Eigenstructure of polynomial and rational matrices: perturbation theory and nearness problems

Polynomial and rational matrices are important algebraic objects that have many engineering applications, most notably in control theory. As with more classical matrices over the field of real or complex numbers, many of the properties of polynomial and rational matrices can be described in terms of their eigenstructure. This thesis considers questions related to the eigenstructure of polynomial and rational matrices. In particular, we study how variations in the entries affect the eigenstructure, and vice versa. These questions generally fall under either perturbation theory or matrix nearness problems, and are important in, for example, understanding how much error there can be in a numerical computation of the eigenstructure. This thesis begins with an overview on the theory of polynomial and rational matrices and their eigenstructures and continues by providing new scientific results in the form of four attached articles. In the first article, we derive new results for the conditioning of zeros of rational matrices. This is achieved by using the transfer function representation for rational matrices together with Rosenbrock's theorem, which relates the eigenstructure of a rational matrix with that of the associated polynomial system matrix. These results have significance in control theory, where rational matrices appear as transfer function matrices in a natural way. The remaining three articles deal with nearness problems associated with the eigenstructure of polynomial matrices. These include the problem of finding the distance to singularity and the problem of finding the distance to stability. For these problems, we use a novel approach that relies on Riemannian optimization. This approach leads to computational algorithms that often outperform the competitors in terms of the running time as well as the quality of the output.

Department dc:contributor.department

Matematiikan ja systeemianalyysin laitos

Grantor dc:publisher

Aalto University

Year dc:date.issued

2025

Author dc:creator

• Nyman, Lauri

Advisor dc:contributor.advisor

• Noferini, Vanni, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland

Contributors dc:contributor

• Aalto-yliopisto
• Aalto University

Language dc:language.iso

en

Repository record dc:identifier.uri

https://aaltodoc.aalto.fi/handle/123456789/135593

OAI record oai:identifier

https://aaltodoc.aalto.fi/server/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Aaaltodoc.aalto.fi%3A123456789%2F135593