Book Review: Nonlinear Systems: The Parameter Analysis and Design

Books on nonlinear systems are always welcome, especially when they contain as much useful and interesting information as this one. It is particularly concerned with parameter mapping techniques for system analysis and synthesis. This technique originated in Vishnegradsky's famous study of Watt's flyball governor and was subsequently extended by Neimark, Meerov and Mitrovic. The parameter mapping technique is first explained with the aid ofa number of simple examples; the explanation is well supplemented by a pair of Appendices on basic theorems and computer applications. This generous use ofappendices is typical of the whole text; indeed rather more than one third of the whole book consists of a set of eight very useful appendices. This is followed by a description of the use of parameter mapping techniques for linear systems. The next chapter provides a helpful introduction to some approximate methods for the analysis of nonlinear systems, particularly the describing function method and the Krylov-Bogoliubov asymptotical method for the analysis of nonlinear oscillations. A minor error in the index and in the references to chapter 4 gives]. C. West's initials incorrectly as U. C. The absence ofreferences to several well-known British papers on the describing function method seems to show that the author has concentrated heavily on the Russian and American literature in his studies. Chapter 4 is concerned with the stability and sensitivity analyses of symmetrical oscillations and contains material on systems with two nonlinearities and a very brieflook at multivalued nonlinearities. This is followed by a chapter dealing with the use of the parameter mapping and describing function techniques for the analysis of asymmetrical nonlinear oscillations. The parameter plane method enables certain techniques developed by Popov and Grensted to be extended to high order (or at least highe1 than second order) systems; this is the subject of a chapter on transient oscillations which also contains some material on systems with two nonlinearities. Chapter 7 is concerned with the analysis of forced oscillations, and gives treatments of periodic solutions and jump phenomena. The final chapter deals with absolute stability analyses for nonlinear systems. It is mostly concerned with Liapanov stability and the Lure problem, and contains a discussion of Popov's method together with Yakubovitch's extension to forced systems. This chapter also contains a short discussion of counter-examples to Aizerman's conjecture. The text is very well printed with many excellent diagrams. There is a generous use of footnotes to clarify points in the main text, an excellent bibliography, a good index and an extensive set of problems (without answers). It should be of great interest to anyone concerned with engineering approaches to the design and analysis of nonlinear dynamical systems. A. G. J. MACFARLANE