State Feedback Design Methods: Eigenstructure Assignment and LQR Cost Functional Synthesis
Gerth, Richard Arthur
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https://hdl.handle.net/2142/71999
Description
Title
State Feedback Design Methods: Eigenstructure Assignment and LQR Cost Functional Synthesis
Author(s)
Gerth, Richard Arthur
Issue Date
1993
Doctoral Committee Chair(s)
Perkins, William R.
Department of Study
Electrical Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Electronics and Electrical
Abstract
The research documented in this thesis involves two distinct but related topics in state feedback design methodology: (i) the assignment of a system's closed-loop eigenvectors and eigenvalues ("eigenstructure"), and (ii) the synthesis of Linear Quadratic Regulator (LQR) cost functional weighting matrices.
The first topic concerns the explicit specification by the designer of a system's closed-loop eigenstructure, and a particular method by which the eigenstructure might be chosen. It is desirable for robustness reasons to have eigenvectors which are mutually orthogonal, or as close to this ideal as possible. However, once a system's closed-loop eigenvalues have been assigned, the freedom in selecting the eigenvectors is restricted. Indeed, for a single-input system the eigenvectors are uniquely determined by the choice of eigenvalues. Algorithms introduced in this thesis allow some or all of the closed-loop eigenvalues to be chosen from a designer-specified set of candidates, in such a way that the assigned eigenvectors will be as nearly orthogonal as possible. Alternatively, gradient algorithms are developed which introduce incremental changes in the closed-loop spectrum, in such a way that the eigenvectors are shifted toward orthogonality. For each type of algorithm, both single-input and multi-input systems are considered.
The second topic considered is the synthesis of LQR cost weighting matrices, starting from a given state feedback matrix; this is essentially the "reverse design" of a linear quadratic regulator. Algorithms are introduced for the synthesis of symmetric matrix bases, both for the weighting matrices and for the solution to the associated algebraic Riccati equation. Conditions are given for the existence of these bases, and for the existence of positive definite solutions to this synthesis problem.
Also included is a case study of the design of an arc current controller for a gas-metal arc welder, using the design methods introduced in this thesis. Analytical results and experimental data from the physical system are presented.
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