华科电气自动控制理论m25

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1、Module25State-SpaceObserverDesignObjectives:&TheConceptoftheobserver&TheConditionofConstructingtheobserverofasystem&DesignofanobserverPlotthestate-variableblockdiagramofaclosed-loopsystemwithanobserver.25.1Observer–AmodelofthesystemunderstudyStateVariableFeedbackControlSystemx∈Rn,u∈Rp,y∈Rqr+ux+xIy

2、BCsA,B,C,Gn×nn×pq×np×n-+u=r−GxAx=Ax+Bu=(A−BG)x+BrGy=Cx用于极点配置Notallofthestatevariablesareavaible,Somedon’tcorrespondtophysicalvariables,andcostprohibitsinprovidingasensorforeachstatevariable.⎧X=AX+Buu⎨⎩y=CXWhere,xareassumedtobeunmeasureddirectly.Theapproachtakentosolvetheproblemisasfollowing:(1)To

3、constructamodelofthesystemunderstudy;(2)Assume(subjecttocertainrestrictions)thatthecomputedstatevariablesaregoodapproximationstothetruestatevariables;(3)Fromthesecomputedstatevariables,asuitablecontrollerfortheactualsystemmaybeconstructedusingthetechniquesdescribedinModule24.⎧X=AX+Bu⎨⎩y=CX+xyruIBC

4、s-+AG+xyIBCs+ANow,weconstructamodel{A,B,C}tosimulatetheactualsystem,{A,B,C}andassumetheparametermatrixA,B,CaregoodapproximationstoA,B,C⎧⎪X=AX+Bu⎧X=AX+BuModel{}A,B,C=⎨Model{}A,B,C=⎨⎪⎩y=CX⎩y=CXuButinxuxxmodelisBCydifferentfromxintheactualAsystem{A,B,C}xiscalledtheestimatedvalueof.x{A,B,C}≠{A,B,C}x

5、−x≠0xois/areunmeasureddirectly,xo≠xox−x⇒y−y←measurableuThiserrorsignal(y−y)isfedbacktotheobserversummingjunctiontocompensateforsuchdiscrepancies.——usingtheprincipleoffeedbackcontrolTodecreasetheerror,(thatiserroy−yr),wetaketox−xy−ycorrecttomakewellapproach:xxxu⎧X=AX+Bu∵⎨(1)⎩y=CX⎧X=AX+Bu+K(y−y)⎪⎪⎨

6、=AX+Bu+KCX−KCX(2)⎪⎪y=CX⎩whenA=A;B=B;C=C(1)−(2):∴X−X=AX−AX−KCX+KCXnamee=X−X,=(A−KC)(X−X)thene=(A−KC)eSelectthematrixKtomakethesolutionofthisequationonerrorbeconvergent(收敛的),then,e→0,orX→XObserver:X−X=(A−KC)(X−X)ThegainmatrixKisAfterthetransientresponseoftheobserverwrittenas:hasdisappeared,x→x⎡k

7、1⎤xchangs⇒xwillchange⎢k⎥K=⎢2⎥⎢#⎥⎢⎥y−y⎣kn⎦Conclusion:Thepolesoftheobserverofasystemmustbeplacedontheleftofthepolesofthissystemins-plane.Observer:X−X=(A−KC)(X−X)sI−()A−KC=0ThegainmatrixKiswrittenas:⎡k1⎤⎢⎥kK