H ∞-norm for the

Appliation of the

H ∞

^{-norm for the} identiation of linear time-invariant and linear parameter-varying

models

Ph.D. dissertation

prepared under the joint supervisor sheme between

University of Poitiers

and

Budapest University of Tehnology and Eonomis

Author :

Dániel Vizer

Advisors :

Guillaume Merère, Ph.D., HdR.

Laboratoired'Informatique etd'Automatique pour les Systèmes,LIAS

(Université de Poitiers)

and

Bálint Kiss, Ph.D.

Irányítástehnika ésInformatikaTanszék

(Budapesti M¶szaki ésGazdaságtudományi Egyetem)

2015

Université de Poitiers

Travaux Sientifiques

présenté

l'Universitéde Poitiers en vue de l'obtention du

Diplme de Dotorat

par

Dániel Vizer

Appliation of the

H ∞

^{-norm for the}

identiation of linear time-invariant and

linear parameter-varying models

Soutenuepubliquement le10 déembre2015

Jury

Président :

Rapporteurs : Xavier Bombois Direteur dereherhe

Laboratoire Ampère, Group ACM

Levente Adalbert Kovás Maître de onférenes, Hdr.

Óbudai Egyetem,

Examinateurs : Marion Gilson Professeur des universités

Université de Lorraine,CRAN

Thierry Poinot Professeur des universités

Université de Poitiers, LIAS

József Kázmér Tar Professeur des universités

Université d'Óbuda

Direteurs dethèse : Guillaume Merère Maître de onférenes, Hdr.

Université de Poitiers, LIAS

BálintKiss Maître de onférenes

Université des Sienes Tehniques et

Éonomiques de Budapest,IIT

"A warrior does not give up what he loves, he nds the love in

what he does."

Dan Millman - The way of the peaeful warrior

Aknowledgements

Here, I would like to mention several people in my life who have ontributed

somehow, eitherdiretly orindiretly,tothe birthof thismanusript. Letme todo

this by using the orrespondinglanguages. But rst of all, thanks for the existene

of the oee that has been the mostly used drug during these years of researh. It

has gave melotsof exellentideas and motivation.

Je voudrais tout d'abord remerier le Dr. Guillaume Merère qui m'avait na-

lement onvainu de hoisir la thèse au lieu de travaillerdans l'industrie et qui est

devenu mon direteur de thèse. Puis, au l du temps, ave ses onseils et ses avis,

ilm'a faitprogresser professionnellementainsi quepersonnellement.Il fautpartiu-

lièrement souligner ses eorts pour me pousser à rédiger orretement. Celui était

un vrai dé pour moi et également pour lui. Enn, je pense que je ommençais à

aimer ladoumentation aussi. Tu es un supermentor pour la reherhe. C'était un

vraiplaisir de travaillerave toi.J'espère que nous pourrons ontinuer lareherhe

ensemble n'importe oùje me trouverai dans l'avenir.

Jedois égalementmentionerleprof.EdouardLarohequim'avaitaueillitelle-

menthaleureusement haque fois sur le site de l'ICUBE etl'IRCAD à Strasbourg

et qui m'avait beauoup aidé ave ses onseils et ave nos disussions très inté-

ressantes lors de mon travail. Il ne faut pas oublier non plus le Dr. Olivier Port,

étantmathématiien, gràeàquije peuxdésormaisvoirlesproblèmesposés un peu

diéremment.

Ctéoeur,quoiquel'onseonnaisseilyapeudetemps,ilfautquejesolliitema

famillefrançaise"aussi.Plus préisément,ChantaletGeorgesquim'avaitaueilli

hezeuxommeleurlsetave quij'aipasséde bons moments.Jene trouvepasde

mots à exprimer mes sentiments vers vous. Meri d'être! J'espère que nous restons

en ontat enore pendant quelques dizaines d'années.

Jevoudraiségalement iterles trentaines de personnes que j'airenontrées pen-

dant mon séjour à Poitiers et qui ont tous ontribué en quelque sorte à mon déve-

loppement personnel. Vous êtes tous dans mon oeur (ainsi que dans mon livredes

mémoires) pour toujours!

Etbien sur,je parlede toiapart espèede Cédri.Alors,tuétais toujourslàau

ldu temps pour m'aider, disuter, s'amuser,uisiner etm'embêter... Tout simple-

ment,meripour tout. Lemondeest sansn et ela e quiest génial.Ne t'inquiéte

pas, onsera toujoursdans le oin.

ésMS. tanulmányaimatvégigkísérveszakmailagelvezetettadoktoritanulmányaim

küszöbéig. Ezen felül azelmúlt három évben is mindig számíthattama tanásaira,

segítségére. Köszönet illeti továbbá Dr. Kiss Bálintot, aki konzulensemként sokat

tett azért, hogy ajelen munkaelkészülhessen.

Szeretném megköszönni a saládomnak, hogy felneveltek, hogy id®r®l-id®re fel-

töltenek energiával és akik miatt olyan vagyok, amilyen. Különösen szerensésnek

érzem magam, mert úgy gondolhatok rátok, mint a nagyon közeli barátaimra. Itt

szeretnémkülönkiemelniaszüleimet,agénekért,azérthogybeneveztekazÉletnev¶

játékba, ésazért hogy támogattakmindenben és tudat alatt igazgattákaz utamat,

ami nem kis teljeseítmény tekintve, hogy mennyire önfej® voltam és vagyok. Nem

tudom, hogyansináltátok, de sak hafele olyan sokattudok adni agyerekeimnek,

mintTinekem, akkorisavilágegyiklegjobbszül®jeleszek. Köszönömagyökereket.

Köszönöm a szárnyakat. Szeretlek titeket! Maradjatok ilyenek még nagyon sokáig

nekünk!

Végül, de nem utolsósorban, hálás köszönet az életem Párjának, Feleségmnek,

Böbinek

♥

^{is. Köszönöm, hogy vagy és leszel nekem, nekünk. Mert hiszen tudod,}

hogy legértékesebb dolog, amit egy emberadhat a másiknak azazideje.

Boldogvagyok. Itt és most. Jesuis heureux. Ii etmaintenant.

Résumé

Depuis près de 20 ans, les modèles linéaires à paramètres variants (LPV) font

l'objet d'une étude très poussée en identiationpuisqu'ils peuvent être onsidérés

omme le hénon manquant entre la modélisation linéaire à temps invariant et la

modélisationnonlinéaireet/ouàparamètrestempsvariant. Cettethèse,réaliséeen

o-tutelle entre l'Université de Poitiers et l'Université Siene Tehnique de Buda-

pest, s'insrit dans ette mouvane et herhe à développer de nouvelles méthodes

de modélisationet d'identiation des modèles linéaires à paramètres variants (et,

par extension, des modèles linéaires à temps invariant (LTI)) sous forme de rep-

résentation d'état, à l'aide de la norme

H ^∞

^{, ave une attention} partiulière aux strutures méatroniquesexibles. Ene quionerne lesmodèles LPV, dans ette

thèse, l'approhe dite loale est plus partiulièrement onsidérée. Elle onsisteplus

préisément (i) à obtenir des données d'entrées-sorties dites loales aquises en un

