Structure of Kron's polyhedron model and the scattering problem

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Kron proposed a polyhedron network model or ‘automaton’ to represent a wide range of physical system problems. The model includes a sequence of higher-dimensional structures, each formulated by an orthogonal electrical network with closed and open paths. A set of electromagnetic waves is assumed to propagate across the polyhedron, and many other physical concepts are introduced intuitively into the model, including those concerned with multidimensional ‘generalised’ machines, thermodynamics, fluid flow and statistical phenomena. The present work highlights the significance of the orthogonal electrical network in the polyhedron structure, and it is shown that a similar structure exists in the general scattering representation of a flow process. The formulation of the scattering problem would then appear to provide an analytical basis for many of the physical concepts discussed by Kron. It is also shown to be associated with the system problems concerned with the multimachine power system, least-squares estimation and linear optimal control and filtering. Kron's polyhedron model and the scattering problem thus incorporate similar basic features, which appear to be of fundamental importance in general system theory.