Index Reduction for Degenerated Differential-Algebraic Equations by Embedding

To find consistent initial data points for a system of differential-algebraic equations, requires the identification of its missing constraints. An efficient class of structural methods exploiting a dependency graph for this task was initiated by Pantiledes. More complete methods rely on differential-algebraic geometry but suffer from other issues (e.g. high complexity). In this paper we give a new class of efficient structural methods combined with new tools from numerical real algebraic geometry that has much improved completeness properties. Existing structural methods may fail for a system of differential-algebraic equations if its Jacobian matrix after differentiation is still singular due to symbolic cancellation or numerical degeneration. Existing structural methods can only handle degenerated cases caused by symbolic cancellation. However, if a system has parameters, then its parametric Jacobian matrix may be still singular after application of the structural method for certain values of the parameters. This case is called numerical degeneration. For polynomially nonlinear systems of differential-algebraic equations, numerical methods are given to solve both degenerated cases using numerical real algebraic geometry. First, we introduce a witness point method, which produces at least one witness point on every constraint component. This can help to ensure constant rank and detection of degeneration on all components of such systems. Secondly, we present a Constant Rank Embedding Lemma, and based on it propose an Index Reduction by Embedding (IRE) method which can construct an equivalent system with a full rank Jacobian matrix. Thirdly, IRE leads to a global structural differentiation method, to solve degenerated differential-algebraic equations on all components numerically. Application examples from circuits, mechanics, are used to demonstrate our method.