现代控制理论chapter5-Linear-Feedback-Control-System

ModernControlTheory沈阳航空航天大学自动化学院Chapter5LinearFeedbackControlSystemModernControlTheory沈阳航空航天大学自动化学院outline5-1Introduction5-2Statefeedbackandoutputfeedback5-3Pole-Placement5-5StateEstimatorModernControlTheory沈阳航空航天大学自动化学院5.1.1AnalysisandDesign5-1IntroductionGiventhesystem,designofthestructureandparametersofcontrolleranalysisResponse,controllability,observabilitystabilityquantitativeandqualitativeanalysisAnalysisandDesignControltheperformanceofthesystemtomeettheexpectedrequirementsgoalfoundationModernControlTheory沈阳航空航天大学自动化学院5-2Statefeedbackandoutputfeedback5.2.1statefeedbackGeneralexpression()(),()utfrxtLITSexpression()()()utLrtKxtor()()utrtKx()()xAxBLrKxABKxBLrModernControlTheory沈阳航空航天大学自动化学院Theintroductionofstatefeedbacknotonlychangestheinputmatrix,butalsochangesthestatematrix.Asresult,itchangestheeigenvaluesofthesystemandthepolesoftransferfunction,anditisclearlyimportantandmeaningful.()()xAxBLrKxABKxBLrModernControlTheory沈阳航空航天大学自动化学院5.2.2outputfeedbackBylinearfeedbacklawurHyrHCxStateandoutputequationsas()xABHCxBuyCxClosed-looptransferfunctionmatrix:1()HGCsIABHCBModernControlTheory沈阳航空航天大学自动化学院5.2.3DiscussiononStateFeedbackandOutputFeedback1Theintroductionoffeedbackdoesnotaddanewstatevariable,namelytheclosed-loopsystemandopen-loopsystemhavethesamedimension.2Twoformscanmaintainthecontrollability,butnottheobservability.Forthestatefeedback,thesystemmaynotbeabletomaintaintheoriginalobservability;fortheoutputfeedback,theclosed-loopsystemwillbeabletomaintaintheobservability.ModernControlTheory沈阳航空航天大学自动化学院3Outputfeedbackiseasilyrealizedinproject.Butstatefeedbackhasbetterfeatures.butwiththedevelopmentofthetheoryabouttheobserverandtheKalmanfilter,thephysicalrealizationofstatefeedbackhasbeenbasicallysolved.Sogenerallyspeaking,statefeedbackisamoreadaptivefeedback.Ofcourse,weselecttheformaccordingtotheactualsituation.5.2.3DiscussiononStateFeedbackandOutputFeedbackModernControlTheory沈阳航空航天大学自动化学院explainFeedbacksystemwithstateestimatorWecannotmeasurethestatesphysicallyWhy?AnswerWecanconstructthestatesbasedoninputandoutput-----stateestimatorFeedbackcontrolsystemwithstateestimatorModernControlTheory沈阳航空航天大学自动化学院Note1.Reconstructionvaluesofstatevariablesaregraduallyequaltotheirvalue.2.Theorderofobserver-statefeedbacksystemisequaltosumofthecontrolledsystemorderandtheestimatororder,thatis,thesystemstatevariablesofestimatorcomposetheclosed-loopsystemstatevariables.ModernControlTheory沈阳航空航天大学自动化学院5-3Pole-placement5.3.1questionThedynamiccharacteristicsofasystemisdeterminedbythepoles,sowecanplacethepolestomeettheresponse.principlestoselectpolesModernControlTheory沈阳航空航天大学自动化学院（1）n-dimensionalcontrolsystem–npoles。（2）PluralpolesmustbeConjugatepoles（3）weneedtostudytheimpactofquality,aswellasdistributionandtherelationshipbetweenthezerowhenselectthedesiredpoleposition.（4）Musttakeintoaccountthesensitivityofanti-jammingprinciplestoselectpolesModernControlTheory沈阳航空航天大学自动化学院5.3.2SISOpole-placementThesystemisasfollowingxAxBuyCxItiscontrollable，whereA、B、Care,1,1nnnnrealconstantmatrices,respectively.Alinearstatefeedbackisdefinedby()()()utrtKxtDVxDKCy)(BVxBKAKxVBAxx)()(Thedynamicequationsofthefeedbacksystemsisas:ModernControlTheory沈阳航空航天大学自动化学院OutputfeedbackHyVuVDDHIBHBxCDHIBHAHyVBAxx])([])([)(11DVDHICxDHIy11)()(ModernControlTheory沈阳航空航天大学自动化学院proveThesystem’scharacteristicpolynomial:111()nnnnfssIAsasasaclosed-loopsystemcharacteristicpolynomialbythestatefeedback:111()nnnnnfssIABKsasasaFindnon-singulartransformationmatrixP，ˆxPxModernControlTheory沈阳航空航天大学自动化学院11110ˆ0nnnIAPAPaaaˆ[001]TBPB　　　　　　111ˆ[]nCCPn　　1ˆˆˆurKPxrKx1ˆKKPˆˆˆˆˆxAxBuyCx--controllablecanonicalformModernControlTheory沈阳航空航天大学自动化学院Havebeentransformed,closed-loopsystemcharacteristicpolynomialissuppose12ˆˆˆˆ[]nKkkkThestatematrixofClosed-loopsystem1110ˆˆˆ0ˆˆ()()nnnIABKakak11121ˆˆˆˆˆˆ()()()()nnnnnsIABKsaksaksak****112121ˆˆ()()()nnnnakaakaaka　　　　　　ModernControlTheory沈阳航空航天大学自动化学院Puttheeigenvaluesoftheclosed-loopsystemtothedesiredlocationbasedonnon-singulartransformation.*11*122*1ˆˆˆnnnnkaakaakaa　　ModernControlTheory沈阳航空航天大学自动化学院Specificsteps（1）ForSISO(controllablecanonicalform),thestatefeedbackmatrixKis***121111nnnnnKkkkaaaaaa(1,2,,)iainarecharacteristicpolynomialcoefficients*(1,2,,)iainarehopefulcharacteristicpolynomialcoefficientsModernControlTheory沈阳航空航天大学自动化学院（2）non-singulartransformation1ˆˆˆAPAP-1　　B=PB　　C=CP1ˆˆˆˆˆˆABKP-1-1（A-BK）P=PAP-PBKP***1111ˆnnnnKKPaaaaaaPModernControlTheory沈阳航空航天大学自动化学院（3）Thenon-singulartransformationdoesnotchangethetransferfunction121121***11ˆˆˆ()()nnnnnnnsssGsCsIABsasasaThetransferfunctionbasedonstatefeedbackis121121***11ˆˆˆˆˆ()()nnnnKnnnsssGsCsIABKBsasasa在按状态反馈组成的闭环系统中，其闭环零点和开环零点是相同的。ModernControlTheory沈阳航空航天大学自动化学院（4）supposethesystemiscontrollableandobservable,henceweget1111100001nnn