Refine
Year of publication
- 2005 (6) (remove)
Language
- English (6) (remove)
Keywords
- controllability (1)
- descriptor system (1)
- model reduction (1)
- observability (1)
- optimal control (1)
- stability (1)
Project
- C4 (2)
Application Area
- C (2)
Descriptor systems present a general
mathematical framework for the modelling, simulation and control of complex dynamical systems arising in many areas of mechanical,
electrical and chemical engineering. This
paper presents a survey of the current theory of descriptor systems,concerning
solvability, stability, model reduction, controllability, observability and optimal control.
We present structure preserving algorithms for the numerical com-
putation of structured staircase forms of skew-symmetric/symmetric
matrix pencils along with the Kronecker indices of the associated skew-
symmetric/symmetric Kronecker-like canonical form. These methods
allow deflation of the singular structure and deflation of infinite eigenvalues with index greater than one. Two algorithms are proposed: one
for general skew-symmetric/symmetric pencils and one for pencils in
0
which the skew-symmetric matrix is a direct sum of 0 and J = −I I .
0
We show how to use the structured staircase form to solve boundary
value problems arising in control applications and present numerical
examples.
We present the mathematical theory of general over- and underdetermined
hybrid (switched) systems of differential-algebraic equations
(HDAEs). We give a systematic formulation of HDAEs and discuss existence
and uniqueness of solutions, the numerical computation of the switch
points and how to perform consistent initialization at switch points. We
show how numerical solution methods for DAEs can be adapted for HDAEs
and present a comparison of these methods for the real world example of
simulating an automatic gearbox.
The classical approach to investigating polynomial eigenvalue problems is linearization, where the
polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely
many linearizations with widely varying properties, but in practice the companion forms are typically used. However,
these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes
be required.
In this paper we develop a systematic approach to generating large classes of linearizations for matrix polynomials.
Given a polynomial P, we show how to simply construct two vector spaces of pencils that generalize the companion
forms of P, and prove that almost all of these pencils are linearizations for P. Eigenvectors of these pencils are
shown to be closely related to those of P. A distinguished subspace is then isolated, and the special properties of
these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured
linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations, issues to be
addressed in further work.
Palindromic polynomial eigenvalue problems and related classes of structured eigenvalue problems are
considered. These structures generalize the concepts of symplectic and Hamiltonian matrices to matrix polynomials.
We discuss several applications where these matrix polynomials arise, and show how linearizations can be derived that
re
ect the structure of all these structured matrix polynomials and therefore preserve symmetries in the spectrum.
We discuss the state of the art in numerical solution methods for large scale polynomial or
rational eigenvalue problems. We present the currently available solution methods such as
the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties.
We briefly introduce a new linearization technique and demonstrate how it can be used to
improve structure preservation and with this the accuracy and efficiency of linearization based
methods. We present several recent applications where structured and unstructured nonlinear
eigenvalue problems arise and some numerical results.