#### Title

### Robust and minimum norm partial quadratic eigenvalue assignment problems : theory and computations

#### Publication Date

2006

#### Document Type

Dissertation/Thesis

#### First Advisor

Datta, Biswa Nath

#### Degree Name

Ph.D. (Doctor of Philosophy)

#### Legacy Department

Department of Mathematical Sciences

#### LCSH

Eigenvalues

#### Abstract

The partial eigenvalue assignment problem for a second-order control system, called the partial quadratic eigenvalue assignment problem (PQEVAP), is one of reassigning a few “troublesome” eigenvalues by using feedback while leaving the remaining large number of eigenvalues unchanged. The problem naturally arises in controlling dangerous vibrations, such as resonance in vibrating structures modeled by multi-input second-order control systems and in stabilizing control systems. One way to solve this problem is to transform the problem to a standard first-order state-space system and then apply one of the methods currently available for partial eigenvalue assignment in the first-order system. However, this approach has some computational drawbacks. To overcome these drawbacks the problems are solved using numerical algorithms that work directly with the second-order model without requiring transformation to a first-order system, which are implementable using the knowledge of a small number of eigenvalues and eigenvectors of the associated quadratic matrix pencil and which do not require model reduction. These features make the algorithms practically applicable to even very large real-life structures. Now, since the eigenvalues of a matrix may be highly sensitive to small perturbations, it is not enough to find the feedback matrices only. Attention must be given to finding them in such a way that they have minimum norms and the conditioning of the closed eigenvalues is as good as possible. The latter variation of the PQEVAP is known as the robust partial quadratic eigenvalue assignment problem ( RPQEVAP); the former is called the minimum norm partial quadratic eigenvalue assignment problem (MNPQEVAP). This dissertation is devoted to the study of these problems. Three new algorithms, one for solving the MNPQEVAP, one for solving the RPQEVAP, and one for simultaneously reducing the magnitude of the feedback norms and improving the conditioning of the closed-loop eigenvalues, have been developed and numerically tested.

#### Recommended Citation

Brahma, Sanjoy Kumar, "Robust and minimum norm partial quadratic eigenvalue assignment problems : theory and computations" (2006). *Graduate Research Theses & Dissertations*. 4775.

https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/4775

#### Extent

vi, 84 pages

#### Language

eng

#### Publisher

Northern Illinois University

#### Rights Statement

In Copyright

#### Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

#### Media Type

Text

## Comments

Includes bibliographical references (pages [82]-84).