Computer-Aided Control System Design:

Theory, Algorithms and Software Tools

In this research
program funded by the Swedish Foundation for Strategic Research (SSF), we
develop theory, algorithms and software tools for matrix
pencil computations in *Computer-Aided Control System Design*
**
(**CACSD**)**.
This includes new and improved algorithms and software for computing
subspaces and canonical structure information and for solving important
matrix equations, as well as theory and software tools for handling
complicated nearness problems. The problems considered are motivated by
our collaboration within the thematic network NICONET **
(**for
numerics in control and systems theory**)**.

We consider descriptor (or generalized state-space) systems,
which, e.g., arise from modelling interconnected systems **
(**electrical
circuits**)**
and mechanical systems ** (**multi-body
contact problems**).
** Computing
staircase forms of the associated system pencil using orthogonal transformations
most reliably solves important ill-posed computational problems encountered
in the analysis and synthesis of descriptor systems. Examples include computation
of system poles at infinity with their multiplicity, solution of non-standard
Riccati equations, computation of rational inverses, and minimal descriptor
realization ** (**observability/controllability
forms).

We develop robust algorithms for computing canonical structures (Jordan,
Kronecker, Brunovsky) of matrix pencils. In several CACSD applications it
is important to know which are the nearby canonical structures that
explain the qualitative behavior of the system under small perturbations.
Our algorithms and software tools will be delivering such information
(qualitative as well as quantitative) based on the combinatorial
relationships among the various Jordan and Kronecker structures known as
strata. Building on our earlier and more recent contributions regarding
algorithms, software, applications and mathematical theory, we now aim at
solving these problems for both dense and sparse matrices and matrix
pencils, including small to medium-sized problems as well as large-scale
problems.
For
large-scale systems, the problems that we address are fundamental in
model reduction applications.

We also
develop algorithms for challenging problems in the design and analysis of
periodic systems, including the computation of periodic invariant
subspaces with specified spectra, the Kronecker structure of
high-dimensional structured matrix pencils, cycling products and quotient
products of square or rectangular matrices, and the solution of various
periodic matrix equations. Moreover, we develop algorithms for problems
relating to indefinite inner product spaces using hyperbolic
transformations.**.
**Such problems appear
in several signal processing and control applications.

An integral part is to develop
theory, algorithms and software for computing reliable error bounds based
on new perturbation results and condition estimators.**.
**Finally, in this project we develop state-of-the-art library
software and tools for high-performance computing systems.