Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation

217 Pages·2008·4.246 MB·English

by  Rafael Vazquez, Miroslav Krstic (auth.)

Systems & Control: Foundations & Applications
SeriesEditor
TamerBas¸ar,UniversityofIllinoisatUrbana-Champaign
EditorialBoard
KarlJohanA˚stro¨m,LundUniversityofTechnology,Lund,Sweden
Han-FuChen,AcademiaSinica,Beijing
WilliamHelton,UniversityofCalifornia,SanDiego
AlbertoIsidori,UniversityofRome(Italy)and
WashingtonUniversity,St.Louis
PetarV.Kokotovic´,UniversityofCalifornia,SantaBarbara
AlexanderKurzhanski,RussianAcademyofSciences,Moscow
andUniversityofCalifornia,Berkeley
H.VincentPoor,PrincetonUniversity
MeteSoner,Koc¸ University,Istanbul
Rafael Vazquez
Miroslav Krstic
Control of Turbulent and
Magnetohydrodynamic
Channel Flows
Boundary Stabilization and State Estimation
Birkha¨user
Boston • Basel • Berlin
RafaelVazquez MiroslavKrstic
EscuelaSuperiordeIngenieros DepartmentofMechanicaland
DepartamentodeIngenier´ıaAeroespacial AerospaceEngineering
UniversidaddeSevilla UniversityofCalifornia,SanDiego
Avda.delosDescubrimientoss.n. LaJolla,CA92093-0411
41092Sevilla U.S.A.
Spain
MathematicsSubjectClassification:76D05,76D55,76F70,93C20,93D15
LibraryofCongressControlNumber:2007934431
ISBN-13:978-0-8176-4698-1 e-ISBN-13:978-0-8176-4699-8
Printedonacid-freepaper.
(cid:1)c2008Birkha¨userBoston
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Preface
This monographpresentsnew constructive designmethods for boundary stabi-
lizationand boundary estimationfor severalclassesof benchmark problems for
flowcontrol,withpotentialapplicationsinturbulencecontrol,weatherforecast-
ing, and plasma control. The basis of our approach is the recently developed
continuousbacksteppingmethodforparabolicPDEs[90]. Weexpandtheappli-
cability of boundary controllersfor flow systems from low Reynolds number [1]
to high Reynolds number conditions.
Effortsinflowcontroloverthelastfewyearshaveledtoawiderangeofdevelop-
ments in many different directions, reflecting the interdisciplinary character of
the research community (composed of control theorists, specialists in fluid me-
chanics, mathematicians, and physicists). However,most implementable devel-
opments so far have been obtained using discretized versionsof the plant mod-
els and finite-dimensional control techniques. In contrast, our design method
is based on the “continuum” version of the backstepping approach, applied to
the PDE model of the flow. The postponement of the spatial discretization
until the implementation stage offers advantages that range from numerical to
analytical. In fact, our methods offer a rather unparalleled physical intuition
by forcing the closed-loop systems to dynamically behave as well-damped heat
equation PDEs.
Thisconstructivedesignphilosophyisparticularlyrewardedintermsoftheob-
serverandcontrolgainsthatwederive. Inallcasesthesegainsarepresentedas
explicitly computable formulas such as rapidly convergentsymbolic recursions,
solutions to well-posed solvable linear PDEs, or even directly as closed-form
analytical expressions. For all designs we state and prove mathematical results
guaranteeing closed-loop stability and observer convergence. This constructive
approach has allowed us to obtain the first nontrivial closed-loop explicit solu-
tion for the 2D Navier–Stokes channel flow model.
vi Preface
The material presented in this monograph is based on the first author’s disser-
tation work with the second author, his PhD advisor.
Acknowledgments
We owe great gratitude to our coauthors in works leading to this book: Jennie
Cochran, Jean-Michel Coron, Eugenio Schuster, Andrey Smyshlyaev, and
Emmanuel Tr´elat. In addition, we have benefited from support from or in-
teraction with Bassam Bamieh, Tom Bewley, Enrique Ferna´ndez-Cara, George
Haller,J´erˆomeHoepffner,MihailoR.Jovanovic,PetarKokotovi´c,JuanLasheras,
Roberto Triggiani, and Enrique Zuazua.
