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

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Virginia Tech


Balanced truncation model reduction for linear systems yields reduced-order models that satisfy a well-known error bound in terms of a system's Hankel singular values. This bound is known to hold with equality under certain conditions, such as when the full-order system is state-space symmetric.

In this work, we derive more general conditions in which the balanced truncation error bound holds with equality. We show that this holds for single-input, single-output systems that exhibit a generalized type of state-space symmetry based on the sign parameters corresponding to a system's Hankel singular values. We prove an additional result that shows how to determine this state-space symmetry from the arrowhead realization of a system, if available. In particular, we provide a formula for the sign parameters of an arrowhead system in terms of the off-diagonal entries of its arrowhead realization.

We then illustrate these results with an example of an arrowhead system arising naturally in power systems modeling that motivated our study.



Model Reduction, Balanced Truncation, Error Bound, Arrowhead Systems, Power Systems