Datta, Biswa Nath
Ph.D. (Doctor of Philosophy)
Department of Mathematical Sciences
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.
Brahma, Sanjoy Kumar, "Robust and minimum norm partial quadratic eigenvalue assignment problems : theory and computations" (2006). Graduate Research Theses & Dissertations. 4775.
vi, 84 pages
Northern Illinois University
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