A smooth compactification of the space of transfer functions with fixed McMillan degree

Ravi, M.S.; Rosenthal, Joachim

doi:10.1007/bf00998684

Cited by 52 publications

(43 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…(See, e.g., the survey article [17] or [25] [4], [8], [11], [12], [20], [26], [29])considered the problem of compactifying the space Sp,,. In this section we will describe a compactification of the space Sp,m, which turns out to be suitable for the study of dynamic feedback compensation.…”

mentioning

confidence: 99%

On Dynamic Feedback Compensation and Compactification of Systems

Rosenthal¹

1994

SIAM J. Control Optim.

View full text Add to dashboard Cite

This paper introduces a compactification of the space of proper $p \times m$ transfer functions with a fixed McMillan degree $n$. Algebraically, this compactification has the structure of a projective variety and each point of this variety can be given an interpretation as a certain autoregressive system in the sense of Willems. It is shown that the pole placement map with dynamic compensators turns out to be a central projection from this compactification to the space of closed-loop polynomials. Using this geometric point of view, necessary and sufficient conditions are given when a strictly proper or proper system can be generically pole assigned by a complex dynamic compensator of McMillan degree $q$. ©1994 Society for Industrial and Applied Mathematics SIAM J. CONTROL AND OPTIMIZATION Vol. 32, No. 1, pp. 279-296, January 1994 Abstract. This paper introduces a compactification of the space of proper p x m transfer functions with a fixed McMillan degree n. Algebraically, this compactification has the structure of a projective variety and each point of this variety can be given an interpretation as a certain autoregressive system in the sense of Willems. It is shown that the pole placement map with dynamic compensators turns out to be a central projection from this compactification to the space of closedloop polynomials. Using this geometric point of view, necessary and sufficient conditions are given when a strictly proper or proper system can be generically pole assigned by a complex dynamic compensator of McMillan degree q.

show abstract

mentioning

confidence: 99%

On Dynamic Feedback Compensation and Compactification of Systems

Rosenthal¹

1994

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…This work was revived by Rosenthal, who introduced the quantum Grassmannian into systems theory in his 1990 PhD thesis [43]. See [39] for a discussion and further references.…”

Section: Dynamic Control Of Linear Systemsmentioning

confidence: 99%

Rational curves on Grassmannians: systems theory, reality, and transversality

Sottile¹

2001

Advances in Algebraic Geometry Motivated by Physics

View full text Add to dashboard Cite

Abstract. We discuss a particular problem of enumerating rational curves on a Grassmannian from several perspectives, including systems theory, real enumerative geometry, and symbolic computation. We also present a new transversality result, showing this problem is enumerative in all characteristics.While it is well-known how this enumerative problem arose in mathematical physics and also its importance to the development of quantum cohomology, it is less known how it arose independently in mathematical systems theory. We describe this second story.

show abstract

“…More concretely, a point in Q q may be (non-uniquely) represented by a p × nmatrix M of forms in s, t with homogeneous rows and whose maximal minors have degree q [10]. The image of such a point under the Plücker map is the collection License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use…”

Section: The Quantum Grassmannianmentioning

confidence: 99%

Real rational curves in Grassmannians

Sottile

1999

J. Amer. Math. Soc.

View full text Add to dashboard Cite

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real. Similarly, given any problem of enumerating p-planes incident on some general fixed subspaces, there are real fixed subspaces such that each of the (finitely many) incident p-planes are real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions be real.

show abstract

A smooth compactification of the space of transfer functions with fixed McMillan degree

Cited by 52 publications

References 10 publications

On Dynamic Feedback Compensation and Compactification of Systems

On Dynamic Feedback Compensation and Compactification of Systems

Rational curves on Grassmannians: systems theory, reality, and transversality

Real rational curves in Grassmannians

Contact Info

Product

Resources

About