Bilinear H 2-optimal Model Order-Reduction with applications to thermal parametric systems

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Bilinear

H

2

-optimal

Model Order Redu tion

with appli ations

to thermal parametri systems

Dissertation

zurErlangungdesakademis henGrades

do torrerumnaturalium

(Dr.rer.nat.)

vonDipl.-Math. AngelikaSusanneBruns

geb. am06.01.1986in Freudenstadt

genehmigtdur hdieFakultätfürMathematik

derOtto-von-Gueri ke-UniversitätMagdeburg

Guta hter: Prof. Dr. PeterBenner

Prof. Dr. TobiasDamm

eingerei htam20.03.2015

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Danksagung v

Zusammenfassung vii

Summary ix

ListofFigures xi

ListofTables xiii

ListofAlgorithms xv Notations xvii Chapter1. Introdu tion 1 1.1. Motivation 1 1.2. Dissertationoverview 2 1.3. Thesis ontributions 3

Chapter2. Mathemati alprerequisites 5

2.1. LinearAlgebra 5

2.2. Dierentialgeometry 8

2.3. Systemstheory 10

Chapter3. Modelingofheattransferproblems 25

3.1. ThermalModeling 26

3.2. Theheatequation 29

3.3. BoundaryandInterfa e onditions 30

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3.5. Thermalmodelingofanele tri almotor 32

Chapter4. Modelparametrization 35

4.1. Dis retizationoftheheatequation 35

4.2. Physi alparametrization 37

4.3. Geometri variations 37

Chapter5. ModelOrderRedu tion 49

5.1. Proje tion-basedMORandtheerrorsystem 50

5.2. MORoflinearsystems 52

5.3. Parametri ModelOrderRedu tion(pMOR) 60

5.4. BilinearModelOrderRedu tion 69

5.5.

H

2

-optimalbilinearModelOrderRedu tion 73

Chapter6. Challengeswhen applyingBIRKAto thermalindustrial

models 101

6.1. Krone kerprodu tappproximation 101

6.2. Stability 105

6.3. Singularstinessmatrix

A

andlargenormmatri es

N

k

115

Chapter7. Redu tionofphysi allyparametrizedthermalmodels 121

7.1. Resultsforthe

H

2

-optimalredu tiononGrassmannmanifolds121

7.2. Resultsfortheredu tionusingBIRKA 131

Chapter8. Redu tionofthermalmodelswithgeometri variations 145

8.1. Reformulationofthelinearparametri asbilinearsystems 146

8.2. Methodsfortheinterpolationoftheredu edmodels 149

8.3. Redu tionandinterpolationusingreformulationone 154

8.4. Redu tionandinterpolationusingthese ondreformulation 159

Chapter9. Con lusionsandOutlook 171

9.1. SummaryandCon lusions 171

9.2. Futureresear h 173

AppendixA. Derivationofthebilinear

H

2

-optimal onditions 175

A.1. Wilson onditions 175

A.2. Derivationoftheoptimality onditionsbyBennerandBreiten177

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Bibliography 187

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Diese Arbeit wäre ohne die Beteiligung vielerPersonenin dieser Art

undWeiseni htmögli hgewesen. Natürli hgiltmeinDankanersterStelle

meinemBetreuerProf. Dr. PeterBenner,dersi hbeimeinenBesu henin

MagdeburgimmerZeitfürmi hnahm,mirhilfrei heHinweiseundTippsgab

undinsbesonderedasWagnisIndustrie-undNumerik-fremdeDoktorandin

einging.Herzli henDank!

Auÿerdembedankei h mi hherzli hbeiProf. Dr. TobiasDamm fürdie

ÜbernahmederZweitkorrektur.

DieCSCGruppeamMPIinMagdeburghatmi hbeimeinenBesu hen

im-merfreundli haufgenommen. Danke füreureHilfsbereits haftunddiverse

s höneAbendeinMagdeburg,S hlossRingberg,imHarzundin Kroatien.

EinspeziellerDankgehtanmeineMentorinDr. UlrikeBaur.

MeinePromotionszeitbeiBos hwäreni htdasGlei hegewesenohnemeine

KollegenausderMathematikere ke. I hdankehierinsbesondereDr.

Ka-trinS huma herundDr. RudyEidfürdiehervorragendeBetreuung-ni ht

nurfa hli h,au hpersönli hhabei hvielvoneu hgelernt.Ebensobedanke

i hmi hbeiallenanderenKollegenausderCR/ARH,diemirvorallembei

Fragenzuthermis henSimulationenweitergeholfenhaben,zusätzli hdanke

i hDr. KilianKrienerundThomasHeidvonED/ESY3.

I h danke auÿerdemmeinen Elternund meiner S hwesterfür ihre

Unter-stützungwährendnuns honüber10JahrenMathematik. DankeanSophie

fürüber6JahreHorizonterweiterung.

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WirdinderIndustrieeineneueKomponenteentwi kelt,sospielen

Com-putersimulationenmittlerweileeinewi htigeRolle. Immers hnellereund

im-mergenauereSimulationsmodellewerdengewüns ht,damitZeitundKosten

gespart werden können. Mit Hilfe von Modellordnungsreduktion(MOR)

kann man aus groÿen, mit der FiniteElemente Methode erstellten

Mod-ellenkleineundgenaueModelleerhalten,diedann inkurzer Zeitsimuliert

werdenkönnen. Immerhäugerwirdau hgefordert,dieVariationvon

Pa-rameternimgroÿenFiniteElementeModellaufdiekleinenreduzierten

Mod-ellezuübertragen. DieseParameterbes hreibenbeispielsweisevers hiedene

Randbedingungen,dieimModellabgebildetwerden,genausowie

Änderun-geninderGeometrie(z.B.VariationvonLängen). MitHilfevonMethoden

ausderparametris henModellordnungsreduktion(pMOR)könnendiese

Pa-rameterabhängigkeitenau himreduziertenModellerhaltenundzur

Simu-lationvonunters hiedli henSzenariengenutztwerden.

Anstattdie heute übli hen Verfahren zur pMOR zu benutzen, werden in

dieserArbeit die parametris henModelle, die eine spezielle

Parameterab-hängigkeitzeigen, in bilineareModelle umges hrieben. Nun können au h

Verfahrenzur bilinearenModellordnungsreduktionangewandt werden,

ins-besondereVerfahrenzur

H

2

-optimalenReduktion.Zieldieser

H

2

-optimalen Verfahren ist es, den Fehler zwis hen dem Ausgangsmodellund dem

re-duziertenModellinder

H

2

-Normzuminimieren. Wirverwendenzumeinen densogenanntenBilinearInterpolatoryRationalKrylovAlgorithm(BIRKA)

vonBennerundBreiten[12℄. Auÿerdementwi kelnwirneuebilineare

H

2

-optimaleAlgorithmen,dieaufOptimierungsverfahrenauf

Grassmann-Man-nigfaltigkeitenberuhen.

Dietheoretis henGrundlagenderthermis henModellierungwerdenerklärt

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ModellekönnenausdenFiniteElementeModellendur heineAnalyse der

Glei hungenabgeleitetwerden. DieParametersindeinerseitsGröÿen, die

das thermis heVerhalten währenddes Betriebserklärenundandererseits

Gröÿen, dieVariationenin der Geometriedes Motors bes hreiben. Diese

ParametersollenindenreduziertenModellenerhaltenbleiben.

