Chapter-6-The-Stability-of-Linear-Feedback-System-

CurriculumSystemBasicconceptsSystemmodelingPerformanceissuesanalysiscorrection•Timedomain•Complexdomain•Frequencydomain•Definitionofstability•Howtodeterminethestabilityofasystem•TheRouth-HurwitzStabilityCriterion•DesignExamplesChapter6TheStabilityofLinearFeedbackSystemIntroductionTheproblemofstabilityofthecontrolsystemiscentraltocontrolsystemdesignAnunstableclosed-loopsystemisgenerallyofnopracticalvalueTwoconcepts:ThestabilityofafeedbacksystemitselfThebounded-inputandbounded-outputstabilityThischaptersolvestwoproblems:DefinitionofstabilityHowtodeterminethestabilityofasystem6.1TheConceptofStabilityAstablesystemisdefinedasasystemwithabounded(limited)systemresponsetoaboundedinput6.1TheConceptofStabilityThestabilityofadynamicsystemisdefinedinasimilarmanner.Theresponsetoadisplacement,orinitialcondition,willresultineitheradecreasing,neutral,orincreasingresponse.Linearsystemisstableifandonlyiftheabsolutevalueofitsimpulseresponse,g(t),integratedoveraninfiniterange,isfinite.Thelocationinthes-planeofthepolesofasystemindicatestheresultingtransientresponse.Thepolesintheleft-handportionofthes-planeresultinadecreasingresponse-stablesystemThepolesofthejω-axisandintheright-handplaneresultinaneutralandanincreasingresponse-unstablesystem6.1TheConceptofStabilityIntermsofthelinearsystems,weknowthatthestabilityrequirementmaybedefinedintermsofthelocationofthepolesoftheclosed-looptransferfunctionTheclosed-loopsystemtransferfunctioniswrittenas1.6)](2[)()()()()(112221QkRmmmmknMiissszsKsqspsTs6.1TheConceptofStabilityWhereq(s)=Δ(s)=0isthecharacteristicequationwhoserootsarethepolesoftheclosed-loopsystemTheoutputresponseforanimpulsefunctioninput(whereN=0)is:2.6)sin(1)(11mmtQkRmmmtkteBeAtymk6.1TheConceptofStabilityToobtainaboundedresponse,thepolesoftheclosed-loopsystemmustbeintheleft-handportionofthes-planeAnecessaryandsufficientconditionforafeedbacksystemtobestableisthatallthepolesofthesystemtransferfunctionhavenegativerealpartsThesystemisnotstableifnotallpolesareintheleft-hands-plane6.1TheConceptofStabilityIfthecharacteristicequationhassimplerootsontheimaginaryaxis(jω-axis)andallotherrootsintheleft-hands-plane,thesteady-stateoutputwillbesustainedoscillationsforaboundedinputIftheinputisasinusoidwhosefrequencyisequaltothemagnitudeofthejω-axisroots,theoutputbecomeunbounded6.1TheConceptofStabilityToascertainthestabilityofafeedbackcontrolsystem,onecoulddeterminetherootsofthecharacteristicpolynomialq(s)Therearethreeapproachestothequestionofstability1.Thes-planeapproach2.Thefrequencyplane(jω)approach3.Thetime–domainapproach6.1TheConceptofStability6.2TheRouth-HurwitzStabilityCriterionTheRouth-HurwitzstabilitymethodprovidesananswertothequestionofstabilitybyconsideringthecharacteristicequationofthesystemThecharacteristicequationintheLaplacevariableiswrittenastoascertainthestabilityofthesystem,itisnecessarytodeterminewhetheranyoneoftherootsofq(s)liesintheright-halfofthes-plane3.60)()(0111asasasasqsnnnnIfEq.(6.3)iswritteninfactoredform,wehaveWewillfinethatWenotethatallthecoefficientsofthepolynomialmusthavethesamesignifalltherootsareintheleft-handplaneAllthecoefficientsarenonzero4.60)())((21nnrsrsrsa0)1()()()()(2134213212323121121nnnnnnnnnnnnrrrasrrrrrrasrrrrrrasrrrasasq6.2TheRouth-HurwitzStabilityCriterionTheRouth-Hurwitzcriterionisbasedonorderingthecoefficientsofthecharacteristicequationintoanarrayorscheduleasfollows13121006.8nnnnnnasasasasa531421nnnnnnnnaaaaaass6.2TheRouth-HurwitzStabilityCriterionFurtherrowsoftheschedulearethencompletedasfollows1531531531420321nnnnnnnnnnnnnnnnnhcccbbbaaaaaasssss6.2TheRouth-HurwitzStabilityCriterionWhereand5141331211321111)())((nnnnnnnnnnnnnnnnnaaaaabaaaaaaaaaab3131111nnnnnnbbaabc6.2TheRouth-HurwitzStabilityCriterionTheRouth-Hurwitzcriterionstatesthatthenumberofrootsofq(s)withpositiverealpartsisequaltothenumberofchangesinsignofthefirstcolumnoftheRoutharrayForthestablysystem,theremustbenochangesinsigninthefirstcolumnTherearefourdistinctcasesreconsidered1.noelementinthefirstcolumniszero2.thereisazero,butsomeotherinthesamerowarenonzero3.thereisazero,buttheotherinthesamerowarealsozero4.asin3withrepeatedrootsonthejω-axis6.2TheRouth-HurwitzStabilityCriterionSomeexample:case1second-ordersystem0122)(asasasq001102012baaasss010211201101)0(aaaaaaaaab6.2TheRouth-HurwitzStabilityCriterionThird-ordersystem012233)(asasasasq001102130123cbaaaassss01011230121ababcandaaaaab6.2TheRouth-HurwitzStabilityCriterionCase2:1011422)(2345ssssssq000010110010642102111012345dcssssss121241c6106111ccd6.2TheRouth-HurwitzStabilityCriterionunstablesystemksssssq234)(0000001111101234kkkcssssskkc16.2TheRouth-HurwitzStabilityCriterionCase3:Zerosinthefirstcolumn,andotherelementsoftherowcontainingthezeroarealsozeroThisconditionoccurswhenthepolynomialcontainssingularitiesthataresymmetricallylocated