 ThispaperwasnotpresentedatanyIFACmeeting.Thispaperwas recommended for publication in revised form by Associate Editor HeinzUnbehauenunderthedirectionofEditorMituhikoAraki.This research was supported by the Australian Research Council and the CentreforIntegratedDynamicsandControl(CIDAC). * Corresponding author. Tel.: #61-2-49216030; fax: #61-2- 49216993. E-mail addresses: reza@ee.newcastle.edu.au (S. O. R. Moheimani), wheath@ee.newcastle.edu.au(W.P.Heath). Automatica38(2002)147}155 BriefPaper Modelcorrectionforaclassofspatio-temporalsystems  S.O.R.Moheimani *,W.P.Heath Department of Electrical and Computer Engineering, University of Newcastle, Callaghan, NSW 2308, Australia Centre for Integrated Dynamics and Control, University of Newcastle, Callaghan, NSW 2308, Australia Received16June2000;revised25January2001;receivedin "nalform2July2001 Abstract Modalanalysishasbeenusedinmodelingofalargenumberofphysicalsystemssuchasbeams,plates,acousticenclosures,strings, etc.Thesemodelsareoftensimpli"edbytruncatinghigherfrequencytermsthatlieoutofthebandwidthofinterest.Truncationcan introducealargeerror.Thispapersuggestsamethodofminimizingthee!ectoftruncatedmodesonspatiallow-frequencydynamics ofthesystembyaddingaspatialzerofrequencytermtothetruncatedmodel.Thefeed-throughtermisfoundsuchthatthespatial H  normoftheerrorsystemisminimized.  2001ElsevierScienceLtd.Allrightsreserved. Keywords: Modelreduction;Spatiallydistributedsystems;Distributedparametersystems;Spatialnorms;Optimization 1. Introduction The modal analysis technique has been widely used throughouttheliteraturetomodelthedynamicsofspa- tio-temporal systems such as #exible beams and plates (Meirovitch, 1986), slewing beams (Fraser & Daniel, 1991; Book & Hastings, 1987), piezoelectric laminate beams(Alberts&Colvin,1991)andacousticenclosures (Hong et al., 1996). These systems have the common propertythatdynamicsofeachoneofthemisdescribed by a particular partial di!erential equation. The modal analysisisconcernedwithexpandingthesolutionofthis partialdi!erentialequationintheformofanin"niteseries usingtheeigenvaluesandeigenfunctionsofthesystem. Thecontroldesignerisoftenonlyinterestedindevis- ingacontrollerforaparticularbandwidth.Asaresult,it iscommonpracticetoremovethemodeswhichcorres- pond to frequencies that lie out of the bandwidth of interest.Theremovedmodes,however,docontributeto the low-frequency dynamics of the system. If the trun- catedmodelisthenusedtodesignacontrollerwhichis implemented on the system, say in the laboratory, the closedloopperformanceofthesystemcanbeconsider- ably di!erent from the theoretical predictions. This is mainly due to the fact that although the poles of the truncatedsystemareatthecorrectfrequencies,thezeros canbefarawayfromwheretheyshouldbe.Therefore,itis naturaltoexpectthatacontrollerdesignedforthetrun- catedsystem,maynotperformwellwhenimplementedon therealsystemsincetheclosedloopperformanceofthe systemislargelydictatedbytheopenloopzeros. This issue is addressed in Clark (1997) and a model correction technique is presented which results in amodelthatisclosertotherealsystemthanthetrun- cated model. The technique of (Clark, 1997), however, appliesonlytoSISOmodelsandisnotaimedatcorrect- ingthespatio-temporalcharacteristicsofthesystem.In Moheimani(1999,2000b)an H  optimalmodelcorrec- tiontechniqueisproposedthatappliestomultivariable modelsofspatio-temporalsystems.InZhuandAlberts (1998),thetruncationerrorisreducedviaaddingasyn- thetic out-of bandwidth mode to the truncated model andminimizingasimilarcostfunction.Allofthesetech- niques are limited to correcting point-wise models of suchsystems.InMoheimani(2000a),thismethodologyis extended to allow for model correction of multi-input models of spatio-temporal systems while minimizing aspatial H  normoftheerrorsystem.Inthispaper,we 0005-1098/02/$-seefrontmatter  2001ElsevierScienceLtd.Allrightsreserved. PII:S0005-1098(01)00178-9