# Algorithmic aspects of algebraic system theory

- Algorithmische Aspekte der algebraischen Systemtheorie

Schindelar, Kristina; Zerz, Eva (Thesis advisor)

*Aachen : Publikationsserver der RWTH Aachen University (2010)*

Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2010

Abstract

The mathematical roots of system and control theory date back to the paper On Governors by J. C. Maxwell published 1868 in Proceedings of the Royal Society of London. The seminal work of R.E. Kalman established system theory as a mathematical discipline in the 1950s. The contribution of U. Oberst, which appeared in 1990, gives fundamental insight for algebraic system theory. A very important algebraic property of the signal space is realized to be highly copious for signals and systems there, namely the property of the signal space to be an injective cogenerator over the underlying operator ring. The goal of algebraic system theory is the structural analysis of dynamical systems using algebraic tools. These systems may arise from various practical problems settled for instance in a scientific, technical or economical area. Classically linear time-invariant systems with field coefficients are studied. In the recent past variations of these systems have proved to be worthy for extended studies. From the applied point of view, there is obviously the interest to consider corresponding generalizations. From the algebraic point of view, some particular settings are very interesting for further investigations since ring theory and homological algebra provide a deep insight. Beyond theoretical studies, the computer algebra machinery allows the enormous benefit of constructive analyses. This thesis elaborates both aspects,the theoretical and the computational, in parallel. It is organized as follows. Chapter 1 and Chapter 2 serve for an extended introduction. System theoretical aspects are provided in Chapter 1. Basic concepts and definitions are presented and furthermore the following chapters are motivated from the system theoretical point of view. Chapter 3 studies systems with coefficients in a finite ring, in contrast to the classical case. The general motivation for this framework stems mainly from communication theory. However, the extension leads to problems like zero-divisors and the principal ideal domain property is lost. Therefore concepts useful for coding fail to generalize straightforwardly. In the field case the so-called predictable degree property is useful for many areas of system theory, ranging from controller parameterization to minimal realizations of linear systems over fields. This property does not carry over directly to the ring case. The paper The predictable degree property and row reducedness for systems over a finite ring by M. Kuijper, R. Pinto, J. W. Polderman and P. Rocha establishes a new framework which allows the adoption of that classical result in a novel setting. By the tool of Gröbner bases these results are extended to a more general framework which additionally allows concrete calculations. For this purpose the notion of the so-called minimal Gröbner p-basis is established and the connection to known results is pointed out. Chapter 4 is focused on one-dimensional systems with time-varying rational coefficients. This leads to the non-commutative operator ring called rational Weyl algebra which is a principal ideal domain. Therefore the non-commutative analogon to the Smith form, the so-called Jacobson form, exists. This normal form can be used to obtain a decomposition into a controllable and an autonomous subsystem of the corresponding linear abstract system. Furthermore the order of the underlying ordinary differential equation system is obtained directly. But computational problems known from the commutative counterpart even increase due to the non-commutative structure, namely the explosive growth of the coefficients. A novel approach which can be applied in a completely fraction free framework is presented in this chapter. This approach shows first how to obtain a decoupled form. It should be stressed that this decoupled form may even be interesting by itself. Further we show how to obtain a normal form from the decoupled form. The implementation is realized as a library called jacobson.lib for the computer algebra system Singular::Plural. This implementation is compared with all implementations which are available to the best of our knowledge. A behavioral approach to linear exact modeling is formulated for one-dimensional systems with constant coefficients by J.C. Willems. This problem of system identification is extended to polynomial-exponential signals in a multi-dimensional time-varying model class in Chapter 5. These model classes are summarized in the so-called Ore algebras. The idea of this approach is to derive a model describing the observed data and containing as much information as possible. It turns out that the particular model classes yield a very precise description, as pointed out in the case of continuous systems. Two alternative possibilities to calculate the models will be presented, one of them working in a purely commutative framework.

Institutions

- Chair of Algebra and Number Theory [114710]
- Department of Mathematics [110000]

### Identifier

- URN: urn:nbn:de:hbz:82-opus-32462
- RWTH PUBLICATIONS: RWTH-CONV-112604