Über die Autorin bzw. den Autor

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.

Von der hinteren Coverseite

"Unique and excellent, this book systematically and rigorously develops design and analysis tools and clearly explains technical concepts. As the first book to cover its topics, it significantly expands the scope of adaptive control knowledge. I strongly recommend this book as either a reference or an advanced textbook for researchers and graduate students who study and work in engineering and applied sciences."--Gang Tao, University of Virginia

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Adaptive Control of Parabolic PDEs

PRINCETON UNIVERSITY PRESS

Contents

Preface.................................................................................................ixChapter 1. Introduction.................................................................................1PART I NONADAPTIVE CONTROLLERS..........................................................................11Chapter 2. State Feedback...............................................................................13Chapter 3. Closed-Form Controllers......................................................................35Chapter 4. Observers....................................................................................55Chapter 5. Output Feedback..............................................................................63Chapter 6. Control of Complex-Valued PDEs...............................................................73PART II ADAPTIVE SCHEMES................................................................................109Chapter 7. Systematization of Approaches to Adaptive Boundary Stabilization of PDEs.....................111Chapter 8. Lyapunov-Based Designs.......................................................................125Chapter 9. Certainty Equivalence Design with Passive Identifiers........................................150Chapter 10. Certainty Equivalence Design with Swapping Identifiers......................................166Chapter 11. State Feedback for PDEs with Spatially Varying Coefficients.................................176Chapter 12. Closed-Form Adaptive Output-Feedback Contollers.............................................198Chapter 13. Output Feedback for PDEs with Spatially Varying Coefficients................................226Chapter 14. Inverse Optimal Control.....................................................................261Appendix A. Adaptive Backstepping for Nonlinear ODEs—The Basics...................................277Appendix B. Poincaré and Agmon Inequalities........................................................305Appendix C. Bessel Functions............................................................................307Appendix D. Barbalat's and Other Lemmas for Proving Adaptive Regulation.................................310Appendix E. Basic Parabolic PDEs and Their Exact Solutions..............................................313References..............................................................................................317Index...................................................................................................327

Chapter One

1.1 PARABOLIC AND HYPERBOLIC PDE SYSTEMS

This book investigates problems in control of partial differential equations (PDEs) with unknown parameters. Control of PDEs alone (with known parameters) is a complex subject, but also a physically relevant subject. Numerous systems in aerospace engineering, bioengineering, chemical engineering, civil engineering, electrical engineering, mechanical engineering, and physics are modeled by PDEs because they involve fluid flows, thermal convection, spatially distributed chemical reactions, flexible beams or plates, electromagnetic or acoustic oscillations, and other distributed phenomena. Model reduction to ordinary differential equations (ODEs) is often possible, but model reduction approaches suitable for simulation (in the absence of control) may lead to control designs that are divergent upon grid refinement.

While ODEs represent a class of dynamic systems for which general, unified control design can be developed, at least in the linear case, this is not so for PDEs. Even in dimension one (1D), different classes of PDEs require fundamentally different approaches in analysis and in control design. Three basic classes of PDEs are often considered to be the fundamental, distinct classes:

? parabolic PDEs (including reaction-diffusion equations),

? first-order hyperbolic PDEs (including transport equations),

? second-order hyperbolic PDEs (including wave equations).

In addition to these basic classes, many other classes and individual PDEs of interest exist, some formalized in the Schrödinger, Ginzburg-Landau, Korteweg–de Vries, Kuramoto-Sivashinsky, Burgers, and Navier-Stokes equations.

Designing adaptive controllers for PDEs requires not only mastering the nonadaptive designs for PDEs (with known parameters) but also developing a design approach that is parametrized in such a way that the controllers can be made parameter-adaptive. This requirement excludes most design approaches. For example, optimal control approaches require solving operator Riccati equations, which cannot be done continuously in real time, whereas pole placement–based approaches are parametrized in terms of the open-loop system's eigenvalues and not in terms of the plant parameters (as required for indirect adaptive control) or in terms of controller parameters (as required for direct adaptive control). In this book we build on the backstepping approach to control of PDEs [70], as it leads to feedback laws that are explicitly or nearly explicitly parametrized in terms of the plant parameters.

At the present time the backstepping approach [70] is well developed for a rather broad set of classes of PDEs, including the three basic classes of PDEs—parabolic, first-order hyperbolic, and second-order hyperbolic. The backstepping approach allows adaptive control development for PDEs in all of these classes. However, owing to the vast number of design possibilities, the development of adaptive controllers for PDEs using the backstepping approach has so far been mostly limited to parabolic PDEs. The reason for this is that this class is complex enough to be representative of many (but not all) of the mathematical challenges one faces when dealing with PDEs but does not include the idiosyncratic challenges such as those arising from second-order-in-time derivatives in second-order hyperbolic PDEs, where the adaptive issue may be secondary to the analysis issues specific to this PDE class.

Though our focus in this book is on parabolic PDEs, which we have chosen as the benchmark class for the development of adaptive controllers for PDE systems, adaptive control designs for parametrically uncertain hyperbolic PDE systems with boundary actuation are starting to emerge. Examples of such designs are the design in [63] for an unstable wave equation, as a representative of second-order hyperbolic PDEs, and in [18, 19] for ODE systems with an actuator delay of unknown length, as a representative of first-order hyperbolic PDEs.

1.2 THE ROLES OF PDE PLANT INSTABILITY, ACTUATOR LOCATION, UNCERTAINTY STRUCTURE, RELATIVE DEGREE, AND FUNCTIONAL PARAMETERS

The field of adaptive control of infinite-dimensional systems is a complex landscape in which problems of vastly different complexity can be considered, depending on not only the class of PDE systems but also the following three properties of the system:

? Stability of the open-loop system. Open-loop unstable systems present greater challenges than open-loop stable systems. In this book we focus on PDEs that are open-loop unstable. ? Actuator location. An immense difference exists between problems with distributed control, where independent actuation access is available at each point in the PDE domain, and...