John M. Davis, Ian A. Gravagne, Billy J. Jackson, Robert J. Marks II
Abstract:
We develop a linear systems theory that coincides with the existing
theories for continuous and discrete dynamical systems, but that
also extends to linear systems defined on nonuniform time scales.
The approach here is based on generalized Laplace transform methods
(e.g. shifts and convolution) from the recent work [13].
We study controllability in terms of the controllability Gramian and
various rank conditions (including Kalman's) in both the time invariant
and time varying settings and compare the results. We explore
observability in terms of both Gramian and rank conditions and
establish related realizability results. We conclude by applying this
systems theory to connect exponential and BIBO stability problems in
this general setting. Numerous examples are included to show the
utility of these results.
Submitted January 23, 2009. Published March 3, 2009.
Math Subject Classifications: 93B05, 93B07, 93B20, 93B55, 93D99
Key Words: Systems theory; time scale; controllability; observability;
realizability; Gramian; exponential stability; BIBO stability;
generalized Laplace transform; convolution.
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John M. Davis Department of Mathematics, Baylor University Waco, TX 76798, USA email: John_M_Davis@baylor.edu |
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Ian A. Gravagne Department of Electrical and Computer Engineering Baylor University Waco, TX 76798, USA email: Ian_Gravagne@baylor.edu |
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Billy J. Jackson Department of Mathematics and Computer Science Valdosta State University Valdosta, GA 31698, USA email: bjackson@valdosta.edu |
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Robert J. Marks II Department of Electrical and Computer Engineering Baylor University Waco, TX 76798, USA email: Robert_Marks@baylor.edu |
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