Incremental Models of Bond Graph Elements

In order to simplify the presentation, it is assumed that energy sources, energy storage elements and resistors are linear 1-port elements and that transformers and gyrators have got two ports. 

The underlying idea of incremental bond graphs is that if a parameter 0 of a component model varies, then both power variables at its port are perturbed due to its interaction with the ports of other elements in the model. That is, an effort (f) in a bond graph with nominal parameters becomes e t) = en t) + Ae t). The same holds for the conjugate power variable /(r). In incremental bond graphs, bonds carry the increments Aeif), Af(l) of power variables. In other words, they represent energy flows carrying the amount of power Ae t) Af(t). 

An incremental bond graph of an LTI system is constructed from an initial bond graph with nominal parameters by replacing those bond graph elements by their incremental model for which a parameter variation has taken place. Clearly, sources that do not depend on a parameter become null sources. Furthermore, the incremental model of a l(0)-junction remains a l(0)-junction. As to the incremental models of storage elements, resistors, transformers and gyrators, it turns out that they differ from the respective element just by modulated sinks added to junctions. The sinks 

In order to see how an incremental bond graph model for a bond graph element is obtained, a linear 1-port C-element with the nominal capacitance Cn is considered. In the following, an index n indicates a parameter or a variable of the non-faulty bond graph model with nominal parameters. In the case of a time constant parameter variation AC the linear constitutive equation of a 1-port C element in derivative causality takes the form 

The power variable fc controlling the modulated source is an output variable of the original bond graph model. If fc has been obtained by measurements of the real system, then the contribution to the output of the incremental bond graph model of the C element may contain sensor noise. In any case, the outputs of the incremental bond graph of a bond graph element indicate a parameter variation. 

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