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

Electronic Journal of Differential Equations (Impact Factor: 0.52). 01/2009; 2009.
Source: DOAJ


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.

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    • "Here we examine controllability, reachability, and observability in the time-invariant case. It should be noted that there have been other excellent attempts to do so, e.g., in [2], [3], [7], [8]. They all examine the linear system "
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    • "For dynamical systems on these general time domains T, understanding the structure of S plays a key role in various aspects of both the control theory and control applications [2], [3], [9], [21]. However, since S can be difficult to compute on general time scales, other more tractable sufficient conditions for the exponential stability of (II.1) have been explored [25], [14], [10]. For each fixed t ∈ T, define the (open) "
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