Währenddieneuentwi keltenAlgorithmenno hni htreiffürdieReduktion

vongroÿenModellensind,wirdinderArbeitgezeigt,dassdieReduktionmit

BIRKAzuguten reduziertenModellenführt. Allerdingsmüssen dazu

ver-s hiedeneNa hbesserungenanderReduktionsmethodikvorgenommen

wer-den, beispielsweisemüssen Methoden zur Stabilitätserhaltung angewandt

werden. InModellen mitVariationen in derGeometrie,werden zusätzli h

zumursprüngli henBIRKAna hderReduktionno hInterpolationsverfahren

verwendet,umreduzierteModellemitderParameterabhängigkeitdes

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The design pro ess of a new omponent in industry is nowadays

al-mostalwaysa ompaniedby omputersimulations. In order to savetime

andmoney,fastanda uratemodelsforthesimulationofthe omponent

arerequired. UsingModelOrderRedu tion(MOR)largemodelsobtained

by Finite Elementsimulations an beredu edto smallmodels possessing

thesame behavioras the original. Oftenit isrequiredto obtain redu ed

models,wherethe dependen e in one orseveralparameters (for example

thelengthorwidth ofapart)oftheoriginalmodel ispreserved. Usingso

alledparametri ModelOrder Redu tion(pMOR) the parameters in the

redu edmodel anbevariedandthemodels anbeusedforfastsimulation

ofseverals enarios.

Insteadof using the ommonlyemployedmethods from pMOR,methods

frombilinearModelOrderRedu tionwillbeusedwithinthiswork,as

para-metri modelswitha ertainformofparameterdependen e anberewritten

asbilinearmodels. We fo uson methodsfrombilinear

H

2

-optimalModel OrderRedu tion,astheirobje tiveistominimizetheerrorbetweenthe

orig-inaland theredu edmodel measuredinthe

H

2

-norm. First, theBilinear InterpolatoryRationalKrylovAlgorithm(BIRKA)developedbyBennerand

Breiten[12℄isused. Se ond,wederivenewbilinear

H

2

-optimalalgorithms basedonoptimizationonGrassmannmanifolds.

Thefoundationsofthermalmodelingandtheirappli ationtothermal

sim-ulationsofele tri almotorsusingFiniteElementsoftwarewillbeexplained.

Parametri modelssuitableforpMOR anbederivedfromaFiniteElement

softwareanalyzingtheunderlyingequations. Two lassesofparameterswill

be onsidered:Constantsinuen ingthethermalbehaviorofthemodeland

hangesinthegeometryofthemodel.

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we ndthat they are not yet readyforthe redu tionof large parametri

modelsasen ounteredinourthermalsimulations. In ontrast,theBIRKA

performswellfortheredu tionofthesemodels. However,several

modi a-tionsontheredu tionmethodsneedtobeperformedtoassure,forexample,

thepreservationofstabilityduringtheredu tion. Fortheredu tionof

mod-els with parameters resulting from hangesin the geometry, interpolation

pro eduresneedtobeappliedaftertheredu tiontotransfertheparameter

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3.1 Temperatureontheinterfa ebetweentwosolidsin onta t. 27

3.2 Drawingofasli ethroughanele tri almotor. 31

3.3 Example: Bos hgenerators. 32

3.4 Modelforsimulatingtheheattransferinastator. 33

4.1 Simplelinears alingofare tangle. 38

4.2 S alingsforthegeometryvariationofanele tri almotor. 39

4.3 Dierents alingsshowninthemotormodel. 39

4.4 Modelparametrizedingeometry,topview. 40

4.5 S alingofatriangularmeshelementintheannulus. 41

4.6 Simulationoflargemodelnos alingfun tionwasapplied. 46

4.7 Simpliedmotormodel. 47

4.8 Simpliedmotormodelafterthes alingandashortsimulation. 48

6.1 Proposedworkowforstabilization. 112

6.2 Redu tionwithstabilizationviamirroringofpoles. 114

7.1 Redu tionwithbilGFA,bilFGFAandbilSQA. 124

7.2 Des entinfun tion

J (U)

. 125

7.3 ResultsbilGFA,(I1),dierentstopping riteria. 127

7.4 ResultsbilFGFA,(I1),dierentstopping riteria. 128

7.5 ResultsbilSQA,(I1),dierentstopping riteria. 129

7.6 Comsol R

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7.7 Temperatureproleoriginalandredu edordermodel. 133

7.8 Resultsfordierentheattransfer oe ients. 135

7.9 One-sidedmethods. 137

7.10Temperatureprolesfordierentheattransfer oe ients. 140

7.11Redu tionwithdierentapproa hes. 141

8.1 Interpolationofredu edordermodelrstoutput. 156

8.2 Interpolationofredu edordermodelfourthoutput. 157

8.3 Interpolationwithdierentmethods,modelwithfourparameters.160

8.4 Interpolationwithdierentmethods,vesamplingpoints,

p

new

1

. 162

p

new

2

. 163

p

new

3

. 164

8.7 Interpolationusingtheapproa hes(P1)and(P2). 167

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5.1 Exponentialandlogarithmmappingsfordierentmanifolds. 66

7.1 ResultsforbilGFA,bilFGFAandbilSQA,initialization(I1). 126

7.2 ResultsforbilGFA,bilFGFAandbilSQA,initialization(I2). 126

7.3 Comparisonofsimulationandredu tiontimes. 136

8.1 Tworeformulationmethodsshortsummary. 150

8.2 One-stepmethodsfortheinterpolationofredu edordermodels. 153

8.3 Two-stepmethodsfortheinterpolationofredu edordermodels. 154

8.4 Costsmodelwithoneaneparameter. 158

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1 IRKAasgivenin[6℄. 60

2 GeneralizedSylvesteriteration( f. [12℄). 77

3 BilinearIRKAforsystemswith

E

6= I

,

E

nonsingular( f. [12℄). 79

4 GFAforbilinearsystems(bilGFA). 91

5 FGFAforbilinearsystems(bilFGFA). 93

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R

eldofrealnumbers

C

eldof omplexnumbers

Re(z )

realpartof

z

∈ C

Im(z )

imaginarypartof

z

∈ C

A

T

transposeofthematrix

A

∈ C

n×m

A

∗

onjugate transpose of the matrix

A

∈ C

n×m

A

−1

inverseofthematrix

A

∈ C

n×n

A

−∗

inverse, onjugatetranspose ofthe

matrix

A

∈ C

n×n

I

n

identitymatrixofdimension

n

× n

σ

i

(A)

i

-thsingularvalue ofamatrix

A

∈

C

n×m

, singular values ordered by magnitude

λ

i

(A)

,

λ

i

(A, E)

i

-theigenvalueofamatrix

A

∈ C

n×n

insomeordering,generalized

eigen-valueofthepen il

A

− λE

insome

ordering

||A||

2 = σ

max

(A)

spe tralnormofamatrix ve

(A) =

a

11 , . . . , a

1n

, a

21 , . . . , a

mn

T

the ve tor formed by sta king the

olumnsofthematrix

A

∈ C

n×m

det(A)

determinantofamatrix

A

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tr(A) =

P

n

_k=1

a

kk

tra eofamatrix

A

span(A)

spa espannedbythe olumnsofthe

matrix

A

⊗ B =







a

11 B

. . .

a

1m

B

. . . . . .

a

n1

B

a

nm

B







Krone ker produ tof two matri es

A

∈ C

n×m

and

B

∈ C

k×l

diag(a

11 , . . . , a

nn

) =







a

11

0

. . .

