Adaptive Control of Parabolic PDEs

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

"The text is easy to read due to ubiquitous remarks, examples and explanations before and after the rigorous mathematical derivations. There are plenty of numerical simulations and figures that illustrate the control designs and compare them to each other. This book is recommended for everybody interested in control systems, system identification for PDEs, or just PDEs in general. From students to researchers, and from engineers to mathematicians everybody can find interesting new results in it."--Andras Balogh, Mathematical Reviews