Bond Graphs II: Causality and Singularity

Abstract The concepts of causality and singularity for bond graphs are defined, and related through the idea of a singular (or nonsingular) causal assignment, which is one with zero (or nonzero) discriminant. Necessary and sufficient conditions are given for a bond graph to have a causal assignment. It is proved that every acausal bond graph is acausally equivalent to a nonsingular bond graph, and that every nonsingular acausal bond graph has at least one nonsingular causal assignment for each consistent choice of input variables. The strong components of a causal bond graph B are defined, and it is shown that the discriminant of B is the product of the discriminants of its strong components; in particular, a causal assignment is singular if and only if there is at least one singular strong component. The reduced bond graph R(B) of B is defined and, if B is nonsingular, R(B) is proved to be acausally equivalent to B. Mason's determinant rule and the Mason gain formula are proved, and a simpler way of calculating the discriminant is derived.

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