Inhalt des Dokuments
Preprint 05-2003
Sensitivity of Computational Control Problems
Author(s) :
Nicholas J. Higham
,
Mihail Konstantinov
,
Volker Mehrmann
,
Petko Petkov
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 05-2003
MSC 2000
- 65F15 Eigenvalues, eigenvectors
-
93B40 Computational methods
Abstract :
It is well-known that many factors contribute to the accurate and
efficient numerical solution of mathematical problems such as
those arising in computational control system design. In simple
terms these are the arithmetic of the machine on which the
calculations are carried out, sensitivity (or conditioning) of the
mathematical model to small changes of the data and the numerical
stability of the algorithms. It happens quite often that these
concepts are confused. We define these concepts and demonstrate
some of the subtleties that often lead to confusion. In particular
we demonstrate with several examples what may happen when a
problem is modularized, i.e., split into subproblems for which
computational modules are available.
For three classical problems in computational control, pole
placement, linear quadratic control and optimal $H_\infty$
control, we then discuss the conditioning of the problems and
point out sources of difficulties. We give some ill-conditioned
examples for which even numerically stable methods fail.
We also stress the need for condition and error estimators that
supplement the numerical algorithm and inform the user about
potential or actual difficulties, and we explain what can be done
to avoid these difficulties.
Keywords :
Sensitivity and conditioning, numerical stability, machine arithmetic, pole placement, linear quadratic control,algebraic Riccati equation, $H_\infty$ control.
Notes :
This preprint has also appeared as
Preprint 424, 2003, Dept. of Math. Univ. Of Manchester,
url: http://www.maths.man.ac.uk/~nareports/narep424.pdf