Abstract by Charles Johnson
Graph Theoretic Foundations of Cyclic and Acyclic Linear Dynamic Networks
Dynamic Networks are signal flow graphs explicitly partitioning structural information from dynamic or behavioral information in a dynamic system. We explicitly develop the mathematical foundations underlying this class of models, revealing structural roots for system concepts such as system behavior, well-posedness, causality, controllability, observability, minimality, abstraction, and realization. This theory of abstractions uses graph theory to systematically and rigorously relate LTI state space theory, developed by Kalman and emphasizing differential equations and linear algebra, to the operator theory of Weiner, emphasizing complex analysis, and Willem's behavioral theory. New systems concepts, such as net effect, complete abstraction, and extraneous realization, are introduced, and we reveal conditions when acyclic abstractions exist for a given network, opening questions about their use in network reconstruction and other applications.