Inverse Eigenvalue Problems for Multivariable Linear Systems

Rosenthal, Joachim; Wang, X. A.

doi:10.1007/978-1-4612-4120-1_16

Cited by 8 publications

(5 citation statements)

References 37 publications

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“…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%

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

Kim¹,

Rosenthal²,

Wang³

2004

SIAM J. Control Optim.

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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

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Section: If One Defines Hsupporting

confidence: 55%

“…For the convenience of the reader we summarize the basic notions. More details can be found in [12,16] and [6,Chapter 14].…”

Section: N(2m+n)mentioning

confidence: 99%

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

Kim¹,

Rosenthal²,

Wang³

2004

SIAM J. Control Optim.

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“…is surjective. A dimension argument shows the necessity of mp ≥ n. This question has been studied intensively and we refer to the survey articles [4,14] and the recent papers [9,10,12,13,19,20] for details. We summarize some of the most important results.…”

Section: Preliminariesmentioning

confidence: 99%

Some remarks on real and complex output feedback

Rosenthal

Sottile

1998

Systems & Control Letters

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Abstract. We provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers. We exhibit a non-trivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real.

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“…The techniques which we use in this paper have been developed by the authors in the context of the additive inverse eigenvalue problem [2,10,13] and in the context of the pole placement problem [12].…”

Section: Det(λi − Mz) = P(λ)?mentioning

confidence: 99%

The Multiplicative Inverse Eigenvalue Problem over an Algebraically Closed Field

Rosenthal¹,

Wang²

2001

SIAM J. Matrix Anal. & Appl.

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The multiplicative inverse eigenvalue problem over an algebraically closed field Abstract Let M be an n × n square matrix and let $p(\lambda)$ be a monic polynomial of degree n. Let $\mathcal{Z}$ be a set of n × n matrices. The multiplicative inverse eigenvalue problem asks for the construction of a matrix $Z\in\mathcal{Z}$ such that the product matrix MZ has characteristic polynomial $p(\lambda)$. In this paper we provide new necessary and sufficient conditions when $\mathcal{Z}$ is an affine variety over an algebraically closed field. ©2001 Society for Industrial and Applied Mathematics Abstract. Let M be an n×n square matrix and let p(λ) be a monic polynomial of degree n. Let Z be a set of n × n matrices. The multiplicative inverse eigenvalue problem asks for the construction of a matrix Z ∈ Z such that the product matrix MZ has characteristic polynomial p(λ).In this paper we provide new necessary and sufficient conditions when Z is an affine variety over an algebraically closed field.

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Inverse Eigenvalue Problems for Multivariable Linear Systems

Cited by 8 publications

References 37 publications

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

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

Some remarks on real and complex output feedback

The Multiplicative Inverse Eigenvalue Problem over an Algebraically Closed Field

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