ertainnombrede pointsde fontionnement hoisispar l'utilisateur, (ii)àidentier

desmodèlesloauxboite noires,(iii)àinterpoleresmodèlesloauxan dealuler

lemodèleLPVnal. Dansettethèse,de nouvellessolutionsontétéproposéespour

améliorer haune des étapes sus-itées. Le hoix optimal des points de fontion-

nement a ainsi été étudié et une nouvelle méthode utilisant la nu-gap métrique a

été proposée. Leproblème de ohérene des basesinhérentà l'approhe loaleaété

onsidéré etdeux approhes utilisantlanorme

H ^∞

^{ont ainsi été} développées. Dans un premier temps, en supposant l'aès, loalement, aux strutures boîte grise des

modèles LTI à identier, le problème de restruturation des modèles loaux boite

noire en modèle boite grise est résolu à l'aide de l'algorithme "proximity ontrol"

avant d'appliquer une interpolation, de type moindres-arrés, des paramètres lo-

aux restruturés. Dans un deuxième temps,une interpolationdirete des modèles

loaux boite noire est développée en minimisant, à l'aide toujours de l'algorithme

proximityontrol,ladistane entre leomportemententrée-sortiedu modèle LPV

à identier et les modèles boite noire loaux pré-estimés. Cette dernière tehnique

est développée pour fontionner ave des modèles LPV ayant une représentation

linéairefrationnaire(LFR).Lesdiérentes tehniques développées dansette thèse

sontnalementtestéesenutilisantàlafoisdesdonnéesdesimulationetdesdonnées

expérimentales réelles d'un robotexible.

Kivonat

A lineáris paraméterváltozós (LPV) modellek identikáiója kiemelt gyelmet

kapott az elmúlt 20 esztend® során. Ezek a modellek lényegében lánszemnek te-

kinthet®k a lineárisid®invariáns(LTI)ésa nemlineáris vagylineáris id®benváltozó

(LTV) modellek között. Jelen doktori disszertáió egy kett®s-témavezetés¶, nem-

zetköziegyüttm¶ködés (o-tutelle)kereténbelülvalósult mega Budapesti M¶szaki

ésGazdaságtudományiEgyetemésaPoitiers-iEgyetemközött. Akutatásf® irány-

vonala az LPV és LTI modellek állapottérben történ® identifkáiója volt, exiblis

mehatronikai rendszerek számára, mely a

H ^∞

^norma bevonásával valósult meg.

Az LPV estben, az ún. lokális eljárást vizsgáltam behatóbban, mely a következ®

lépésekb®lépül fel : (i) ki- ésbemenetiadatsorok gy¶jtése az identikáióhoz el®re

meghatározottmunkapontokban,(ii)lokálisfeketedobozmodellekidentikáiójaaz

el®z®lépésbenmértadatokalapján,(iii)alokálismodellekinterpoláiójaazütemez®

változóm¶ködésitartományamentén. Doktoridisszertáiómban, afentebbemlített

lépésekmindegyikére javaslokjavításilehet®ségeket. A lokálisidentikáióbanrészt

vev®munkapontokoptimáliskiválasztásaérdekébenegyújalgoritmustdolgoztamki

a nu-gap mérték alkalmazásával. A lokális modellek interpoláiója során felmerül®

koherens bázis problémát két

H ∞

^{norma alapú módszer} kifejlesztésével, oldottam meg. Els®ként a fekete-szürke doboz áttérést valósítottam meg, lokálisan, az ún.

proximity ontrol algoritmussegítségével. Ezt azátstruktúrálást követ®en a loká-

lismodellekinterpoláiójátalegkisebbnégyzetekmódszersegítségévelvalósítottam

meg. A második módszer a lokális fekete doboz modellek direkt interpoláióját

végzi elszinténaproximityontrol algoritmusalkalmazásával. Ezutóbbimódszer

lineáris törtalakban felírt LPV modelleket ad eredményül. A disszertáióban ki-

dolgozott módszerek hatékonyságát szimulált ésvalósadatok együttes bevonásával

demonstráltam.

Abstrat

During the last two deades, the linear parameter varying (LPV) models have

been inthe main fousin the eld ofsystem identiationtheory sinethey an be

onsideredas themissing linkbetween the linear time-invariant(LTI)and thenon-

linear and/or parameter time-varying modeling approahes. This thesis has been

performed in the adre of a bi-direted researh projet o-tutelle between the

University ofTehnologyand Eonomisandthe University ofPoitiers. Duringthis

researh the main fous has been plaed to the identiation of state spae LPV

and LTI models for exible non-linear strutures by using the

H ^∞

^{-norm. As far}

as the LPV models are onerned, the so-alled loal approah is, more preisely,

onsidered whihontains the following steps: (i) I/O data sequenes are gathered

in ertain working points. (ii) loal blak-box LTI models are estimated in eah

working point, (iii) then, the loal models are interpolated in order to obtain the

nal LPV model. In this thesis, novel methods have been proposed for eah of the

above ited steps. A method able to determine a set of operating points by ap-

plying the nu-gap metri non-linearity measurement has been developed. In order

toensurethe oherent basis representation, whih isneessary forthe interpolation

step, two

H ∞

-norm-based methods have been developed. First, the loally esti- mated blak-box state-spae models are re-strutured into gray-box ones by using

the proximityontrolalgorithmfollowedby the least-squares-based interpolation of

the obtained loal gray-box models. Seond, a behavioral interpolation (BI) has

been performed by minimizing the frequeny domain distane between eah or-

responding loal blak-box LTI and frozen LPV models by using one global ost

funtion. This latter tehnique is designed to operate with LPV models written as

linear frationalrepresentations (LFRs). Finally,the performane of the developed

methodshasbeentestedby using,atsametime,simulatedandreal datasequenes.

