The DAE System

Assume we have a system of differential and algebraic equations

$$\begin{eqnarray} \html{eqn20}f(\dot{x}, x, y) = 0 \\ h(x, y) = 0\end{eqnarray}$$

The algebraic equations enforce constraints on $\dot{x}$which results in dependencies between rate variables, $\dot{x}_i$.This dependency can be made explicit by solving the algebraic variables, y, from the system of equations. To find independent expressions for $\dot{x}_i$ in terms of x, the algebraic equations that determine the DAE manifold can be differentiated to produce additional constraints that have to be satisfied by the rates $\dot{x}_i$ to evolve on the prescribed manifold. Often only a subset of the algebraic equations constitutes rate constraints, and, therefore, form the manifold. In these cases, it is not required to differentiate the entire system, h, but a subset, $\bar{h}$, suffices.

For DAEs of index 2 or less, one differentiation step results in a system of DAEs that can be solved for all its rates, $\dot{x}_i$, to derive a system of explicit ordinary differential equations (ODEs). For DAEs of index higher than two, multiple differentiations may be required and even new rate variables may be introduced. This paper deals with DAEs of index two or less. Therefore, the assumption is made that algebraic equations need not be differentiated more than once, and that no new rate variables are introduced, i.e., $\bar{h}(\bar{x})$ is no function of y. Though this restriction limits the DAE systems that can be handled, the assumption holds true for a large class of physical system models, a.o., those based on the unifying bond graph modeling formalism [4,15]. The remaining algebraic equations, $\hat{h}$, are a function $\hat{h}(\bar{x}, \hat{x}, \hat{y})$ where $\hat{x}$ and $\hat{y}$are subsets of $x = \{\bar{x}, \hat{x}\}$ and $y = \{\hat{y}, \tilde{y}\}$that are not in $\bar{h}$.The subset $\tilde{y}$ are algebraic variables in f that are not in h.