On Dynamic Feedback Compensation and Compactification of Systems

Rosenthal, Joachim

doi:10.1137/s036301299122133x

Cited by 49 publications

(35 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As a consequence the number of solutions in the critical dimension, i.e., in the situation where dim L = n, is equal to degL when counted with multiplicities and when some possible "infinite solutions" are taken into account. The results in section 2 generalize mathematical ideas which have been developed for the static pole placement problem by Brockett and Byrnes [2] and for the dynamic pole placement problem by Ravi, Rosenthal, and Wang [14], Rosenthal [15], and Rosenthal and Wang [16].…”

Section: If One Defines Hsupporting

confidence: 55%

“…Following [14,15,18] we identify a closed loop characteristic polynomial ϕ(s) with a point in P n . In analogy to the situation of the static pole placement problem considered in [2,18] (compare also with [15, section 5]) one has a well-defined characteristic map…”

Section: Compactification Of the Problemmentioning

confidence: 99%

“…If Z ∩L = ∅ and dim L = n, then one says that χ describes a finite morphism from the projective varietyL onto the projective space P n . In analogy to the situation of the static pole placement problem [2,18] and the dynamic pole placement problem [15] we introduce a definition for this important situation.…”

Section: Compactification Of the Problemmentioning

confidence: 99%

See 2 more Smart Citations

Pole Placement and Matrix Extension Problems: A Common Point of View

Kim¹,

Rosenthal²,

Wang³

2004

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

Abstract. This paper studies a general inverse eigenvalue problem which generalizes many wellstudied pole placement and matrix extension problems. It is shown that the problem corresponds geometrically to a so-called central projection from some projective variety. The degree of this variety represents the number of solutions the inverse problem has in the critical dimension. We are able to compute this degree in many instances, and we provide upper bounds in the general situation.Key words. pole placement and inverse eigenvalue problems, matrix completion problems, Grassmann varieties, degree of a projective variety AMS subject classifications. 14M15, 15A18, 93B55, 93B60

show abstract

Section: If One Defines Hsupporting

confidence: 55%

Section: Compactification Of the Problemmentioning

confidence: 99%

See 1 more Smart Citation

Pole Placement and Matrix Extension Problems: A Common Point of View

Kim¹,

Rosenthal²,

Wang³

2004

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…With this result it was then shown in [8] that the set of non-degenerate systems inside the quasi-projective variety of proper transfer functions contains a dense Zariski-open set as soon as n ≥ mp.…”

Section: Remarkmentioning

confidence: 90%

Pole Placement with Fields of Positive Characteristic

Gorla

Rosenthal

2010

Three Decades of Progress in Control Sciences

Self Cite

View full text Add to dashboard Cite

The pole placement problem belongs to the classical problems of linear systems theory. It is often assumed that the ground field is the real numbers R or the complex numbers C. The major result over the complex numbers derived in 1981 by Brockett and Byrnes states that arbitrary static pole placement is possible for a generic set of m-inputs, p-outputs and McMillan degree n system as soon as mp ≥ n. Moreover the number of solutions in the situation mp = n is an intersection number first computed by Hermann Schubert in the 19th century. In this paper we show that the same result with slightly different proofs holds over any algebraically closed field.

show abstract

“…We started this paper with a view of understanding the space X, considered as a compactification of the space of all m-input, p-output transfer functions of McMillan degree q ( [RR94], [Ros94]) in Systems theory. A substantial portion of this paper has been in circulation for some time under the title "Degree of the Generalized Plücker embedding of a Quot scheme".…”

mentioning

confidence: 99%

Degree of the generalized Pl�cker embedding of a quot scheme and quantum cohomology

Ravi¹,

Rosenthal

Wang

1998

Mathematische Annalen

View full text Add to dashboard Cite

We compute the degree of the generalized Plücker embedding κ of a Quot scheme X over IP 1 . The space X can also be considered as a compactification of the space of algebraic maps of a fixed degree from IP 1 to the Grassmanian Grass(m, n). Then the degree of the embedded variety κ(X) can be interpreted as an intersection product of pullbacks of cohomology classes from Grass(m, n) through the map ψ that evaluates a map from IP 1 at a point x ∈ IP 1 . We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomol-. We arrive at the degree by proving a version of the classical Pieri's formula on the variety X, using a cell decomposition of a space that lies in between X and κ(X).Let X ′ be the space of all algebraic maps, of a fixed degree q, from the projective line IP 1 to Grass(m, n), the Grassmannian of all m-dimensional subspaces of a fixed n-dimensional vector space V . In various papers, physicists have been discussing so called correlation functions on this space ([Int91] [Vaf92] [Wit93]) which specify the intersection products on X ′ of pullbacks of cohomology classes from the Grassmannian (a precise formulation can be found in Section 4). Further, based on certain physical arguments, there have been some conjectured formulas for these correlation functions (actually these conjectures deal with the more general case of maps from any Riemann surface to Grass(m, n), but we will restrict our attention to maps from IP 1 ). In [BDW93] the authors give a mathematically rigorous proof of the conjectured formula in the case of maps from a Riemann surface of genus one to Grass(2, n). In

show abstract

On Dynamic Feedback Compensation and Compactification of Systems

Cited by 49 publications

References 27 publications

Pole Placement and Matrix Extension Problems: A Common Point of View

Pole Placement and Matrix Extension Problems: A Common Point of View

Pole Placement with Fields of Positive Characteristic

Degree of the generalized Pl�cker embedding of a quot scheme and quantum cohomology

Contact Info

Product

Resources

About