Contents

1 General Introdution 1

I Theoretial developments 7

2 Appliation of the linear frational representation for model de-

sription 9

2.1 Introdution . . . 9

2.2 Notions and denitions . . . 10

2.3 Linear system modelingby usinglinear frational representations . . 11

2.3.1 Linear time-invariantmodels . . . 11

2.3.2 Multi-dimensionallineartime-invariantmodels . . . 12

2.3.3 Linear parameter-varying models . . . 13

2.3.4 Comments onwell-posedness . . . 15

2.4 Blak and gray-box state-spae models . . . 15

2.4.1 Identiability . . . 17

2.5 Conlusion . . . 19

3 Non-smooth and non-onvex optimization: the proximity ontrol algorithm 21 3.1 Introdution . . . 21

3.2 The proximity ontrol algorithm . . . 22

3.2.1 Denitions . . . 22

3.2.2 Basiidea . . . 23

3.2.3 Outerloop. . . 25

3.2.4 Inner Loop . . . 26

3.2.5 The algorithm . . . 29

3.3 Conlusion . . . 30

4 Gray-boxstate-spaeLTImodelidentiationbyusingre-struturing tehniques 31 4.1 Introdution . . . 31

4.2 Problem statement . . . 32

4.3 Output-error framework . . . 33

4.4 From blak-box state-spae models to gray-box ones . . . 35

4.4.1 An

H ^∞

-norm-based approah . . . 37

4.4.2 Convergene of the proximity ontrolalgorithm . . . 40

4.5 Conlusion . . . 42

5 State-spae LPV model identiation from loal experiments 43 5.1 Introdution . . . 43

5.2 Problem statement . . . 44

5.3 Operatingpointseletionfor LPV modelidentiation . . . 45

5.3.1 Nu-gap metri . . . 47

5.3.2 Seletion algorithm . . . 48

5.4

H ^∞

-norm-basedloalre-struturing method . . . 49

5.5

H ^∞

-norm-basedbehavioral interpolation (

H ^∞

^{-BI) . . . . . . . . . . 51}

5.6 Conlusion . . . 53

II Appliations 55 6 Simulation examples 57 6.1 Introdution . . . 57

6.2 Gray-boxlineartime-invariantmodelidentiationoftheprinter-belt system . . . 57

6.2.1 Printer-belt . . . 57

6.2.2 Data Generation . . . 60

6.2.3 Identiationproedure . . . 60

6.2.4 Results . . . 61

6.2.5 Conlusion. . . 76

6.3 LPV modelidentiationof the mass-spring-dampersystem . . . 76

6.3.1 Mass-spring-dampersystem . . . 76

6.3.2 Working point seletion . . . 80

6.3.3 Estimationof the nalLPV model . . . 84

6.3.4 Conlusion. . . 91

7 Real data driven examples 93 7.1 2-DoF exible roboti manipulator . . . 94

7.1.1 Nonlinear and linearized dynami models . . . 96

7.2 Experimentdesign . . . 101

7.2.1 Loal experiments. . . 101

7.2.2 Global experiments . . . 101

7.2.3 To sum up . . . 103

7.3 Blak-box linear time-invariantmodelidentiation . . . 107

7.4 Linear parameter-varying modelidentiationand validation . . . 107

7.4.1 Gray-box and blak-box LPV modelidentiation . . . 110

7.4.2 LPV modelvalidation . . . 110

7.5 Conlusion . . . 112

8 Summary of the obtained results and future researh objetives 115 8.1 Thesis Points . . . 115

8.2 Possible researh diretions . . . 116

8.2.1 Improvements of the null-spae-based tehnique . . . 117

8.2.3 Handlingthe possible blak-box modelestimation errors . . . 118

8.2.4 ImprovementsoftheLPVmodelidentiationtehniquesbased onloalexperiments . . . 118

8.3 Furtherindustrialappliations . . . 119

9 Appendix 121 9.1 Nonlinear dynami modelof a2-DOF roboti arm . . . 121

9.2 Summary of the PO-MOESP algorithm . . . 123

9.3 Summary of the SRIVCalgorithm. . . 125

9.4 Norms onmatries, signals and systems . . . 126

9.4.1 Matrixnorms . . . 127

9.4.2 Signalnorms . . . 127

9.4.3 System norms . . . 128

Chapter 1

General Introdution

In the rapidly developing world, not only the available omputational apa-

ity,but alsothe demands for morereliableteleommuniation onnetions,optimal

and intelligentenergy distribution and generation systems, better robot pilots, au-

tonomous systems to explore new plaes, for seurity and resue purposes or for

military appliations..., et. just to name a few, have ontinuously been growing

fromday-to-day. Theseappliations,one-by-one, doneessitatethe adequateappli-

ationofthemostreentresultsofseveraldierentelds,suhas,appliednumerial

mathematis, physis, hemistry, omputer sienes..., et. The olletive applia-

tion of the previously enumerated sienti priniples, with the intention to reah

better and autonomous behaviour of a ertain proess, are joined together in the

so-alled ontrol theory. On top of that, in order to be able to develop reliable

ontrollerswhih meet the strit requirementsstated by the onsidered appliation

eld,itisindispensabletohaveaorretmodeloftheproessorsystemunderstudy.

Thisisthe momentwhensystem identiationenters intothe piture. Beingavery

important omponent of the modern ontrol theory, system identiation aims at

derivingmodelsforphysialproessesandestimatingitsparameters. Averyimpor-

tant requirement at the end of the identiation proedure is that the parametri

modelestimationistratablebynumerialproessingunits,suhas,e.g.,embedded

omputers,PCs ormorereentlyeven by smartphones. Despite the fatthat there

is anenormous amount of ontributions in this eld whih an be overed by some

benhmark publiations, for instane, [78, 113, 103, 47, 135, 81℄ (see as well the

referenes therein), there remain lots hallenging open questions waiting to be an-

swered, mainlyonerning the determination of reliable linear models of nonlinear

and/ortime-varyingproesses. Thislatter pursuitinvolves thedevelopmentof spe-

i model strutures in whih the nonlinear behavior has been broken down into

several loallinear models, in order to obtain amore tratable modelstruture for

the original nonlinear system under study. This interest an mainly be explained

by the following reasons. On the one hand, the resulting model is often designed

to be lose to the standard linear time-invariant (LTI) one but with an embedded

strutural exibility able to ope with time-varying, even highly nonlinear behav-

iors. Ontheotherhand,thedevelopmentofsuhasetofmultiplemodelsisdiretly

linked toontrolengineering, wherea ontroller must bedesigned inorder to meet

thelosed-looprequirementsforagivenplantindierentoperatingonditions,e.g.,

and onquer basiideaisthegainshedulingapproah[3,124,75℄(see alsotheref-

erenes therein for further informationabout gain sheduling) whihan be briey

summarized as follows:

nd one or more sheduling variables whih an ompletely parameterize the

operating spae of interest forthe system toontrol,

dene a parametri family of linearized models for the plant assoiated with

the set of operatingpointsof interest,

design a parametri ontroller whih an both ensure the desired ontrol ob-

jetivesineahoperatingpointandanaeptablebehaviorduring(slow)tran-

sients between one operatingondition and the other.

A wide body of eient ontroller design tehniques sharing this basi idea is now

available in the literature (see, e.g., [126, 64, 2℄), whih an be solved reliably,

provided that a suitable model in a parameter-dependent form has been derived

beforehand. By onstrution, the reliability of these tehniques highly depends on

the availabilityof a suitable and onsistentmodel. In order tosupportthis kind of

multiple model-based ontroller design, several methods have been developed dur-

ing the past two deades, in order to derive reliable multiple model strutures for

nonlinear dynamialsystems, suhas,e.g.,theswithedlinearsystems [110,111℄or

the multiple-model adaptive estimation (MMAE) tehniques [55℄. One of the mul-

tiple model-basedsolutionsfound inthe literatureisthe so-alledlinearparameter-

varying(LPV) model struture. More preisely, the modelingand estimation prob-

lem of LPV models are reently among the most popular researh topis in system

identiation theory [135, 81, 83℄. This is the main reason why the identiation

of (state-spae) LPV models is prinipallyonsidered in this thesis. These spei

strutures anbeseenasaombinationofloalmodelswithparametersevolvingas

a funtion of measurable time-varying signals, alled the sheduling variables, sig-

nals whihan berelated todierent operatingpointsof the system toidentify. As

far asthe determinationof LPV modelsis onerned,twobroad lassesof methods

an befound inthe literature [24℄:

(i) rst, the analyti methods onsisting exlusively in onverting the available

nonlinear equations governing the behavior of the system into an LPV rep-

resentation by resorting to extensions of the familiar notions of linearization

[58, 118, 123, 128, 127, 125, 74, 10, 89, 66,112,135,1℄;

(ii) seond, the experimentalmethodsinlinedtodetermining LPV models of the

plantunder study from the available input-outputdata [84, 30,32,133, 23℄.

The rst lass gathers the solutions using rst-priniples modeling onsiderations.

