3R23. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. - PD Christofides (Dept of Chem Eng, UCLA, Los Angeles CA 90095-1592). Birkhauser Boston, Cambridge MA. 2001. 248 pp. ISBN 0-8176-4156-4. $69.95.
Reviewed by J Chow (Adv Tech Center, Org L9-24, Lockheed Martin, 3251 Hanover St, Palo Alto CA 94304-1191).
The author has written a book that seeks to present practical, general nonlinear, and robust control methods for hyperbolic and parabolic PDE systems, and to illustrate their application to transport-reaction processes that are found in the chemical industry. There is also an attempt to compare their effectiveness with respect to traditional control methods for PDE systems. This book is written for process control engineers, researchers, and students at the graduate level.
The field of chemical engineering contains many different examples of hyperbolic and parabolic PDE systems. As the author rightly states in his preface, “The interest in control of nonlinear partial differential equation (PDE) systems has been triggered by the need to achieve tight distributed control of transport-reaction processes that exhibit highly nonlinear behavior and strong spatial variations.” As a result, there is a requirement to provide the chemical process community with an overview of the most recent advances in nonlinear PDE control theory.
In this book, general and practical methods for synthesizing nonlinear controllers for hyperbolic and parabolic PDE systems are systematically developed, and then extended to include robustness. Geometric and Lyapunov-based control techniques are used to synthesize nonlinear and robust controllers that use a finite number of measurement sensors and control actuators to achieve a stable closed-loop system in the face of model uncertainty. All the PDE systems considered in this text are assumed to have unique solutions that are sufficiently smooth. Readers are assumed to have a basic knowledge about PDE systems and control theories.
The first two chapters focus first on quasi-linear, first-order hyperbolic PDE systems, and then on the same type of hyperbolic PDE systems, but those which include time-varying uncertain variables and unmodeled dynamics. These control methods are based on geometric control concepts and are applied to nonisothermal plug-flow reactor examples. The next two chapters focus on developing general methods for quasi-linear parabolic PDE systems using Galerkin’s method and approximate inertial manifolds, both with and without time-varying uncertain variables. Examples of the methodology’s successful application to the control of temperature profiles for catalytic rods and nonisothermal reactors are presented for illustration. The following chapter deals with the nonlinear and robust control of parabolic PDE systems with moving boundaries, ie, time-dependent spatial domains. General methods for synthesizing nonlinear and robust time-varying output feedback controllers are presented, again using a combination of Galerkin’s method and the approximate inertial manifold. The final chapter presents case studies in which the control methods derived in earlier chapters for parabolic PDE systems are applied. In particular, the control of a rapid thermal chemical vapor deposition process and a Czochralski crystal growth process are illustrated and simulated. The book ends with an extensive set of mathematical proofs to support the results developed in the preceding chapters. It also has an extensive bibliography and is sprinkled throughout with clear figures and examples.
Nonlinear and Robust Control of PDE Systems is a well-written book that succeeds in its objectives and is possibly the first one to do so. The control theories and synthesis methodologies are described in exhaustive mathematical detail. This book can also be recommended to researchers and engineers in other fields who are faced with the task of developing nonlinear feedback and robust controllers for hyperbolic and parabolic PDE systems. They will find this book to be very useful because it provides the reader with the general framework for nonlinear feedback control based on detailed mathematical models.