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trol: A Neural Network Approach by G.P. Liu, Springer, New York, 2001, 210 pp., Euro 79.95, ISBN 1-85233-342-1. Reviewed by Victor M. Becerra, University of Reading, U.K. The field of neural networks is vast, with many different known network architectures and training algorithms. This monograph deals with the application of neural networks to nonlinear identification and control. The book focuses on radial basis functions of the Gaussian type and Volterra polynomial basis functions, with a chapter based on wavelet networks. It introduces neural networks and provides details of different architectures. It also provides background on learning and approximation theory, including some classical training algorithms such as backpropagation. One of the central neural network paradigms used in the book, which is presented quite convincingly by the author, is the variable-structure neural network, wherein the number of basis functions of the network can be increased or reduced over time to avoid overfitting or underfitting and considering the novelty of the observations. Techniques based on Lyapunov stability theory are used to guarantee the stability of estimation. The variable-structure networks based on radial basis functions are introduced in chapter 2, together with a method for sequential identification for continuous-time nonlinear systems. These networks are used later in the book to derive a continuous-time adaptive control scheme with guaranteed stability. The book also presents a recursive identification scheme for nonlinear discrete time systems based on Volterra polynomial basis function networks, with techniques for offline initial structure selection and online structural adaptation of the networks. In nonlinear system identification, multiple objectives usually need to be considered (for example, approximation accuracy and model complexity). These multiple objectives are often conflicting, so some tradeoff is inevitable. The book presents methods based on genetic algorithms and multiobjective optimization techniques for model selection and parameter estimation. In recent years, wavelet functions have become an important tool for localized function approximation at different scales, and they can be viewed as a basis for representing functions. Wavelet networks, which are inspired by feedforward networks, have found application in nonlinear system identification. The book introduces a wavelet network nonlinear identification scheme. Once again, techniques are provided to adapt the structure of the network over time and also to guarantee the stability of the learning algorithms. A nonlinear predictive control scheme based on affine neural network predictors is also presented. This scheme avoids the use of nonlinear programming techniques and therefore is easy to implement and computationally inexpensive, but loses appeal by not considering inequality constraints on inputs, states, or outputs. The neural network predictors are adjusted in structure and parameters using an algorithm that guarantees weight and estimation error convergence. However, the stability of the closed-loop system is not guaranteed. Variable-structure control based on sliding modes is a well-known nonlinear control technique that has attractive robustness properties. A variable-structure control technique based on neural-network-based affine predictors is introduced in the book. The technique involves adjustable weight parameters and network structure and exhibits guaranteed convergence of the weights and estimation errors. The book is the result of many years of research and publications by Dr. Liu. Overall, it is well written and presented and will be useful for researchers in the field of nonlinear identification and control. Methods are illustrated with relevant simulation examples, and the final chapter presents a simulated and experimental case study based on a combustion process wherein a network with sinusoidal basis functions is used as an output predictor. Unfortunately, the experimental case study did not involve the identification and control techniques presented in previous chapters, particularly those techniques based on variable-structure networks. The book would have benefited from a more careful writing of the introductory section of some of the chapters to avoid redundancies. Also unfortunate is that Lyapunov’s second stability theorem for continuous-time systems is presented for the first time in chapter 6, while Lyapunov’s second method is used previously, without introduction and for both continuousand discrete-time systems, in chapters 2, 3, and 5. Nevertheless, the book will be a good addition to the libraries of those interested in the subject.