Book Reviews 1621 Identification of Linear Systems: A Practical Guideline to Accurate Modeling* J. Schoukens and R. Pintelon Reviewer: B. WAHLBERG Automatic Control, Royal Institute of Technology, S-100 44 Stockholm, Sweden. ADVANCED ENGINEERING applications need useful mathe- matical models. System identification (SI) deals with the problem of obtaining models of dynamical systems from measured data. Software packages have made SI an everyday tool for many engineers and virtually a myriad of successful applications have been reported in the literature. As indicated by Ljung (1991), although system identification has developed into a mature, practical tool, it is not dead as a research area. The reason is that progress and new application areas require increasingly advanced modeling concepts. System identification is often divided into non-parameteric and parameteric methods. As the reviewer was brought up in the Swedish school of SI, starting with the classical work of AstrGm and Bohlin (1965), he may tend to show a bias in favor of time domain parameteric SI methods; this should be taken into account when reading the review. The standard references Ljung and S/SderstrGm (1983), Ljung (1987) and SGderstr~im and Stoica (1989) thus act as the bible when reviewing new books in the area of SI. This is not completely fair when judging work that is motivated by application areas slightly outside the mainstream. It should be noted that the authors of the book are active in the area of instrumentation and measurements. The aim is to obtain dynamical models from noisy measurements, leading to physical interpretation. Con~quently, a frequency domain parametric approach is a natural choice. In order to relate the book to the current status of SI, ~me relevant research issues will first be discussed. The recent progress in robust control design has triggered frantic research activity around modeling of uncertainty in control systems; see e.g. the IEEE Trart~. Aut. Control Special Issue on System Identification for Robust Control Design, July 1992. A result of this effort is a renewed interest in frequency domain SI methods. Most of the frequency domain methods give non-parametric estimates. In Ljung and Glover (1981) frequency domain versus time domain methods in system identification are discussed. The idea of fitting parametric models to estimated frequency responses is classical. Recently, there has been extensive interest in such methods. For more details and relevant references see e.g. Parker and Bitmead (1987), LaMarie et al. (1991), Sidman et al. (1991) and Hemicki et al. (1991). The Empirical Transfer Function Estimate studied in Ljung (1985) gives insight into the connection between time domain and frequency domain SI methods. The same idea was used by Peter Whittle at the beginning of the 50s to derive the so-called Whittle estimator for time series analysis, see e.g. Chapter 6.2 in Hannan and Deistler (1988). The Whittle estimator can be derived by the maximum likelihood principle using statistical properties of Fourier coefficients. This is also the starting point for the main method of the book under review. The input and output signals are transformed into the frequency domain. A parameteric model of the system is then fitted to the corresponding spectral lines using a cost-function motivated by the maximum likelihood principle. The first two chapters of the book introduce system identification in general and the maximum likelihood * Identification of Linear Systems: A Practical Guideline to Accurate Modeling by J. Schoukens and R. Pintelon. Pergamon Press, Oxford, U.K. ISBN 0-08-040734-X. £55.00. estimator in particular. The presentation feels a bit "rough" compared with the standard books in the area. A fundamental problem in estimation theory is errors-in- variables. In a control framework this corresponds to the fact that both the input and the output measurements are corrupted by noise. A difficulty is the identifiability of such systems, see e.g. Andersson and Deistler (1984). This problem can only be solved under rather restrictive assumptions on the noise, the most simple one being that the errors can be modeled as white noise with known variances. In Chapter 3 of the book the so-called ELiS (Estimation of Linear Systems) method for transfer function estimation under these conditions is presented. As mentioned above, the idea is to apply the maximum likelihood principle to measured Fourier coefficients (with known variances). The result is a very flexible frequency domain method, which works for both discrete and continuous time systems. The method has many practical advantages and would be attractive for engineers working with, for example, frequency response analysers and modal analysis. The approach seems very suitable for identification of a resonant (poorly damped) system. It is also possible to take slight nonlinearities into account when designing the experimental conditions. From a statistical point of view some properties and design variables of the ELiS methods are less clear; for example, how to find (choose) the variances of the disturbances. Since a model of the noise is not estimated, the statistical efficiency of the method is, in general, not optimal. Better models can be obtained by more accurate modeling of the noise. System identification also includes experimental design and model validation. The book gives a rather complete answer on these issues from a practical engineering point of view. The freedom in designing the experiments may vary considerably with the application. In case the choice of input signal is not restricted and the experimental time is quite free, it is often a good idea, as is recommended in the book, to use periodical input signals and wait for the transients to die off before starting to collect the data. Combining this with the ELiS approach often gives good and reliable models. Finally, several sucessful applications of the approach taken are reported. These chapters are less pedagogical than the first part of the book and seem to be condensed versions of journal papers, which have been added afterwards. Sucessful application of experimental modeling does not depend on the use of only one system identification method. Instead one should try to use as many methods as possible, and then use model validation and aspects related to the intended application of the model to decide on the most appropriate estimate. This book presents one very flexible, practical and sound approach which should be complemented with traditional SI methods. I see this book as a complement, focusing on a specific method, rather than a basic treatment of system identification in general. The prime use of the book is perhaps not as a graduate text, instead it is more a research monograph. The book is very well written, and the timing is good due to renewed interest in the control community in frequency domain SI methods. It contains a lot of valuable experiences and insights in practical experimental modelling. Hopefully, the book also acts as an introduction of advanced system identification concepts to the instrumen- tation and measurements community. References Andersson, B. D. O. and M. Deistler (1984). ldentifiability in dynamic error-in-variables models. J. of Time Series Analysis, 5 1-13.