Directed bondgraphs and integral matroids

The presentation of a theory for the mathematical foundations of the bondgraph modelling method is continued by this analysis of the combinatorial properties of directed bondgraphs (di-bondgraphs). This theory also provides a means to explore the relationship between a bondgraph model and a graph-theoretic model of the same system, as well as establishing a framework for developing a generalized coordinate-free approach which encompasses both methods for modelling a spatially discrete physical system. A di-bondgraph, B¯, is formed by adding a half-arrow on each bond of a bondgraph, called the underlying bondgraph. Only junctions and bonds are included, and no physical components or effort and flow variables. The junctions in B¯ define a set of elementary integral chains which generate a chain group; its restriction to external bonds, the cycle chain group, provides a set of combinatorial constraints with integral coefficients. These are essential for representing polarities for physical measurements in expressing the spatial constraints (generalized Kirchhoff laws) that relate the physical variables in the model. The cycle chain group also provides an integral representation of the cycle matroid of the di-bondgraph, which represents the interconnections between components. Duals of these structures, the cocycle chain group and matroid, can be constructed for a di-bondgraph in the same way. Orthogonality over the integers is established for the cycle and cocycle chain groups, a crucial property which ensures that generalized energy, the product of each pair of dual variables, will be a conserved quantity over the model (Tellegen’s theorem or equivalent). Causality for di-bondgraphs is analyzed using pseudo-base colouring of the bonds, a procedure equivalent to the addition of causal strokes in a conventional bondgraph model. It is established that the pseudo-base set obtained from a pseudo-base colouring with no even causal loop always provides a base of the cycle matroid of a simple di-bondgraph with no redundant junction. Formulas for the rank and corank of such a di-bondgraph in terms of junction and external bond counts are provided.