Controllability, Observability, Realizability, and Stability of Dynamic Linear Systems

TL;DR

This paper extends linear systems theory to nonuniform time domains, analyzing controllability, observability, realizability, and stability using generalized Laplace transforms, with numerous examples demonstrating practical applications.

Contribution

It introduces a unified linear systems framework for nonuniform time domains, extending classical controllability, observability, and stability concepts with new rank conditions and Gramian analyses.

Findings

Controllability characterized by Gramian and rank conditions.

Observability analyzed through Gramian and rank criteria.

Connections established between exponential and BIBO stability.

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 domains. The approach here is based on generalized Laplace transform methods (e.g. shifts and convolution) from our recent work \cite{DaGrJaMaRa}. 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 also explore observability in terms of both Gramian and rank conditions as well as 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.