Dynamic causality in hybrid bond graphs

Hybrid models are those containing continuous and discontinuous behaviour. In constructing dynamic systems models, it is frequently desirable to abstract rapidly changing, highly nonlinear behaviour to a discontinuity. Bond graphs lend themselves to systems modelling by being multi-disciplinary and reflecting the physics of the system. The causally dynamic hybrid bond graph is therefore suitable for simulation as well as providing engineering insight through analysis. There is a distinction between structural and parametric switching. The controlled junction is used for the former, and gives rise to dynamic causality. A controlled element is developed for the latter. Dynamic causality is unconstrained so as to aid insight. The junction structure matrix (JSM) for the hybrid bond graph features Boolean terms to reflect the controlled junctions in the graph structure. This hybrid JSM is used to generate a mixed-Boolean state equation. When storage elements are in dynamic causality, the resulting system equation is implicit. Control properties can be seen in the structure and causal assignment. An impulsive mode may occur when storage elements are in dynamic causality, but otherwise there are no energy losses associated with commutation because this method dictates the way discontinuities are abstracted. An impulsive model can be derived if required to further aid simulation. The mixed-Boolean mathematical model can be implemented in an environment such as Matlab.