They more preisely try to transform the original nonlinear model into a reliable

LPV representation. Aording to the literature, more speially, the following

lasses of methodsan be distinguished:

extended linearization[123℄;

pseudolinearization[118℄;

global linearization[58℄;

Jaobian linerization[128, 125℄;

state transformation [127℄;

funtion substitution [89℄;

veloity-based approah[74℄;

tensor-produt polytopi deomposition [10, 112℄;

dierentialgeometri approah [1℄.

Theseondfamilyofmethodsan besplitup intotwofurthersub-lasses, generally

alled, the global approah and the loal approah,respetively. On the one hand,

theglobalapproahfousesontheglobalproedureandassumesthatoneglobalex-

perimentanbeperformedduringwhihthe ontrolinputsaswellasthesheduling

variablesan be both exited[73, 140℄. Hereby, allthe non-linearitiesofthe system

are exited simultaneously by passing through a large number of operating points.

On the other hand, reent methods [51, 84, 30, 32, 23℄ are based on a multi-step

proedurewhere

1. loalexperimentsarearriedoutinwhihtheoperatingpoints(orresponding

to xed values of the sheduling variables) are held onstant and the ontrol

inputsare (persistently) exited,

2. loalLTImodelsareestimatedbyusingthesesetsofloalinput/output(I/O)

measurements,

3. aninterpolationphaseisperformedinordertoderiveanalglobalparameter-

dependent model.

It isobvious that this multi-stepapproah isa lotloserto the standard proedure

used for nonlinear system identiation or the one dediated to gain sheduling.

Both lasses of methods have advantages and drawbaks (see e.g., [135, Chapter

1℄ for more details about the state-of-the-art of LPV modeling and identiation).

Fromapratialpointofview, theglobalapproahmay suerfromthe diultyto

satisfytherih exitationof theontrolinputsand theshedulingvariablessimul-

taneously. Itisobviousthatsuhanexperimentalproeduremay notbereasonable

for spei appliations mainly for safety or eonomi reasons. On the ontrary,

applying small variations around partiular operating points, as onsidered by the

loalapproah,ismore oneivablein many pratialases. Furthermore,the iden-

tiationof LTI modelsis wellestablishedand implementationsare availableinthe

Matlab environment [78, 79℄. That is the reason why during this thesis, the de-

velopmentsfous on the loal approah. Notie also that, inthis thesis, duringthe

identiationproedure, state-spae LPV models are sought to be estimated. This

hoie an be explained by the following reasons. First, when ompared with the

standard input-output LPV representations, a state-spae model often provides a

moreparsimoniousrepresentationofthesystem(fewerparametersandlowerMMil-

landegree)withonlyonetuningparameter: theorderofthesystem. Seond,thanks

totheintrodution(withautionintheLPVframework(see[136,65,138,23℄foran

importantdisussionabouttheoherentbasisissue))ofspeiuser-denedsimilar-

itytransformation,itispossibletoyieldanumeriallybetteronditionedframework

for parameter estimation. Last, but not least, state-spae representations are often

favored when ontroller designis the reason why the modelis built.

Remark 1.1. In many pratial ases, the systems to identify are modeled by a

so-alled quasi-LPV model [24℄. In this ase, the sheduling variables are somehow

relatedtothe statevariablesof themodelrepresentation. Thus,undersuhpratial

onditions, it is diult to guarantee that the sheduling variables are kept exatly

onstant during the rst step of the loalapproah. However, as shown hereafterin

Chapter 7, this diulty an be handled and reliable LPV models an be estimated

In this thesis, the loal approah is revisited by suggesting, for step 1 and 3

mainly, innovative solutions. As far as the seletion of the loal working points

is onerned, it is important to point out rst that, in the literature, most of the

time, the Authorshoose equidistantworkingpointsby seletinga user-dened but

xed step between twoloalexperimentsin terms of the shedulingvariables. It is

pointed out in Chapter 5 that suh a simple hoie may lead to a set of working

points where the loal information about the nonlinear system under study is far

frombeingoptimal. Inorder toaddressthis problem,anovel approahisproposed

in this thesis to selet a better established set of loal operating points [148, 149℄.

Asfarastheinterpolationstepisonerned,itisnowwell-known thatinterpolating

properlyloalonsistent LTIstate-spaemodelsisfarfrombeingeasy,espeiallyin

the blak-box framework, beause, for many LPV state-spae representations, the

similarity transformation onverting the initial, even stati-dependent state-spae

models, yields equivalent LPV models depending on the time-shifted or derivative

versionsof the shedulingvariables[135,65,137℄. Thus, asfarasinterpolationstep

is onerned, the loal blak-box state-spae LTI models must be transformed into

a ommonoherent basis representation. Interesting solutions have been proposed

intheblak-boxframework,i.e.,whennopriorinformationaboutthe systemunder

study isavailable[139,54, 84, 105, 31,99,32℄. Among them, the most numerially

reliable solutions are the ones published in [84, 32℄. In a nutshell, as pointed out

rst in[136℄, then in [65℄, the interpolation step involved inthe loalapproah an

lead to a global LPV model with an inaurate dynami behavior even if the loal

LTI models are onsistent. These diulties are takled hereafter by onsidering

two omplementarysolutions.

When spei strutural prior information about the system to identify is

available,i.e.,whengray-boxLPVmodelsarehandled,thesolutiondeveloped

in this thesis onsists in resorting to the knowledge available from the study

of the non-linear equations governing the system behavior in order to x the

strutureof theglobalLPVmodel,thenusingthe availableexperimentaldata

sets in order to estimate the unknown parameters and to rene the analyti

model omposed of unknown values. More speially, we onvert the loal

input-output data-sets, translated initially into (fully-parameterized) blak-

boxmodels,intore-parameterizedgray-boxstate-spaeformsderivedfromthe

frozen struture of the LPV representation and alulated for the onsidered

working points.

When blak-boxstatidependentLPVmodelsareonerned,aspeiatten-

tion is paid to the preservation of the input-output dynamis of the models

instead of doing a diret interpolation of the system matries. More spei-

ally,by onsidering a user-dened tting measure (hereafterthe

H ^∞

^-norm),

given a set of loalLTI models as well as a fully-parameterized stati depen-

dentLPVmodelstruture,our solutiononsistsinestimatingtheLPV model

parameters so that the nal LPV isoptimal with respet toa global measure

of the error between the loalmodels and the LPV representation. As shown

inChapter5,suhaninput-outputbehaviorapproahallowsustoirumvent

the diult oherent basis issue.

Thedissertationisbuiltup asfollows. The rstpart ofthismanusriptpresents

the maintheoreti developments. InChapter 2,the LFRs,usedoriginallyinthe ro-

resentations anbe appliedfor systemmodelingand system identiation. Chapter

3introduesanon-smoothandnon-onvexoptimizationtehniqueusedhereafterin

thisthesistominimize

H ^∞

-norm-basedostfuntions. Notieindeedthat,asshown inChapter 4,the solutionsdeveloped tobypass the diultproblemofLTImodels

interpolationare mainlybased on

H ^∞

-norm-baseddisrepany measurements. The identiationofLPV models from loalexperiments istakled inChapter 5 where,

more speially, a novel method is proposed to determine the set of loal operat-

ingpointsinvolved in the loalapproah followed by the introdutionof two novel

methods aiming at estimating a reliable LPV state spae model. The developed

methods support the identiation of both blak- and gray-box LPV models. The

rst Part is then losed by Chapter 8 where, rst, the developed novel tehniques

areenumerated intheformofthesis points. Then,severalideas have beengathered

so that tomotivate and support further researh pursuits. The seond Part of this

thesis demonstratesthe eetiveness of the developed methods. To reahthis goal,

rst,simulationsbyonsideringdierentkindsof systemsareperformedinChapter

6. Seond, the proposed LPV identiation approahes have been tested on a real

testbenhaswell,onsistingina2-DoF exiblerobotimanipulatorusedinardia

surgery. Finally, in the Appendix, mathematial preliminaries, the desriptions of

several existing identiationtehniques, system modeling aspets are provided.

