretingtheBasicRouthTheRouth-Hurwitzcriteriondeclaresthatthenumberofrootsofthepolynomialthatareintherighthalf-planeisequaltothenumberofsignchangesinthefirstcolumn.Thus,thesystemisunstablesincetwopolesexistintherighthalf-plane.1-1.7068+1.7068-21234Oneminute(1)Isthissystemstable1234Oneminute(1)Isthissystemstableor(A)Stable(B)HowmanyrootslocatedintherighthalfOne(B)Two(C)ThreeD)SpecialCases1:ZeroOnlySpecialCases1:ZeroOnlyintheFirstIfthefirstelementofarowiszero,divisionbyzerowouldberequiredtoformthenextrow.Toavoidthisphenomenon,anextremelysmallpositiveε,isassignedtoreplacethezerointhefirstExample:DeterminethestabilityofExample:Determinethestabilityoftheclosed-looptransferfunctionThetableshowsasignThetableshowsasignchangefromthes3rowtothes2row,andtherewillbeanothersignchangefromthes2rowtothes1row.Hence,thesystemisunstableandhastwopolesintherighthalf-plane.SpecialCases2:EntireRowSpecialCases2:EntireRowisSometimeswhilemakingaRouthtable,wefindthatanentirerowconsistsofzerosbecausethereisanevenpolynomialthatisafactoroftheoriginalpolynomial.Thiscasemustbehandleddifferentlyfromthecaseofazeroinonlythefirstcolumnofarow.AnentirerowofzeroswillappearAnentirerowofzeroswillappearintheRouthtablewhenapurelyevenorpurelyoddpolynomialisafactoroftheoriginalpolynomial.+5s2+Evenpolynomialsonlyhaverootsthataresymmetricalabouttheorigin.Thissymmetrycanoccurunderthreeconditionsofrootposition:–––TherootsaresymmetricalandTherootsaresymmetricalandimaginary;Therootsarequadrantal.ItisthisevenpolynomialItisthisevenpolynomialthatcausestherowofzerostoappear.Therowprevioustotherowofzeroscontainstheevenpolynomialthatisafactoroftheoriginalpolynomial.everythingfromtherowcontainingtheevenpolynomialdowntotheendoftheRouthtableisatestofonlytheevenroots([1,1,-6,0,-1,----0.0000+roots([1,1,-6,0,-1,----0.0000+-0.0000-StabilitydefinitionforlinearsystemsfromStabilitydefinitionforlinearsystemsfromtheviewpointofnatural–––Stable:Naturalresponsedecaystozeroastimeapproachesinfinity;Unstable:Naturalresponseincreaseswithoutbound;Margin