The mathematical foundations of bond graphs—IV. Matrix representations and causality

Abstract

The development of a mathematical theory for bond graphs continues with an analysis of two areas crucial to the derivation of system equations from a bond graph model. Matrix representations of bond graph matroids are examined and used to provide a rigorous proof of the mathematical equivalence of the linear graph and bond graph modelling methods. The procedure of selecting causality by the method of causal strokes is discussed. This is shown to be a device for choosing a base of the cycle matroid of a bond graph, and a sufficient condition is proved for when the method gives a base. An example is given of a bond graph with a causal loop which corresponds to a valid causal assignment. Various combinatorial formulae are proved concerning ranks and dimensions. Formulae for the ranks of effort and flow matrices and the number of independent equations obtained from a bond graph are consequences of the results.