3 Decoupling.ppt

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1、Thetransferfunctionmatrixofthesystemis1).Definitionofdecouplingsystems(4-32)whereA,BandCaren×n,n×pandp×nmatrices,respectively.Sincep=q,thisisasquarematrixdecouplingproblem.1.ProblemformulationConsideradynamicsystemgivenby§4-2DecouplingbyStateFeedback1Definition.System(4-32)issaidtobedecouple

2、difthetransferfunctionmatrixG(s)p=qisnonsingularanddiagonal.2Example.Considerthefollowingsystem:Itisclearthatthesystemisdecoupled.Nowthecontrolproblembecomesasinglevariableproblem,whosecontrollawiseasytodesign.Fromy=G(s)u,wehave3Thedecouplingofmultivariablesystemsisoneofthemostimportantprobl

3、emsinmultivariablesystemscontrol.Considerthefollowingsystem:Becauseofthecoupling,itisdifficulttodesignstablecontrollawsformultivariablesystemswithanicedynamicperformance.42).Decouplingbystatefeedbackp=q,i.e.,thesystemissquare;Thestatefeedbackcontrollawisu=Kx+Hv(Hisnonsingular)(4-34)whereisth

4、estatefeedbackmatrix.isthenonsingularinputtransformationmatrix.5Hence,theclosed-looptransferfunctionmatrixwiththestatefeedbackcontrollawisu=Kx+Hv(Hisnonsingular)(4-34)6Problemstatement:FindmatricesKandH,suchthatisnonsingularanddiagonal,i.e.,71).Relationshipbetweentheopen-looptransferfunction

5、matrixandtheclosed-looptransferfunctionmatrix.2.PreliminarylemmaswhereGf(s)andG(s)aretheclosed-loopandopen-looptransferfunctionmatrices,respectively.Lemma1:Therelationshipbetweentheopen-looptransferfunctionmatrixandtheclosed-looptransferfunctionmatrixisasfollows:82).Nonnegativeintegersdiandn

6、onzerovectorsEioftheopen-looptransferfunctionmatrixG(s)Example.Arealizationofg(s)=1/(s3+3s2+2s+1)isExpressg(s)asItcanbeverifiedthatthefirsttwoelementsoftheaboveprogressionarezerosandthethirdelementisnonzero.9Inthegeneralcase,lettheithrowofCbedenotedasciandtheithrowofG(s)bedenotedasGi(s).Then

7、,Gi(s)canbeexpressedas10thenwegetanonzerovectorsEiandanonnegativedi0diistheleastintegersuchthatIfbut11Fromthedefinitionofand,wehave(4-39)TheaboveanalysisindicatesthatitispossibletocomputediandEifromtwodifferentexpressionsG(s)and(A,B,C).12Example1.