Stratification of Matrix Pencils

To design a modern control system is a complex problem which requires high qualitative software. This software must be based on robust algorithms and numerical stable methods which both can provide quantitative as well as qualitative information. In this Licentiate Thesis, the focus is on the qualitative information. The aim is to grasp the underlying advanced mathematical theory and provide algorithms and tools for their implementation. Using a unifying terminology and notation, Paper I gives an introduction to stratification for orbits and bundles of matrices, matrix pencils, and system pencils with applications in systems and control. An extensive part of the paper is dedicated to the underlying theory and to introduce the reader to the subject. The theory is throughout the paper illustrated with several examples and the differences between the terminology from mathematics, systems and control theory, and numerical linear algebra are highlighted. The introduction includes a presentation of different canonical forms which reveal the system characteristics of the model under investigation. A stratification provides the qualitative information of which canonical structures of matrix (system) pencils are near each other in the sense of small perturbations. Fundamental concepts in systems and control, like controllability and observability of a system, are considered and it is shown how these system characteristics can be investigated with the use of the stratification theory. New results are presented in the form of the cover relations for controllability and observability pairs. Moreover, the permutation matrices which take a matrix pencil in the Kronecker canonical form to the corresponding system pencil in (generalized) Brunovsky canonical form are derived. Two novel algorithms for determining these two permutation matrices are provided. Paper II gives a short introduction to stratification of orbits and bundles of controllability and observability pairs. The underlying theory is introduced and it is shown how the results are used in the software tool StratiGraph.