On the Tightness of the Balanced Truncation Error Bound with Applications to Arrowhead Systems

Sean Reiter, Mark Embree, Serkan Gugercin

Balanced truncation model reduction for linear dynamical systems yields a reduced-order model that satisfies a well-known error bound involving the system's Hankel singular values. We show that this bound holds with equality for single-input, single-output systems that exhibit a generalized state-space symmetry in the truncated part of the model. This result extends to singular perturbation balancing. The sign parameters associated with the Hankel singular values are the fundamental ingredients in revealing the underlying symmetry structure. We prove an additional result showing how to determine these sign parameters for a class of systems having arrowhead realizations. We describe how these systems arise naturally in certain applications of network modeling, and illustrate these results with one such example that motivated our study.

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