0 a

nn







matrixwithdiagonal

(a

11 , . . . , a

nn

)

O

n

⊂ R

n×n

orthogonalrealmatri esi.e.

matri- eswith

A

T

_A

_{= AA}

T

_{= I}

n

P

rea habilityGramian

Q

observabilityGramian

κ

2 (A) = ||A||

2 ||A

−1

||

2 =

σ

max

(A)

σ

min

(A)

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Introdu tion

1.1. Motivation

Inindustry,simulationsareanimportant toolin thedesignpro essof

a new omponent. In order to save time and money, fast and a urate

modelsforsimulationareneeded. ModelOrderRedu tion(MOR)isa

pow-erfulmethodtoobtainsmallanda uratemodelsfromlargeFiniteElement

models.Moreandmoreoften,FiniteElementmodelsareused,whi h

on-tainseveralparameters. Su h parameters anbe lengths and heights as

wellasphysi albehavior. Theseparametrizedmodelswilloftenbeusedto

ndoptimaldesignsbyusingoptimizationw.r.t. thegiven parameters. As

theFiniteElementmodelsarelarge,optimizationruns aneasilyex eedthe

omputation apa ities.Itishen edesirabletoredu emodelswhile

preserv-ingthe parameterdependen y. This istheobje tive ofparametri Model

OrderRedu tion(pMOR). Re ently,Bennerand Breiten[11℄ presenteda methodtorewritelinearparametri modelsintobilinearmodels.Thisallows

bilinearModel OrderRedu tionmethods tobeused forparametri Model

OrderRedu tion. Theresultingredu edordermodelshouldbeagood

ap-proximation of the original model. Within the framework of

H

2

-optimal ModelOrder Redu tion,the error anbemeasuredand minimizedin the

H

₂

-norm. In this work, we will examinebilinear

H

2

-optimalmethods for theredu tionoflinearparametri systems,whi hhavebeenappliedtoand

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1.2. Dissertationoverview

InChapter2,wereviewresultsfromLinearAlgebra,Dierential

Geom-etryandSystemsTheory. The on eptswillbestatedforlinearandbilinear

systems.

Chapter3 providesthereaderwiththefoundationsofheattransfer

mod-eling. Theunderlyingphysi alee ts(heat ondu tan e, onve tiveheat

transfer, radiation) will be reviewedand the mode of operation and the

thermalmodelingofanele tri almotorwillbedes ribed. Threedierent

ele tri almotor modelshavebeen builtand will bepresented. Chapter4

gives anoverview over theequations that aresolved duringheat transfer

modeling,andthepro eduretoobtainparametri modelsby arefulanalysis

oftheseequations.

InChapter5,methodsforModelOrderRedu tion(MOR)willbedis ussed.

First, methods for linearMOR will be reviewed, followed by a dis ussion

ofmethods for theredu tionofparameterdependentmodels(parametri

MOR).Itispossibletorewriteparametri modelswitha ertainparameter

dependen yasbilinearmodels,andhen emethodsfrombilinearMORwill

be onsidered. Of parti ularinterest are methods from the lassof

H

2

-optimal bilinearMOR,as their obje tiveis to minimizetheerrorbetween

originalandredu edmodel. First,wereviewexistingmethodsandstatethe

Bilinear Interpolatory Rational Krylov Algorithm(BIRKA) [12℄. Se ond, wedevelopalgorithmsfortheredu tionofbilinearsystemsviaoptimization

on Grassmannmanifolds. Thesemethodsareofinterest,as theypreserve

stabilityduringtheredu tionpro ess.

The obje tive of Chapter 6 is the dis ussionof several issues that were

en ountered while applying BIRKA to thermal models. Theseissues are

examined,and strategiesfortheirmitigationwillbedeveloped. Espe ially

preservationofstabilityduringthe al ulationis ru ial. ResultsforBIRKA

andthenew

H

2

-optimalmethodswillbegiveninChapters7and8.Whereas thenewmethodsarenotyetappli ableto largesystems,BIRKAperforms

wellonbilinearsystemsthathavebeenobtainedfromlinearparametri

sys-tems. First, onlyphysi alparametersare onsidered. Se ond, wepresent

resultsforsystemswithaparameterdependen yresultingfrom hangesin

geometry, whi h anonly be rewritten partiallyas bilinear systems. For

su hsystems,parametri redu edordermodels anthenbeobtainedbyan

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1.3. Thesis ontributions

Themain ontributionsofthisthesisare:

•

Oneobje tive ofthisthesis isMORof thermalele tri almotor models. Hen e,itisshownhowmatri essuitableforpMOR an

beobtainedfromComsol R

,aFiniteElementSoftware. Todoso,

theequationswhi haresolvedbytheSoftwareareusedto

theo-reti allyre onstru tthedependen einparametersofthemodel

( f.Chapter4).

•

In ontrasttootherworksaboutpMOR,inthisthesisthe redu -tionoftheparametri modelsisdoneusingBIRKA[12℄. Several issueswhereen ounteredwhen thealgorithmwas applied: One

lassofparametersleadstoanon-singularstinessmatrix,in

sev-eral asesthereistheneedtos aleothersystemmatri estofulll

aKrone kerprodu tapproximationandinaddition,BIRKAdoes

notpreservestability.Alltheseissueshavebeenresolved,andwe

showresultsfortheredu tionofamotormodelfrom

n

= 41, 199

degreesoffreedomtoaredu edorderof

r

= 300

. Thishasbeen

donefor

13

physi alparameters.

•

In addition, models with geometri al variations are onsidered. Aftertheredu tionwithBIRKA,severalinterpolationstrategies

betweentheredu edordermodelsobtainedin severalparameter

pointshavebeen ompared.

•

Finally, we develop new

H

2

-optimal bilinearmethods for MOR usingoptimizationonGrassmannmanifolds. Thesemethods an

preservestabilityforsymmetri systemsmatri es,andtheir

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Mathemati al prerequisites

2.1. LinearAlgebra 5

2.2. Dierentialgeometry 8

2.3. Systemstheory 10

In this rst theoreti al hapter, some results from dierent areasof

mathemati sarereviewed.First,generalresultsfromLinearAlgebrawillbe

presented, followedby a loser look on some denitions fromDierential

Geometry. The last se tion provides the reader with an introdu tion to

linearandbilinearsystemstheory.

2.1. LinearAlgebra

Withinthisse tionwereviewthede ompositionofmatri es,the

prop-ertiesoftheKrone kerprodu tandprovidethereaderwithbasi knowledge

onmatrixpertubationtheory.

2.1.1. Matri es and their de ompositions. Most of thematri es in

thisworkaresymmetri ,whi hiswhywestatethedenitionhere.

Denition 2.1.1. A matrix

A

∈ R

n×n

is alled symmetri if

A

= A

T

. A

symmetri matrixispositive(semi)denite,denotedby

A >

(≥)0

,if

x

T

_{Ax >}

(≥)0

forall ve tors

0 6= x ∈ R

n

. Itisnegative(semi)denite,denotedby

A <

(≤)0

,if

x

T

_{Ax <}

_(≤)0

forallve tors

0 6= x ∈ R

n

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We will often refer to the following two matrixde ompositions, the

eigenvalueandthesingularvaluede omposition.

Denition2.1.2 (Generalizedeigenvaluede omposition[38,Se tion7.7℄). If

A, B

∈ C

n×n

,thenthesetofallmatri esoftheform

A

− λB

with

λ

∈ C

isapen il. Thegeneralizedeigenvaluesof

A

− λB

areelementsoftheset

λ(A, B)

denedas

λ(A, B) = {z ∈ C : det(A − z B) = 0}.

