Mathematical systems theory (third edition), G. J. Olsder and J. W. van der Woude, VSSD, Delft, The Netherlands, 2005, 208pp., ISBN 90‐71301‐40‐0

Abstract

The book Mathematical Systems Theory by G. J. Olsder and J. W. van der Woude is essentially based on courses given by the authors at Delft University during more than 20 years at an undergraduate level. This is the third edition of these lecture notes which is now available. As a result it is a very good reference book, which reflects the feedback of students and benefits from a number of improvements and corrections compared to the previous editions. Of course, the content of this book, mainly centred on the theory of linear time invariant systems, leads to define it as a classical book. This also leads to put it in a category where a huge amount of literature is already available. The main question is then inevitably what is the usefulness of such a new reference for the automatic control community? The book is short, about 200 pages, which is in my opinion a major quality for a text book. Although presented as a mathematical book, it does not contain long and technical preliminaries in which we usually do not know what is really important to be understood. Here, the necessary mathematical tools are introduced just when needed and with just the level of abstraction which is needed. The book contains a chapter on modelling principles, with a lot of classical examples from electrical and mechanical engineering and also some applications in other fields. These examples and some others appear further in the book to illustrate new concepts or new results. Each chapter is ended by an important set of exercises which a good mix between direct numerical application of the results of the chapter and questions with a theoretical flavour in the style ‘Show that…’. These two types of exercises are important in the complete understanding of a topic, but they are not so often encountered together in classical references. In the same spirit a chapter is devoted to some MATLAB exercises with solutions. A brief description of the content of the book follows. The book begins by a short introduction chapter which introduces the ideas of system theory and control and a brief history. The Chapter 2 is dedicated to modelling principles with application in various fields. Chapters 3–5 treat in detail (with, in general, the proof of the theorems) the standard aspects of linear systems in state-space representation. Chapter 3 which considers the solution of the state equation begins by some necessary considerations on the linearization problem. Chapter 4 deals with the classical properties of linear systems: stability, controllability and observability but with some attempt to see them with a ‘modern’ or geometric point of view, in particular by using the notion of A-invariant subspace. Chapter 5 is devoted to state and output feedback, pole placement until the separation theorem. Chapter 6 is on the transfer matrix approach, it contains the main results on this point of view but is treated with less details than the previous chapters. Chapter 7 presents rapidly the discrete-time state-space models. Chapter 8 which is called ‘Extensions and some related topics’ proposes some further directions of studies as the behavioural models, polynomial matrix representations, nonlinear, distributed parameter systems and discrete event systems. Some pages are also devoted to optimal control and filtering. This book is mainly written for an undergraduate course on system theory for a department of mathematics, but is also suitable for an electrical engineering audience. Both types of students will find in it familiar subjects and ways of thinking and also more exotic aspects and interests. This book may also be useful for graduate students and researchers whose main field is not in control (say from mechanics, chemistry, biology, etc.). They will find in it a rapid overview of the classical results in control theory, without too difficult mathematics but with a real rigour of presentation. Compared with well-known and excellent references 1-3 it is certainly much less encyclopedic. Although the titles are very similar it is far from the level of mathematical abstraction which may be found in Reference 1 or even in Reference 4. For this book the usual assertion ‘it is readable with a basic knowledge in linear algebra and differential equations’ is really true. It is also certainly much less engineering oriented as References 2, 3. In conclusion, this book may find a very useful place in the literature on linear system theory because of its qualities of presentation and writing, its concision and its good trade-off between rigour and accessibility. It can be considered as an appealing entrance in system theory which will encourage students, and also new researchers in the field, to pursue by the study of more complete books as, for example, those cited before.