Passivity preserving model reduction via interpolation of spectral zeros

Abstract

An algorithm is developed for passivity preserving model reduction of linear time invariant systems. Implementation schemes are described for both medium scale (dense) and large scale (sparse) applications. The algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros. These interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum. Explicit interpolation is avoided and passivity of the reduced model is established, instead, through satisfaction of the necessary conditions of the Positive Real Lemma. It is also shown that this procedure indirectly solves the associated controllability and observability Riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations. From these, a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable.