Supporting the generation of a state space model by adding tearing information to the bond graph

Abstract

For many engineering systems, the mathematical model derived from a graphical model description takes the form of a system of Differential Algebraic Equations (DAEs) of index one which can be passed directly to a DAE solver. If the algebraic constraints are linear with regard to the so-called tearing variables, an alternative can be to solve them symbolically, and in that way reduce the initial DAE system into a state space model. Bond graphs well suited for a unified graphical representation of multi-disciplinary engineering systems across energy domains clearly indicate algebraic dependencies before any equations are set up. It is shown how this feature can help identify possible tearing variables and equations that determine them without having to inspect (automatically) generated model equations. Moreover, if the bond graph model is described in a modeling language like Dymola, the corresponding model processor can exploit the tearing information, solve the algebraic dependencies symbolically provided they are linear with respect to the tearing variables, and output assignment statements in a simulation language like ACSL. The proposed method is heuristic and provides a small number, not necessarily a minimal set of tearing variables. For didactic reasons, it is illustrated by means of fairly small and linear examples containing different types of algebraic dependencies. However, the method works just as well when applied to large and non-linear systems. In the latter case, tearing allows for a less costly numerical solution compared to a non-torn system.