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Contrôle actif des vibrations dans un système couplé fluide
Bogdan Robu
To cite this version:
Bogdan Robu. Active vibration control of a fluid/plate system. Contrôle actif des vibrations dans un système couplé fluide. Mathematics [math]. Université Paul Sabatier - Toulouse III, 2010. English.
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THÈSE THÈSE
En vue de l'obtention du
DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE
Délivré par l'Université Toulouse III - Paul Sabatier Discipline ou spécialité : Systèmes Automatiques
JURY
M. Jean-Pierre RAYMOND, President M. Miroslav KRSTIC, Rapporteur M. Yann LE GORREC, Rapporteur Mme. Lucie BAUDOUIN, Membre M. Gildas BESANCON, Membre Mme. Valérie BUDINGER, Membre
M. Edouard LAROCHE, Membre M. Christophe PRIEUR, Membre
Ecole doctorale : EdSYS Unité de recherche : LAAS - CNRS
Directeur(s) de Thèse : M. Christophe PRIEUR, Directeur de thèse Mme. Lucie BAUDOUIN, Co-Directeur de thèse
Rapporteurs : M. KRSTIC and Y. LE GORREC Présentée et soutenue par Bogdan ROBU
Le 03.12.2010 Titre :
Active vibration control of a fluid/plate system
"R bdare, r bdare, r bdare, r bdare, r bdare ...
r bdare, r bdare, r bdare, r bdare ..."
Arhim. IlieCLEOPA
"That whih does not kill you
makes you stronger."
FriedrihNietzshe
I wouldrst liketoexpressmygratitude towards myadvisors Mme. LuieBaudouin
etM.Christophe Prieurfortheir helpand onstantenouragementthroughoutthese
years. They provided mewith freedomand supporttopursue my researhideas and
I amgratefulI hadthe hanetobenetfromtheirexperiene. Thank youespeially
for the motivationnotes during the writing of the nal part of the thesis.
I also like to express my thanks to M. Miroslav Krsti and M. Yann Le Gorre
for making methe honorto review my thesis and for givingmeuseful ommentsfor
improving it's quality. I equally thank to M. Jean-Pierre Raymond for aepting to
be the president of my thesis ommittee. My thanks also go towards Mme. Valérie
Budinger for the great help with the experimental setup and for aepting to be a
memberofmythesisommittee. Ialsothank toM.GildasBesançonandM.Edouard
Larohe fortheirpartiipationinmythesisommitteeandfortheiruseful omments.
I also want to thank to Mme. Isabelle Queinne, head of the MAC group, for
welomingmehereatLAAS-CNRS.MygratitudegoesalsotowardsM.DenisArzelier
for guiding me during the master projet and for allowing me to benet from his
experiene during my thesis. I also wish to thank to all the members of the MAC
group, my eduation has been enhaned beause of your advies and presene.
I would alsolike to thank to all my friends, to the ones that are stillnext to me
and to the ones that are far away. I'm glad that you entered my life, it's been a
wonderful growing experiene. I learnedmany things from eah and any one of you
... I just hope that I learned allthe things I had to.
A speial thanks to all my high shool friends. Thank you very muh guys for
all your help and support in the diultmoments, I ould not done it without you.
Thank youthat after allthese years we an still get together and have agreat time.
Finally, my deepest and sinere thanks go to my parents and espeially to my
brotherCos. Thebest brotherandthe bestparentsthatouldeverexist. Thankyou
for always being here for me- I apologize for the times that I was not there for you.
I am glad I have youand I loveyou very muh. This thesis isdediated to you !!!
