控制工程3(英文).ppt

Ch6TheStabilityofLinearFeedbackSystems TheconceptofstabilityTheRouth HurwitzstabilitycriterionTherelativestability 6 1Theconceptofstability Astablesystemisadynamicsystemwithaboundedoutputtoaboundedinput BIBO Theissueofensuringthestabilityofaclosed loopfeedbacksystemiscentraltocontrolsystemdesign Anunstableclosed loopsystemisgenerallyofnopracticalvalue absolutestability relativestability Absolutestability Wecansaythataclosed loopfeedbacksystemiseitherstableoritisnotstable Thistypeofstable notstablecharacterizationisreferredtoasabsolutestability Relativestability Giventhataclosed loopsystemisstable wecanfurthercharacterizethedegreeofstability Thisisreferredtoasrelativestability 6 2TheRouth Hurwitzstabilitycriterion where Anecessaryandsufficientconditionforafeedbacksystemtobestableisthatallthepolesofthesystemtransferfunctionhavenegativerealparts Anecessarycondition Allthecoefficientsofthepolynomialmusthavethesamesignandbenonzeroifalltherootsareinleft handplane LHP Thecharacteristicequationiswrittenas HurwitzandRouthpublishedindependentlyamethodofinvestigatingthestabilityofalinearsystem Thenumberofrootsofq s withpositiverealpartsisequaltothenumberofchangesinsignofthefirstcolumnoftheRoutharray Routh Hurwitzstabilitycriterion CASE1Noelementinthefirstcolumniszero CASE2Zerointhefirstcolumnwhilesomeotherelementsofrowcontainingazerointhefirstcolumnarenonzero CASE3Zerosinthefirstcolumn andotherelementsoftherowcontainingthezeroarealsozero Considerthecharacteristicpolynomial TheRoutharrayis Case3 Considerthecharacteristicpolynomial TheRoutharrayis Theauxiliarypolynomial Designexample weldingcontrol 6 3Therelativestability Therelativestabilityofasystemcanbedefinedasthepropertythatismeasuredbytherelativerealpartofeachrootorpairofroots Axisshiftandexamples Considercontrolsystem DeterminetherangeofKsatisfyingthestabilityandallpoles 1 TheRoutharrayis TheRoutharrayis Let weobtain Designexample Trackedvehicleturningcontrol Summary Inthischapter wehaveconsideredtheconceptofthestabilityofafeedbackcontrolsystem Adefinitionofastablesystemintermsofaboundedsystemresponsewasoutlinedandrelatedtothelocationofthepolesofthesystemtransferfunctioninthes plane TheRouth Hurwitzstabilitycriterionwasintroduced andseveralexampleswereconsidered Therelativestabilityofafeedbackcontrolsystemwastransferfunctioninthes plane Assignment E6 1E6 4E6 5E6 8 Ch7TheRootLocusMethod Maincontent TheRootLocusConceptTheRootLocusProcedureParameterDesignbytheRootLocusmethodSensitivityandtheRootLocusThree term PID ControllersTheRootLocususingMATLAB 7 1TheRootLocusConcept Theresponseofaclosed loopfeedbacksystemcanbeadjustedtoachievethedesiredperformancebyjudiciousselectionofoneormoreparameters Thelocusofrootsinthes planecanbedeterminedbyagraphicalmethod TherootlocusmethodwasintroducedbyEvansin1984andhasbeendevelopedandutilizedextensivelyincontrolengineeringpractice Therootlocusisthepathoftherootsofthecharacteristicequationtracedoutinthes planeasasystemparameterischanged Closed loopcontrolsystemwithavariableparameterK unityfeedbackcontrolsystem thegainKisavariableparameter 7 2TheRootLocusProcedure Step1 Writethecharacteristicequationas1 F s 0Andrearrangetheequation Ifnecessary sothatthepolynomialintheformpolesandzerosfollows 1 KP s 0Step2FactorP s ifnecessary andwritethepolynomialintheformofpolesandzerosasfollows Step3Locatethepolesandzerosonthes planewithselectedsymbols Thelocusoftherootsofthecharacteristicequation1 KP s 0beginsatthepolesofp s andendsatthezerosofp s asKincreasesfrom0toinfinity withnpolesandMzerosandn