Systems of differential-algebraic equations (DAEs) are generated routinely bysimulation and modeling environments such as Modelica and MapleSim. Before asimulation starts and a numerical solution method is applied, some kind ofstructural analysis is performed to determine the structure and the index of aDAE. Structural analysis methods serve as a necessary preprocessing stage, andamong them, Pantelides's algorithm is widely used. Recently Pryce's $\Sigma$-method is becoming increasingly popular, owing toits straightforward approach and capability of analyzing high-order systems.Both methods are equivalent in the sense that when one succeeds, producing anonsingular system Jacobian, the other also succeeds, and the two give the samestructural index. Although provably successful on fairly many problems of interest, thestructural analysis methods can fail on some simple, solvable DAEs and giveincorrect structural information including the index. In this report, we focuson the $\Sigma$-method. We investigate its failures, and develop twosymbolic-numeric conversion methods for converting a DAE, on which the$\Sigma$-method fails, to an equivalent problem on which this method succeeds.Aimed at making structural analysis methods more reliable, our conversionmethods exploit structural information of a DAE, and require a symbolic toolfor their implementation.
展开▼
