Canonical forms and stratification of orbits and bundles of system pencils∗

Using a unifying terminology and notation an introduction to the the-ory of stratification for orbits and bundles of matrices, matrix pencilsand system pencils with applications in systems and control is presented.Canonical forms of such orbits and bundles reveal the important systemcharacteristics of the models under investigation. A stratification providesthe qualitative information of which canonical structures are near eachother in the sense of small perturbations. We discuss how fundamentalconcepts like controllability and observability of a system can be studiedwith the use of the stratification theory. New results are presented inthe form of the cover relations for controllability and observability pairs.Furthermore, different canonical forms are considered from which we canderive the characteristics of a system. Specifically, we discuss how theKronecker canonical form is related to the Brunovsky canonical form andits generalizations. Concepts and results are illustrated with several ex-amples throughout the presentation.