M Step4Therootlocusontherealaxisalwaysliesinasectionoftherealaxistotheleftofanoddnumberofpolesandzeros Step5Determinethenumberofseparateloci SL thenumberofseparatelociisequaltothenumberofpoles Example7 1Second ordersystem Step6Therootlocimustbesymmetricalwithrespecttothehorizontalrealaxiswithangles Step7Therootlociproceedtothezerosatinfinityalongasymptotescenteredatandwithangles TheselinearasymptotesarecenteredatapointontherealaxisgivenbyTheangleoftheasymptoteswithrespecttotherealaxisis Example7 2Fourth ordersystem Step8Determinethepointatwhichthelocuscrossestheimaginaryaxis ifitdoesso usingtheRouth Hurwitzcriterion TheactualpointatwhichtherootlocuscrossestheimaginaryaxisisreadilyevaluatedbyutilizingtheRouth HurwitzCriterion Step9Determinethebreakawaypointontherealaxis ifany LetorStep10TheangleoflocusdeparturefromapoleisTheangleoflocusarrivalfromazerois Step11Determinetherootlocationsthatsatisfythephasecriterionatroot Thephasecriterionisq 1 2 Step12Determinetheparametervalueataspecificrootusingthemagnituderequirement Themagnituderequirementatis Example7 4Fourth ordersystem 7 3ParameterDesignbytheRootLocusmethod Thismethodofparameterdesignusestherootlocusapproachtoselectthevaluesoftheparameters Theeffectofthecoefficienta1maybeascertainedfromtherootlocusequation 7 4SensitivityandtheRootLocus TherootsensitivityofasystemT s canbedefinedas thesensitivityofasystemperformancetospecificparameterchanges wehave 7 5Three term PID Controllers Thecontrollerprovidesaproportionalterm anintegrationterm andaderivativeterm Summary Inthischapter wehaveinvestigatedthemovementofthecharacteristicrootsonthes planeasthesystemparametersarevariedbyutilizingtherootlocusmethod Therootlocusmethod agraphicaltechnique canbeusedtoobtainanapproximatesketchinordertoanalyzetheinitialdesignofasystemanddeterminesuitablealterationsofthesystemstructureandtheparametervalues Furthermore weextendedtherootlocusmethodforthedesignofseveralparametersforaclosed loopcontrolsystem Thenthesensitivityofthecharacteristicrootswasinvestigatedforundesiredparametervariationsbydefiningarootsensitivitymeasure Assignment E7 4E7 8 Ch8FrequencyResponseMethods BasicconceptoffrequencyresponseFrequencyresponseplotsDrawingtheBodediagramPerformancespecificationinthefrequencydomain 8 1Basicconceptoffrequencyresponse Thefrequencyresponseofasystemisdefinedasthesteady stateresponseofthesystemtoasinusoidalinputsignal Theresultingoutputsignalforalinearsystem isalsoasinusoidalinthesteadystate itdiffersfromtheinputwaveformonlyinamplitudeandphaseangle Letinput TheLaplacetransformation Theoutput undeterminedcoefficient iscomplexvector FrequencyCharacteristics TransferfunctionandLaplacetransformFrequencycharacteristicsandFouriertransform Frequencycharacteristic Transferfunctionanddifferentialequationareequivalentinrepresentationofsystem FrequencycharacteristicandTransferfunction Computationoffrequencyresponse 8 2Frequencyresponseplots PolarplotBodediagramNicholschartFrequencyresponseplotsoftypicalelements frequencyresponseofanRCfilter Theprimaryadvantageofthelogarithmicplotistheconversionofmultiplicativefactorintoadditivebyvirtueofthedefinitionoflogarithmicgain BodediagramofanRCfilter Nicholschart 0o 180o 180o w 0 20dB 20dB Frequencyresponseplotsoftypicalelements GainPoleatoriginZeroatorigin Poleontherealaxis jwT 1 Zeroontherealaxis jwT 1 TwocomplexpolesTwocomplexzeros Bodediagramofatwin Tnetwork 8 3DrawingtheBodediagram DrawingBodediagram 1 2 DrawtheasymptoticapproximationofL inthelowfrequencyrange 3 Changetheslopeatthebreakfrequency 4 Thi