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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 1, JANUARY 2004
Differential Neural Networks for Robust Nonlinear Control—A. S. Pozniak, E. N. Sanchez, and W. Yu (Singapore: World Scientific Ltd., 2001, Hardcover, 456 pp., ISBN 981-02-4624-2) Reviewed by F. Camastra
prehension. Appendix A deals with some useful mathematical facts; Appendix B contains the basis required to understand the Lyapunov-like approach used to derive the results in the book; in Appendix C are discussed some definitions and properties concerning to the locally optimization technique required to derive the optimal control law.
Neural networks have stimulated the interest of more and more scientists and engineers who have to cope with the control of nonlinear systems. The appeal is based on theoretical capabilities of neural networks to approximate arbitrary well continuous functions in compact sets. The books devoted to the control by neural networks are few. Therefore the arrival of new books devoted to neural networks for control has to be considered welcome. The book, object of the review, covers a particularly specific branch of neural networks for control that is Dynamic (Differential in authors’ terminology) Neural Networks applied to nonlinear robust control, namely to a high stable control. This review is organized as follows: in the Overview the contents of the book are examined; in the Conclusion some of its important features are discussed. Overview The authors discuss in the book the application of dynamic neural networks for identification, state estimation of nonlinear systems and present their theoretical results on these topics. In the authors’ aim, "the book is intended to familiarize the reader with the new field of the dynamic neural networks applications for robust nonlinear control". The authors intend the book as a textbook for graduate students or as an handbook for pratictioner engineers who could use it for updating their professional skills. The potential reader of the book should have a background in differential equations, nonlinear systems analysis and optimization techniques. Besides, it is recommended a familiarity with Lyapunov approach to the analysis of nonlinear systems. The book discusses two main control problems: the construction of the identifier (or state estimator) of a nonlinear system and of its controller. Regarding to the identifier, a dynamic neural network is used to build a model of the plant. Two possible cases exist. In the first case, the dimension of the neural network state is the same of the nonlinear system, therefore, the neural network is a system identifier. In the second case, the neural network allows to estimate the system state by a neural observer implementation. Regarding to the controller, after the implementation of the neural identifier (or observer), a local optimal control law is developed. The book can be divided in four principal parts: 1) An introductory chapter (the first chapter) on the basic facts about neural networks. 2) A part, from the second chapter to the fourth chapter, related to the neural identification and estimation. 3) A part on the passivation and the neurocontrol (fifth and sixth chapters). 4) Last part is devoted to the application of the previously presented techniques to the solution of applicative problems such as the control of chaotic systems (seventh chapter), the neurocontrol for a robot manipulator (eighth chapter), the identification of a multicomponent nonstationary ozonization process (ninth chapter) and the neurocontrol for a multicomponent distillation column (tenth chapter). Three appendices conclude the book containing some auxiliary mathematical result that could help the reader in the book comThe reviewer is with INFM-DISI, University of Genova, Genova 16146, Italy (e-mail:
[email protected]). Digital Object Identifier 10.1109/TNN.2004.823449
Conclusion The most striking feature of the book is the clarity. The authors explain very well notions such as observability and system identification, that can be not familiar to neural network researchers unless they have a solid knowledge in control theory. Each chapter of the book is provided with several examples that illustrate the theory previously presented and by a rich bibliography. Nevertheless the book presents some drawbacks. First, the book has some trivial spelling mistakes that could be eliminated by a careful editorial review. In the introductory part devoted to neural networks, powerful neural techniques for function approximation as Support Vector Machines (SVMs) [1] and Gaussian Processes (GPs) [2] are neither described nor mentioned. Since SVMs and GPs are more and more used for their effectiveness in function approximation problems, the authors’ choice is arguable. With regard to the seventh chapter, whose topic is the application of the neural control to chaotic systems, it is necessary to point out some remarks. Dynamics Reconstruction techniques [3], that allow to fix the order of dynamical neural network that implements the neuro-identifier, are completely ignored in the chapter. Moreover there are not mentioned the results obtained in nonlinear dynamics theory on controlling chaos. Therefore, it is not clear if the methodologies proposed in the book, compared with the techniques for controlling the chaos, developed in nonlinear dynamics, are an improvement. Finally, the potential reader could be lead to dangerous misunderstandings since there is omitted that chaotic system in some cases cannot be controlled, as pointed out in [4]. Despite the drawbacks previously mentioned, the book presents a lucid exposition of the application of dynamic neural network to the solution to classical problem such as Nonlinear System Identification and Nonlinear System Control. Moreover the authors devote a large part of the book to the discussion of a practical applications. Last feature makes the book particularly interesting for engineers that, in their professional daily practice, have to cope with control problems. In spite of authors’ aim, to design the work as a textbook, the lack of exercise sections, in each chapter, seems to discourage its usage as textbook for graduate students. On the contrary the book can be a useful monography for students, who can deepen their knowledge in the neural network for control, and for researchers and engineers interested to neurocontrol applications. In summary, despite a few shortcomings, the authors’ effort is commendable. ACKNOWLEDGMENT I wish to acknowledge D. Ugolini and A. Vinciarelli for commenting on the draft. REFERENCES [1] N. Cristianini and J. Shawe-Taylor, Introduction to Support Vector Machines. Cambridge, U.K.: Cambridge Press, 2000. [2] “Tech. Rep.,” Cambridge University, Cambridge, U.K., 1997. [3] E. Ott, Chaos in Dynamical Systems. Cambridge, U.K.: Cambridge University Press, 1993. [4] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett., vol. 64, no. , p. 1196, 1990.
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