Many devices (we say dynamical systems or simply systems) behave like
black boxes: they receive an input, this input is transformed following
some laws (usually a differential equation) and an output is observed.
The problem is to regulate the input in order to control the output,
that is for obtaining a desired output. Such a mechanism, where the
input is modified according to the output measured, is called feedback.
The study and design of such automatic processes is called control
theory. As we will see, the term system embraces any device and control
theory has a wide variety of applications in the real world. Control
theory is an interdisci- plinary domain at the junction of differential
and difference equations, system theory and statistics. Moreover, the
solution of a control problem involves many topics of numerical analysis
and leads to many interesting computational problems: linear algebra
(QR, SVD, projections, Schur complement, structured matrices,
localization of eigenvalues, computation of the rank, Jordan normal
form, Sylvester and other equations, systems of linear equations,
regulariza- tion, etc), root localization for polynomials, inversion of
the Laplace transform, computation of the matrix exponential,
approximation theory (orthogonal poly- nomials, Pad6 approximation,
continued fractions and linear fractional transfor- mations),
optimization, least squares, dynamic programming, etc. So, control
theory is also a. good excuse for presenting various (sometimes
unrelated) issues of numerical analysis and the procedures for their
solution. This book is not a book on control.
