# Time-varying linear systems

Belayneh, Berhanu Bekele (Author); Zerz, Eva (Thesis advisor)

*Aachen / Publikationsserver der RWTH Aachen University (2007)* [Dissertation / PhD Thesis]

*Page(s): 178 S.*

Abstract

The main goal of this thesis is to examine linear dynamical systems, both in the continuous- and discrete-time case, with time-varying coefficients using module theory and homological methods and to generalize results, such as the realization problem and matrix fraction descriptions, from the time-invariant case to the time-varying case. After a short introduction to behavioral systems theory we briefly describe the mathematical background of the algebraic approach. We then study structural properties such as autonomy, controllability, and observability. A realization problem of time-varying systems specified by a transfer matrix is also addressed. It is shown that realizability is equivalent to properness and minimality is equivalent to both controllability and observability. The notions of coprimeness of a matrix fraction description and row-properness of a skew polynomial matrix are introduced and used to determine the size of a minimal realization. An algorithm to transfer a given skew polynomial matrix into row-proper form is given and it is used to compute one sided greatest common divisors and least common multiples.

### Identifier

- URN: urn:nbn:de:hbz:82-opus-19723
- REPORT NUMBER: RWTH-CONV-123984