“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

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.

Adaptive Control of Parabolic PDEs

By Andrey Smyshlyaev Miroslav Krstic

PRINCETON UNIVERSITY PRESS

Copyright © 2010 Princeton University Press
All right reserved.
ISBN: 978-0-691-14286-9

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

Introduction

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 problems with localized actuation, such as boundary control, where the dimensionality of the state is higher than the dimensionality of the actuation. A similar difference exists between problems with distributed sensing and boundary sensing. In this book we focus on PDEs with boundary control. In cases where we extend our designs from the full-state feedback case to the output-feedback case, we consider boundary sensing. ? Uncertainty structure. One of the key challenges in adaptive control is compensating parametric uncertainties that are not matched by the control, namely, uncertainties that cannot be directly canceled by the control but require some state transformation, or the solution of some Bezout equation, or some other nontrivial construction through which the control gains access to uncertainties that are not collocated with the control input. In this book we focus exclusively on problems with unmatched parametric uncertainties. ? Relative degree. Historically, the greatest challenge in adaptive control has been associated with extending adaptive control from systems of relative degree one to systems of relative degree two, and then to systems of relative degree three and higher. In PDE control problems the relative degree can be infinite. We aim for problems of this type in this book, as they introduce issues that are beyond what is encountered in adaptive control of ODE systems. Our benchmark system in this regard is the unstable reaction-diffusion PDE of dimension one, where control is applied at one boundary and sensing is conducted at the opposite boundary. ? Spatially constant or functional parameters? Even when a PDE has a finite number of spatially constant parameters, the designer may face a challenging problem if the PDE is unstable and the actuation is applied from the boundary. However, an entirely different level of challenge arises when the parametric uncertainties are functional, namely, when the unknown parameters depend on the spatial variables. This book contains solutions to such problems for unstable reaction-diffusion plants with boundary control and sensing.

1.3 CLASS OF PARABOLIC PDE SYSTEMS

As we have indicated earlier, this book focuses on parabolic PDEs as a good benchmark class for adaptive control design for PDEs with boundary actuation, openloop instability, possibly functional parametric uncertainties, and the possibility of occurrence of infinite relative degree when the input and output are non-collocated.

A good representative example of such a parabolic PDE is the reaction-diffusion equation. However, in this book we develop designs that are applicable to the following class of linear parabolic partial integro-differential equation (P(I)DE):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

This system is supplied with two boundary conditions. The x = 0 boundary of the domain is uncontrolled and is of either Dirichlet or Robin type:

u(0, t) = 0 or u_x(0, t) = –qu(0, t). (1.2)

The other end, x = 1, is controlled, either with Dirichlet or with Neumann actuation:

u(L, t) = U(t) or u_x(L, t) = U(t), (1.3)

where U(t) is the control input.

The PDE (2.1)–(2.3) serves as a model for a variety of systems with thermal, fluid, and chemically reacting dynamics. Applications include, among others, chemical tubular reactors, solid propellant rockets, thermal-fluid convection problems, flows past bluff bodies, steel casting, and cardiac pacing devices. In general, spatially varying coefficients come from applications with non-homogeneous materials or unusually shaped domains, and can also arise from linearization.

The g- and f -terms are not as common as reaction, advection, and diffusion terms but are important nonetheless (see Chapter 13) and have the “spatially causal” structure, which makes them tractable by our method.

Most of the book covers 1D, real-valued plants. However, in Chapters 6 and 13 we extend the approach to complex-valued plants, and in Chapter 9 we deal with an example of a 3D problem.

Two main characteristic features of the plant (1.1)–(1.3) are:

? The plant is unstable for large ratios ?(x)/e(x), [g(x)/e(x), f(x, y)/e(x) or large positive q.

? The plant is actuated from the boundary. In many applications this is the only physically reasonable way of placing actuators.

In this book we consider two types of measurement scenarios, either full-state feedback or boundary sensing, where the measured output can be one of the following four:

u(0, t), u_x(0, t), u(L, t), u_x(L, t).

1.4 BACKSTEPPING

The backstepping approach in control of nonlinear ODEs “matches” nonlinearities unmatched by the control input by using a combination of a diffeomorphic change of coordinates and feedback. In this book we pursue a continuum equivalent of this approach and build a change of variables, which involves a Volterra integral operator (which has a lower-triangular structure, just as backstepping transformations for nonlinear ODEs do) that “absorbs” the destabilizing terms acting in the domain and allows the control to completely eliminate their effect acting only from the boundary.

