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# Reviewing the closure hierarchy of orbits and bundles of system pencils and their canonical forms

Using a unifying terminology and notation an introduction to the theory of stratification for orbits and bundles of matrices, matrix pencils and system pencils with applications in systems and control is presented. Canonical forms of such orbits and bundles reveal the important system characteristics of the models under investigation. A stratification provides the qualitative information of which canonical structures are near each other in the sense of small perturbations. We discuss how fundamental concepts like controllability and observability of a system can be studied with the use of the stratification theory. Important results are presented in the form of the closure and cover relations for controllability and observability pairs. Furthermore, different canonical forms are considered from which we can derive the characteristics of a system. Specifically, we discuss how the Kronecker canonical form is related to the Brunovsky canonical form and its generalizations. Concepts and results are illustrated with several examples throughout the presentation.

### Keywords

Brunovsky canonical form, Jordan canonical form, Kronecker canonical form, Stratification, bundle, closure relations, cover relations, matrix pair., matrix pencil, orbit, state-space system, system pencil

### Authors

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Entry responsible: Stefan Johansson

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Page Responsible: Frank Drewes
2022-12-08