Analysis, Stability, and Control

Springer-Verlag, New York (1999)

ISBN: 0387985131

Reviewing books across disciplines is a difficult task, in particular if the book itself spans more than one field. Having a critical look at Sastry's ``Nonlinear Systems'', a mathematics text for engineers, was quite an experience for me, a theoretical physicist. I had not expected such a density of presentation, such a rigour of the derivations and such a clarity of argument.

In the preface, the author mentions that he has taught the same material as first year graduate courses at MIT and Berkeley. He doesn't say exactly what subjects the students were specialising in, but the author himself is in a robotics department. This placement of the text seems correct to me, though I wished I had had such a course during my own graduate studies. It contains so many things that I should have known as a nonlinear scientist but, I confess, I didn't.

After motivating the nonlinear approach, the book reviews the necessary mathematical background in as far as it is likely to exceed standard curricula. We learn about vector fields, different types of ordinary differential equations, including discontinuities, smooth manifolds, also including boundaries, and a number of other things.

Input-output systems get much more attention than they do in physics or mathematics texts, and that makes a lot of sense in a book whose main issues include control theory. Lyapunov theory is developed and illustrated with several applications. The chapter on dynamical systems and bifurcations was of course much more familliar for me. After all this we are well prepared for the parts on control, including feedback control.

In its scope, the volume has a substantial overlap with what is generally presented in dynamical systems books, also including the chaos control parts. Of course, other issues are much less of interest here, like fractal geometry and information theory. It is even the more striking how little overlap there is in the cited material. Many protagonists of the recent progress in nonlinear dynamics are never mentioned, nor cited. Ott, Grebogi, and Yorke are actually given credit in the text for having introduced chaos control, first in the physics literature, but then explicit reference is made to a number of later articles by control people. Maybe Sastry felt that linking two disciplines, applied mathematics and engineering, is enough trouble for a single book.

Personally, I enjoyed reading about nonlinear control without having to cope with the laboratory slang often mistaken for expert terminology. Any non-obvious term that is used is also defined and most often motivated. As a physicist, I sometimes missed some more intuitive insight. I felt I had to work out for myself why what we just proved is really so exciting. For that reason, it is not quite the book I would give to a student as introductory reading, but there are plenty of those already. In summary, Sastry has written a highly recommendable book. It provides a wealth of precise information, is well structured and very readable.

`UK Nonlinear News` thanks Springer-Verlag for providing a
copy of this book for review.

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