上交自控讲义第九讲.ppt

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1、Module9Routh’sMethod,RootLocus:MagnitudeandPhaseEquationsRequirementsontheModuleGrasptherelationshipofSystemStabilitywithRootsofSystemCharacteristicEquationGraspthenecessaryandsufficientrequirementsforRouth’sCriterionUnderstandsomespecialsituationsGrasptheprincipleofRootLocus2DefinitionofLyapunov

2、stabilityThefundamentalconditionforanormallyworkingcontrolsystemisthatitisstableinclinetoanequilibriumstateinafinitetimeundernoinputbutinitialconditions(ordisturbances)MarginallystableAsymptoticallystable3LinearsystemstabilityanditsnecessaryandsufficientrequirementsThesystemwillbecameunstableasso

3、onasoneclosed-looppoleislocatedintheright-handhalfofthecomplexplaneReIm[s]4Systemresponsescorrespondingtoclosed-looppoles’location5DiversemethodsonsystemstabilitydeterminationforlinearsystemsMethodsdirectlydependedonroot:SolvingDifferentialequationRootlocusmethodsbydeterminingtherootscopeRouth’sm

4、ethodNyquistcriterionNyquistcriteriononBodediagramLyaponuvStabilityCriterion6IntroductionofRouth’sMethodTheclosed-looppolesaretherootsofthesystemcharacteristicequationWhythemethodsolvingequationisnottakenintouse?Solvingahigh-ordercharacteristicequationistediousorevenimpossiblewithoutusingnumerica

5、lmethodsItisunnecessarytocalculateexactvalueofeveryrootfordeterminingifthereisanyoneontheright-handhalfofthecomplexplanePrincipleofRouth’sStabilityCriterionRelationshipofrootswithcoefficientsforhigh-orderalgebraequations7Routh’scriterion:NecessaryrequirementThenecessaryrequirementsforastablesyst

6、emarethat,NozeroexistsamongthecoefficientsofD(s)AllcoefficientsofD(s)possessthesamesignSamesignNozero8Routh’scriterion:SufficientrequirementEstablishtheRoutharrayinthefollowingmanner:ThereminderoftherowelementsshouldbefilledwithzerosThenumberofclosed-looppolesintheright-handhalfcomplexplaneisequ

7、altothenumberofsignchangesoftheelementsofthefirstcolumnofRouth’sarray9Example:determinethestabilitySolution:Necessaryrequirementsaresatisfied:NozerocoefficientexistsAllcoefficientspossessthesamesignSufficientrequiremen