On perturbations and the equivalence orbit of a matrix pencil

Pokrzywa, Andrzej

doi:10.1016/0024-3795(86)90144-8

Cited by 42 publications

(20 citation statements)

References 5 publications

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“…For orbits of matrix pencils, the closure ordering was published by Pokrzywa in 1986 [15], later reformulated by De Hoyos [16] and Elmroth [17]. Recently, Gracia and De Hoyos presented safety neighbourhoods for the necessary conditions in the change of the Jordan and Kronecker canonical structures under small perturbations [18,19].…”

Section: Related Stratification Workmentioning

confidence: 97%

Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures

Elmroth

Johansson

Kågström

2001

Numerical Linear Algebra App

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SUMMARYStratiGraph, a Java-based tool for computation and presentation of closure hierarchies of Jordan and Kronecker structures is presented. The tool is based on recent theoretical results on stratiÿcations of orbits and bundles of matrices and matrix pencils. A stratiÿcation reveals the complete hierarchy of nearby structures, information critical for explaining the qualitative behaviour of linear systems under perturbations. StratiGraph facilitates the application of these theories and visualizes the resulting hierarchy as a graph. Nodes in the graph represent orbits or bundles of matrices or matrix pencils. Edges represent covering relations in the closure hierarchy. Given a Jordan or Kronecker structure, a user can obtain the complete information of nearby structures simply by mouse clicks on nodes of interest. This contribution gives an overview of the StratiGraph tool, presents its main functionalities and other features, and illustrates its use by sample applications.

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Section: Related Stratification Workmentioning

confidence: 97%

Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures

Elmroth

Johansson

Kågström

2001

Numerical Linear Algebra App

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“…For a given matrix A, den Boer and Thijsse [3] and, independently, Markus and Parilis [17] described the set of all Jordan canonical matrices such that for each J from this set there exists a matrix that is arbitrarily close to A and is similar to J. Their description was extended to Kronecker's canonical forms of pencils by Pokrzywa [18]. Edelman, Elmroth, and Kågström [7] developed a comprehensive theory of closure relations for similarity classes of matrices, for equivalence classes of matrix pencils, and for their bundles.…”

Section: Elamentioning

confidence: 99%

Change of the *-congruence canonical form of 2-by-2 matrices under perturbations

Futorny¹,

Klimenko²,

Sergeichuk³

2014

ELA

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“…The necessary conditions for an orbit or a bundle of two matrix pencils to be closest neighbours in a closure hierarchy were derived in [3,8,50], where the orbit is the manifold of strictly equivalent matrix pencils:…”

Section: Closure and Cover Relationsmentioning

confidence: 99%

Stratification of Controllability and Observability Pairs—Theory and Use in Applications

Elmroth¹,

Johansson²,

Kågström³

2009

SIAM J. Matrix Anal. & Appl.

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Abstract. Cover relations for orbits and bundles of controllability and observability pairs associated with linear time-invariant systems are derived. The cover relations are combinatorial rules acting on integer sequences, each representing a subset of the Jordan and singular Kronecker structures of the corresponding system pencil. By representing these integer sequences as coin piles, the derived stratification rules are expressed as minimal coin moves between and within these piles, which satisfy and preserve certain monotonicity properties. The stratification theory is illustrated with two examples from systems and control applications, a mechanical system consisting of a thin uniform platform supported at both ends by springs, and a linearized Boeing 747 model. For both examples, nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system pencils.Key words. Stratification, matrix pairs, controllability, observability, robustness, Kronecker structures, orbit, bundle, closure hierarchy, cover relations, StratiGraph.1. Introduction. Computing the canonical structure of a linear time-invariant (LTI) system,ẋ(t) = Ax(t) + Bu(t) with states x(t) and inputs u(t), is an ill-posed problem, i.e., small changes in the input data matrices A and B may drastically change the computed canonical structure of the associated system pencil A − λI B (e.g., see [13]). Besides knowing the canonical structure, it is equally important to be able to identify nearby canonical structures in order to explain the behavior and possibly determining the robustness of a state-space system under small perturbations. For example, a state-space system which is found to be controllable may be very close to an uncontrollable one, and can therefore by only a small change in some data, e.g., due to round-off or measurement errors, become uncontrollable. If the LTI system considered and all nearby systems in a given neighborhood are controllable, the system is called robustly controllable (e.g., see [46]).The qualitative information about nearby linear systems is revealed by the theory of stratification for the corresponding system pencil. A stratification shows which canonical structures are near to each other (in the sense of small perturbations) and their relation to other structures, i.e., the theory reveals the closure hierarchy of orbits and bundles of canonical structures. A cover relation guarantees that two canonical structures are nearest neighbours in the closure hierarchy.For square matrices, Arnold [1] examined nearby structures by small perturbations using versal deformations. For matrix pencils, Elmroth and Kågström [23] first investigated the set of 2-by-3 matrix pencils and later extended the theory, in collaboration with Edelman, to general matrices and matrix pencils [17,18]. In line of this work, the theory has further been developed in [21], and for matrix pairs in [20,42]. Several other people have worked on the theory of stratifications and similar topics, and we refer...

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On perturbations and the equivalence orbit of a matrix pencil

Cited by 42 publications

References 5 publications

Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures

Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures

Change of the *-congruence canonical form of 2-by-2 matrices under perturbations

Stratification of Controllability and Observability Pairs—Theory and Use in Applications

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