Electron. J. Diff. Eqns., Vol. 2009(2009), No. 37, pp. 1-32.

Controllability, observability, realizability, and stability of dynamic linear systems

John M. Davis, Ian A. Gravagne, Billy J. Jackson, Robert J. Marks II

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
Ian A. Gravagne
Department of Electrical and Computer Engineering
Baylor University
Waco, TX 76798, USA
email: Ian_Gravagne@baylor.edu
Billy J. Jackson
Department of Mathematics and Computer Science
Valdosta State University
Valdosta, GA 31698, USA
email: bjackson@valdosta.edu
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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