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1、MassachusettsInstituteofTechnologyDepartmentofElectricalEngineeringandComputerScience6.245:MULTIVARIABLECONTROLSYSTEMSbyA.Megretski�StandardLTIFeedbackOptimizationSetup1Thetwomostfrequentlyusedtechniquesofoptimization-basedfeedbackdesignareH2andH-Infinityoptimization.Botharenormallyappliedtowhat
2、canbereferredtoasthestandardLTIfeedbackoptimizationsetup,tobethemainsubjectofdiscussioninthislecture.1.1BasicConceptsTherearethreebasicconceptsbehindthestandardfeedbackoptimizationsetup:thenotionofamulti-input,multi-output(MIMO)lineartime-invariant(LTI)finiteordersystem,thenotionofaninternallyst
3、ablefeedbackinterconnectionoftwoMIMOLTIsystems,andthenotionofasystemnorm.Inaddition,thereisaconceptofwell-posednessoftheresultingoptimizationproblem,whichhastobesatisfied(otherwisetheavailableoptimizationalgorithmsareguaranteedtobreakdown).1.1.1ComponentsofastandardsetupInordertodefineastandardLT
4、Ifeedbackoptimizationproblem,sketchedonFigurefig:PKstandard,onehastospecifytheplantP,andaperformancemeasure.�cA.Megretski,20041VersionofFebruary4,20042wzPuyK�Figure1.1:StandardLTIfeedbackoptimizationsetupTheplantisafiniteorderlineartime-invariantsystemwithtwoinputs,wandu,andtwooutputs,zandy.Input
5、sandoutputscanbemulti-variable.Thefirstinputwiscalleddisturbance,andistypicallyusedtodescribeexternalnoisesandinternalperturbationscausedbynonlinearityanduncertainty.Thesecondinputuiscalledactuator,anddescribestheoutputofthefeedbackcontrollerK(tobedesignedviaoptimization).Thefirstoutputziscalledc
6、ost,andrepresentssignalswhichthedesignerwantstobesmall.Finally,thesecondoutputy,usuallyreferredtoassensor,ormeasurement,representsinputofthecontrollerK(tobedesigned).Theperformancemeasurespecifiesaparticularquantitivemeasureof“smallness”forthecostoutputz.TheoptimizationprocesswillaimatfindinganLT
7、IfeedbacksystemKwhichmakesthefeedbacksystemonFigurefig:PKstandardstable,and“minimizes”theclosedloopsystemfromwtoz.Twomeasures(alsocallednorms)ofhow“large”astableLTIsystemisaremostpopularinfeedbackoptimization:H2normandH-Infinitynorm
