Qualitative features of matrix pencils and DAEs arising in circuit dynamics

The dynamical behaviour of nonlinear electrical circuits is usually modelled in the time domain by differential–algebraic equations (DAEs). The differential–algebraic formalism drives qualitative analyses based on linearization to a matrix pencil setting. In this context, the present paper performs a spectral analysis of matrix pencils and DAEs arising in nonlinear circuit theory. Specifically, the non-singularity, hyperbolicity and asymptotic stability of equilibria are addressed in terms of circuit topology. The differential–algebraic framework puts the results beyond those already known for state-space models, unfeasible in many actual problems. The topological conditions arising in this qualitative study are proved independent of those supporting the index, and therefore they apply to both index-1 and index-2 configurations. The approach illustrates how graph theory, matrix analysis and DAE theory interact in the dynamical study of nonlinear circuits. ¶Preliminary results of this research were presented at the SCEE'04 Workshop in Capo D'Orlando, Sicilia, Italy 1.

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