Orthogonality and partial pole assignment for the symmetric definite quadratic pencil

Datta, Biswa Nath; Elhay, Sylvan; Ram, Yitshak M.

doi:10.1016/s0024-3795(96)00036-5

Cited by 158 publications

(93 citation statements)

References 9 publications

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“…The problems are taken from Refs. [2,24]. Table 1, Table 2, Table 3, Table 4, Table 5 contain, respectively, the results of Problem 1, Problem 2 (i), Problem 2(ii), Problem 2 (iii), and Problem 3.…”

Section: Results Of Numerical Experimentsmentioning

confidence: 99%

“…[20]), three orthogonality relations for the quadratic matrix pencil were derived by Datta, Elhay and Ram (see Ref. [2]). The relations were further modified by Datta and Sarkissian (see Ref.…”

Section: Orthogonality Properties Of the Eigenvectors Of A Quadratic mentioning

confidence: 99%

“…(1), are governed by the eigenvalues and eigenvectors of the quadratic matrix polynomial P (λ) (see Refs. [1][2][3]) where…”

Section: Introductionmentioning

confidence: 99%

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An optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures

Brahma

Datta

2009

Journal of Sound and Vibration

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Section: Results Of Numerical Experimentsmentioning

confidence: 99%

Section: Orthogonality Properties Of the Eigenvectors Of A Quadratic mentioning

confidence: 99%

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An optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures

Brahma

Datta

2009

Journal of Sound and Vibration

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“…To meet these practical engineering and computational difficulties, several ''direct, partial-modal and no spill-over'' methods for the PQEVAP have been developed in recent years [12][13][14][15]20,34,8].…”

Section: Q2mentioning

confidence: 99%

Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems: A new optimization approach

Bai

Datta

Wang

2010

Mechanical Systems and Signal Processing

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a b s t r a c tThe partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and eigenvectors unchanged (no spillover). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop eigenvector matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant ''no spill-over property'' of the closed-loop eigenvalues and eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.& 2009 Published by Elsevier Ltd.

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“…In most applications, the matrices M, K and D are symmetric, furthermore, M is positive definite and K positive semidefinite. The dynamics of the structures modeled by equation (1) are governed by the eigenvalues and eigenvectors of the quadratic matrix polynomial ( [12], [10]) P (λ) ≡ M λ 2 + Dλ + K. If M is non-singular, P (λ) has 2n eigenvalues which are the roots of the equation det(P (λ)) = 0. * The research of this author was supported by NSF Grant #DMS-0505784.…”

Section: Introductionmentioning

confidence: 99%

A sylvester-equation based parametric approach for minimum norm and robust Partial Quadratic Eigenvalue Assignment Problems

Brahma

Datta

2007

2007 Mediterranean Conference on Control &Amp; Automation

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Abstract-The Partial Quadratic Eigenvalue Assignment Problem is the problem of reassigning a small number of undesirable eigenvalues of a quadratic matrix pencil using feedback. The problem arises in controlling resonance in vibrating structures and also in stabilizing control systems. The solution involves the computation of a pair of feedback matrices. For practical effectiveness, the magnitudes of the feedback norms need to be reduced and the conditioning of the closed-loop eigenvalues needs to be improved.In this paper, we propose new optimization methods for solving these problems. An important practical aspect of these methods is that the gradient formulas needed to solve the underlying unconstrained optimization problems are computed using only a small number of eigenvalues and eigenvectors of the quadratic pencil, which are all that can be computed using the state-of-theart computational techniques. *

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Orthogonality and partial pole assignment for the symmetric definite quadratic pencil

Cited by 158 publications

References 9 publications

An optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures

An optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures

Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems: A new optimization approach

A sylvester-equation based parametric approach for minimum norm and robust Partial Quadratic Eigenvalue Assignment Problems

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