While the backstepping method is certainly not the first solution to the problems of boundary control of PDEs, it has several distinguishing features. First of all, it takes advantage of the structure of the system, resulting in a problem of solving a linear hyperbolic PDE for the gain kernel, an object much easier, both conceptually and computationally, than operator Riccati equations arising in linear quadratic regulator (LQR) approaches to boundary control. Second, the well-posedness difficulties are circumvented by transforming the plant into a simple heat equation. The control problem is solved essentially by calculus, making a design procedure clear and constructive and the analysis easy, in contrast to standard abstract approaches (semigroups, etc.). Third, for a number of physically relevant problems, the observer/controller kernels can be found in closed form, that is, as explicit functions of the spatial variable. Finally, unlike other methods, the backstepping approach naturally extends to adaptive control problems.

1.5 EXPLICITLY PARAMETRIZED CONTROLLERS

One of the most significant features of the design approach presented in this book is that for a certain class of physically relevant parabolic 1D PDEs, the control laws can be obtained in closed form, which is not the case with other methods (LQR, pole placement). One of the subclasses for which explicit controllers can be obtained is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.4)

u_x(0, t) = –qu(0, t), (1.5) u(L, t) = U(t) or u_x(L, t) = U(t), (1.6)

for arbitrary values of six parameters e, b, ?, g, a, and q. The explicit controllers, in turn, allow us to find explicit closed-loop solutions. Some of the PDEs within the class (1.4)–(1.6) are not even explicitly solvable in open loop; however, we solve them in closed form in the presence of the backstepping control law. While we prove closed-loop well-posedness for the general class separately from calculating the explicit solutions for the subclasses, these solutions serve to illustrate how well-posedness issues can be pretty much bypassed by a structure-specific control approach, allowing the designer to concentrate on control-relevant issues such as the feedback algorithm and its performance.

Since the backstepping observers are dual to the backstepping controllers, they are also available in closed form for the same class of plants (1.4)–(1.6).

1.6 ADAPTIVE CONTROL

In many distributed parameter systems the physical parameters, such as the Reynolds, Rayleigh, Prandtl, or Péclet numbers, are unknown because they vary with operating conditions. While adaptive control of finite-dimensional systems is an advanced field that has produced adaptive control methods for a very general class of linear time-invaria systems, adaptive control techniques have been developed for only a few classes of PDEs restricted by relative degree, stability, or domain-wide actuation assumptions. There are two sources of difficulties in dealing with PDEs with parametric uncertainties. The first difficulty, which also exists in ODEs, is that even for linear plants, adaptive schemes are nonlinear. The second difficulty, unique to PDEs, is the absence of parametrized families of controllers. The backstepping approach removes this second difficulty, opening the door for the use of a wealth of certainty equivalence and Lyapunov techniques developed for finite-dimensional systems.

In this book we develop the first constructive adaptive designs for unstable parabolic PDEs controlled from a boundary. We differentiate between two major classes of schemes:

? Lyapunov schemes,

? certainty equivalence schemes.

Within the certainty equivalence class, two types of identifier designs are pursued:

? passivity-based identifiers,

? swapping-based identifiers.

Each of those designs is applicable to two types of parametrizations:

? the plant model in its original form (which we refer to as the u-model),

? a transformed model to which a backstepping transformation has been applied (which we refer to as the w-model).

Hence, a large number of control algorithms result from combining different design tools—Lyapunov schemes, w-passive schemes, u-swapping schemes, and so on.

Our presentation culminates in Chapter 13 with the adaptive output tracking design for the system with scalar input and output, infinite-dimensional state, and infinitely many unknown parameters.

1.7 OVERVIEW OF THE LITERATURE ON ADAPTIVE CONTROL FOR PARABOLIC PDES

Early efforts to design adaptive controllers for distributed parameter systems used tuning of a scalar gain to a high level to stabilize some classes of (relative degree one) infinite-dimensional plants (see also the survey by Logemann and Townley [84] for additional references).

Early identifiability results include. A monograph by Banks and Kunisch focuses on approximation methods for least squares estimation of elliptic and parabolic equations. Online parameter estimation schemes were developed by Demetriou and Rosen, Baumeister, Scondo, Demetriou, and Rosen, and Orlov and Bentsman.

Model reference adaptive control (MRAC)–type schemes were designed by Hong and Bentsman, Bohm, Demetriou, Reich, and Rosen, and Bentsman and Orlov. These schemes successfully identify functional, spatially varying parametric uncertainties under the assumption that control is distributed in the PDE domain. A sliding mode observer of a spatial derivative has been designed in to get a state-derivative free MRAC scheme. Demetriou and Rosen developed robust parameter estimation schemes using parameter projection and sigma modification. Variable structure MRAC is considered in Demetriou and Rosen.

(Continues…)

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