Part I

Theoretial developments

Chapter 2

Appliation of the linear frational

representation for model desription

2.1 Introdution

The main purpose of this Chapter istoexplain howa largelass of linear state-

spae models an be transformed into a linear frational representation (LFR) in

order toset the stage for the system identiation. Originally,this modelrepresen-

tation isintrodued inrobust ontrol theory [164℄ asa speial formof the so-alled

Redheer star produt [119℄. More preisely, LFRs are widely used to represent

any feedbak interonneted losed-loop model having dierent kind of unertain-

ties during the ontrollersynthesis [164℄.

Asfarasitsappearane insystemidentiationisonerned,LFRshavealready

been employed in[71,70,73℄,toestimateblak-box linearparameter-varying state-

spaemodelswhereanoutput-error(OE)set-upisonsideredandtheresultingost

funtion is minimized by applying nonlinear programming. More reently, among

others, LFRs have been used in [108, 141, 142, 102℄ (see the referenes therein as

wellforotherreentappliations)toestimatemodelshavinginnerstrutured stati

non-linearities. In [108℄, more preisely, a pieewise ane proedure is developed,

while in [141, 142, 102℄, the so-alled Best-Linear-Approximation (BLA) approah

is onsidered. On top of that, in [34℄, the Modelia software tool is applied to

generate LFRs, in order to model nonlinear systems having unertain parameters

foridentiationpurposes. AsitisshownhereafterinthisChapter,LFRshavesome

spei properties whih make its appliation in the identiation of LPV models

reallyinteresting. InthefollowingChapters,theLFRsareutilized,morespeially,

for the identiation of LPV models by using a spei

H ^∞

^-norm optimization framework.

Thus, in the sequel, some neessary notations and denitions are given followed

by denitions related to the identiability onept of the LFRs. Then, it is shown

howalarge lassof linearstate-spae models anbetransformed intoanLFR.The

urrentChapterisonludedbyadisussiononthewell-posednessissueofthelinear

frationalrepresentations.

M

∆ _u ^✛

✲

✲

✲

u ₁₂ u ₁₁

y ₁₂ y ₁₁

(a)The upperLFR.

∆ l

M

✛

✲

✲

✲

u 21

u 22

y 21

y 22

(b)The lowerLFR.

Figure2.1: Thelinearfrationalrepresentationbyusinganupperandalowerlinear

frational transformation (

F u (M, ∆ _u )

^and

F l (M, ∆ _l )

^{) of two matries}

M

^and

∆ _u

or

∆ _l

^having appropriatedimensions.

2.2 Notions and denitions

Asmentionedabove, inthis Setion, someimportantdenitionsare introdued.

So, let us rst dene what a linear frational representation is and how it an be

alulated [164℄.

Denition 2.2.1. A linear frational representation is a feedbak interonnetion

of two appropriately partitioned matries denoted hereafter by

∆ _u ∈ R ^{n u 11 ×n y 11}

or

∆ _l ∈ R ^{n u22 ×n y22}

and

M

^{(see Fig. 2.1),}

M =

M ₁₁ M ₁₂ M ₂₁ M ₂₂

∈ R ^{n M ×n M} .

^(2.1)

If

(I − M ₁₁ ∆ _u )

^or

(I − M ₂₂ ∆ _l )

^is invertible, the so-alled upper or lower LFR an be omputed by

F u (M, ∆ _u ) = M ₂₂ + M ₂₁ ∆ _u (I − M ₁₁ ∆ _u ) ⁻¹ M ₁₂ ,

^(2.2a)

F l (M, ∆ _l ) = M ₁₁ + M ₁₂ ∆ _l (I − M ₂₂ ∆ _l ) ⁻¹ M ₂₁ .

^(2.2b)

Eah form omposing Eq. (2.2) is, in fat, the input/output representation be-

tween

u 12

^and

y 12

^{, when an upper LFR is handled, or between}

u 21

^and

y 21

^when

a lower LFR is onsidered (see Fig. 2.1). The operation denoted by

F ^u ( • , ⋆)

^and

F ^l ( • , ⋆)

^{isthe soalledupperand lowerlinear frational}transformation(LFT) of

•

and

⋆

^,respetively. To larifythe ambiguoususageof the notationsLFT and LFR, by LFT, the operation itself is meant, while LFR stands for the model representa-

tion. Now, in order toset the stage for a generi LFR-based modeling framework,

let usonsider a spei blok-matrixstruture by followingthe lines found in [70,

Chapter 8℄,

K ∆ = { diag(δ 1 I _n 1 , · · · , δ S I _{n S} ), δ i ∈ O ^Lin , i = { 1, · · · , S }}

^(2.3)

and a vetor

Υ =

δ 1 · · · δ S

⊤

∈ O ^S Lin ,

^(2.4a)

where,

O ^Lin

^{denotes the set of linear operators [22℄ (for instane, the derivative}

operator

d

dt

^{ortheforwardshiftoperator}

q

^)whilethe

K _∆

blok-diagonalsetontains

n

^{of them, where}

n = Σ ^S _i=1 n _i

^{isthe sum of the blok dimensions given by Eq. (2.3).}

[70℄ beause, in this thesis, only blok-diagonal

∆ ∈ K ∆

^{matries are employed.}

Furthermore,onsider

d =

n 1 · · · n S

⊤

∈ N ^S .

^(2.4b)

Then, by using Eq. (2.2), by fousing on the upper LFT 1

(see Fig. 2.1a), the fol-

lowing input-output (between

u 12

^and

y 12

^{(see Fig. 2.1a))} representation an be alulated

F ^u ( M , ∆(d, Υ)) = D + C ∆(d, Υ)(I n − A ∆(d, Υ)) ⁻¹ B ,

^(2.5)

where

∆(d, Υ) ∈ K _∆ ,

^(2.6)

isastruturedblok-diagonalmatrix parameterizedby the vetors

d

^and

Υ

^respe-

tively, and with

M =

A B C D

∈ R ^{(n+n y )×(n+n u )} ,

^(2.7)

where

n u

^and

n y

^{are the dimensions of the input and output hannels involved in}

Fig. 2.1a while

n = Σ ^S _i=1 n _i

^{. In Eq. (2.7), the system matries}

( A , B , C , D )

^{an be}

partitioned asfollows [70℄

A =

" _A

1,1 ··· A 1,S

.

.

. .

.

.

A S,1 ··· A S,S

# , B =

B 1

.

.

.

B S

,

^(2.8a)

C = [ ^{C 1 ··· C S} ] , D = D .

^(2.8b)

NotiethatthenotationsoftheinnermatriesdenedabovebyEq.(2.8)aregeneri.

