Adaptive control of parabolic PDEs [electronic resource] / Andrey Smyshlyaevand Miroslav Krstic.

ISBN:

9781400835362

9781282569126

Title:

Adaptive control of parabolic PDEs [electronic resource] / Andrey Smyshlyaevand Miroslav Krstic.

Author:

Smyshlyaev, Andrey, author.

Personal Author:

Smyshlyaev, Andrey, author.

Publication Information:

Physical Description:

1 online resource (xiii, 328 pages) : illustrations

Contents:

Introduction -- Nonadaptive controllers. State feedback -- Closed-form controllers -- Observers -- Output feedback -- Control of complex-valued PDEs -- Adaptive schemes. Systemization of approaches to adaptive boundary stabilization of PDEs -- Lyapunov-based designs -- Certainty equivalence design with passive identifiers -- Certainty equivalence design with swapping identifiers -- State feedback for PDEs with spatially varying coefficients -- Closed-form adaptive output-feedback controllers -- Output feedback for PDEs with spatially varying coefficients -- Inverse optimal control -- Appendex A. Adaptive backstepping for nonlinear ODEs -- the basics -- Appendix B. Poincaré and Agmon inequalities -- Appendix C. Bessel functions -- Appendix D. Barbalat's and other lemmas for proving adaptive regulation -- Appendix E. Basic parabolic PDEs and their exact solution.

Local Note:

eBooks on EBSCOhost

Subject Term:

Differential equations, Parabolic.

Distributed parameter systems.

Adaptive control systems.

Added Author:

Krstić, Miroslav,

Format:

Electronic Resources

Electronic Access:

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Publication Date:

2010

Publication Information:

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Summary

This book introduces a comprehensive methodology for adaptive control design of parabolic partial differential equations with unknown functional parameters, including reaction-convection-diffusion systems ubiquitous in chemical, thermal, biomedical, aerospace, and energy systems. Andrey Smyshlyaev and Miroslav Krstic develop explicit feedback laws that do not require real-time solution of Riccati or other algebraic operator-valued equations. The book emphasizes stabilization by boundary control and using boundary sensing for unstable PDE systems with an infinite relative degree. The book also presents a rich collection of methods for system identification of PDEs, methods that employ Lyapunov, passivity, observer-based, swapping-based, gradient, and least-squares tools and parameterizations, among others.

Including a wealth of stimulating ideas and providing the mathematical and control-systems background needed to follow the designs and proofs, the book will be of great use to students and researchers in mathematics, engineering, and physics. It also makes a valuable supplemental text for graduate courses on distributed parameter systems and adaptive control.

Author Notes

Andrey Smyshlyaev is assistant project scientist at the University of California, San Diego. Miroslav Krstic is the Sorenson Distinguished Professor and the founding director of the Cymer Center for Control Systems and Dynamics at the University of California, San Diego. Smyshlyaev and Krstic are the authors of Boundary Control of PDEs .

