Mode switching in causally dynamic hybrid bond graphs

The causally dynamic hybrid bond graph is extended to the case of mode-switching behaviour. Mode-switching 'trees' of switches and elements are historically used by bond graph practitioners to represent elements with piecewise-continuous functions. This case is defined as 'parametric switching' for the purposes of the hybrid bond graph, since the switching is internal to the element, as opposed to 'structural switching' which alters the model structure. This mode-switching 'tree' is concatenated into a new controlled element which features Boolean switching parameters in the constitutive equation, removing unnecessary complexity from the model. Mixed-Boolean state equations can be derived from the model, which are nonlinear and/or time-varying (and hence not in the familiar Linear Time Invariant Form). It can be seen that controlled elements often have a static causality assignment and leave the model structure unchanged. The result is a concise method for representing nonlinear behaviour as a piecewise-continuous function in the bond graph modelling framework.