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 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit- tenpermissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. 987654321 www.birkhauser.com (KeS/SB) 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