Matrix equations and the separation of matrix eigenvalues

Howland, James Lucien

doi:10.1016/0022-247x(71)90088-6

Cited by 24 publications

(9 citation statements)

References 3 publications

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“…In that sense, this could be seen as a special case of [17]. However, although [17, Theorem 1] seems suitable to prove the first statement in theorem 3.1, we do not agree with the proof of [17, Theorem 2] related to the inertia of P .…”

Section: Discussioncontrasting

confidence: 68%

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Robust root-clustering of a matrix in intersections or unions of regions: Addendum

Bachelier

Henrion²,

Pradin

et al.

Proceedings of the 44th IEEE Conference on Decision and Control

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This paper considers robust stability analysis for a matrix affected by LFT-based complex uncertainty (LFT for linear fractional transformation). A method is proposed to compute a bound on the amount of uncertainty ensuring robust root-clustering in a combination (intersection and/or union) of several possibly nonsymmetric half planes, discs, and exteriors of discs. In some cases to be detailed, this bound is not conservative. The conditions are expressed in terms of (linear matrix inequalities) LMIs and derived through Lyapunov's second method. As a distinctive feature of the approach, the Lyapunov matrices proving robust root-clustering (one per subregion) are not necessarily positive definite, but have prescribed inertias depending on the number of roots in the corresponding subregions. As a special case, when rootclustering in a single half plane, disc or exterior of a disc is concerned, the whole clustering region reduces to only one convex subregion and the corresponding unique Lyapunov matrix has to be positive definite as usual. The extension to polytopic LFT-based uncertainty is also addressed.

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Section: Discussioncontrasting

confidence: 68%

“…The proofs basically require simple algebraic manipulations that might help extension to other curves. In that sense, it is closer to the result of [17]. Nevertheless, some differences are pointed out in the forthcoming discussion.…”

Section: Root-distribution and The Inertia Of Psupporting

confidence: 47%

Robust root-clustering of a matrix in intersections or unions of regions: Addendum

Bachelier

Henrion²,

Pradin

et al.

Proceedings of the 44th IEEE Conference on Decision and Control

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“…Transformability of D ensures the existence of a unique and non-singular solution P. This conjecture is not available for DˆH described in (10) with a < b. The arbitrary choice of Q cannot guarantee the existence of a single and non-singular solution P. The reader may ® nd more details about those problems of existence in the papers by Howland (1971), Ahn (1974) and Gutman and Jury (1981). &…”

Section: Conjecture Of Gutman and Jurymentioning

confidence: 95%

“…To enable this choice of Q, a restriction to transformable regions is assumed. Further details about that problem can be found in the work of Howland (1971), Jury and Ahn (1974) and Gutman and Jury (1981).…”

Section: Lemma 1: Given a 2 N£n And D 2 P¸ A Is D-stable If And Onlymentioning

confidence: 98%

Robust matrix root-clustering analysis in a union of ?-subregions

Bachelier¹,

Pradin²

2001

International Journal of Systems Science

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This paper addresses the research of robust D-stability bounds. A matrix is D-stable when its eigenvalues lie in a speciWed region D of the complex plane. Such a property, which is easily testable for some regions, is of practical interest for linear systems analysis in terms of pole location. Ensuring the state matrix D-stability can guarantee some performances on the transient response of this system, but the matrix D-stability is not testable any longer when this matrix is subject to an additive uncertainty. A bound on the uncertainty domain can be computed. This bound is established either on the 2-norm of the uncertainty matrix in the case of an unstructured uncertainty or on the maximal perturbation in the entries of the matrix in the case of a structured uncertainty. It guarantees that the D-stability of the uncertain matrix is preserved when the nominal matrix is already D-stable. Most of the results in this area of work are relevant to connected regions. In this work, because of a linear matrix inequality approach, D is a union of possibly disjoint O-subregions.

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“…It must be noted that T includes half planes, circles, and hyperbolas but not ellipses and parabolas. In [3], Howland obtained the matrix equation for a more generalized region than F. However, he showed only that for a particular positive definite hermitian Q, there exists a positive definite hermitian H (see Eq. (2)).…”

Section: Introductionmentioning

confidence: 99%

Remarks on the root-clustering of a polynomial in a certain region in the complex plane

Jury¹,

Ahn²

1974

Quart. Appl. Math.

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Abstract.A general formulation for the root clustering of a polynomial is given. An attempt has been made to answer an open question raised by Kalman. Introduction.Since 1852, when Hermite first established the connection between the number of the roots of a polynomial in an arbitrary half-plane and the signature of a certain quadratic form, the root-clustering problem has been investigated by many mathematicians, physicists and engineers. Recently Kalman [1] showed a general formulation which includes all the previously known results as particular cases. His formulation applies to the region r in the complex plane given by

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Matrix equations and the separation of matrix eigenvalues

Cited by 24 publications

References 3 publications

Robust root-clustering of a matrix in intersections or unions of regions: Addendum

Robust root-clustering of a matrix in intersections or unions of regions: Addendum

Robust matrix root-clustering analysis in a union of ?-subregions

Remarks on the root-clustering of a polynomial in a certain region in the complex plane

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