If

λ

∈ λ(A, B)

and

0 6= x ∈ C

n

satises

Ax

= λBx,

(2.1)

then

x

is an eigenve tor of

A

− λB

. The problem of nding nontrivial

solutionsto(2.1)isthegeneralizedeigenvalueproblem.If

B

isnonsingular,

λ(A, B) = λ(B

−1

_A)

holds.

Theorem 2.1.3 (The singularvalue de omposition (SVD) [38, Theorem 2.4.1℄). If

A

∈ R

m×n

,thenthereexistorthogonalmatri es

U

= [u

1 , . . . , u

m

] ∈

R

m×m

and

V

= [v

1 , . . . , v

n

] ∈ R

n×n

su hthat

U

T

AV

= diag(σ

1 , . . . , σ

p

) ∈ R

m×n

,

(2.2) with

p

= min (m, n)

where

σ

1 ≥ σ

2 ≥ · · · ≥ σ

p

≥ 0

.

The

σ

j

will be alled singularvalues. If it shallbe lariedthat they resultfromasingularvaluede ompositionofthematrix

A

,wedenotethem

by

σ

j

(A)

. Let

r

besu hthat

σ

1 ≥ σ

2 ≥ · · · ≥ σ

r

> σ

r +1

= · · · = σ

p

= 0

. Then

rk(A) = r

and

A

anbede omposedinthefollowingway:

A

=

r

X

i =1

σ

i

u

i

v

i

T

.

Usingmatri es,wewillwritethisde ompositionasfollows:

A

= U

r

Σ

r

V

r

T

,

(2.3) with

U

r

∈ R

m×r

,

Σ

r

∈ R

r ×r

and

V

r

∈ R

n×r

andreferto it asthe ompa t

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2.1.2. Properties of the Krone ker produ t. The following matrix

produ tisreferredtoastheKrone kerprodu t:

Denition2.1.4. Fortwomatri es

A

∈ C

n×m

and

B

∈ C

k×l

,theKrone ker

produ t isdenedas:

A

⊗ B =







a

11 B

. . .

a

1m

B

. . . . . .

a

n1

B

a

nm

B







.

The Krone ker produ thasthe followingproperties(see for example

[38℄,Se tion12.3):

(A ⊗ B)

T

= A

T

⊗ B

T

,

with

A

∈ C

n×m

_{, B}

_{∈ C}

k×l

_,

(A ⊗ B)

−1

= A

−1

⊗ B

−1

,

with

A

∈ C

n×m

_{, B}

_{∈ C}

k×l

_,

(A ⊗ B) ⊗ C = A ⊗ (B ⊗ C),

with

A

∈ C

n×m

_{, B}

_{∈ C}

k×l

and

C

∈ C

s×q

_,

(AC ⊗ BD) = (A ⊗ B)(C ⊗ D),

with

A

∈ C

n×m

_{, B}

_{∈ C}

k×l

_{, C}

_{∈ C}

m×s

and

D

∈ C

l ×q

_,

butingeneral

A⊗B 6= B ⊗A

! Inadditiononeobtaines(with

A

∈ C

n×m

_{, B}

_∈

C

k×l

):

rk(A ⊗ B) = rk(A) · rk(B),

det(A ⊗ B) = det(A)

n

· det(B)

m

for

A

∈ R

m×m

and

B

∈ R

n×n

_,

tr(A ⊗ B) = tr(A) · tr(B),

||A ⊗ B||

2 = ||A||

2 · ||B||

2 .

If

C

= AXB

for

C

∈ R

n×m

_{, A}

_{∈ R}

n×k

_{, X}

_{∈ R}

k×l

and

B

∈ R

l ×m

then

oneobtainsfortheKrone kerprodu tandtheve operator:

ve

(C) = (B

T

_{⊗ A)}

ve

(X).

(2.4)

2.1.3. Matrixpertubationtheory. The onne tionbetweenthe

eigen-valuesoftwomatri eswillbeneededwithinthiswork. Thefollowingresults

havebeenestablishedin the ontextofmatrixpertubationtheory, the

re-lationoftheeigenvaluesofapertubedMatrix

M

+ S

andtheunpertubed

(29)

Theorem2.1.5(Bauer-Fike,[38,Theorem7.2.2℄). If

µ

isaneigenvalueof

M

+ S ∈ C

n×n

and

X

−1

_MX

_{= diag(λ}

1 , . . . , λ

n

)

,then

min

i =1,...,n

|λ

i

− µ| ≤ κ

2 (X)||S||

2 .

(2.5) Corollary2.1.6. Let

X

−1

_MX

_{= diag(λ}

1 , . . . , λ

n

)

,and

M

+ S ∈ C

n×n

. For

everyeigenvalue

λ(M + S)

aneigenvalue

λ

i

(M)

existssu hthat

|λ

i

(M) −

λ(M + S)| ≤ κ

2 (X)||S||

2 .

Thenextresultsshowthe onne tionbetweentheeigenvaluesoftwo

realsymmetri matri es

A

and

B

.

Proposition 2.1.7 (Weyl,[60, Theorem 4.8, Corollary 4.9℄). Let

A, B

∈

R

n×n

betwosymmetri matri es. Let

λ

i

(A)

and

λ

i

(B)

for

i

= 1, . . . , n

be theeigenvaluesof

A

and

B

with

λ

1 (A) ≥ · · · ≥ λ

n

(A)

and

λ

1 (B) ≥ · · · ≥

λ

n

(B)

. Thenitholds:

λ

i

(A + B) ∈ [λ

i

(A) + λ

n

(B), λ

i

(A) + λ

1 (B)]

for

i

= 1, . . . , n.

(2.6)

Corollary2.1.8 ([60, Corollary4.10℄). Undertheassumptionsof Proposi-tion2.1.7itholds

|λ

i

(A + B) − λ

i

(A)| ≤ ||B||

2

for

i

= 1, . . . , n.

(2.7)

2.2. Dierentialgeometry

InSe tion5.5.4,severalalgorithmsbasedonoptimizationonmanifolds

willbederived. Foramoredetailedpresentationofthistopi , wereferto

[1℄and[30℄. Let

O

r

denotethesetoftheorthogonalmatri esin

R

r ×r

_.

Denition2.2.1(Stiefelmanifold[1,Se tion3.3.2℄). For

r

≤ n

,theStiefel manifoldisdenedasthesetofall

n

× r

orthonormalmatri es:

St

(r, n) := {X ∈ R

n×r

_|X

T

_X

_{= I}

r

}.

Clearly,St

(r, n) ⊂ R

n×r

. It anbeshownthat St

(r, n)

is a ompa t

submanifoldof

R

n×r

( f.[1,Se tion3.3.2℄). Thetangentspa eofaStiefel manifoldat

X

∈

St

(r, n)

isdenedasfollows( f. [1,Example3.5.2℄):

T

X

St

(r, n) = {Z ∈ R

n×r

_|X

T

_Z

_{+ Z}

T

_X

_{= 0}.}

Headingfor analgorithmforthegradientow, thegradientofafun tion

on themanifoldhastobe al ulated. Therefore, werst haveto provide

(30)

denitionofaninnerprodu tonthetangentspa e.ForaStiefelmanifold,

theinnerprodu t isdenedas

hξ, ηi = tr(ξ

T

η)

with

ξ, η

∈ T

X

St

(r, n).

