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Vertex and edge theorems which simplify classical analyses of linear systems with uncertain parameters
Abstract
This dissertation addresses the problem of analyzing both continuous and discrete-time, linear time-invariant systems with uncertain parameters. The investigation considers four classifications of uncertain systems that are loosely called interval families, affine uncertainties, polytopes, and multiaffine uncertainties. The focus is on classical analyses: stability, pole locations, frequency response, and time response. The goal of each analysis is to determine the worst-case behavior of the system over all possible values of the parameter vector. To simplify these analyses, the existence of relationships between the extreme behavior of the system and the "prominent" values of the parameters is investigated. Because the "prominent" parameter values are analogous to the vertices or the edges of a box, results which show that the worst-case behavior can be determined using only "prominent" parameters are referred to as vertex and edge theorems. For each class of systems and each analysis problem, the existence of vertex and edge theorems is reviewed. For stability and pole location analyses, edge theorems are presented for interval families, affine uncertainities, and polytopes. These edge theorems are a contribution of this dissertation. For multiaffine uncertainties, no stability or pole location edge theorems exist. In general, stability and pole location vertex theorems do not exist for any of the four system classes. The main exception is Kharitonov's stability vertex theorem for continuous-time interval families. For frequency response determination, the contribution of this dissertation is an edge theorem for interval families, affine uncertainities, and polytopes. For multiaffine uncertainties, no frequency response edge theorem exists. For all four classes of systems, frequency response vertex theorems also do not exist. A steady state time response vertex theorem is presented for all four classes of uncertain systems. For affine uncertainties, polytopes, and multiaffine uncertainties, it is shown that a transient response vertex theorem does not exist. These two results are contributions of this dissertation. The existence or absence of a transient response vertex theorem for interval families is still an open question. The availability of transient response edge theorems remains an open problem for all four classes of uncertain systems.
Subject Area
Electrical engineering
Recommended Citation
Bartlett, Andrew C, "Vertex and edge theorems which simplify classical analyses of linear systems with uncertain parameters" (1990). Doctoral Dissertations Available from Proquest. AAI9110102.
https://scholarworks.umass.edu/dissertations/AAI9110102