9R11. Model Reduction for Control System Design. - G Obinata (Dept of Mech Eng, Akita Univ, 1-1 Tegatagakuen Univ, Akita City, 010-8502, Japan) and BDO Anderson (Res Sch of Info Sci and Eng, Australian Natl Univ, Canberra, ACT 0200, Australia). Springer Verlag London Ltd, Surrey, UK. 2001. 168 pp ISBN 1-85233-371-5. $99.00.

Reviewed by FH Lutze (Dept of Aerospace and Ocean Eng, VPI, Blacksburg VA 24061-0203).

This book is one of many in the series Communications and Control Engineering, published by Springer Verlag. It was originally printed in Japanese in 1999. Those of you familiar with this series know that it is not for those faint at heart. This book is no exception to this norm. The material in the book is presented in a logical, well-thought-out progression of methods starting with dominant modes ideas developed prior to 1960, to more modern techniques which take advantage of the modern control theory developments of the 1980s and 1990s. The problem addressed is that of designing low order controllers for high order plants. Generally, using conventional controller design techniques, the controller has the same (or nearly the same) order as the plant. For practical reasons (hardware-software limitations, or expense), one would like to use a lower order controller to provide similar performance as the higher order controller. This book approaches this problem along two paths, 1) reducing a high order system to an approximating low order system and then designing the controller, and 2) designing a high order controller for the high order system and then reducing the controller. Dealing with the first approach depends on the definition of what is meant by “approximating” the higher order system, while dealing with the second approach requires retaining the closed-loop behavior, including stability, of the higher order controller using the reduced order controller. These problems, along with their nuances, are the subject of this monograph.

The book consists of four chapters. Loosely, the first two deal with system reduction methods and error approximations, and the latter two deal with the controller reduction. The titles of the chapters are, respectively, Methods for Model Reduction (54 pages), Multiplicative Approximation (29 pages), Low Order Controller Design (31 pages), and Model and Controller Reduction Based on Coprime Factorizations (30 pages). The ideas presented in the first chapter are essentially the foundations of much (but not all) of the material in subsequent chapters. Hence it is useful to list some of these topics as indicated by the titles of the subsections: Model reduction by truncation, Singular perturbation, Reduced order models based on balanced realization truncation, Methods involving minimizing an approximation error norm, Hankel norm approximations, and Markov and covariance parameter matching.

The second chapter defines the multiplicative approximation problem and its treatment using balanced stochastic truncation. The third chapter deals with the problems of reduced order controller design and the importance of including plant information in the developments. To this end, the topics include, Controller and plant reduction via frequency weighted approximation, Frequency weighted balanced truncation, Frequency weighted Hankel norm reduction, Frequency weighted reduction using partial fractions, and Multicative approximation with frequency weighting. In addition, at the end of each chapter are examples showing the comparison of the results of applying the different methods presented in the chapter. These examples present figures showing the amplitude and phase angle characteristics of each system as they depend on frequency.

This book is written primarily for the practitioner in the field. It has no exercises for the student or worked problems. The material is presented, for the most part, in a concise fashion with details left to the references, of which there are 90. This “concise fashion” requires the reader to be current in the modern methods of control theory and in some cases makes the reading difficult. Occasionally when new material is presented, a theorem-proof scenario is used. In several of the sections, one can directly code the presented results to produce software to attack the problem at hand. In other sections, reducing the results to an algorithm would be a non-trivial exercise. At the end of each section, there is a summary of that section under a heading, Main points of the section. These summaries allow the reader to step back and look at the big picture and are quite helpful.

Overall, Model Reduction for Control System Design presents an overview of the most recent methods for reducing the order of controllers for higher order systems. It gives several techniques, and presents results indicating how these techniques perform on a specified system. Translating the results presented into a computer code is non-trivial. However, the material is well documented. The practical implementation of feedback controllers for controlling high order systems requires such model and/or controller reduction. This book is the only place where one would find all this material in one place and certainly would be a good place to start if entering to this area of control implementation.