(2.8)

Thegradientin

X

ofafun tion

F

onaStiefelmanifoldisdenedtobethe

tangentve tor

∇F

su hthat

tr(F

X

T

Y

) = tr((∇F )

T

(I −

1

2 XX

T

_{)Y ),}

(2.9)

holdsforalltangentve tors

Y

∈ T

X

St

(r, n)

. Here,

F

X

isthematrixofall partialderivativesof

F

withrespe tto

X

,i.e.:

(F

X

)

i j

=

∂F

∂X

i j

.

(2.10)

Solvingequation(2.9)leadstothefollowingexpressionforthegradient:

∇F = F

X

− XF

X

T

X.

(2.11)

TheGrassmann manifold

Gr(r, n)

,

r

≤ n

, is dened as the set of all

r

-dimensionalsubspa esof

R

n

. Following[30℄,it anbeseenasaquotient manifoldinthefollowingway: Twomatri es

U

1

and

U

2

inSt

(r, n)

are equiv-alent,iftheyspanthesame

r

-dimensionalsubspa e. Thisholdsifandonly

if

U

1 = U

2 Q

foranorthogonalmatrix

Q

∈ R

r ×r

. Theequivalen e lass

[U]

ofapoint

U

∈

St

(r, n)

anbedenedas:

[U] = {UQ|Q ∈ O

r

} .

Themap

G

: Gr(r, n) →

St

(r, n)/O

r

isabije tion. We will therefore onsiderthe Grassmannmanifoldas this

quotientmanifold ofSt

(r, n)

. A matrix

U

∈ St(r, n)

represents a whole

equivalen e lassin

Gr(r, n)

. Thetangentspa eoftheGrassmannmanifold

anbedes ribedasfollows[30,Se tion2.5℄:

T

X

Gr

(r, n) = {Z ∈ R

n×r

_|X

T

Z

= 0}.

(2.12)

Onamanifold,theshortest onne tionbetweentwopointsis alleda

ge-odesi . Let

X(0) = X

and

X(0) = H

˙

. Let

H

= W ΣV

T

bethe ompa t

singularvalue de omposition ( f. equation(2.3)) of

H

with

W

∈ R

n×r

,

Σ, V ∈ R

r ×r

_.

Thegeodesi anbedes ribedas[30,Se tion2.5.1℄:

X(t) =

XV

W

cos Σt

sin Σt

V

T

.

(2.13)

(31)

ForaGrassmannmanifold,theinnerprodu tisdenedas

hξ, ηi = tr(ξ

T

η),

with

ξ, η

∈ T

X

Gr

(r, n).

(2.14)

Thegradientin

X

ofafun tion

F

ontheGrassmannmanifoldisdenedto

bethetangentve tor

∇F

su hthat

tr(F

X

T

Y

) = tr((∇F )

T

Y

),

(2.15)

holdsforalltangentve tors

Y

∈ T

X

Gr

(r, n)

. Solvingequation(2.15)leads tothefollowingexpressionforthegradient[30,Se tion2.5.3℄:

∇F = F

X

− XX

T

F

X

.

(2.16)

Wewillalsoneedthefollowingdenition:

Denition 2.2.2 ([1,Denition4.2.1℄). Givenafun tion

F

on St

(r, n)

or Gr

(r, n)

,asequen e

{η

k

}

,

η

k

∈ T

x

k

St

(r, n)

or

η

k

∈ T

x

k

Gr

(r, n)

is gradient-related if,forany subsequen e

{x

k

}

k∈K

of

{x

k

}

that onvergesto a non- riti alpointof

F

,the orrespondingsubsequen e

{η

k

}

k∈K

isboundedand satises

lim

k→∞

sup

_k∈K

h∇F (x

k

), η

k

i < 0.

(2.17)

2.3. Systemstheory

Manyphysi alphenomena, hemi alrea tions,biologi alpro essesor

models for the fore ast of nan ial pro esses anbe mathemati ally

de-s ribedbythesame lassofsystems,so alleddynami alsystems. External

inuen esthathaveadire timpa tonthebehaviorofthesystemare alled

inputs. Thebehaviorofthesystemswillbemonitoredwithina ertaintime

rangeandat ertainpoints,thesystem'soutputs. The onne tionbetween

the inputsandthe outputswill oftenbemeasuredand referredto as the

system'sinput-output-relationship.Adynami alsystem anbedes ribedby

adierentialequation.Inthiswork,twokindsofdynami alsystemswillbe

(32)

2.3.1. LinearSystems. Inthefollowingse tionsomebasi knowledge

onlineardynami alsystemswillbereviewed,su hasstability,observability,

ontrollability, balan ed systems, norms of systems and the input-output

relationship.

Denition 2.3.1. A linear system

Σ

lin

of order

n

is a system ofordinary dierentialequationsofthefollowingform:

Σ

lin

:

_E

_{˙x (t) = Ax(t) + Bu(t),}

y

(t) = Cx(t), x(0) = x

0 ,

(2.18) where

E, A

∈ R

n×n

,

B

∈ R

n×m

,

C

∈ R

p×n

. Theinput

u(t) ∈ R

m

anbe

time-dependentjustasthestates

x(t) ∈ R

n

andtheoutput

y

(t) ∈ R

p

are.

Thevalueof

x(0) = x

0

is alled initialvalue. Thespa e

X

ontainingall states

x(t)

is alledstatespa e.

2.3.1.1. Stability. Systemswithboundedsolutiontraje tories

x(t)

are

of spe ial importan e. This hara teristi of a system is referredto as

stability.Forlinearsystems( .f. system(2.18) )withnonsingular

E

,stability isdenedasfollows:

Denition2.3.2( .f. [63℄Chapter2.7,[5℄Chapter5.8,[61℄Chapter3.2.1). Thesystem

E

˙x (t) = Ax(t), E

nonsingular, isasymptoti allystableif

(i) Forall

x

0 _{∈ R}

n

theinitialvalueproblem

E

˙x (t) = Ax(t), x(0) =

x

0 _,

hasasolutionandforevery

ε >

0

thereexistsa

δ >

0

su h

that

||x(t)||

2 < ε

forall

t

≥ 0

andforall

||x(0)||

2 < δ

(Lyapunov stability).

(ii) Thereexists

δ >

0

su hthat

x(t) → 0

as

t

→ ∞

if

||x(0)||

2 < δ.

Theorem2.3.3([63℄Corollary2.11,[61℄Theorem3.7). Thesystem

E

˙x (t) = Ax(t), E

nonsingular,

isasymptoti allystableifandonlyifalltheeigenvaluesof

λE

− A

lieinthe

openlefthalf-plane.

We will therefore speak of a stable system, if all the eigenvalues of

λE

− A

,

E

nonsingular, lie in the open left half-plane. In this ase, the eigenvaluesofthepen il

λE

− A

arethoseofthematrix

E

−1

_A

(33)

2.3.1.2. Controllability, Observability and Balan ed Systems. During

the analysis of a linear system (2.18) one might ask how the system is

ae tedbytheinput

u(t)

. Thefollowingtwo hara terisationswillbe

on-sidered.

Denition 2.3.4 ([5℄).

x

∗

_{∈ R}

n

isrea hable(from theorigin

x

(0) = 0

)if

thereexistan admissibleinputfun tionand

t

e

<

∞

su hthat

x

(t

e

) = x

∗

holds (andhen e

x(t

e

) = x

∗

belongstothestatespa eofalinearsystem

(2.18)).

Denition2.3.5([5,46℄). Anonzerostate

x(0) = x

0

is ontrollableifthere existsanadmissibleinputfun tionsu hthatthesystem anbetransformed

from

x

0

toanygivenendstate

x

(t

e

)

withinanitetime

[0, t

e

]

.

