Structured linear algebra problems and applications to system identification
Stewart, Michael Alan
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https://hdl.handle.net/2142/20219
Description
Title
Structured linear algebra problems and applications to system identification
Author(s)
Stewart, Michael Alan
Issue Date
1996
Department of Study
Electrical and Computer Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Engineering, Electronics and Electrical
Language
eng
Abstract
This thesis considers problems of stability, rank estimation and conditioning for structured matrices. The ideas are developed with attention to potential applications in control and signal processing where such matrices arise routinely. A stability result for the factorization of the broad class of positive definite Toeplitz-like matrices is given. For nearly semidefinite Toeplitz matrices, it is proven that the Cholesky factor has a limited rank-revealing property. This property has a close connection with a stability result for the Schur algorithm for the factorization of a positive definite Toeplitz matrix. An attempt is made to extend the connection between Cholesky factors and conditioning to block-Toeplitz matrices by considering fundamental properties that govern the conditioning of transformations used in fast algorithms for the factorization of such matrices. A quotient URV decomposition is introduced and applied to block Toeplitz matrices to provide an on-line algorithm for the solution of the multi-input/multi-output (MIMO) state space identification problem. Finally, theoretical results are given that relate to the problem of determining the distance of a state space model from a state space model that is non-minimal. This may be interpreted as an attempt to show that the problem of determining when a state-space model is nearly uncontrollable or unobservable is well-posed.
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