h2-NORM OPTIMAL MODEL REDUCTION FOR LARGE-SCALE DI

Zentrumf¨urTechnomathematikFachbereich3–MathematikundInformatikh2-normoptimalmodelreductionforlarge-scalediscretedynamicalMIMOsystemsAngelikaBunse-GerstnerDorotaKubali´nskaGeorgVossenDanielWilczekReport07–04BerichteausderTechnomathematikReport07–04August2007h2-NORMOPTIMALMODELREDUCTIONFORLARGE-SCALEDISCRETEDYNAMICALMIMOSYSTEMSA.BUNSE-GERSTNER,D.KUBALI´NSKA,G.VOSSEN,ANDD.WILCZEKAbstract.Modelingstrategiesoftenresultindynamicalsystemsofveryhighdimension.Itisthendesirabletofindsystemsofthesameformbutoflowercomplexity,whoseinput-outputbehaviorapproximatesthebehavioroftheoriginalsystem.Hereweconsiderlineartimeinvariant(LTI)discretetimedy-namicalsystems.Thecornerstoneofthispaperisarelationbetweeenoptimalmodelreductionintheh2-normand(tangential)rationalHermiteinterpola-tion.Firstordernecessaryconditionsforh2-optimalmodelreductionarepre-sentedfordiscreteMultiple-Input-Multiple-Output(MIMO)systems.Theseconditionsleadtoanoptimalchoiceofinterpolationdataandanewefficientalgorithmforh2-optimalmodelreductionforMIMOsystems.ItisalsoshownthattheconditionsareequivalenttotwoknownLyapunovbasedfirstordernecessaryconditions.Numericalexperimentsdemonstratetheapproximationqualityofthemethod.1.IntroductionThepurposeofmodelorderreductionistoreplacealargemodelbyasmallerone,whichpreservestheessentialbehavioroftheoriginalmodel.Forthesystemsconsideredinthispaper,itcanbestatedasfollows:Problem:GiventheLinearTimeInvariant(LTI),discretetimedynamicalsysteminstate-spacerepresentation:Σ:xk+1=Axk+Buk,yk=Cxk(1.1)orequivalently,inthefrequencydomain,representedbyitstransferfunctionH(s):=C(sIN−A)−1B(1.2)whereA∈CN×N,B∈CN×mandC∈Cp×N.Thevectorsxk∈CN,yk∈Cpanduk∈Cmarethestate,outputandtheinputofthesystemattimetk,respectivelyandNisverylarge.Itwillbeassumedthroughoutthepaperthatthesystemisstable,thatis,alleigenvaluesofAlieinsidetheunitcircle,observableandreachable.Constructareduced-ordersystemˆΣ:ˆxk+1=ˆAˆxk+ˆBuk,ˆyk=ˆCˆxk(1.3)withtransferfunctionˆH(s)=ˆC(sIn−ˆA)−1ˆB(1.4)Keywordsandphrases.ModelReduction,RationalInterpolation,TangentialInterpolation,discreteSylvesterEquation,h2Approximation.12A.BUNSE-GERSTNER,D.KUBALI´NSKA,G.VOSSEN,ANDD.WILCZEKwhereˆA∈Cn×n,ˆB∈Cn×m,ˆC∈Cp×nandnN,whoseinput-outputbehaviorapproximatestheinput-outputbehaviorofthelargesystem.Thequalityofthisapproximationcouldbemeasuredbytheclosenessofthetransferfunctions,i.e.kH(s)−ˆH(s)kεforagivenaccuracyεandasuitablenorm.Discretesingle-input-single-output(SISO)andmultiple-input-multiple-output(MIMO)dynamicalsystemsarisequitefrequentlyinvariusfieldsofapplicationsinwhichthephysicalortechnicalsystemsaremodelledbysuitablesystemsofdifferentialordifferenceequations.Forasimpleexample,considerthediscretizationofthe1-Dheatequationyt=yxx.Itiswell-knownthatasemi-discretizationusingthemethodoflinescanleadtostabilityproblemsifthediscretizationinspaceistoofine.Hence,oneoftenprefersafulldiscretizationviaaCrank-Nicolsonschemewhichprovidesadiscretesystem.Onecouldimagineaboundarytimedependentcontrolwhichinourcontextwouldleadtosingleordoubleinput,oradistributedcontrolwhichimpliesasmanycontrolsasstates.Furthermore,themeasuredstate(oroutput)canbeononeorbothsidesofthespacedomainbutalsothetemperaturedistributiononthewholedomaincanbeinterestinginthemodel.FormorechallengingexampleswereferforinstancetoVerlaan[19]orLawlessetal.[12].Inallcases,modelingleadstosystemswithaveryhigh–dimensionalstatewhichmakesmodelreductionanimportantandessentialtask.Existingreductionmethodscanbedividedintotwogroups.Ontheonehand,therearetruncationmethods,usingsingularvaluedecompositionstoselecttheimportantpartofthesystemandneglectingtherest.Awell-knownmethodinthisgroupisbalancedtruncation(seee.g.Mullis/Roberts[16]orMoore[15]).Theadvantageofthistechniqueisthatitpreservesstabilityandthatglobalerrorboundscanbederived.ThecomplexityofthecomputationhoweverisoforderN3.Thereforethemethodisbyfartooexpensiveforverylargesystems.Newapproachestrytoapproximatelycomputethetransformationmatrixwithlowercosts(seeforexample[5],[4],[17]andreferencestherein),butthentheglobalerrorboundsarelost.Ontheotherhand,wehaveKrylov-interpolationbasedmethodswhichcanhandlelargedimensionsinthecomputationofthereducedsystem,butoftencannotguaranteethepreservationofstabilityanddonothavecomputableglobalerrorbounds.Therearealsomethods,whichareacombinationofthesetwoapproximationmethods.ArecentandrathercompletestudyofmodelreductiontechniqueswithanemphasisonKrylovandSVD-Krylovmethodscanbefoundin[1]andreferencestherein.Inthispaperwewillfocusonthequestion”Whichreducedordersystemmini-mizestheapproximationerrorH−ˆHinanappropriatemeasure?”Hereweinves-tigatethisproblemtakingtheh2-normofthesystemasthemeasure.Thisnormisdefinedas||Σ||2h2=||H||h2=12π2πZ0traceH(eiw)∗H(eiw)
dw1/2(1.5)(cf.[1]).WewillshowthatalocalminimizerforthisproblemsatisfiescertainHermiteinterpolationconditions.Hence,thismethodisaninterpolationbasedmodelreduction.Westressthatalthoughsuchmethodsarewidelyconsideredasprovidingreducedsystemswithonlyagoodlocalapproximationaroundtheapriorih2-NORMOPTIMALMODELREDUCTIONFORLARGE-SCALESYSTEMS