Forlinear ontinuoustimesystemsthe on eptsof ontrollabilityand

rea hability oin ide( f. [5℄,Theorem4.18). Hen e,thefollowing on epts willbedevelopedforthe ontrollabilityofalinearsystem. Inthefollowing

hapterswewillneedthe on eptofthe ontrollabilityGramian.

Denition2.3.6([61℄Lemma4.57). Considerastablelinearsystem(2.18) with

E

nonsingular. The ontrollabilityGramian anbedenedasfollows:

P

=

1 2π

Z

∞

−∞

(iωE − A)

−1

BB

∗

(iωE − A)

−∗

d ω.

(2.19)

If one onsiders the eigenvaluede omposition of

P

, the eigenvalues

measurethedegreeof ontrollability,whereastheeigenve tors

orrespond-ingtothelargesteigenvalues anbeunderstoodasthedire tionsinwhi h

thesystemiseasyto ontrol.

Proposition 2.3.7 ([61℄ Corollary 4.58). Consider a stable linear system (2.18)with

E

nonsingular.The ontrollabilityGramian

P

(2.19)existsand istheuniqueHermitiansolutiontothefollowingLyapunovequation:

AXE

T

+ EXA

T

+ BB

T

= 0.

(2.20)

Inaddition,

P

ispositivedeniteifandonlyifthesystemis ontrollable.

Inpra ti e,wewilloftenbeabletomeasuretheoutput

y

(t)

ofalinear

system (2.18) . Iftheinput

u(t)

andtheoutput

y

(t)

areknown,wewant tore onstru tthestates

x

(t)

. Thisleadstothe on eptofobservability.

Denition2.3.8([46℄). Alinearsystem(2.18)is ompletelyobservable,if theinitialstate

x

0

anbere onstru tedfromthebehavioroftheinput

u(t)

andtheoutput

y

(t)

withinanitetimeinterval

[0, t

e

]

.

(34)

Again,wewillneedthe on eptofthesystemsobservabilityGramian.

Denition2.3.9. Considerastablelinearsystem(2.18)with

E

nonsingular. TheobservabilityGramian

Q

isdenedasfollows:

Q

= E

T

QE,

˜

with

˜

Q

=

1 2π

Z

∞

−∞

(iωE − A)

−∗

C

∗

C

(iωE − A)

−1

d ω.

(2.21)

Theinterpretationissimilartothe ontrollability ase: Ifone onsiders

theeigenvaluede ompositionof

Q

,theeigenvaluesmeasurethedegreeof

observability, whereas the largest eigenve tors anbe understood as the

dire tionsinwhi hthesystemiseasyto observe.

Proposition 2.3.10([61℄ Corollary4.58). Considera stablelinearsystem (2.18)with

E

nonsingular. Thematrix

Q

˜

(seeDenition2.3.9)existsand istheuniqueHermitiansolutionto thefollowingLyapunovequation:

A

T

XE

+ E

T

XA

+ C

T

C

= 0.

(2.22)

In addition,

Q

˜

and therefore also the observabilityGramian

Q

is positive

deniteifandonlyifthesystemisobservable.

Abalan edrepresentationofalineardynami alsystemisa

representa-tionofthesysteminwhi heverystateisequally"rea hableandobservable.

Thisse tionintrodu esthe on eptswhi hwillbeneededfortheBalan ed

Trun ation Model Order Redu tionin Se tion 5.2.1. The reader should

notethat thereexistseveralotherbalan edrepresentationsbesidetheone

presentedhere. They anbefoundintheworkbyGuger inandAntoulas

[40℄andthereferen estherein.

Denition 2.3.11 ([61, Denition7.5℄). The Hankel singularvalues, de-noted by

ς

j

, of a stablelinearsystem (2.18) with

E

nonsingularare the square-rootsoftheeigenvaluesof

P Q

.

Proposition2.3.12([61,Corollary7.7℄). Astablelinearsystem(2.18)with

E

nonsingularis ontrollableandobservableifandonlyifitsHankelsingular valuesarenon-zero.

Denition2.3.13([61,Denition7.10℄). Astablelinearsystem(2.18)with

E

nonsingularis alledbalan ed,ifthe ontrollabilityandtheobservability Gramiansareequalanddiagonal.

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Everystable, ontrollableandobservablelinearsystem with

E

nonsin-gular anbe transformed intoa balan ed representation. To do so, one

omputestheCholeskyfa torizationoftheGramians

P

= RR

T

and

Q

˜

= L

T

_L,

whi hexistsdueto thepositivedenitenessof

P

and

Q

˜

( f. Propositions

2.3.7and2.3.10). ComputingtheQRde ompositionoftheCholesky

fa -tors

L

and

R

leadstothefollowingde ompositionwithorthogonalmatri es

Q

c

and

Q

o

:

R

T

= Q

c

R

˜

T

and

L

= Q

o

˜

L.

It isobvious that

P

= RR

T

_{= ˜}

_{R ˜}

_R

T

and

Q

˜

= L

T

_L

_{= ˜}

_L

T

_L

_˜

. TheHankel

singularvalues annowbe omputedviathesingularvaluesof

LE ˜

˜

R

:

ς

j

2 = λ

j

(P E

_{| {z }}

T

QE

˜

Q

) = λ

j

( ˜

R ˜

R

T

E

T

L

˜

T

LE) = λ

˜

j

( ˜

R

T

E

T

L

˜

T

LE ˜

˜

R) = σ

2 j

(˜

LE ˜

R),

withthesingularvaluede omposition

˜

LE ˜

R

= U

b

ΣV

b

T

,

andorthogonal

U

b

, V

b

and

Σ =

diag

(ς

1 , . . . , ς

n

).

Thematri esofthelinear system annowbetransformedtoabalan edsystemrepresentation:

W

b

T

ET

b

, W

b

T

AT

b

, W

b

T

B, CT

b

,

where

W

b

= ˜

L

T

U

b

Σ

−1/2

,

T

b

= ˜

RV

b

Σ

−1/2

,

W

b

−1

= T

b

T

E

T

,

T

b

−1

= W

b

T

E.

TheGramian(astheobservabilityandthe ontrollabilityGramian oin ide

f. Denition2.3.13)ofthebalan edsystemisobtainedfromthoseofthe

originalsysteminthefollowingway:

T

_b

−1

P T

_b

−T

= Σ = W

_b

−1

QW

˜

_b

−T

= T

b

T

QT

b

.

2.3.1.3. Systemsnormsandspa esandinput-outputrelationship. As

theobje tiveisto approximatethegiven originalmodels,oneneedstobe

abletoquantifythedieren ebetweentheoriginalandtheredu edsystem,

or generallyspeaking, between twodynami alsystems. To do so,several

dierent spa es and their norms, both in the time and in the frequen y

(36)

Denition2.3.14([5, Se tion5.1.2℄). Let

f

: I → R

n

,with

I ∈ {R, R

−

,

R

+

,

[a, b]}

beave torvaluedfun tion. TheLebesguespa e

L

n

2 (I)

isdened as:

L

n

2 (I) =

(

f

: I → R

n

:

Z

t∈I

||f (t)||

2

1

2 <

∞

)

.

(2.23)

Inourmodels,inputandoutputwillbe onsideredasfun tionsinthese

spa es:

u(t) ∈ L

m

2 (I)

and

y(t) ∈ L

p

2 (I)

with

t

∈ I

( f. thedenitionof alinearsystem(2.18) ). Usually,oneisinterestedinarelationshipbetween inputand output. As su ha relationshipin the timedomainis des ribed

by a onvolutionwhi his oftendi ultto al ulate,the relationisoften

examinedinthefrequen ydomain. There,it aneasilybedeterminedbya

produ tofmatri es,as wewillseeinthisse tion. Forthetransformation

fromtimetofrequen ydomaintheLapla etransformationisused.