In the next Setion, the above introdued tool, namely the LFR, is applied to

represent some spei lassesof linear models.

2.3 Linear system modeling by using linear fra-

tional representations

In this Setion, it is shown how a large lass of linear models an be trans-

formed into an LFR [70, Chapter 8℄. From Eq. (2.5)-(2.7), it an beseen that any

linear state-spae form an be represented aording to the operators plaed into

the blok-diagonal struture denoted by

∆(d, Υ)

^{. This results, basially, in the}

I/Orepresentation orthetransferfuntion formof onsideredthelinear state-spae

models.

2.3.1 Linear time-invariant models

Let us rst onsider an

n _x

^{order LTI model given by the following state-spae}

form

γx(t) y (t)

=

A B C D

x(t) u (t)

,

^(2.9)

1. Notie that, hereafter in this Chapter, the upper LFR is applied. However, this an be

replaed bylowerLFRs as well by modifying theinner struture of thematrix

M

^aordingto

where

u(t) ∈ R ^{n u}

is the input signal vetor,

y(t) ∈ R ^{n y}

is the outputsignal vetor,

x(t) ∈ R ^{n x}

isthe statevetorand

t ∈ R

or

Z

. Herein,

γ

^{standsfor theforward shift}

operator (

q

^{) when} disrete-time systems are onsidered or for the dierential (

d dt

⁾

operator when ontinuous-time systems are handled. This modelan be desribed

by anLFR satisfying

S = 1, d = n x ,

Υ = 1

η , ∆(d, Υ) = 1

η I _{n x} , M _{LT I} =

A B C D

,

where

η

^{istheLaplaetransform}(ontinuous-timease)orthe

z

^{transform(disrete-}

time ase) of

γ

^{. Indeed, the transfer funtion form of the onsidered linear state-}

spae modelan be alulatedby using anupperLFT as follows,

G _{LT I} (η) = D + C (ηI _{n x} − A ) ⁻¹ B = D + C 1

η (I _{n x} − A 1

η ) ⁻¹ B = F u

A B C D

, 1 η I _{n x}

= F u ( M _{LT I} , ∆(d, Υ)),

^(2.10)

where the system matriesdenoted by

( A , B , C , D )

^have appropriatedimensions.

2.3.2 Multi-dimensional linear time-invariant models

Theproposedmodelstrutureanalsodealwithaspeimulti-dimensionalLTI

(MDLTI) model, namely,a disrete-time

2D

^{Roesser model[121℄, whih satises}





x ₁ (k 1 + 1, k 2 ) x ₂ (k 1 , k 2 + 1)

y(k 1 , k 2 )



 =





A ^1,1 A ^1,2 B ¹ A ^2,1 A ^2,2 B ² C ¹ C ² D









x ₁ (k 1 , k 2 ) x ₂ (k 1 , k 2 ) u(k 1 , k 2 )



 .

^(2.11)

This 2D model an be writtenas anLFR with

S = 2, d =

n x 1 n x 2

⊤

, Υ =

z ₁ ⁻¹ z ₂ ⁻¹ ⊤

, ∆(d, Υ) =

z ₁ ⁻¹ I _{n x1}

z ₂ ⁻¹ I _{n x2}

,

as follows

G _{M D} (z ₁ , z ₂ ) = F u ( M _{M D} , ∆(d, Υ)),

^(2.12a)

M _{M D} =





A ^1,1 A ^1,2 B ¹ A ^2,1 A ^2,2 B ² C ¹ C ² D



 ,

^(2.12b)

∆(d, Υ) =

z ₁ ⁻¹ I _{n x1} 0 0 z ₂ ⁻¹ I _{n x2}

.

^(2.12)

Here,

z 1

^and

z 2

^{are the}

z

^{transform variables. Again, the system matries found}

in

M _{M D}

^have appropriate dimensions. The identiation problem of this speial model written as anLFR is addressed in [40℄. Notie that this formulation an be

extended toany Roesser

nD

^{-models without any} restritions.

2.3.3 Linear parameter-varying models

As farasLPVmodelsare onerned,throughoutthis thesis,the followingstate-

spae representation is onsidered

γx(t) = A (p(t))x(t) + B (p(t))u(t),

^(2.13a)

y(t) = C (p(t))x(t) + D (p(t))u(t),

^(2.13b)

where

u(t) ∈ R ^{n u}

is the input signalvetor,

y(t) ∈ R ^{n y}

is the output signal vetor,

x(t) ∈ R ^{n x}

is the state vetor and

t ∈ R

or

Z

. In the LPV framework, the system matries

( A , B , C , D )

^{are rational or ane funtions of measurable} time-varying signals as

A : P 7−→ R ^{n x ×n x} B : P 7−→ R ^{n x ×n u} C : P 7−→ R ^{n y ×n x} D : P 7−→ R ^{n y ×n u} .

The shedulingvariablesare, more preisely,gathered intothe vetor

p(t) ∈ P ⊆ R ^{n p} ,

where

P =

p (t) ∈ R ^{n p}

p _i (t) ≤ p i (t) ≤ p _i (t)

˙

p i (t) ≤ p ˙ _i (t) ≤ p ˙ _i (t) ∀ i ∈ { 1, · · · , n p }

,

^(2.14)

isthe so-alledsheduling"spae"[134℄whihisaompatset [22℄. Inthe sequel,it

isfurthermoreassumed thatthesystemmatriessatisfyastatidependenyon

p(t)

[135℄, i.e., they do not depend on the time-shifted versions or the time-derivatives

of the sheduling variables (

γp(t), γ ² p(t), · · ·

^{) when disrete or} ontinuous-time modelsareonsidered,respetively. Notiehoweverthat,ifthe LPVmodeldepends

also on the derivatives of some sheduling variables whih are measurable, then

the sheduling variable vetor an be augmented by onsidering these signals as

shedulingvariables as well.

Now, by using again the upperlinear frationaltransformation presented previ-

ously (see Fig. 2.1a), any arbitrarily LPV model an be transformed into an LFR

intwosteps.

First, the parameter-dependent state-spae matriesintrodued by Eq. (2.13)an

be derived by

A (p(t)) B (p(t)) C (p(t)) D (p(t))

=

A 0 B 0

C ⁰ D ⁰

+ B ^w

D yw

∆ _p(t) (I n ∆ − D ^zw ∆ _p(t) ) ⁻¹

C ^z D ^zu

= F ^u ( M _{LP V} , ∆ _p(t) ),

^(2.15a)

with

∆ _p(t) =



 

p 1 (t)I r 1

.

.

.

p n p (t)I r _np



  ∈ R ^{n ∆} ,

^(2.15b)

n _∆ =

n p

X

i=1

r _i ,

^(2.15)

and

M _{LP V} =





D zw C z D zu

B ^w A ⁰ B ⁰ D ^yw C ⁰ D ⁰



 ,

^(2.15d)

and where

A ⁰ , B ^w , · · · , D ⁰

are time-invariantmatrieshavingappropri-

ate dimensions. In this partiular ase,

S = n p , d = [r 1 , · · · , r n p ] ^⊤ , Υ = [p 1 (t), · · · , p n p (t)] ^⊤ .

