This is an excellent handbook on the behavioral approach to modeling linear systems. To the uninitiated, a behavioral approach may sound like a social science approach to mathematics. While clearly the applications of linear systems can include social science applications, the term used here follows from a fundamental group of papers by J.C. Willems [1,2,3]. In that series, Willems provides a thorough approach for deriving mathematical models, termed system identification.

In 1995, Roorda and Heij [4] incorporated global total least squares within this framework. Roorda [5] followed up with concrete algorithms to solve a system of these equations. The authors of this book feel this approach did not make its way into mainstream mathematical practice, despite its rigorous basis for deriving models and solving problems. They demonstrate that there are advantages as well; the behavioral approach can handle both static and dynamic models and both linear and nonlinear systems. It is for these reasons that this book was written.

The book is based on the doctoral thesis of the first author, Ivan Markovsky. The text contains 12 chapters and two appendices, covering a variety of models and approximate and exact identification systems. Most of the examples presented in the book are formulated in a deterministic setting, although it would not be hard to apply a stochastic interpretation to these problems. Most of the chapters (except two) deal with total least squares. The approach of this book differs from the classic latency approach, which includes unobserved latent variables. The authors term the unified framework of this book “the misfit approach.”

The book begins in chapter 1 with an introduction to system modeling and the differences between the misfit and latency approaches. Classical versus behavioral methodologies are discussed with data-fitting examples provided. Distinctions are made between deterministic and stochastic views of a given problem. Chapter 2 continues with the introduction and concentrates on approximations through misfit minimizations. Issues regarding model representation and parameterization are presented. Nonlinear static models are fitted with ellipsoids and structured total least squares involving a block-Hankel structured matrix.

The subsequent chapters are viewed as the main parts of the book. Part 1 comprises four chapters and deals with static problems. Chapter 3 presents weighted total least squares, and chapter 4 deals with structured total least squares. Chapters 5 and 6 concentrate on errors in variables. Bilinear models are estimated by fundamental matrices (chapter 5), and ellipsoid modeling (chapter 6) includes algorithms for adjusted least squares estimation. Part 2 is dedicated to dynamic problems, and consists of five chapters. Chapter 7 introduces dynamical models. Exact identification algorithms are presented in chapter 8, balanced model identification algorithms (basic and alternative) are found in chapter 9, and chapter 11 considers approximate system identification problems. Smoothing and filtering techniques are simulated in chapter 10. Conclusions are found in chapter 12, followed by appendices discussing advanced proofs and MATLAB software.

The text is a thorough presentation of problems, their representations, and suggested solutions using total least squares-based approaches. The analysis of the book assumes an undergraduate level of linear algebra and systems theory. Prior knowledge of system identification modeling will facilitate reading it. The authors delegate the topics of statistical and numerical robustness of the algorithms presented to future work, and indicate that these subjects are not currently well developed within the context of misfit approaches. The book would greatly benefit from an accompanying CD-ROM containing MATLAB files for all algorithms included in the text. The authors have published an extensive text on the behavioral approach to the exact and approximate modeling of linear systems.