A norm-minimizing parametric algorithm for quadratic partial eigenvalue assignment via Sylvester equation

“…In case of the problems under consideration here a further computational challenge is to develop such gradient formulas using only a few eigenvalues and eigenvectors of the associated quadratic eigenvalue problem, since it is impossible to compute in practice the entire spectrum and the eigenvectors of a large quadratic matrix pencil even with the state-of-the-art computational techniques. In the present paper and in another recent one [5], meeting these challenges, (i) parametric expressions for feedback matrices via Sylvester equations have been derived, and (ii) appropriate gradient formulas both for both minimumnorm and robust eigenvalue assignment problems have been developed in terms of only a small number of eigenvalues that need to be reassigned and the associated eigenvectors, without reducing the order of the model.These techniques are, therefore, implementable in practice even for large-scale structures. However, some more work still needs to be done.…”

Section: Resultsmentioning

confidence: 99%

“…The following theorem, proved in [4] and [5] shows how to do this evaluation in terms of the known quantities Λ 1 , Λ 1 , X 1 , and B . Theorem 2 (Gradient Formula for I)…”

Section: Minimizing the Feedback Normsmentioning

confidence: 99%

“…A parametric Sylvester-equation approach was first developed in [3]for the complete pole assignment problem in the standard first-order state space form and using this, minimum-norm and robust pole assignment problems in first-order state space form have been solved in [6], [16], [21]. In a recent paper [5], the authors have generalized this to solve MNPQEVAP in quadratic setting, that is, without requiring any transformation to the standard first-order form. Note that computationally, such a transformation is not desirable, because it might need inversion of the ill-conditioned mass matrix and furthermore, the nice structures offered by a practical problem, such as the symmetry, sparsity, definiteness, etc., which are often assets for large-scale computations, are totally destroyed.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Only two papers that deals with the robustness of the PQEVAP have been published so far. They are: the paper by Qian and Xu [33] and the recent papers by Brahama and Datta [3,4]. The method proposed by Qian and Xu is not an optimization-based and has limitations.…”

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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A norm-minimizing parametric algorithm for quadratic partial eigenvalue assignment via Sylvester equation

Cited by 14 publications

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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

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

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

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