Chained Aggregation and Control System Design: A Geometric Approach
Lindner, Douglas Kent
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https://hdl.handle.net/2142/69239
Description
Title
Chained Aggregation and Control System Design: A Geometric Approach
Author(s)
Lindner, Douglas Kent
Issue Date
1982
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
This thesis is an indepth study of the Generalized Hessenberg Representation (GHR) of a linear time-invariant control system. It is shown that the GHR explicitly exhibits a sequence of observability subspaces, { (,i)}. By studying these subspaces in this specific basis, a number of results follow.
Having defined the subspace { (,i)} algebraically, we introduce a topology into the subspaces of state space. Using the GHR we are able to estimate distances between key subspaces. This leads to a measure of the degree of observability, called here near unobservability, which formalizes the intuitive geometric notion that a system is "nearly unobservable" if it has an invariant subspace near the null space of C. The relationship to other measures of observability is discussed as well as its role in model reduction.
The behavior of the subspaces { (,i)} under the action of an input is also discussed. The connection to the supremal (A,B)-invariant subspace in the nullspace of C is made, but other (A,B)-invariant subspaces are also described. In addition, the GHR is used to identify (C,A)-invariant subspaces. Both of these subspaces play a fundamental role in compensator design. Thus, the GHR leads to a state feedback design scheme, called Three Control Component Design, based on (A,B)-invariant subspaces produced by the GHR. This control is hierarchical in that it gives priority to the primary design goals. Furthermore, it explicitly identifies a reduced order model used to meet the design goals. This results in an interactive design procedure which allows for a trade-off between model order and computational complexity. Furthermore, by using (C,A)-invariant subspaces, observer design is carried out in the same framework. This leads directly to dynamic compensator design. The results are applied to decentralized control problems, noninteractive control, and nonlinear systems.
Implicit in this discussion is the decomposition of a system into subsystems based on the underlying geometric structure. We investigate this aspect of the GHR and show how the information and control structures are related to physical subsystems in several types of interconnections. The role of system decomposition in reduced order modeling and compensator design is discussed.
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