Introdution 1
1 Experimental devie presentation 5
1.1 Charateristis of the experimentaldevie . . . . . . . . . . . . . . . 6
1.2 Data aquisitionhain . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Atuators and sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Presentation of the piezoeletri phenomenon . . . . . . . . . 14
1.3.2 Optimalplaement of atuators and sensors . . . . . . . . . . 15
1.3.3 Dynamiof piezoeletripathes . . . . . . . . . . . . . . . . 18
1.4 Conlusion of the hapter . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Mathematial modeling of the system 21 2.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Beam model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Plate innite dimensionalmodel . . . . . . . . . . . . . . . . . 32
2.2.3 Plate nite dimensional approximation . . . . . . . . . . . . . 35
2.2.3.1 Computationof the dynamiplate matrix Ap . . . . 36
2.2.3.2 Computationof the plateinput matrix Bp . . . . . . 40
2.2.3.3 Computationof the plateoutput matrix Cp . . . . . 45
2.3 Tank model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.1 Sloshingof liquids -state of the art . . . . . . . . . . . . . . . 48
2.3.2 Tank approximation . . . . . . . . . . . . . . . . . . . . . . . 54
2.3.3 Tank innitedimensional model . . . . . . . . . . . . . . . . . 57
2.3.3.1 Generalequations . . . . . . . . . . . . . . . . . . . 57
2.3.3.2 Determination of fores and moments. . . . . . . . . 67
2.3.4 Tank nite dimensionalmodel . . . . . . . . . . . . . . . . . . 71
2.3.4.1 Generalpresentationoftheequivalentmehanialmodel 71
2.3.4.2 Determination of parameters for the mass-pendulum
model . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4 Complete modelrepresentation . . . . . . . . . . . . . . . . . . . . . 80
2.4.1 Innite dimensional oupling. . . . . . . . . . . . . . . . . . . 80
2.4.1.1 Inuene of the liquidsloshing onthe plate movement 80 2.4.1.2 Inuene of plate deformationon the liquidsloshing 81 2.4.2 Finitedimensional oupling . . . . . . . . . . . . . . . . . . . 81
2.4.2.1 Liquidsloshing inuene onthe retangularplate . . 82
2.4.2.2 Plate deformationinuene on tankliquidsloshing . 84 2.4.2.3 Compat writing of omplete model . . . . . . . . . 85
2.5 Conlusion of the hapter . . . . . . . . . . . . . . . . . . . . . . . . 86
3 Controller synthesis - Theoretial approah 87 3.1 Energy omputation . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Pole plaement and full state observer . . . . . . . . . . . . . . . . . 90
3.3 H∞ ontroller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.4 Conlusion of the hapter . . . . . . . . . . . . . . . . . . . . . . . . 103
4 Experimental results 105 4.1 Inuene ofthe atuator dynamis . . . . . . . . . . . . . . . . . . . 105
4.2 Choieof the suitableamount of modes . . . . . . . . . . . . . . . . . 107
4.3 Model adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.1 Computationof the naturalfrequeny . . . . . . . . . . . . . 111
4.3.1.1 Computationof plate natural frequenies. . . . . . . 111
4.3.1.2 Computation of the natural frequenies of sloshing modes . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3.1.3 Naturalfrequeniesoftheompletesystem: plateand tank . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.2 Computationof modaldamping . . . . . . . . . . . . . . . . . 125
4.3.3 Modelmathing problem . . . . . . . . . . . . . . . . . . . . . 125
4.4 Pole plaement ontroller . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.5 H∞ robust ontroller . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.5.1 Synthesis of a H∞ ontrollerwithout lters . . . . . . . . . . . 134
4.5.2 Synthesis of a H∞ ontrollerwith lters . . . . . . . . . . . . 140
4.5.2.1 Matlab Robust ControlToolbox ontroller . . . . 141
4.5.2.2 HIFOO ontroller . . . . . . . . . . . . . . . . . . . . 143
4.5.2.4 Simultaneous redued-order HIFOO ontroller . . . . 150
4.6 Comparison of the ontrolmethods . . . . . . . . . . . . . . . . . . . 153
4.7 Conlusion of the hapter . . . . . . . . . . . . . . . . . . . . . . . . 155
General onlusion 157
Bibliography 162
1.1 Experimentaldevie ISAE-ENSICA . . . . . . . . . . . . . . . . . . . 5
1.2 Experimentaldevie, detailedpresentation of main omponents . . . 6
1.3 Deformation ofthe retangularplate (1st mode) . . . . . . . . . . . . 6
1.4 Retangular plate withoutylindrialtank . . . . . . . . . . . . . . . 7
1.5 Equipped experimentalsetup . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Detailview of harge amplier . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Detailview of aquisitionsystem and xPCTarget . . . . . . . . . . . 10
1.8 Detailview of high voltage amplier . . . . . . . . . . . . . . . . . . 11
1.9 Atuators onneted tothe plate . . . . . . . . . . . . . . . . . . . . 12
1.10 Sensorsonneted tothe plate . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Beamwith a exion movement . . . . . . . . . . . . . . . . . . . . . 24
2.2 Plate bending along xaxis . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Theplateand thetwobeamsseletedforthe hoieofthe Ritzfuntions 34 2.4 Qualityfator Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Neutral berof the retangular plate . . . . . . . . . . . . . . . . . . 43
2.6 Cylindrialtank onneted tothe plate . . . . . . . . . . . . . . . . . 50
2.7 Horizontalylindrialtank . . . . . . . . . . . . . . . . . . . . . . . . 51
