Near aggregation: a time and frequency domain analysis using state trajectories and transfer function residues

Abstract

In this thesis we investigate concepts associated with aggregation. The basic idea of aggregation is that there exists a reduced order model such that, for an appropriate initial condition, the trajectories of the reduced-order model are linear combinations of the trajectories of the ful 1-order model. We study systems which do not aggregate exactly, but which "nearly aggregate". It is shown that for "nearly aggregable" systems there exists a reduced-order model such that, for an appropriate initial condition, the trajectories of the reduced-order model are near a linear combination of the trajectories of the full-order model.

Under certain conditions it has also been shown that near-aggregation is equivalent to near-unobservability (roughly, an invariant subspace close to the null space of C). Here we establish a relationship between near-unobservability and modal measures of observability as suggested by Selective Modal Analysis. With this result we then obtain an upper bound on the norm of the transfer function residue using near-unobservability measures. The Generalized Hessenberg Representation (GHR) and Dual GHR are examined throughout this analysis. It is finally shown that for SISO systems, the residue norm may be expressed in terms of certain parameters of the Dual GHR.