Denition2.3.15([18,Se tion15.2℄). TheLapla etransformofafun tion

f

: R

+

_{→ R}

isdenedas

F

(s) = L{f (t)}(s) =

Z

∞

0 f

(t)e

−st

dt,

(2.24) with

L{f

′

(t)}(s) = sF (s) − f (0).

(2.25) Forave tor,theLapla etransformhasto beseenelementwise. We

transformthelinearsystem(assuming

x(0) = x

0 = 0

):

L{E ˙x (t)}(s) = L{Ax(t) + Bu(t)}(s)

⇒

EL{ ˙x (t)}(s) = AL{x(t)}(s) + BL{u(t)}(s)

⇒

sEX(s) = AX(s) + BU(s)

⇒

X(s) = (sE − A)

−1

BU

(s),

and

Y

(s) = CX(s)

. Thisleads to thefollowing onne tion between the

inputandtheoutput:

Y

(s) = C(sE − A)

−1

BU(s).

Denition2.3.16. Thetransferfun tion

H

: C → C

p×m

ofthelinearsystem

(2.18)isdenedas

(37)

Fun tions in frequen ydomain will oftenbe interpreted as fun tions

of a omplexvariable. A detaileddes riptionof frequen ydomainspa es

forlinearsystems anbefound in[5℄. Hereweuse Hardyspa es

H

2

and

H

_∞

. Thefollowingsystemnorms anthenbeestablishedusingthetransfer fun tion

H(s)

andthe orrespondingHardyspa enorms:

Denition2.3.17([5,Se tion5.1.3℄). The

H

2

normofastablesystemis denedas

||Σ

lin

||

H

₂

:=

Z

∞

−∞

tr(H

∗

(−iy )H(iy ))dy

1

2 .

(2.27)

The

H

∞

normofastablesystemisdenedas

||Σ

lin

||

H

_∞

:= sup

y ∈R

(σ

max

(H(iy ))) ,

(2.28)

withmaximalsingularvalue

σ

max

. Proposition2.3.18([5℄). Itholds:

||Σ

lin

||

H

₂

=

p

tr(B

∗

_{QB) =}

p

_{tr(CP C}

∗

_),

(2.29)

forthesystemsGramiansasdenedin(2.21)and (2.19).

2.3.2. BilinearSystems. These ond lassofdynami alsystemswhi h

willbe onsideredinthisthesisarebilinearsystems.Anoverviewand

exam-ples anbefoundin[49℄.

Denition 2.3.19. A bilinearsystem of order

n

is asystem ofdierential

equationsofthefollowingform:

Σ

bil

:











E

˙x (t) = Ax(t) +

m

X

k=1

N

k

u

k

(t)x(t) + Bu(t),

y

(t) = Cx(t),

x(0) = x

0 ,

(2.30) where

E, A, N

k

∈ R

n×n

,

B

∈ R

n×m

,

C

∈ R

p×n

. Theinput

u(t) ∈ R

m

anbe

time-dependentjustasthestates

x(t) ∈ R

n

andtheoutput

y

(t) ∈ R

p

are.

Thevalueof

x

(0) = x

0

is alledinitialvalue.

Inthis se tion,onlysystemswith

E

6= I

n

,

E

nonsingular,willbe on-sidered.

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2.3.2.1. Volterraseriesrepresentation. A onne tionbetweenthe

sys-tems inputand output anbeestablished by using thefollowing Volterra

seriesrepresentationforthestatesofbilinearsystemsestablishedbyMohler

[49℄. Wewill onsidersystemswith

E

nonsingular.

x(t) =

∞

X

i =1

Z

∞

0 · · ·

Z

∞

0 m

X

k

1 ,k

2 ,...,k

i

=1

e

E

−1

A(τ

1 )

_E

−1

_N

k

1

· · e

E

−1

A(τ

2 −τ

1 )

_E

−1

_N

k

2 e

E

−1

A(τ

3 −τ

2 )

_{· · · E}

−1

_N

k

_i−1

e

E

−1

A(τ

i

−τ

i−1

)

_E

−1

_b

_k

i

· · u

k

1 (t − τ

1 ) · · · u

k

i

(t − τ

i

)dτ

1 . . . d τ

i

.

(2.31)

Theinput-outputrelationshipofthesystem anthenbedenedas:

y

(t) =

∞

X

i =1

Z

∞

0 · · ·

Z

∞

0 m

X

k

1 ,k

2 ,...,k

i

=1

Ce

E

−1

A(τ

1 )

_E

−1

_N

k

1

· · e

E

−1

A(τ

2 −τ

1 )

_E

−1

_N

k

2 e

E

−1

A(τ

3 −τ

2 )

_{· · · E}

−1

_N

k

_i−1

e

E

−1

A(τ

i

−τ

i−1

)

_E

−1

_b

_k

i

· u

k

1 (t − τ

1 ) · · · u

k

i

(t − τ

i

)dτ

1 . . . d τ

i

,

·

(2.32) with olumns

b

k

i

of

B

andVolterrakernelsdenedas:

h

(k

1 ,...,k

i

)

i

(τ

1 , . . . , τ

i

) = Ce

E

−1

Aτ

1 _E

−1

_N

k

1 e

E

−1

A(τ

2 −τ

1 )

_{· . . .}

(2.33)

. . .

· E

−1

N

k

_i−1

e

E

−1

_A(τ

_i

_−τ

i−1

)

_E

−1

_b

_k

i

,

where

i

= 1, 2, . . . , k

i

= 1, . . . , m,

and

τ

i +1

≥ τ

i

≥ 0

. Theinput-output relation annowbewrittenas:

y

(t) =

∞

X

i =1

Z

∞

0 · · ·

Z

∞

0 m

X

k

1 ,k

2 ,...,k

i

=1

h

(k

1 ,...,k

i

)

i

(τ

1 , . . . , τ

i

)

(2.34)

· i

Y

j=1

u

k

j

(t − τ

j

)

!

d τ

1 . . . d τ

i

.

Inpra ti e,theVolterrakernels

h

(k

1 ,...,k

i

)

i

(τ

1 , . . . , τ

i

)

needto be exam-ined in the frequen y domain as well. Therefore we need a multivariate

(39)

Denition2.3.20([24℄). Givenafun tion

f

(t

1 , . . . , t

n

)

denedon

R

n

dene

itsLapla etransform

F

(s

1 , . . . , s

n

)

by:

F

(s

1 , . . . , s

n

) =

Z

∞

−∞

· · ·

Z

∞

−∞

f

(t

1 , . . . , t

n

)exp

−

n

X

k=1

t

k

s

k

!

d t

1 . . . d t

n

.

(2.35)

We annowtransformtheVolterrakernels.

Denition2.3.21. The

i

-thordertransferfun tionoftheVolterrakernel

h

(k

1 ,...,k

i

)

i

(τ

1 , . . . , τ

i

)

= Ce

E

−1

Aτ

1 _E

−1

_N

k

1 e

E

−1

A(τ

2 −τ

1 )

_{. . . E}

−1

_N

k

_i−1

e

E

−1

A(τ

i

−τ

i−1

)

_E

−1

_b

_k

i

,

isdenedas

H

(k

1 ,...,k

i

)

i

(s

1 , . . . , s

i

)

= C(s

i

E

− A)

−1

N

k

1 (s

i −1

E

− A)

−1

_{. . . N}

k

_i−1

(s

1 E

− A)

−1

b

k

i

.

