Time‐Scale and Eigenstructure Assignment L S
Time‐ScaleandEigenstructure Assignment LINEAR SYSTEMS &CONTROL ~PAGE 131 BEN M.CHEN,NUSECE ThekeyideabehindtheATEAdesignistodecomposethegivensysteminto varioussubsystemsusingthespecialcoordinatebasis(SCB)techniqueand thentacklethesubsystemsonebyoneinaccordancewiththeirstructural propertiesthroughtheselectionofappropriatecontrolgains… LINEAR SYSTEMS &CONTROL ~PAGE 132 BEN M.CHEN,NUSECE badsubsystems goodsubsystems fastsubsystems nonexistentfor strictlyproper systems LINEAR SYSTEMS &CONTROL ~PAGE 133 BEN M.CHEN,NUSECE badsubsystems LINEAR SYSTEMS &CONTROL ~PAGE 134 BEN M.CHEN,NUSECE goodsubsystems LINEAR SYSTEMS &CONTROL ~PAGE 135 BEN M.CHEN,NUSECE fastsubsystems highgains LINEAR SYSTEMS &CONTROL ~PAGE 136 BEN M.CHEN,NUSECE oneofthepricespaid forcontrollingbad subsystemsishighgain TheOverall StateFeedbackGain LINEAR SYSTEMS &CONTROL ~PAGE 137 BEN M.CHEN,NUSECE stability ? LINEAR SYSTEMS &CONTROL ~PAGE 138 BEN M.CHEN,NUSECE H2 Control,H Control,Disturbance DecouplingProblems LINEAR SYSTEMS &CONTROL ~PAGE 139 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 140 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 141 BEN M.CHEN,NUSECE Thedisturbancew canbeattenuatedfromthecontrolledoutputifandonly if Es 0 Im(E ) S ( A, B , C , D) spanofX a X c X d Thisconditionisautomaticallysatisfiedifthegivensystem(A,B,C,D)isof minimumphaseandrightinvertible. LINEAR SYSTEMS &CONTROL ~PAGE 142 BEN M.CHEN,NUSECE H2 Control 道 LINEAR SYSTEMS &CONTROL ~PAGE 143 BEN M.CHEN,NUSECE H Control disturbance rejection 道 LINEAR SYSTEMS &CONTROL ~PAGE 144 BEN M.CHEN,NUSECE 本书的内容充分表明这种系统结构分解方 法在适用领域上的广泛性和系统分析与综合上 的方便性，如在结构分解的框架下，本书中定 理 10.2.2 就简单明了地揭示了H2 控制和H 控 制以及干扰解耦控制之间的内在关系。。。 郑大钟 教授 2007年7月于北京清华大学 LINEAR SYSTEMS &CONTROL ~PAGE 145 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 146 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 147 BEN M.CHEN,NUSECE H2 Control LINEAR SYSTEMS &CONTROL ~PAGE 148 BEN M.CHEN,NUSECE H2 Control LINEAR SYSTEMS &CONTROL ~PAGE 149 BEN M.CHEN,NUSECE H Control LINEAR SYSTEMS &CONTROL ~PAGE 150 BEN M.CHEN,NUSECE Anexplicitly parameterized gainmatrix intermsof and optimalvalue* H Control LINEAR SYSTEMS &CONTROL ~PAGE 151 BEN M.CHEN,NUSECE Measurementfeedbackcases… Considerastabilizable anddetectablelineartime‐invariantsystem withaproper controllerC w z u x A x B u E w : y C1 x 0 u D1 w z C x D u D w 2 2 22 y C Wewillonlyfocusonthe casewhenD22 eitheris0or canbemadeto0. v AC v BC y C : u CC v DC y x R n statevariable p y R measurement z R q controlledoutput LINEAR SYSTEMS &CONTROL ~PAGE 152 m u R controlinput & l w R disturbance k v R controllerstate BEN M.CHEN,NUSECE H2 controlwithmeasurementfeedback… D22 =0 1. FindastatefeedbackgainmatrixF forthequadruple(A,B,C2,D2)using theATEAmethodforH2 control(orotherappropriatemethods); 2. FindanobservergainmatrixK forthequadruple(A’,C’1,E’,D’1)usingthe ATEAmethodforH2 control(orotherappropriatemethods); 3. TheH2 controloutputfeedbacklawisthengivenby v A BF KC1 v K y C : F v u H2 Control LINEAR SYSTEMS &CONTROL ~PAGE 153 BEN M.CHEN,NUSECE H Control fairlycommon assumptions LINEAR SYSTEMS &CONTROL ~PAGE 154 BEN M.CHEN,NUSECE H Control LINEAR SYSTEMS &CONTROL ~PAGE 155 BEN M.CHEN,NUSECE H Control LINEAR SYSTEMS &CONTROL ~PAGE 156 BEN M.CHEN,NUSECE If(A,E,C1,D1)isleftinvertibleandofminimumphase,thenitcanbeprovedthat Q =0andH controllawcanbemuchsimplified… H Control LINEAR SYSTEMS &CONTROL ~PAGE 157 BEN M.CHEN,NUSECE H disturbancedecouplingwithmeasurementfeedback… disturbance rejection LINEAR SYSTEMS &CONTROL ~PAGE 158 BEN M.CHEN,NUSECE H disturbancedecouplingproblem… disturbance rejection LINEAR SYSTEMS &CONTROL ~PAGE 159 BEN M.CHEN,NUSECE H disturbancedecouplingsolvabilityconditions… common assumptions itisentirespace ifnoj axiszero itisemptyspace ifnoj axiszero couplingterm …… W.M.Wonham LINEAR SYSTEMS &CONTROL ~PAGE 160 JanC.Willems CarstenScherer BugsBunny BEN M.CHEN,NUSECE Aspecialbutveryusefulcase… IfD22 =0,thencondition3isautomaticallysatisfied… disturbance rejection LINEAR SYSTEMS &CONTROL ~PAGE 161 BEN M.CHEN,NUSECE Exercise LINEAR SYSTEMS &CONTROL ~PAGE 162 BEN M.CHEN,NUSECE Exercise LINEAR SYSTEMS &CONTROL ~PAGE 163 BEN M.CHEN,NUSECE Exercise LINEAR SYSTEMS &CONTROL ~PAGE 164 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 165 BEN M.CHEN,NUSECE RobustandPerfectTrackingControl LINEAR SYSTEMS &CONTROL ~PAGE 166 BEN M.CHEN,NUSECE Consider thesystem theassumptions thecontroller theproblem LINEAR SYSTEMS &CONTROL ~PAGE 167 BEN M.CHEN,NUSECE ThesolvabilityconditionsforRPTcontrolproblem… Condition1isnecessaryforallcontrolproblem.Condition2isautomatically satisfiedwhenD22=0. LINEAR SYSTEMS &CONTROL ~PAGE 168 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 169 BEN M.CHEN,NUSECE Rewritethegivenreferencesignalinastatespaceformas anartificial system LINEAR SYSTEMS &CONTROL ~PAGE 170 BEN M.CHEN,NUSECE DesignanappropriatestatefeedbackgainusingtheATEA techniqueandthen TheRPTcontrollawisgivenby LINEAR SYSTEMS &CONTROL ~PAGE 171 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 172 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 173 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 174 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 175 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 176 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 177 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 178 BEN M.CHEN,NUSECE LINEAR SYSTEMS &CONTROL ~PAGE 179 BEN M.CHEN,NUSECE
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