A Compositional Framework for Bond Graphs

Abstract

Electrical circuits made only of perfectly conductive wires can be seen as partitions between finite sets. These are also known as "corelations" and are the morphisms in the category $\mathrm{FinCorel}$. The two-element set has two different Frobenius monoid structures in $\mathrm{FinCorel}$. These two Frobenius monoids are related to "series" and "parallel" junctions, which are used to connect pairs of wires. We show that these Frobenius monoids interact to form a "weak bimonoid" as defined by Pastro and Street. We conjecture a presentation for the subcategory of $\mathrm{FinCorel}$ generated by the morphisms associated to these two Frobenius monoids, which we call $\mathrm{FinCorel}^{\circ}$. We are interested in "bond graphs," which are built from series and parallel junctions. Although the morphisms of $\mathrm{FinCorel}^{\circ}$ resemble bond graphs, there is not a perfect correspondence. Since bond graphs and circuits determine Lagrangian relations between symplectic vector spaces, we then consider the category of Lagrangian relations, $\mathrm{LagRel}_k$. Bond graphs pick out a subcategory $\mathrm{LagRel}_k^{\circ}$ with generating morphisms corresponding to those of $\mathrm{FinCorel}^{\circ}$. Thus we define a category $\mathrm{BondGraph}$ with generators and equations that are found in both $\mathrm{FinCorel}^{\circ}$ and $\mathrm{LagRel}_k^{\circ}$. We study the functorial semantics of $\mathrm{BondGraph}$ by giving two different functors from it to the category $\mathrm{LagRel}_k$ and a natural transformation between them. Given a bond graph, the first functor picks out a Lagrangian relation in terms of "effort" and "flow," while the second picks one out in terms of "potential" and "current." The natural transformation arises from the way that effort and flow relate to potential and current.