2.8 Natural angular frequeny ωn of the rst transverse sloshing modes (extrated from [48℄) . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.9 Natural angular frequeny ωn of the rst longitudinal sloshing modes (extrated from [48℄) . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.10 Implementing the rst method(only one retangular tankis shown) . 54 2.11 Equivalenttanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.12 Coordinate system for a partially lled retangular ontainer under external aeleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.13 The mode shape of the rst three symmetriwaves (fromleft to right) 65
2.14 The mode shape of the rst three antisymmetri waves (from left to
right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.15 Masspendulum and mass spring mehanialmodels . . . . . . . . . . 71
2.16 Mehanial model with one xed mass and 3 sloshing masses, repre- senting fuel sloshingunder longitudinalexitation . . . . . . . . . . . 73
3.1 State feedbak ontrol . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Feedbak ontrollaw and observer. . . . . . . . . . . . . . . . . . . . 96
3.3 Standard H∞ problem . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Standard H∞ problem . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1 Bode plot of the system with and without onsidering the atuator dynamis,tank lllevelof 0.9 . . . . . . . . . . . . . . . . . . . . . . 106
4.2 Experimental Bode plot for a tank ll level of 0.9 in the frequeny range [0,200]Hz. #1 is the rst exion mode of the plate, #2 is the rst sloshing mode of the liquid,#3 and #4 are the seondand third sloshingmode ofthe liquid(they are almostinvisibledue totheir very smallamplitude), #5 is the rst torsion mode of the plate, #6 is the seond exion mode of the plate, #7 is the third exion mode of the plate, #8 is the forthexion mode ofthe plate, #9 isthe fthexion mode of the plate, #10 is the sixth exion mode of the plate, #11 is the seond torsionmode of the plate, #12 isthe seventh exion mode of the plate, #13 is the eightexion mode of the plate . . . . . . . . 108
4.3 First threemodaldisplaements of the free-free beam . . . . . . . . . 112
4.4 First ve modaldisplaements of the lamped-freebeam . . . . . . . 113
4.5 Plate rst exion mode at2.301Hz, η1(y, z) = Y1(y)Z1(z) . . . . . . . 114
4.6 Plate seond exion mode at 14.413Hz,η2(y, z) =Y2(y)Z1(z) . . . . . 115
4.7 Plate third exion mode at 40.3583Hz,η3(y, z) =Y3(y)Z1(z) . . . . . 115
4.8 Plate rst torsion mode at 49.2027Hz, η4(y, z) = Y1(y)Z2(z) . . . . . 115
4.9 Plate seond torsion mode, η8(y, z) = Y3(y)Z2(z) (not taken into a- ount during the modeling phase) . . . . . . . . . . . . . . . . . . . . 116
4.10 First 4modaldisplaementsof the plate . . . . . . . . . . . . . . . . 118
4.11 ExperimentalBode plot for the plate and a tankll level of 0.7 . . . 122
4.12 ExperimentalBode plot the plate and a tank lllevel of 0.9 . . . . . 123
4.13 Frequeny mathing forthe tankllinglevele= 0.7(numerialmodel
4.14 Frequeny mathing forthe tankllinglevele= 0.9(numerialmodel
- plainline and experimental set-up-dotted line). . . . . . . . . . . . 127
4.15 Feedbak ontrollaw and observer. . . . . . . . . . . . . . . . . . . . 128
4.16 Pole/zeromap of the open-loopsystem (× for the poles, ◦ forthe zeros)129
4.17 Experimentaloutputof the ofopen-loop(dottedline) and losed-loop
(plain line) systems using a pole plaement ontroller with a tank ll
level of 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.18 Voltagedelivered bythepoleplaementontrollerduringexperiments,
tankll level of 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.19 Frequeny response of the poleplaementontroller,tanklllevelof 0.9131
4.20 Standard H∞ problem . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.21 Temporal response for robust ontrollers using Robust Control Tool-
box,withoutlters; simulationsonasystem with the same amountof
modes; tank ll level equal 0.9. Thin line is obtained with d12 = 0.1,
plain linewith d12 = 0.25and boldlinewith d12 = 1 . . . . . . . . . . 135
4.22 Voltagedeliveredbytherobustontrollers;tanklllevelequal0.9with
d12 = 0.1(thin line), d12 = 0.25(plain line) and d12= 1 (boldline) . . 136
4.23 Bodeplotoftherobustontrollerssimulatedonanaugmentedsystem;
tank ll level equal 0.9. The thin line is for d12 = 0.1, plain line for d12 = 0.25and boldlinefor d12= 1 . . . . . . . . . . . . . . . . . . . 137
4.24 Temporal response for robust ontrollers using Robust Control Tool-
box,withoutltersandwith d12= 0.1; testsonanaugmentedsystem;
tankll level 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.25 Pole/ zero map for the open-loop system augmented with one mode;
tankll level equal0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.26 Pole/ zero map for the losed-loop system augmented with one mode
and withthe ontrolleromputed with d12= 0.1; tanklllevel equal0.9138
4.27 Pole/zero map of the previously omputed ontroller, d12 = 0.1; tank
lllevelequal 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.28 Standard H∞ problemwith lters . . . . . . . . . . . . . . . . . . . . 141
4.29 ExperimentalBodeplotoftheopen-loopsystem(plainline)andofthe
losed-loop system (bold line) omputed with the Robust ontroller
fromMatlab for 2 modes and a xed tank llingof 0.7 . . . . . . . . 142
4.30 ExperimentalBode plot ofthe open-loopsystem (thinline) and ofthe
losed-loop system using a HIFOO ontroller and a Robust ontroller
