Simulation of System Models Containing Zero-order Causal Paths

The existence of zero-order causal paths in bond graphs of physical systems implies the set of state equations to be an implicit mixed set of Differential and Algebraic Equations (DAEs). In the block diagram expansion of such a bond graph, this type of causal path corresponds with a zero-order loop. In this paper the numerical solution of the DAEs by methods commonly used for solving stiff systems of Ordinary Differential Equations (ODEs) is discussed. Apart from a description of the numerical implications of zero-order causal paths, a classification of zero-order causal paths is given with respect to the behavior of the numerical solution method. This behavior is characterized by “the index of nilpotency” (Gear and Petzold, Siam J. Numerical Anal., Vol. 21, No. 4, 1984). Propositions concerning the index of nilpotency and the class of zero-order causal path are formulated. These propositions are illustrated by examples. The concept “essential causal cycle” is introduced as a special, closed, causal path which cannot be eliminated.