WegratefullyacknowledgethesupportthatwehavereceivedfromtheNational
ScienceFoundationandtheEuropeanCommunity(intheformofaMarieCurie
Fellowship under the CTS framework).
Finally, we thank those who always calm our turbulence: Mar´ıa Dolores, Luis,
Mercedes, Angela, Alexandra, and Victoria.
La Jolla, California Rafael Vazquez
September 2007 Miroslav Krstic
Contents
Preface v
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mathematical Preliminaries and Notation . . . . . . . . . . . . . 3
1.2.1 Function spaces and norms . . . . . . . . . . . . . . . . . 3
1.2.2 Stability in the infinite-dimensional setting . . . . . . . . 8
1.2.3 Spatial invariance, Fourier transforms, and Fourier series. 11
1.2.4 Singular perturbation theory . . . . . . . . . . . . . . . . 20
1.2.5 The backstepping method for parabolic PDEs . . . . . . . 23
1.3 Overview of the Monograph . . . . . . . . . . . . . . . . . . . . . 34
1.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Thermal-Fluid Convection Loop: Boundary Stabilization 39
2.1 Thermal Convection Loop Model . . . . . . . . . . . . . . . . . . 39
2.2 ReducedModelandVelocityControllerforLargePrandtlNumbers 41
2.3 Backstepping Controller for Temperature . . . . . . . . . . . . . 42
2.3.1 Temperature target system . . . . . . . . . . . . . . . . . 42
2.3.2 Backstepping temperature transformation . . . . . . . . . 43
2.3.3 Temperature control law . . . . . . . . . . . . . . . . . . . 45
2.3.4 Inverse transformation for temperature . . . . . . . . . . 46
2.4 Singular Pertubation Stability Analysis for the System . . . . . . 46
2.5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
viii Contents
2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Thermal-Fluid Convection Loop: Boundary Estimation and
Output-Feedback Stabilization 55
3.1 A Decoupling Transformation for the Temperature . . . . . . . . 56
3.2 Stabilization of Uncoupled Temperature Modes . . . . . . . . . . 57
3.3 Stabilization of Velocity and Coupled Temperature Modes . . . . 58
3.3.1 Boundary control design using singular
perturbations and backstepping . . . . . . . . . . . . . . . 58
3.3.2 Observer design using singular perturbations and
backstepping . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.3 Output-feedback controller . . . . . . . . . . . . . . . . . 63
3.3.4 Singular perturbation analysis for large Prandtl
numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Stability Properties of the Closed-Loop System . . . . . . . . . . 64
3.5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Observer Convergence and Output-Feedback Stabilization Proofs 66
4 2D Navier–Stokes Channel Flow: Boundary Stabilization 71
4.1 2D Channel Flow Model . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Velocity Boundary Controller . . . . . . . . . . . . . . . . . . . . 75
4.3 Closed-Loop Stability and Explicit Solutions . . . . . . . . . . . 77
4.4 L2 Stability for the Closed-Loop System . . . . . . . . . . . . . . 81
4.4.1 Controlled velocity wave numbers. . . . . . . . . . . . . . 82
4.4.2 Uncontrolled velocity wave number analysis . . . . . . . . 88
4.4.3 Analysis for the entire velocity wave number range . . . . 89
4.5 H1 Stability for the Closed-Loop System . . . . . . . . . . . . . . 90
4.5.1 H1 stability for controlled velocity wave numbers . . . . . 90
4.5.2 H1 stability for uncontrolled velocity wave numbers . . . 91
4.5.3 Analysis for all velocity wave numbers . . . . . . . . . . . 93
4.6 H2 Stability for the Closed-Loop System . . . . . . . . . . . . . . 94
4.6.1 H2 stability for controlled velocity wave numbers . . . . . 94
