 # Adaptive Control of Parabolic PDEs

Andrey Smyshlyaev
Miroslav Krstic
Pages: 342
https://www.jstor.org/stable/j.ctt7s6dx

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-xiv)
Andrey Smyshlyaev and Miroslav Krstic
4. Chapter One Introduction
(pp. 1-10)

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...

5. ### PART I NONADAPTIVE CONTROLLERS

• Chapter Two State Feedback
(pp. 13-34)

In this chapter we introduce the method ofbacksteppingand present nominal designs that will serve as the basis for all the designs in the rest of the book.

We consider the following class of linear parabolic partial integro-differential equations (P(I)DEs):

$\begin{gathered} {u_t}(x,t) = \varepsilon (x){u_{xx}}(x,t) + b(x){u_x}(x,t) + \lambda (x)u(x,t) \\+ g(x)u(0,t) + \int_0^x {f(x,y)u(y,t)dy} \\ \end{gathered}$(2.1)

for$x \in (0,L)$,$t > 0$, with boundary conditions

${u_x}(0,t) = - qu(0,t)$(2.2)

$u(L,t) = U(t)$, (2.3)

where$U(t)$is the control input. We make the following assumptions:

$\varepsilon (x) > 0{\text{forall}}x \in [0,L],\varepsilon \in {C^3}[0,L],q \in \mathbb{R}$,

$b \in {C^2}[0,L],\lambda ,g \in {C^1}[0,L],f \in {C^1}([0,L] \times [0,L])$. (2.4)

For large positiveq,$\lambda \left( x \right),$or$f\left( {x,y} \right),$

the plant (2.1) with$U\left( t \right) = 0$is unstable; therefore, the objective is to stabilize he equilibrium u(x, t) ≡ 0.

Before we proceed with the control design, a...

• Chapter Three Closed-Form Controllers
(pp. 35-54)

In this chapter we present a collection of problems for which one can obtain explicit stabilizing controllers.

One of the striking features of the backstepping control design for PDEs is that it leads to explicit feedback controllers for many physically relevant problems. Such controllers are important for several reasons. The first and most obvious benefit is that one does not have to numerically compute a solution to the gain kernel PDE. Second, the explicit gain kernels allow us to find explicit solutions to the closed-loop system, offering valuable insight into how control affects eigenvalues and eigenfunctions. Explicit solutions to gain...

• Chapter Four Observers
(pp. 55-62)

The measurements in distributed parameter systems are not always available across the entire domain. They are often not available even at individual points strictly inside the domain. In fact, in some of the most exciting and complex applications, such as those involving fluid flows, sensors can be placed only at the boundaries. This is the situation that we focus on here, designing observers that employ onlyboundary sensing.

The state-feedback controllers developed in Chapters 2 and 3 require information on the state at each point in the domain. The design of state observers depends on the type (Dirichlet or Neumann)...

• Chapter Five Output Feedback
(pp. 63-72)

The exponentially convergent observers developed in Chapter 4 are independent of the control input and can be used with any controller. In this chapter we combine the backstepping observers from Chapter 4 with the backstepping controllers developed in Chapter 2 to solve the output-feedback problems.

First, in Sections 5.1 and 5.2, respectively, we establish closed-loop stability results for observer-based backstepping controllers in anti-collocated and collocated configurations. These results are essentially “separation principle” results for the backstepping approach to output-feedback stabilization.

Then, in Section 5.3 we derive an explicit output-feedback law (in the statespace format) for a reaction-diffusion system with constant...

• Chapter Six Control of Complex-Valued PDEs
(pp. 73-108)

In this chapter we extend the designs developed in Chapters 2 5 to the case of plants with a complex-valued state. Such plants can also be viewed as coupled parabolic PDEs. We consider two classes of complex-valued PDEs, the Ginzburg-Landau equation and its special case, the Schrödinger equation. For the Schrödinger equation, which we treat as a single complex-valued equation, the controllers and observers are obtained in closed form. The Ginzburg-Landau equation is treated as two coupled PDEs and serves as an example of the extension of the backstepping designs to (semi-)infinite domains. This is not trivial, since control and...

6. ### PART II ADAPTIVE SCHEMES

• Chapter Seven Systematization of Approaches to Adaptive Boundary Stabilization of PDEs
(pp. 111-124)

Having presented our underlying nonadaptive control designs in Chapters 2–6, we are now ready to broach the main subject of the book—control of PDEs with parametric uncertainties.

This tutorial chapter is an informal but rather comprehensive catalog of design tools, meant to serve as an entry point for a reader with little background in adaptive control. The chapter presents an introduction to the key ideas employed in the synthesis of adaptive boundary controllers for PDEs. Its purpose is also to help the reader of the subsequent technical Chapters 8–13, which contain the detailed proofs for most of...

• Chapter Eight Lyapunov-Based Designs
(pp. 125-149)

In this section we present the first of our three adaptive design approaches, the Lyapunov design.

The Lyapunov design that we employ for linear PDEs is inspired by an idea Praly  developed for adaptive nonlinear control under growth conditions. Since PDE problems in this book are linear, we significantly simplify Praly’s approach; however, we retain its main feature—a logarithm weight on the plant state in the Lyapunov function. This results in a normalization of the update law by a norm of the plant state, which is uncommon for Lyapunov designs employing backstepping controllers.

Except for some special examples,...

• Chapter Nine Certainty Equivalence Design with Passive Identifiers
(pp. 150-165)

In this chapter we present the first of the estimation-based adaptive design approaches, the design with passive identifiers.

We start our presentation with a benchmark 1D plant with only one uncertain parameter to illustrate the main ideas of the passivity-based approach in a tutorial way without the extensive notation that is needed in higher dimensions, such as 2D and 3D, and with more than one physical parameter.

After the 1D benchmark plant we present the adaptive design for a 3D reactionadvection-diffusion plant. We prove that passive identifiers are stable with all three (reaction, advection, and diffusion) coefficients unknown.

A fundamental...

• Chapter Ten Certainty Equivalence Design with Swapping Identifiers
(pp. 166-175)

In this chapter we present a second estimation-based approach, the design with swapping identifiers. This approach (often called simply the gradient or least squares method) is the most commonly used identification method in finite-dimensional adaptive control . One converts a dynamic parametrization of the problem into a static one using filters of the regressor and of the measured part of the plant. The standard gradient and least squares estimation techniques are then applied.

In this chapter we deal with a 1D reaction-advection-diffusion plant with three unknown constant parameters. As in Chapter 9, there exists a fundamental difficulty in proving closed-loop...

• Chapter Eleven State Feedback for PDEs with Spatially Varying Coefficients
(pp. 176-197)

In Chapters 8–10 we introduced three basic approaches for the design of parametric identifiers for boundary-controlled PDEs with constant coefficients: the Lyapunov, the passive, and the swapping approach. In this chapter we extend two of those designs, the Lyapunov and passive approaches, to plants withspatially varyingparameters. The third approach (swapping) is not addressed here because it will be extended to plants with spatially varying coefficients in Chapter 13 in the output feedback setting.

We consider the plant

${u_t}(x,t) = \varepsilon (x){u_{xx}}(x,t) + b(x){u_x}(x,t) + \lambda (x)u(x,t),$(11.1)

$u(0,t) = 0,$(11.2)

$u(1,t) = U(t),$(11.3)

where the parameters$\varepsilon (x)$,$b(x)$,$\lambda (x)$are unknown. We assume only that$\varepsilon (x)$is positive for...

• Chapter Twelve Closed-Form Adaptive Output-Feedback Controllers
(pp. 198-225)

In Chapters 8–11 we designed adaptive controllers for reaction-advection-diffusion systems under the assumption of availability of state measurements across the domain. In many applications, sensors are typically placed only at the boundary of the domain, for example, at the outlet of a chemical tubular reactor.

In this chapter we introduce output-feedback adaptive controllers for two parametrically uncertain, unstable parabolic PDEs controlled from the boundary. Both PDEs are motivated by a model of thermal instability in solid propellant rockets . Perhaps even more important than the physical motivation behind those systems is the fact that they are conceptually the simplest...

• Chapter Thirteen Output Feedback for PDEs with Spatially Varying Coefficients
(pp. 226-260)

In this chapter we consider the problem of output-feedback stabilization of reactionadvection-diffusion systems with uncertain parameters. We assume that sensing and actuation are performed at the opposite boundaries of the PDE domain. Thus, the open-loop plant has infinite relative degree, infinite-dimensional state, infinitedimensional (functional) parametric uncertainties, and only scalar input and output.

We start the design by introducing the notion of anobserver canonical formfor PDEs. To estimate the state, we use the adaptive observers that are infinitedimensional extensions of Kreisselmeier observers .

The identifiers are designed using the swapping approach. Typically the swapping method requires one filter per...

• Chapter Fourteen. Inverse Optimal Control
(pp. 261-276)

In this chapter we present backstepping designs that not only achieve stability but at the same timeminimizesome meaningful, positive definite, cost functionals. This property of a controller is referred to asinverse optimality.The controllers presented in the previous chapters do not possess this property. In this chapter we present redesigns that achieve inverse optimality. The idea of inverse optimality has a long history in control, starting from Kalman’s paper  for linear systems and a paper by Moylan and Anderson  that extends the idea to nonlinear systems. More recent nonlinear efforts inspired by control Lyapunov functions...

7. Appendix A. Adaptive Backstepping for Nonlinear ODEs—The Basics
(pp. 277-304)
8. Appendix B. Poincaré and Agmon Inequalities
(pp. 305-306)
9. Appendix C. Bessel Functions
(pp. 307-309)
10. Appendix D. Barbalat’s and Other Lemmas for Proving Adaptive Regulation
(pp. 310-312)
11. Appendix E. Basic Parabolic PDEs and Their Exact Solutions
(pp. 313-316)
12. References
(pp. 317-326)
13. Index
(pp. 327-328)