Preface	p. ix
Chapter 1 Introduction	p. 1
1.1 Parabolic and Hyperbolic PDE Systems	p. 1
1.2 The Roles of PDE Plant Instability, Actuator Location, Uncertainty Structure, Relative Degree, and Functional Parameters	p. 2
1.3 Class of Parabolic PDE Systems	p. 3
1.4 Backstepping	p. 4
1.5 Explicitly Parametrized Controllers	p. 5
1.6 Adaptive Control	p. 5
1.7 Overview of the Literature on Adaptive Control for Parabolic PDEs	p. 6
1.8 Inverse Optimality	p. 7
1.9 Organization of the Book	p. 7
1.10 Notation	p. 9
Part I Nonadaptive Controllers	p. 11
Chapter 2 State Feedback	p. 13
2.1 Problem Formulation	p. 13
2.2 Backstepping Transformation and PDE for Its Kernel	p. 14
2.3 Converting the PDE into an Integral Equation	p. 17
2.4 Analysis of the Integral Equation by Successive Approximation Series	p. 19
2.5 Stability of the Closed-Loop System	p. 22
2.6 Dirichlet Uncontrolled End	p. 24
2.7 Neumann Actuation	p. 26
2.8 Simulation	p. 27
2.9 Discussion	p. 27
2.10 Notes and References	p. 33
Chapter 3 Closed-Form Controllers	p. 35
3.1 The Reaction-Diffusion Equation	p. 35
3.2 A Family of Plants with Spatially Varying Reactivity	p. 38
3.3 Solid Propellant Rocket Model	p. 40
3.4 Plants with Spatially Varying Diffusivity	p. 42
3.5 The Time-Varying Reaction Equation	p. 45
3.6 More Complex Systems	p. 50
3.7 2D and 3D Systems	p. 52
3.8 Notes and References	p. 54
Chapter 4 Observers	p. 55
4.1 Observer Design for the Anti-Collocated Setup	p. 55
4.2 Plants with Dirichlet Uncontrolled End and Neumann Measurements	p. 58
4.3 Observer Design for the Collocated Setup	p. 59
4.4 Notes and References	p. 61
Chapter 5 Output Feedback	p. 63
5.1 Anti-Collocated Setup	p. 63
5.2 Collocated Setup	p. 65
5.3 Closed-Form Compensators	p. 67
5.4 Frequency Domain Compensator	p. 71
5.5 Notes and References	p. 72
Chapter 6 Control of Complex-Valued PDEs	p. 73
6.1 State-Feedback Design for the Schrödinger Equation	p. 73
6.2 Observer Design for the Schrödinger Equation	p. 76
6.3 Output-Feedback Compensator for the Schrödinger Equation	p. 79
6.4 The Ginzburg-Landau Equation	p. 81
6.5 State Feedback for the Ginzburg-Landau Equation	p. 83
6.6 Observer Design for the Ginzburg-Landau Equation	p. 98
6.7 Output Feedback for the Ginzburg-Landau Equation	p. 101
6.8 Simulations with the Nonlinear Ginzburg-Landau Equation	p. 104
6.9 Notes and References	p. 107
Part II Adaptive Schemes	p. 109
Chapter 7 Systematization of Approaches to Adaptive Boundary Stabilization of PDEs	p. 111
7.1 Categorization of Adaptive Controllers and Identifiers	p. 111
7.2 Benchmark Systems	p. 113
7.3 Controllers	p. 114
7.4 Lyapunov Design	p. 115
7.5 Certainty Equivalence Designs	p. 117
7.6 Trade-offs between the Designs	p. 121
7.7 Stability	p. 122
7.8 Notes and References	p. 124
Chapter 8 Lyapunov-Based Designs	p. 125
8.1 Plant with Unknown Reaction Coefficient	p. 125
8.2 Proof of Theorem 8.1	p. 128
8.3 Well-Posedness of the Closed-Loop System	p. 132
8.4 Parametric Robustness	p. 134
8.5 An Alternative Approach	p. 135
8.6 Other Benchmark Problems	p. 136
8.7 Systems with Unknown Diffusion and Advection Coefficients	p. 142
8.8 Simulation Results	p. 147
8.9 Notes and References	p. 149
Chapter 9 Certainty Equivalence Design with Passive Identifiers	p. 150
9.1 Benchmark Plant	p. 150
9.2 3D Reaction-Advection-Diffusion Plant	p. 154
9.3 Proof of Theorem 9.2	p. 157
9.4 Simulations	p. 163
9.5 Notes and References	p. 164
Chapter 10 Certainty Equivalence Design with Swapping Identifiers	p. 166
10.1 Reaction-Advection-Diffusion Plant	p. 166
10.2 Proof of Theorem 10.1	p. 169
10.3 Simulations	p. 175
10.4 Notes and References	p. 175
Chapter 11 State Feedback for PDEs with Spatially Varying Coefficients	p. 176
11.1 Problem Statement	p. 176
11.2 Nominal Control Design	p. 177
11.3 Robustness to Error in Gain Kernel	p. 179
11.4 Lyapunov Design	p. 185
11.5 Lyapunov Design for Plants with Unknown Advection and Diffusion Parameters	p. 190
11.6 Passivity-Based Design	p. 191
11.7 Simulations	p. 195
11.8 Notes and References	p. 197
Chapter 12 Closed-Form Adaptive Output-Feedback Contollers	p. 198
12.1 Lyapunov Design--Plant with Unknown Parameter in the Domain	p. 199
12.2 Lyapunov Design--Plant with Unknown Parameter in the Boundary Condition	p. 205
12.3 Swapping Design--Plant with Unknown Parameter in the Domain	p. 210
12.4 Swapping Design--Plant with Unknown Parameter in the Boundary Condition	p. 216
12.5 Simulations	p. 223
12.6 Notes and References	p. 225
Chapter 13 Output Feedback for PDEs with Spatially Varying Coefficients	p. 226
13.1 Reaction-Advection-Diffusion Plant	p. 226
13.2 Transformation to Observer Canonical Form	p. 227
13.3 Nominal Controller	p. 228
13.4 Filters	p. 230
13.5 Frequency Domain Compensator with Frozen Parameters	p. 232
13.6 Update Laws	p. 233
13.7 Stability	p. 235
13.8 Trajectory Tracking	p. 242
13.9 The Ginzburg-Landau Equation	p. 244
13.10 Identifier for the Ginzburg-Landau Equation	p. 246
13.11 Stability of Adaptive Scheme for the Ginzburg-Landau Equation	p. 248
13.12 Simulations	p. 255
13.13 Notes and References	p. 255
Chapter 14 Inverse Optimal Control	p. 261
14.1 Nonadaptive Inverse Optimal Control	p. 262
14.2 Reducing Control Effort through Adaptation	p. 265
14.3 Dirichlet Actuation	p. 267
14.4 Design Example	p. 267
14.5 Comparison with the LQR Approach	p. 268
14.6 Inverse Optimal Adaptive Control	p. 271
14.7 Stability and Inverse Optimality of the Adaptive Scheme	p. 273
14.8 Notes and References	p. 275
Appendix A Adaptive Backstepping for Nonlinear ODEs--The Basics	p. 277
A.1 Nonadaptive Backstepping--The Known Parameter Case	p. 277
A.2 Tuning Functions Design	p. 279
A.3 Modular Design	p. 289
A.4 Output Feedback Designs	p. 297
A.5 Extensions	p. 303
Appendix B Poincaré and Agmon Inequalities	p. 305
Appendix C Bessel Functions	p. 307
C.1 Bessel Function Jn	p. 307
C.2 Modified Bessel Function In	p. 307
Appendix D Barbalat's and Other Lemmas for Proving Adaptive Regulation	p. 310
Appendix E Basic Parabolic PDEs and Their Exact Solutions	p. 313
E.1 Reaction-Diffusion Equation with Dirichlet Boundary Conditions	p. 313
E.2 Reaction-Diffusion Equation with Neumann Boundary Conditions	p. 315
E.3 Reaction-Diffusion Equation with Mixed Boundary Conditions	p. 315
References	p. 317
Index	p. 327

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