Seond, thedynamialstate-spae LPVmodeldened by Eq.(2.13)an bealu-

lated by applyingone again the upperLFT, i.e.,

F ^u

A ( p (t)) B ( p (t)) C (p(t)) D (p(t))

, 1 η I _{n x}

= F ^u ( F ^u ( M _{LP V} , ∆ _p(t) ), 1

η I _{n x} ).

^(2.16)

If the sheduling variables are xed in a ertain working point denoted here-

afterby

p _i

^{,thentheloalfrozentransferfuntionsanbealulatedasfollows}

G _{LP V} (η, p _i ) = F ^u

A (p i ) B (p i ) C (p i ) D (p i )

, 1 η I _{n x}

(2.17)

= F u ( F u ( M _{LP V} , ∆ _{p i} ), 1

η I _{n x} ),

^(2.18)

where

i

^{standsfor the}

i ^th

^{xed valueof the shedulingvariablevetor. Thisis}

basially a standard LTI transfer funtion. Aording tothe notations intro-

dued by Eq. (2.4), similarlytothe LTI ase,

S = 1, d = n x ,

Υ = 1

η , ∆(d, Υ) = 1

η I _{n x} .

As pointed out in [70℄, the obtained modelisI/O equivalent withthe one dened

as follows



 γx(t)

z(t) y(t)



 =





A ⁰ B ^w B ⁰ C ^z D ^zw D ^zu C 0 D yw D 0







 x(t) w(t) u(t)



 ,

^(2.19)

w(t) = ∆ _p(t) z(t).

^(2.20)

The resulting LPV model is referred hereafter as an LPV/LFR one. In order to

highlightthegenerinatureoftheLFRsappliedduringthemodelingofLPVmodels,

it an be onluded fromEq. (2.15) that the parameter dependeny an be

(i) ane when

D ^zw = 0

^;

(ii) rationalwhen

D ^zw 6 = 0

^.

This property is a very important feature of the LFR when it is ompared to the

other LPV model representations found in the literature (see [135, Chapter 3℄ for

a reent overview of the existing LPV modeling approahes), beause, most of the

by Eq. (2.13) depend on the sheduling variable vetor in an ane or polynomial

manneri.e.,

♣ (p(t)) = ♣ ⁰ +

n p

X

i=1

♣ ⁱ p i ,

^(2.21a)

♣ (p(t)) = ♣ ⁰ +

n p

X

i=1 s i

X

j=1

♣ ^i,j p ^s _i ⁱ

^(2.21b)

where

♣ ^•

^{is a onstant matrix standing for}

A _•

^,

B _•

^,

C _•

^and

D _•

^{and where}

p = p 1 · · · p n p

⊤

.

Thanks to the examples presented above, it an be seen that a wide range of

linear systems an be desribed by using LFRs. This is the main reason why the

developments of this thesis utilizethis modelrepresentation tool.

2.3.4 Comments on well-posedness

The LFR an be dened and omputed if and only if it is a well-posed LFR,

i.e., for our study, if

I _n − A∆(d, Υ)

^{is invertible (see Eq. (2.5)). More preisely,}

in the

1D

^or

2D

^{LTI ases, this} well-posedness ondition an be related to the standard stability onditions available, e.g., in [39, 35℄. Thus, if the system to

identify is assumed to be stable, the LFR should be well-posed. However, in the

LPV framework, the problem is muh more ompliated beause, in this ase, the

matrix

∆(d, Υ)

^{expliitly depends on the sheduling variable}

p(t)

^{. Therefore, the}

well-posedness onditions of the problem should be related to the evolution range

of

p(t)

^{dened on the ompat set}

P

(see Eq. (2.14)).

Remark 2.1. When the sheduling variable

p(t)

^{is xed, whih is the ase in the}

loal approah, the resulting frozen LPV model is basially an LTI one, where the

well-posedness an easily be veried.

Aording tothe Author'sknowledge, noneessary and suientonditions are

availableforthisproblemintheliterature. Nevertheless, inthegray-boxframework,

whenthemodelstrutureisobtained fromthe physiallaws governingthe behavior

of the system, the struture of

∆(d, Υ)

^{an be xed a priori and suient LMI}

onditions ould be obtained to ensure the well-posedness of the LFR by applying

a S-proedure tool [21℄. This problem is not solved in this thesis and is referred

to future researh. At the same time, this study, more preisely the link between,

e.g., the ondition number of

I _n − A ∆(d, Υ)

^{and the}

p(t)

^{-trajetory, an be seen}

asaninteresting startingpoint forananalysis ofthe optimal

p(t)

^{-trajetorydesign}

ensuring a onsistentidentiation.

2.4 Blak and gray-box state-spae models

As desribed in the previous Setion, the value of

S

^{inEq. (2.8) is user-dened}

and an be hosen aording to the onsidered linear model (linear time-invariant

(LTI), multi-dimensional linear time-invariant (MDLTI), linear parameter-varying

(LPV), et). In order to emphasize the generi nature of this representation, the

matriesfound in Eq. (2.7) an be blak-box or, aording to the prior knowledge,

is meant that the struture of the model to identify is a parameterized struture

derived from some prior knowledge governing the behavior of the system under

study. In other words, the parameters to estimate are unknown funtions (e.g.,

+, − , ÷ , ×

^{)ofthe realphysialones whilesomematrixentriesanbexed as0or}

1. On theotherhand,theunknownparametersfound inlight-gray-boxmodels have

real physialmeaningbased onrst priniples modelingonsiderations. In orderto

illustrate thesedenitions, let usonsider the following simplemass-spring-damper

system, i.e.,

m¨ z + k z ˙ + dz = F,

where

z

^{denotes the position,}

m, k, d

^{denotethe mass, stinessand damperoe-}

ients, respetively. This simple system an then betransformed intoa state-spae

form by onsidering

F

^and

z

^{as input and output,}respetively,

z ˙

¨ z

=

0 1

− _m ^d − _m ^k z

˙ z

+ 0

1 m

F,

^(2.22a)

y =

1 0 z

˙ z

,

^(2.22b)

while the state variablesare

x = z

˙ z

.

^(2.23)

In the blak-box ase, the modelis fully-parametrizedand has the followingform

ξ ˙ ξ ¨

=

θ ₁ ^b θ ₂ ^b θ ₃ ^b θ ₄ ^b

ξ ξ ˙

+ θ ^b ₅

θ ^b ₆

F, y =

θ ₇ ^b θ ^b ₈ ξ ξ ˙

+ [θ ₉ ^b ]F.



 



 



→ ξ ˙ = A (θ ^b )ξ + B (θ ^b )F,

y = C (θ ^b )ξ + D (θ ^b )F,

^(2.24)

where

θ ^b =

θ ^b ₁ θ ^b ₂ θ ₃ ^b θ ₄ ^b θ ^b ₅ θ ^b ₆ θ ₇ ^b θ ₈ ^b θ ₉ ^b ⊤

,

and

ξ = ξ ξ ˙ ⊤

.

Notie that

ξ

^{denotes the blak-boxstate vetor whihan bealulatedfrom}

x

^by

using any invertible similarity transformation

T

^{, having} appropriate dimension, as follows,

ξ = Tx.

