Table Of ContentEwarystRafajłowicz
OptimalInputSignalsforParameterEstimation
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Ewaryst Rafajłowicz
Optimal Input Signals
for Parameter
Estimation
|
In Linear Systems with Spatio-Temporal Dynamics
MathematicsSubjectClassification2020
Primary:62K05,93C20,35R30;Secondary:35B30,35E99
Author
Prof.Dr.EwarystRafajłowicz
WrocławUniversityofScienceandTechnology
DepartmentofControlSystemsand
Mechatronics
WybrzezeWyspianskiego27
50-370Wroclaw
Poland
[email protected]
ORCID:0000-0001-8469-2910
ISBN978-3-11-035089-0
e-ISBN(PDF)978-3-11-035104-0
e-ISBN(EPUB)978-3-11-038334-8
LibraryofCongressControlNumber:2021951119
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2022WalterdeGruyterGmbH,Berlin/Boston
Coverimage:EwarystRafajłowiczusingtheWolfram’sMathematica®
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Preface
Theaimofthisbookistoprovidesomeresultsonselectinginputsignalsactingonsys-
temswithspatio-temporaldynamics.Theyshouldbeselectedinawaythatensures
themostaccurateestimatesoftheirparameters,undergivenconstraints.Weconfine
ourattentiontosystemsdescribedbylinearpartialdifferentialequations(PDEs)and
theirGreen’sfunctions,sincethetheoryfornonlinearequationsisstillrelativelyun-
developed.
Newoptimalityconditionsarepresentedforsystemsoftheelliptic,parabolic,and
hyperbolictypes.Theyarebasedonearlierresultsonoptimalexperimentdesignfor
regressionfunctions,aswellasonthefrequencydomainapproachtoselectingop-
timal input signals for systems described by linear ordinary differential equations
(ODEs). Additionally, for such systems new necessary conditions for optimality are
derivedinthetimedomain.
Inallcases,computationalalgorithmsarederivedfromtheoptimalityconditions.
Examplesoftheirperformanceareprovidedtogetherwithsometimesfascinating3D
plots.
The results are derived under a number of simplifying assumptions that fre-
quentlyallowustoobtaintheminacompactform.Theycanbeinterpretedaslower
boundsonparameterestimationaccuracywhenmorecomplicatedcasesareconsid-
ered.Inparticular,weadvocatenewparadigms,namelythatatthepresentstateof
engineering,itisfrequentlypossibletoobserveandactuatespatio-temporalsystems
atallpointsofaspatialdomainoccupiedbythem.
Thereasonsforattemptingtoestimateasystem’sparametersasaccuratelyaspos-
siblearetwofold.Inmanycases,theseparametershaveinterpretationsofthemate-
rialconstantsandtheirmoreexactvaluesareofinterestformanybranchesofscience
andengineering.ThesecondreasonisthatPDEsaremoreandmorefrequentlyused
asmodelsforidentificationandthenforcontrolofindustrialprocesses.Thelatteras-
pectsarenotcoveredinthisbook,buttheywillbereferenced.
Asameasureoftheparameterestimationaccuracyweselectthedeterminantof
the inverse of the Fisher information matrix (FIM), due to its important invariance
properties.Extensionsoftheresultstootheroptimalitycriteriaispossible.Inthefinal
chapterwementionseveralopenproblemsthatcanbeofinterestbothfortheoryand
forapplications.
Theintendedaudienceofthisbookincludespost-doctorateresearchersandPhD
and graduate students of applied mathematics, mathematical statistics, and many
branchesofengineeringthatapplyspatio-temporalmodels.
Wrocław,January2022 EwarystRafajłowicz
https://doi.org/10.1515/9783110351040-201
Acknowledgements
IwouldliketoexpressmysincerethankstoProfessorDariuszUcińskiattheUniversity
ofZielonaGóra,Polandforhisverycarefulandcriticalreadingofthedraftofthebook.
He also made many suggestions clarifying the statements of the results, improving
themormakingthemmoreprecise.
ManysincerethanksarealsoaddressedtoProfessorRainerSchwabefromtheOtto
vonGuerrickeUniversity,Magdeburg,Germanyforourlong-lastingcooperationand
discussionsonexperimentdesignthatwereveryilluminatingforme.Myviewsonex-
perimentdesignproblemshavealsobeeninfluencedbytheMODA1community.These
meetingsarealsoacknowledged.
Iappreciatetheco-authorshipofDr.WojciechMyszkaofourearlypapersonex-
perimentdesignandhishelpincorrectingthemanuscriptofthisbook.
Iwouldliketoexpressmysincerethankstomyfamilyfortheirsupportandpa-
tienceduringthewritingofthisbook.ThehelpofDr.WojciechRafajłowiczinproof-
readingandimprovingthedrawingsisalsoacknowledged.
ThismonographwasgrantedhonorarypatronagebytheCommitteeofAutomatic
ControlandRoboticsofthePolishAcademyofSciences.
1 MODAistheacronymfortheveryfamousconferenceonexperimentdesignModelOrientedData
Analysis.
https://doi.org/10.1515/9783110351040-202
Contents
Preface|V
Acknowledgements|VII
ListofTables|XIII
ListofFigures|XV
Frequentlyusedacronyms|XVII
1 Introduction|1
1.1 Overviewandmotivations|1
1.2 Remarksonexperimentsandtheirroleingainingvaluable
knowledge|3
1.3 Experimentsforparameterestimationandmodelbuilding
methodology|4
1.4 Ourlimitations|6
1.5 Weneednewparadigms|7
1.6 Bibliographicalnotes|9
1.7 Notationalconventions|13
2 Optimalexperimentdesignforlinearregressionmodels–aprimer|17
2.1 Thebasiclinearmodel|17
2.2 TheLSM–somebasicfactsonitsaccuracy|20
2.3 Towarddesignoptimalitycriteria|25
2.4 Approximateexperimentdesigns|29
2.5 Optimalityconditions–theKieferandWolfowitztheorem|34
2.6 Casestudies–sensorallocation|39
2.7 WhytheD-optimalitycriterion?|43
2.8 Remarksonoptimalexperimentdesignsfornonlinearmodels|44
3 NumericalsearchforD-optimaldesigns–basicmethodsforalinear
regression|49
3.1 Introductoryconsiderations|49
3.2 Optimalallocationofobservations|50
3.2.1 Whyisthemultiplicativeweightsupdatealgorithmsofast?|55
3.3 ThealgorithmofWynnandFedorov|56
3.4 Thecombinedalgorithm|64
