Eigenvalues of a DAE and stability analysis

dynamical systemsordinary differential equationssystems of equations

I have a nonlinear system of differential algebraic equations (DAE) of index -1, and I was wondering: what is the "meaning" of eigenvalues in that case?

I know they do not indicate stability, as with regular ODEs, unless the Algebraic terms are eliminated, which is not an option in my case.

  1. What do they mean, then?

  2. How do I investigate the stability, in such a DAE?

Best Answer

There are two ways that I can imagine, not necessarily the best ways.

Way I:

A DAE of index one is easily transformable to an ODE system, i.e. one simply eliminates the algebraic equations and substitute them within the differential equations. Then the eigenvalue-based stability analysis can be performed on the resulting ODE system.

Way II:
Differentiate the algebraic equations w.r.t. time, so that the algebraic variables become state variables. Introduce consistent initial conditions using the algebraic equations. Then eigenvalue stability analysis can be applied to the resulting ODE system.

A generalization to DAE systems of arbitrary index can be followed by a combination of Pantelides algorithm and method of dummy derivatives for index reduction.

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