An introduction to the ideas of algebraic geometry in the motivated context of system theory.
This describes this two volume work which has been specifically written to serve the needs of
researchers and students of systems control and applied mathematics. Without sacrificing
mathematical rigor the author makes the basic ideas of algebraic geometry accessible to
engineers and applied scientists. The emphasis is on constructive methods and clarity rather
than on abstraction. While familiarity with Part I is helpful it is not essential since a
considerable amount of relevant material is included here. Part I Scalar Linear Systems and
Affine Algebraic Geometry contains a clear presentation with an applied flavor of the core
ideas in the algebra-geometric treatment of scalar linear system theory. Part II extends the
theory to multivariable systems. After delineating limitations of the scalar theory through
carefully chosen examples the author introduces seven representations of a multivariable
linear system and establishes the major results of the underlying theory. Of key importance is
a clear detailed analysis of the structure of the space of linear systems including the full
set of equations defining the space. Key topics also covered are the Geometric Quotient Theorem
and a highly geometric analysis of both state and output feedback. Prerequisites are the basics
of linear algebra some simple topological notions the elementary properties of groups rings
and fields and a basic course in linear systems. Exercises which are an integral part of the
exposition throughout combined with an index and extensive bibliography of related literature
make this a valuable classroom tool or good self-study resource. The present softcover reprint
is designed to make this classic textbook available to a wider audience. The exposition is
extremely clear. In order to motivate the general theory the author presents a number of
examples of two or three input- two-output systems in detail. I highly recommend this
excellent book to all those interested in the interplay between control theory and algebraic
geometry. -Publicationes Mathematicae Debrecen This book is the multivariable counterpart of
Methods of Algebraic Geometry in Control Theory Part I.... In the first volume the simpler
single-input-single-output time-invariant linear systems were considered and the corresponding
simpler affine algebraic geometry was used as the required prerequisite. Obviously
multivariable systems are more difficult and consequently the algebraic results are deeper and
less transparent but essential in the understanding of linear control theory.... Each chapter
contains illustrative examples throughout and terminates with some exercises for further study.
-Mathematical Reviews
