3 Decoupling.ppt

1 Thetransferfunctionmatrixofthesystemis 1 Definitionofdecouplingsystems 4 32 whereA BandCaren n n pandp nmatrices respectively Sincep q thisisasquarematrixdecouplingproblem 1 Problemformulation Consideradynamicsystemgivenby 4 2DecouplingbyStateFeedback 2 Definition System 4 32 issaidtobedecoupledifthetransferfunctionmatrixG s p qisnonsingularanddiagonal 3 Example Considerthefollowingsystem Itisclearthatthesystemisdecoupled Nowthecontrolproblembecomesasinglevariableproblem whosecontrollawiseasytodesign Fromy G s u wehave 4 Thedecouplingofmultivariablesystemsisoneofthemostimportantproblemsinmultivariablesystemscontrol Considerthefollowingsystem Becauseofthecoupling itisdifficulttodesignstablecontrollawsformultivariablesystemswithanicedynamicperformance 5 2 Decouplingbystatefeedbackp q i e thesystemissquare Thestatefeedbackcontrollawisu Kx Hv Hisnonsingular 4 34 where isthestatefeedbackmatrix isthenonsingularinput transformationmatrix 6 Hence theclosed looptransferfunctionmatrixwiththestatefeedbackcontrollaw is u Kx Hv Hisnonsingular 4 34 7 Problemstatement FindmatricesKandH suchthat isnonsingularanddiagonal i e 8 1 Relationshipbetweentheopen looptransferfunctionmatrixandtheclosed looptransferfunctionmatrix 2 Preliminarylemmas whereGf s andG s aretheclosed loopandopen looptransferfunctionmatrices respectively Lemma1 Therelationshipbetweentheopen looptransferfunctionmatrixandtheclosed looptransferfunctionmatrixisasfollows 9 2 NonnegativeintegersdiandnonzerovectorsEioftheopen looptransferfunctionmatrixG s Example Arealizationofg s 1 s3 3s2 2s 1 is Expressg s as Itcanbeverifiedthatthefirsttwoelementsoftheaboveprogressionarezerosandthethirdelementisnonzero 10 Inthegeneralcase lettheithrowofCbedenotedasciandtheithrowofG s bedenotedasGi s Then Gi s canbeexpressedas 11 thenwegetanonzerovectorsEiandanonnegativedi 0diistheleastintegersuchthat If but 12 Fromthedefinitionofand wehave 4 39 TheaboveanalysisindicatesthatitispossibletocomputediandEifromtwodifferentexpressionsG s and A B C 13 Example1 ComputediandEiforthefollowingG s d1 min 1 2 1 0 d2 min 2 2 1 1 Hence fromthedefinition wehave 14 15 c1B 10 d1 0 E1 10 c2B 01 d2 0 E2 01 Example4 5a Considerthefollowingsystem p33 ComputediandEi 16 Theithrowoftheopen looptransferfunctionmatrixcanbeexpressedas 3 Decouplingoftheclosed looptransferfunctionmatrix 1 Decouplingexpressionoftheopen looptransferfunctionmatrix 17 18 Theopen looptransferfunctionmatrixcanbeexpressedas p21 19 2 Decouplingofclosed looptransferfunctionmatrixFrom 20 Theclosed looptransferfunctionmatrixcanbeexpressedas S 2 p19 21 FromthedefinitionofdiandEi wecangetandoftheclosed loopsystems Notethat 4 Thenonnegativeintegersandnonzerovectorsofclosed loopsystems 22 Hence thereexistandsuchthat Lemma2 Proof Weonlyneedtoprove and 23 Wefirstprovethat 1 When 2 and Q E D 24 where Specially if 25 Theorem4 9 Thesystem canbedecoupledbythestatefeedbacku Kx Hvifandonlyifthematrix 4 32 isnonsingular 3 SufficientandNecessaryConditionforDecoupling 26 Proof Necessity WeonlyneedtoprovethatEisnonsingular Supposethesystemcanbedecoupledbythestatefeedbacku Kx Hv ThenGf s isdiagonalandnonsingular wherearetransferfunctions 27 Eisnonsingular Since Hence thereexistnonnegativeintegersandsuchthat isnonsingular 28 Sufficiency SubstitutingK E 1F H E 1 into S 2 yields K E 1F H E 1 29 4 49 where Q E D 30 Example4 5 TransformthesystemofExample4 5aintoanintegratordecoupledsystemandverifythatifthedecouplingcontradictsthestability c1B 10 d1 0 E1 10 c2B 01 d2 0 E2 01 Fromtheforgoingcomputation diandEiareasfollows p16 FromTheorem4 9 K E 1F H E 1 where 31 Hence 根据例题4 5a的计算可知E是单位阵 故系统可解耦 现采用定理4 9充分性证明中提供的 4 47 式将其化为积分器解耦系统 计算F阵 F1 c1A 001 F2 c2A 1 2 3 故得 32 Thefeedbackcontrollawis Thedynamicequationofclosed loopsystemis Thetransferfunctionmatrixoftheclosed looptransferfunctionmatrixis Theopen loopsystemiscontrollableandobservable buttheclosed loopsystemisunobservable which 33 whichmeansthataninstablemodehasbeencancelledandthedecouplingcontradictsthestability meansthatthestatefeedbackusedfordecouplinginthissystemchangestheobservability Computingtheeigenvaluesoftheclosed loopsystem wehave 34 Theorem4 10 Supposethatthesystemcanbedecoupledbystatefeedback and Then byusingthestatefeedback 4 Adecouplingcontrollaw 35 theclosed looptransferfunctionmatrixcanbetransformedinto wherekijareadjustableparameterswhichcanbeusedtoassignthepolesofthediagonalelements 36 Proof Consider SubstitutingH E 1andK E 1Dintotheaboveequationgives Hence weonlyneedtoprovethat 37 where 38 Theproofisdecomposedintothefollowingsteps 1 Prove A 1 2 Prove A 2 3 39 5 Itfollowsfrom A 1 A 4 that Q E D 4 Generally wehave 40 Theproblemiswhethertheclosed looptransformfunctionmatrixcanbetransformedinto Example Considerthefollowingdynamicequation byusingthestatefeedbacku Kx Hv Ifyes findKandH 41 c1B 0 1 d1 0 c2B 00 c2AB 21 d2 1 d1 d2 p 0 1 2 3 n Hence thematricesinthestatefeedbackcanbechosenas 42 modesunobservable Ifthedecouplingresultsincancelledpolesthatdonotlieinthelefthandsideplane wesaythedecouplingcontradictsthestability Ifthenthereexist 2 Othermethodsarerequiredif 5 Conclusionofdecouplingproblem p q 3 Ifasystemcanbedecoupledbyu Kx Hvand 1 Thesystemcanbedecoupledbyu Kx Hvif thenwecandecoupleitbyusing Theorem4 10 andthepolesofitsdiagonalelementscanbearbitrarilyassigned 43 Whethertheclosed loopsystemcanbeintegratordecoupledbystatefeedback Ifyes pleasegivethetransferfunctionmatrixafterdecoupling Whetherthedecouplingcontradictsthestability Example Considerthefollowingdynamicequation 44 c1B 11 d1 0 c2B 11 d2 0 Hence thesyste