On the balanced truncation error bound and sign parameters from arrowhead realizations

TL;DR

This paper investigates the conditions under which the balanced truncation error bound becomes exact for certain linear systems, linking it to sign parameters derived from arrowhead matrix structures.

Contribution

It establishes that the error bound equality holds for SISO systems when sign parameters are uniform, and provides a method to determine these signs from arrowhead matrix entries.

Findings

Error bound equality for SISO systems with uniform sign parameters

Sign parameters linked to generalized state-space symmetry

Arrowhead matrix structure allows direct sign determination

Abstract

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced-order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling, and illustrate these results with a power system model that motivated this study.