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Time domain model reduction by moment matching [Elektronische Ressource] / Rudy Eid

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¨ ¨TECHNISCHE UNIVERSITAT MUNCHENLehrstuhl fur¨ RegelungstechnikTime Domain Model Reduction By Moment MatchingRudy EidVollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Maschinenwesender Technischen Universit¨ at Munc¨ hen zur Erlangungdes akademischen Grades einesDoktor-Ingenieursgenehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr.-Ing. habil. Heinz UlbrichPrufer¨ der Dissertation:1. Univ.-Prof. Dr.-Ing. habil. Boris Lohmann2. Univ.-Prof. Dr.-Ing. Peter Eberhard, Universit¨ at StuttgartDie Dissertation wurde am 24.11.2008 bei der Technischen Universit¨ at Munc¨ hen eingere-icht und durch die Fakult¨ at fur¨ Maschinenwesen am 30.01.2009 angenommen.ABSTRACTThis dissertation delivers new results in the model reduction of large-scale linear time-invariant dynamical systems. In particular, it suggests solutions for the well-knownproblem of finding a suitable interpolation point in order reduction by moment matching.As a first step, a new time-domain model order reduction method based on matchingsome of the first Laguerre coefficients of the impulse response is presented. Then, theequivalence between the classical moment matching and the Laguerre-based reductionapproaches both in time- and frequency-domain is shown. In addition, this equivalenceis generalized to include a larger family of coefficients known as generalized Markovparameters.

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Published 01 January 2009
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¨ ¨TECHNISCHE UNIVERSITAT MUNCHEN
Lehrstuhl fur¨ Regelungstechnik
Time Domain Model Reduction By Moment Matching
Rudy Eid
Vollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Maschinenwesen
der Technischen Universit¨ at Munc¨ hen zur Erlangung
des akademischen Grades eines
Doktor-Ingenieurs
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr.-Ing. habil. Heinz Ulbrich
Prufer¨ der Dissertation:
1. Univ.-Prof. Dr.-Ing. habil. Boris Lohmann
2. Univ.-Prof. Dr.-Ing. Peter Eberhard, Universit¨ at Stuttgart
Die Dissertation wurde am 24.11.2008 bei der Technischen Universit¨ at Munc¨ hen eingere-
icht und durch die Fakult¨ at fur¨ Maschinenwesen am 30.01.2009 angenommen.ABSTRACT
This dissertation delivers new results in the model reduction of large-scale linear time-
invariant dynamical systems. In particular, it suggests solutions for the well-known
problem of finding a suitable interpolation point in order reduction by moment matching.
As a first step, a new time-domain model order reduction method based on matching
some of the first Laguerre coefficients of the impulse response is presented. Then, the
equivalence between the classical moment matching and the Laguerre-based reduction
approaches both in time- and frequency-domain is shown. In addition, this equivalence
is generalized to include a larger family of coefficients known as generalized Markov
parameters. This allows a first time-domain interpretation of the moment matching
approach which has been until now developed and applied only in the frequency domain.
Moreover, using this equivalence, the open problem of choosing an optimal expansion
point in the rational Krylov subspace reduction methods (moment matching about s =
0
0) is reformulated to the problem of finding the optimal parameter α in the Laguerre-
based reduction methods. Based on the Laguerre representation of the system, two
methods for the choice of the Laguerre parameter and, consequently, the single expansion
point in rational interpolation order reduction are presented. Accordingly, different model
reduction algorithms are suggested. The importance of these approaches lies in the
fact that they try to approximate the impulse response of the original system, have a
simple structure, are numerically efficient, and are suitable for the reduction of large-scale
systems.
To my family and my wife.ACKNOWLEDGMENTS
My deepest gratitude to my advisor Prof. Boris Lohmann for his invaluable support,
advice and guidance in all stages of this work. His detailed and constructive comments
and our fruitful discussions constituted the basis of this dissertation. His ideas and
insights went far beyond the subject of order reduction to being advices for a successful
academic research.
I would like to thank the second examiner Prof. Peter Eberhard whose comments helped
improving this dissertation.
I would like to deeply thank Behnam Salimbahrami for introducing me to the field of
model order reduction and for his valuable comments and long discussions on this work.
I am grateful also to my old and new colleagues at the institute of automatic control
at the Technische Universit¨ at Munc¨ hen for their assistance, scientific and non-scientific
discussions, and for the friendly environment they provided. I am also very thankful for
helping me improve my German language.
A very special acknowledgment to my family for their care and support throughout all my
years of education and to my wife Alena for her love, patience and endless understanding
and encouragement. Having her beside me was the source of all motivations!
The financial support of the ”Deutscher Akademischer Austausch Dienst (DAAD)” is
gratefully acknowledged.TABLE OF CONTENTS
List of Figures vi
Glossary viii
Chapter 1: Introduction 1
1.1 Mainapproachesoforderreduction..................... 2
1.1.1 Balancingandtruncation ........... 2
1.1.2 Krylovsubspacemethods............... 4
1.2 Thesiscontributions .................. 5
1.3 Disertationoverview.................. 6
Chapter 2: Preliminaries 9
2.1 Linearalgebra................................. 9
2.1.1 TheKrylovsubspace.............. 9
2.1.2 Orthogonal projection ........................ 10
2.1.3 PetrovGalerkinprojection........... 1
2.2 Lineardynamicalsystems.......................... 12
2.2.1 TheTustintransformation........... 13
i2.2.2 The Bilinear/M¨obiustransformation................ 14
2.2.3 Stability and passivity ............... 15
2.2.4 Controllability and observability................... 16
2.3 Laguerepolynomialsandfunctions............ 16
2.3.1 Definition.................... 17
2.3.2 Orthonormality................... 18
2.3.3 Timeandfrequency-domainorthonormalbasis........... 19
2.3.4 TheLaguerredifferentialequation......... 20
Chapter 3: Krylov-based order reduction 21
3.1 MomentsandMarkovparameters...................... 21
3.2 MomentandMarkovparametersmatching:theSISOcase........ 23
3.3 MomentandMarkovparametersmatching:theMIMOcase ....... 26
3.4 Rationalinterpolation ............................ 27
3.5 MainpropertiesofKrylov-subspacemethods....... 27
3.6 Numericalalgorithms............................. 29
3.6.1 TheArnoldialgorithm.... 29
3.6.2 TheLanczosalgorithm........................ 30
3.6.3 Thetwo-sidedArnoldialgorithm ......... 31
3.7 OpenproblemsinKrylov-basedMOR.................... 32
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