Control of Turbulent and Magnetohydrodynamic Channel Flows

by Vazquez, Rafael; Krstic, Miroslav

• ISBN13: 9780817646981
• ISBN10: 0817646981
• Format: Hardcover
• Copyright: 2007-12-30
• Publisher: Birkhauser

Summary

Turbulent flows modeled by systems of partial differential equations are often unstable and can typically be controlled or measured only from the boundary or walls of the flow domain. This monograph presents new constructive design methods for boundary stabilization and boundary estimation for several classes of benchmark problems in flow control, with potential applications in turbulence control, aerodynamics, weather forecasting, chemical processes, and plasma control. Topics and Features: * Use of a unique "backstepping" approach to parabolic partial differential equations, which yields not only the stabilization of the flow, but also the explicit solvability of the closed-loop system. * Discussion of hydrodynamics of the turbulent motions modeled by Navier-Stokes partial differential equations. ]* Introduction of control and state estimation designs for flows that include thermal convection and electric conductivity. * Presentation of a new result in fluid dynamics and the analysis of Navier-Stokes equations: closed-form expressions for the solutions of linearized Navier-Stokes equations under feedback. * Extension of the backstepping approach to eliminate one of the well-recognized root causes of transition to turbulence. This monograph is an excellent reference for a broad, interdisciplinary engineering and mathematics audience: control theorists, fluid mechanicists, mechanical engineers, aerospace engineers, chemical engineers, electrical engineers, applied mathematicians, as well as research and graduate students in the above areas. The book may also be used as a supplementary text for graduate courses on control of distributed-parameter systems and on flow control.

Table of Contents

Preface	p. V
Introduction	p. 1
Motivation	p. 1
Mathematical Preliminaries and Notation	p. 3
Function spaces and norms	p. 3
Stability in the infinite-dimensional setting	p. 8
Spatial invariance, Fourier transforms, and Fourier series	p. 11
Singular perturbation theory	p. 20
The backstepping method for parabolic PDEs	p. 23
Overview of the Monograph	p. 34
Notes and References	p. 36
Thermal-Fluid Convection Loop: Boundary Stabilization	p. 39
Thermal Convection Loop Model	p. 39
Reduced Model and Velocity Controller for Large Prandtl Numbers	p. 41
Backstepping Controller for Temperature	p. 42
Temperature target system	p. 42
Backstepping temperature transformation	p. 43
Temperature control law	p. 45
Inverse transformation for temperature	p. 46
Singular Pertubation Stability Analysis for the System	p. 46
Simulation Study	p. 51
Notes and References	p. 54
Thermal-Fluid Convection Loop: Boundary Estimation and Output-Feedback Stabilization	p. 55
A Decoupling Transformation for the Temperature	p. 56
Stabilization of Uncoupled Temperature Modes	p. 57
Stabilization of Velocity and Coupled Temperature Modes	p. 58
Boundary control design using singular perturbations and backstepping	p. 58
Observer design using singular perturbations and backstepping	p. 60
Output-feedback controller	p. 63
Singular perturbation analysis for large Prandtl numbers	p. 63
Stability Properties of the Closed-Loop System	p. 64
Simulation Study	p. 65
Observer Convergence and Output-Feedback Stabilization Proofs	p. 66
2D Navier-Stokes Channel Flow: Boundary Stabilization	p. 71
2D Channel Plow Model	p. 71
Velocity Boundary Controller	p. 75
Closed-Loop Stability and Explicit Solutions	p. 77
L[superscript 2] Stability for the Closed-Loop System	p. 81
Controlled velocity wave numbers	p. 82
Uncontrolled velocity wave number analysis	p. 88
Analysis for the entire velocity wave number range	p. 89
H[superscript 1] Stability for the Closed-Loop System	p. 90
H[superscript 1] stability for controlled velocity wave numbers	p. 90
H[superscript 1] stability for uncontrolled velocity wave numbers	p. 91
Analysis for all velocity wave numbers	p. 93
H[superscript 2] Stability for the Closed-Loop System	p. 94
H[superscript 2] stability for controlled velocity wave numbers	p. 94
H[superscript 2] stability for uncontrolled velocity wave numbers	p. 95
Analysis for all velocity wave numbers	p. 97
Proof of Well-Posedness and Explicit Solutions for the Velocity Field	p. 97
Proof of Properties of the Velocity Boundary Controller	p. 99
Notes and References	p. 101
2D Navier-Stokes Channel Flow: Boundary Estimation	p. 103
Observer with Boundary Sensing of Pressure and Skin Friction	p. 103
Observer Convergence Proof	p. 107
Observed wave number analysis	p. 108
Unobserved wave number analysis	p. 111
Analysis for the entire observer error wave number range	p. 111
An Output-Feedback Stabilizing Controller for 2D Channel Flow	p. 112
Notes and References	p. 114
3D Magnetohydrodynamic Channel Flow: Boundary Stabilization	p. 115
Magnetohydrodynamic Channel Flow Model	p. 115
Hartmann Equilibrium Profile	p. 117
The Plant in Wave Number Space	p. 118
Boundary Control Design	p. 120
Controlled velocity wave number analysis	p. 120
Uncontrolled velocity wave number analysis	p. 127
Closed-loop stability properties	p. 131
Notes and References	p. 133
3D Magnetohydrodynamic Channel Flow: Boundary Estimation	p. 135
Observer Structure	p. 135
Observer Gain Design and Convergence Analysis	p. 138
Observed wave number analysis	p. 140
Unobserved wave number analysis	p. 147
Observer Convergence Properties	p. 148
A Nonlinear Estimator with Boundary Sensing	p. 149
2D Navier-Stokes Channel Flow: Stable Flow Transfer	p. 153
Trajectory Generation and Tracking Error Model	p. 153
Spaces and Transformations for the Velocity Field	p. 158
Periodic function spaces	p. 158
Fourier series expansion in [Omega subscript h]	p. 158
H[superscript 1] and H[superscript 2] functional spaces	p. 159
Spaces for the velocity field	p. 161
Transformations of L[superscript 2] functions	p. 162
Transformations of the velocity field	p. 164
Boundary Controller and Closed-Loop System Properties	p. 165
Proof of Stability for the Linearized Error System	p. 168
Uncontrolled velocity modes	p. 169
Controlled velocity modes. Construction of boundary control laws	p. 175
Stability for the whole velocity error system	p. 178
Well-posedness analysis for the velocity field	p. 179
Proof of Stability for the Nonlinear Error System	p. 180
Proof of Well-Posedness of the Control Kernel Equation	p. 182
Proof for a finite time interval	p. 184
Proof for an infinite time interval	p. 195
Notes and References	p. 195
Open Problems	p. 197
Bibliography	p. 199
Index	p. 209
Table of Contents provided by Ingram. All Rights Reserved.