Passivity preserving model reduction via a structured Lanczos method

Nowadays, modeling dynamical systems often yields state-space models of very high order (that is, 10,000 or more equations). In order to guarantee a numerical simulation in reasonable time, the dynamical system is reduced to one of the same form which allows simulation of and control-design for the reduced-order state-space model in much less computing time. Usually one would like to obtain a reduced-order system that has the same properties as the original system. In this paper, we will consider stable and passive systems. Antoulas suggests in [1] an approach based on positive real interpolation which is modified by Sorensen [36]. The algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced-order 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 Hamiltonian matrix which has the spectral zeros as its spectrum. Here we propose to employ a structure-preserving Lanczos algorithm for this part of the computation in order to make use of the underlying structure and physical properties of the problem.