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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.

We consider a problem of stabilization of the linearized Schrödinger equation using boundary actuation and measurements. We propose two different control designs. First, a simple proportional collocated boundary controller is shown to exponentially stabilize the system. However, the decay rate of the closed-loop system cannot be prescribed. The second, full-state feedback boundary control design, achieves an arbitrary decay rate. We formally view the Schrödinger equation as a heat equation in complex variables and apply the backstepping method recently developed for boundary control of reaction-advection-diffusion equations. The resulting controller is then supplied with the backstepping observer to obtain an output-feedback compensator. The designs are illustrated with simulations. [PUBLICATION ABSTRACT]

We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping. [PUBLICATION ABSTRACT]

We consider a model of the undamped shear beam with a destabilizing boundary condition. The motivation for this model comes from atomic force microscopy, where the tip of the cantilever beam is destabilized by van der Waals forces acting between the tip and the material surface. Previous research efforts relied on collocated actuation and sensing at the tip, exploiting the passivity property between the corresponding input and output in the beam model. In this paper we design a stabilizing output-feedback controller in a noncollocated setting, with measurements at the free end (tip) of the beam and actuation at the beam base. Our control design is a novel combination of the classical \"damping boundary feedback\" idea with a recently developed backstepping approach. A change of variables is constructed which converts the beam model into a wave equation (for a very short string) with boundary damping. This approach is physically intuitive and allows both an elegant stability analysis and an easy selection of design parameters for achieving desired performance. Our observer design is a dual of the similar ideas, combining the damping feedback with backstepping, adapted to the observer error system. Both stability and well-posedness of the closed-loop system are proved. The simulation results are presented.

In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg--Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions on the parameters of the Ginzburg--Landau equation, compatible with vortex shedding modelling on a semi-infinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. In summary, the paper extends previous work in two ways: (1) it deals with two coupled partial differential equations, and (2) under certain circumstances handles equations defined on a semi-infinite domain.

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

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)

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

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