Nonlinear systems analysis. Reprint of the 2nd ed. of the 1993 original. (English) Zbl 1006.93001

Vidyasagar, M.

Classics in Applied Mathematics. 42. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). xvii, 498 p. (2002).

This is a reprint of the 1993 edition which was reviewed by A. Bacciotti (see Zbl 0759.93001). One should add to this review that the emphasis is on an open-loop approach, even if some sections deal with feedback stabilization (FS) and if feedback linearization (FL) is introduced at the end. This clarifies why the theory of ordinary differential equations is used so often. In fact, it constitutes a mathematical background for this book.
One finds however a chapter on differential geometric methods (where FL appears) added to the second edition, which replicates what can be found elsewhere, typically in A. Isidori’s monograph [Nonlinear control systems: An introduction (1985; Zbl 0569.93034)] and the ambiguities between coordinate free notations and representations in coordinates are preserved too.
The functional analytic perspective appears also especially in the treatment of input/output stability (where one finds the parts on FS).
The chapters have been reorganized in the second edition, and some sections on discrete systems have been added.
The treatment on absolute stability is probably the best to be found, and the book is recommendable.

Reviewer: A.Akutowicz (Berlin)

MSC:

93-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
93C15	Control/observation systems governed by ordinary differential equations
93C10	Nonlinear systems in control theory
34H05	Control problems involving ordinary differential equations
93D25	Input-output approaches in control theory
93D10	Popov-type stability of feedback systems
93D30	Lyapunov and storage functions
93B18	Linearizations
93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory

Keywords:

nonlinear systems; Lure’s problem; ODEs in control; Lyapunov function; open-loop; feedback stabilization; feedback linearization; input/output stability; discrete systems; absolute stability

Citations:

Zbl 0759.93001; Zbl 0569.93034