^(2.25)

Asfarasthedark-gray-boxframeworkisonerned,byusingsomepriorinformation

about the struture ofthe system understudy (see Eq. (2.22)), somematrix entries

an bexed as0 or1, i.e.,

z ˙

¨ z

=

0 1 θ ₁ ^g θ ₂ ^g

z

˙ z

+ 0

θ ^g ₃

F, y =

1 0 z

˙ z

,



 



 



→ x ˙ = A(θ ^g )x + B(θ ^g )F,

y = C(θ ^g )x,

^(2.26)

where

θ ^g =

θ ₁ ^g θ ^g ₂ θ ^g ₃ ⊤

,

and

x =

x x ˙ ⊤

.

Here, the parameters to identify are unknown funtions of the real physial ones.

Indeed,

θ ₁ ^g = − d

m , θ ₂ ^g = − k

m , θ ₃ ^g = 1 m .

Finally,thelight-gray-box version isequaltothe originalstate-spae representation

given by Eq. (2.22) by onsidering the physial parameters to be estimated, i.e.,

θ ^g =

d k m ⊤

,

z ˙

¨ z

=

"

0 1

− ^θ _θ ^{g 1 g} ₃ − ^θ _θ ^{g 2 g} ₃

# z

˙ z

+ 0

1 θ ^g ₃

F, y =

1 0 z

˙ z

.



 

 

 

 

→ x ˙ = A(θ ^g )x + B(θ ^g )F,

y = C(θ ^g )x.

^(2.27)

Throughout this thesis, when the modelidentiationproblem isaddressed, blak,

dark-gray and light-gray-box versions of the above presented matries are denoted

asfollows,

M (ϑ) =

A (ϑ) B (ϑ) C (ϑ) D (ϑ)

−→



 

 



 

 



M (θ ^b ) =

A ( θ ^b ) B ( θ ^b ) C (θ ^b ) D (θ ^b )

,

M( θ ^g ) =

A(θ ^g ) B(θ ^g ) C(θ ^g ) D(θ ^g )

,

(2.28)

where

ϑ ∈ R ^{n ϑ}

isthe generalnotationof the unknown parameter vetor,

θ ^b ∈ R ⁿ θb

isavetor ontainingallthe matrixentrieswhen blak-boxmodels aresoughttobe

estimated, and where

θ ^g ∈ R ^{n θg}

is a vetor ontaining the unknown parameters to identifywhihanbeunknown funtionsoftherealphysialparameters(dark-gray-

box) or diretly the physial parameters themselves (light-gray-box). Although, in

the gray-box ases, the same notations are employed, the distintion of these two

types will be apparent during the onrete appliations. For blak- and gray-box

models, the matries found in

M (θ ^b )

^and

M(θ ^g )

^{are assumed to be of} appropriate dimensionsandontinuouslydierentiablew.r.t. theunknownparameters

θ ^b ∈ R ⁿ θb

or

θ ^g ∈ R ^{n θg}

. Furthermore,inthe gray-boxframeworkthemodelstruture,i.e.,the waythematries

A(θ ^g )

^,

B(θ ^g )

^,

C(θ ^g )

^and

D(θ ^g )

^{dependontheunknownparameter}

vetor

θ ^g

^{, is assumed to beknown a priori.}

2.4.1 Identiability

Beause, in this thesis, a partiular attention is paid to the identiation of

state-spaemodels,it isimportanttoexamine whether the parametersof suhrep-

resentations an be estimated uniquely. In order to do so, some properties of the

involved state-spae mappings must be introdued [22℄. The input-output repre-

sentation

F ^u ( M , ∆(d, Υ))

^{, namely the} parameterization

M (θ ^b )

^{(blak-box ase)}

or

M ( θ ^g )

^{(gray-box ase2), over the parameter set}

Θ

^{, is a mapping whih an be}

dened as follows:

2. Hereafter, when gray-box representations are mentioned, it is meant that both dark-and

Denition 2.4.1. A blak-box model struture

χ ^(b) _d,Υ

^{is a} dierentiable map, for a partiular

d

^and

Υ

^{, from a parameter set}

θ ^b ∈ Θ ⊂ R ^{n Θ}

to a set of models (see Eq. (2.28))

χ ^(b) _d,Υ : Θ ⊂ R ^{n Θ} −→ R ^{n y ×n u}

θ ^b 7−→ F ^u ( M (θ ^b ), ∆(d, Υ)).

^(2.29)

Denition 2.4.2. A gray-box model struture

χ ^(g) _d,Υ

^{is a} dierentiable map, for a partiular

d

^and

Υ

^{, from a parameter set}

θ ^g ∈ Θ ⊂ R ^{n Θ}

to a set of models (see Eq. (2.28))

χ ^(g) _d,Υ : Θ ⊂ R ^{n Θ} −→ R ^{n y ×n u}

θ ^g 7−→ F ^u (M(θ ^g ), ∆(d, Υ)).

^(2.30)

Inordertodenethe mostimportantmappropertiesinagenerimanner, letus

onsider two sets denoted by

X

^and

Y

^{and a map}

f : X → Y

^{. Then the following}

denitions an beintrodued[22℄.

Denition 2.4.3. The map

f

^{is surjetive if and only if,}

∀ y ∈ Y

^{there exists an}

f (x), x ∈ X

^{suh that}

y = f (x)

^.

Denition2.4.4. Themap

f

^{isinjetiveifandonlyif,}

∀ x 1 , x 2 ∈ X, f (x 1 ) = f(x 2 )

implies that

x ₁ = x ₂

^.

Denition 2.4.5. The map

f

^{is bijetive if it is both injetive and surjetive.}

Bylookingloser atthe statementgiven by Denition2.4.4,itan beonluded

that the injetivity property isrelated to the uniqueness of the involved parameter

vetor. So, thisisaveryimportantkeypropertyofagivenmodelstruturebeause,

inthegray-boxase, uniqueparametersaresoughttobeestimatedduringtheiden-

tiationproedure. First,letusfousonblak-box,fully-parameterizedstate-spae

models. In this ase, it is well-known that, in the LTI ase, suh representations

are not unique but they an only be determined from the available I/O data sets

up to a similarity transformation [60℄. The LFR framework does not form an ex-

eption eitherto this rule. In other words, blak-box representations are inherently

not identiable [78℄, sine ablak-box parameterizationis not injetive whih gives

rise toa nonunique relationbetween the parameter vetor

θ ^b

^{and the} input-output representation

F ^u ( M ( θ ^b ), ∆ ( d , Υ ))

^{[144℄. In the blak-box ase, it is} interesting to denethe similaritysub-spae involvinginvertible similaritytransformation ma-

tries. This goal an be reahed by following the lines found in [71, 72℄. This an

be seen as an extension of the standard LTI ase [78, 33℄. Thus, the similarity or

indistinguishable sub-spae of the LFRs an be more preisely dened as follows

[71℄,

Denition 2.4.6. Given the blok struture

K _∆

^and

S _∆ = { diag(T ₁ , · · · , T _S ), T _i ∈ R ^{n i ×n i} , i ∈ { 1, · · · , S } ,

^(2.31)

the system realizations

( A ₁ , B ₁ , C ₁ , D ₁ )

^and

( A ₂ , B ₂ , C ₂ , D ₂ )

^{are found in the same}

similarity sub-spae if an invertible similarity transformation matrix

T ∈ S ∆

^exists

suh that

A ₁ B ₁ C ₁ D ₁

T 0 _n×n 0 _{n u ×n u} I _{n u}

=

T 0 _{n y ×n y} 0 _n×n I _{n y}

A ₂ B ₂ C ₂ D ₂

.

^(2.32)