4.6.2 H2 stability for uncontrolled velocity wave numbers . . . 95
Contents ix
4.6.3 Analysis for all velocity wave numbers . . . . . . . . . . . 97
4.7 Proof of Well-Posedness and Explicit Solutions for the Velocity
Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.8 Proof of Properties of the Velocity Boundary Controller . . . . . 99
4.9 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 2D Navier–Stokes Channel Flow: Boundary Estimation 103
5.1 Observer with Boundary Sensing of Pressure and Skin Friction . 103
5.2 Observer Convergence Proof . . . . . . . . . . . . . . . . . . . . . 107
5.2.1 Observed wave number analysis . . . . . . . . . . . . . . . 108
5.2.2 Unobserved wave number analysis . . . . . . . . . . . . . 111
5.2.3 Analysis for the entire observer error wave number range 111
5.3 An Output-Feedback Stabilizing Controller for 2D Channel Flow 112
5.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 3D Magnetohydrodynamic Channel Flow: Boundary
Stabilization 115
6.1 Magnetohydrodynamic Channel Flow Model. . . . . . . . . . . . 115
6.2 Hartmann Equilibrium Profile . . . . . . . . . . . . . . . . . . . . 117
6.3 The Plant in Wave Number Space . . . . . . . . . . . . . . . . . 118
6.4 Boundary Control Design . . . . . . . . . . . . . . . . . . . . . . 120
6.4.1 Controlled velocity wave number analysis . . . . . . . . . 120
6.4.2 Uncontrolled velocity wave number analysis . . . . . . . . 127
6.4.3 Closed-loop stability properties . . . . . . . . . . . . . . . 131
6.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 133
7 3D Magnetohydrodynamic Channel Flow: Boundary
Estimation 135
7.1 Observer Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Observer Gain Design and Convergence Analysis . . . . . . . . . 138
7.2.1 Observed wave number analysis . . . . . . . . . . . . . . . 140
7.2.2 Unobserved wave number analysis . . . . . . . . . . . . . 147
7.3 Observer Convergence Properties . . . . . . . . . . . . . . . . . . 148
x Contents
7.4 A Nonlinear Estimator with Boundary Sensing . . . . . . . . . . 149
8 2D Navier–Stokes Channel Flow: Stable Flow Transfer 153
8.1 Trajectory Generation and Tracking Error Model . . . . . . . . . 153
8.2 Spaces and Transformations for the Velocity Field . . . . . . . . 158
8.2.1 Periodic function spaces . . . . . . . . . . . . . . . . . . . 158
8.2.2 Fourier series expansion in Ωh . . . . . . . . . . . . . . . . 158
8.2.3 H1 and H2 functional spaces . . . . . . . . . . . . . . . . 159
8.2.4 Spaces for the velocity field . . . . . . . . . . . . . . . . . 161
8.2.5 Transformations of L2 functions . . . . . . . . . . . . . . 162
8.2.6 Transformations of the velocity field . . . . . . . . . . . . 164
8.3 Boundary Controller and Closed-Loop System Properties . . . . 165
8.4 Proof of Stability for the Linearized Error System . . . . . . . . 168
8.4.1 Uncontrolled velocity modes. . . . . . . . . . . . . . . . . 169
8.4.2 Controlled velocity modes. Construction of
boundary control laws . . . . . . . . . . . . . . . . . . . . 175
8.4.3 Stability for the whole velocity error system . . . . . . . . 178
8.4.4 Well-posedness analysis for the velocity field. . . . . . . . 179
8.5 Proof of Stability for the Nonlinear Error System . . . . . . . . . 180
8.6 Proof of Well-Posedness of the Control Kernel Equation . . . . . 182
8.6.1 Proof for a finite time interval. . . . . . . . . . . . . . . . 184
8.6.2 Proof for an infinite time interval . . . . . . . . . . . . . . 195
8.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 195
9 Open Problems 197
Bibliography 199
Index 209