(2.36) Bytaking

N

= [N

1 . . . N

m

],

thisdenition anberewritten simultane-ouslyforall

N

k

byusingKrone kerprodu ts:

H

i

(s

1 , . . . , s

i

) =C(s

i

E

− A)

−1

N[I

m

⊗ (s

i −1

E

− A)

−1

](I

m

⊗ N) . . .

· [I

_|

m

⊗ · · · ⊗ I

_{z

m

_}

i −2

times

⊗(s

2 E

− A)

−1

)](I

_|

m

⊗ · · · ⊗ I

_{z

m

_}

i −2

times

⊗N)

· [I

m

⊗ · · · ⊗ I

m

|

{z

}

i −1

times

⊗(s

1 E

− A)

−1

)](I

m

⊗ · · · ⊗ I

m

|

{z

}

i −1

times

⊗B).

(2.37)

Inaddition,Brunietal. [19℄examinedthe onvergen eoftheVolterra seriesandestablishedthefollowingresult:

Proposition2.3.22. IftheVolterraseriesin (2.31) onverges,thenit

uni-formly onvergestothesolutionofthebilinearsystem(2.30) . Forbounded

inputstheVolterraseries(2.31) onvergesonanynitetimeinterval

[0, t

e

]

. The onvergen eoftheVolterraseriesis onne tedto thestabilityof

(40)

2.3.2.2. Stability. The notion of stability for bilinear systems diers

fromthatforlinearsystems. Forboundedinputs,thefollowingdenitionof

stabilityapplies:

Denition2.3.23([72,59℄). Thebilinearsystem(2.30)is alled bounded-input-bounded-output(BIBO)stable,ifforanyboundedinput,theoutput

isboundedon

[0, ∞)

. Aninput/outputis alledbounded ifitsatisesthe

following ondition:

||u||

∞

= max

j

sup

t∈[0,∞)

|u

j

(t)| < M

.

SiuandS hetzen[59℄ ombined onvergen eoftheVolterraserieswith BIBOstability. They showed the followingsu ient ondition forBIBO

stability.

Theorem 2.3.24 ([59℄). Let a bilinearsystem (2.30)with nonsingular

E

begiven,and letthepen il

A

− λE

bestable,i.e. thereexistreals alars

β, α

∈ R

with

β >

0

and

0 < α ≤ − max

i

(Re(λ

i

((A, E))))

su hthat

||e

E

−1

At

||

2 ≤ βe

−αt

, t

≥ 0.

(2.38) Assume

||u(t)|| =

pP

m

k=1

|u

k

(t)|

2 ≤ M

uniformelyon

[0, ∞)

with

M >

0

anddenote

Γ =

P

m

k=1

||E

−1

N

k

||

2

. ThenthesystemisBIBOstableif

Γ <

α

Mβ

.

The bilinearsystem ishen estableifthematri es

N

k

are su iently bounded.

2.3.2.3. Rea hability,observabilityandbalan edrepresentation. Asfor

linearsystems,the on eptsofrea hability,observabilityandbalan ed

rep-resentationexistforbilinearsystems. However, the on epts need to be

generalized,whi hwillbedoneinthefollowingse tion.

Denition 2.3.25([25, 56℄). A state

x(t

e

)

of a bilinearsystem (2.30)is rea hable (from the origin

x

(0) = 0

) if there exists an admissibleinput

fun tionthat mapstheoriginofthestate spa eintothestate

x

(t

e

)

ina niteintervaloftime

[0, t

e

]

.

Denition2.3.26([56℄). Abilinearsystem(2.30)is alled(span)rea hable ifthespa eofallrea hablestates

X

reach

spans

R

n

.

For a bilinear system (2.30) with

E

6= I

nonsingular, the following statementsforrea hability anbederived. Let

P

1 (t

1 ) = e

E

−1

At

1 _E

−1

_B,

P

i

(t

1 , . . . , t

i

) = e

E

−1

At

i

_E

−1

_[N

(41)

Denition2.3.27([72℄). Ifitexists,therea habilityGramianisdenedas

P

=

∞

X

i =1

Z

∞

0 · · ·

Z

∞

0 P

i

P

i

∗

d t

1 . . . d t

i

.

(2.39)

ZhangandLam[72℄establishedthefollowingtheoremfortheexisten e oftherea habilityGramian:

Theorem2.3.28([72℄). Therea habilityGramian(2.39)exists,if (i) thepen il

A

− λE

isstable,with

||e

E

−1

At

||

2 ≤ βe

−αt

, t

≥ 0,

(2.40)

where

β >

0

and

0 < α ≤ − max

i

(Re(λ

i

(A, E))

,

β, α

∈ R.

(ii)

Γ

1 <

√

2α

β

,with

Γ

2

1 = ||

P

m

k=1

E

−1

N

k

N

T

k

E

−T

||

2 .

The onne tionof

P

tothebilinearLyapuonvequationsandthe

rea h-abilityofthesystem annowbeestablished:

Theorem 2.3.29 ([72℄). Suppose

A

− λE

is stable, and the rea hability Gramian

P

exists. Then

(i)

P

satisesthefollowingbilinearLyapunovequation:

AXE

T

+ EXA

T

+

m

X

k=1

N

k

XN

k

T

+ BB

T

= 0.

(2.41)

(ii) Thebilinearsystem(2.30)isrea hableifandonlyif

P

ispositive denite.

Proposition2.3.30([72℄). If (2.41)hasauniquesolution,thenthesolution

P

issymmetri .

Forlinearstablesystems,itisknownthatiftheLyapuonvequationhas

auniquesolutionitistherea hability( ontrollability)Gramian. Forbilinear

systems, however, it is possible that a unique solution to the Lyapunov

equationisnottherea habilityGramian.Considerforexamplethefollowing

bilinearsystem( f. [72℄):

˙x

=

−x + 2xu + u.

This leads to the solution of the Lyapunov equation

p

= −

1

2

. But the integrals

p

˜

i

=

R

p

i

p

i

T

leadto

p

˜

i

= 2

i −2

, whi hgives

p

=

P

∞

i =1

2 i −2

whi h doesnot onvergehen etherea habilityGramiandoesnotexist.

(42)

Theorem2.3.31([72℄). Suppose

A

− λE

isstable.

•

(2.41)hasapositive(semi)denitesolution

X

ifandonlyifthe rea hability Gramian (2.39) exists and onverges to a positive

semidenitematrix

X

ˆ

satisfying(2.41) .

•

If (2.41)hasauniquepositive (semi)denite solution

X

, then (2.39) onvergesto

X

andtherefore

X

istherea habilityGramian. Forabilinearsystem(2.30)with

E

nonsingularthefollowingstatements forobservability anbederived.Let

Q

1 (t

1 ) = Ce

E

−1

At

1 _,

Q

i

(t

1 , . . . , t

i

) = [Q

i −1

E

−1

N

1 Q

i −1

E

−1

N

2 . . . Q

i −1

E

−1

N

m

]

T

e

E

−1

_At

_i

, i

= 2, 3, . . .

Denition2.3.32([72℄). Ifitexists,theobservabilityGramianisdenedas

Q

=

∞

X

i =1

Z

∞

0 · · ·

Z

∞

0 Q

∗

i

Q

i

d t

1 . . . d t

i

.

(2.42)

ZhangandLam[72℄establishedthefollowingtheoremfortheexisten e oftheobservabilitymatrix: