Balanced truncation model reduction with a priori error bounds for LTI systems with nonzero initial value

Schröder, Christian; Voigt, Matthias

doi:10.1016/j.cam.2022.114708

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Journal article

Balanced truncation model reduction with a priori error bounds for LTI systems with nonzero initial value

Schröder, Christian
Voigt, Matthias UniDistance Suisse

2023

Published in:

Journal of Computational and Applied Mathematics. - 2023, vol. 420, p. 114708

English In standard balanced truncation model order reduction, the initial condition is typically
ignored in the reduction procedure and is assumed to be zero instead. However, such
a reduced-order model may be a bad approximation to the full-order system, if the
initial condition is not zero. In the literature there are several attempts for modified
reduction methods at the price of having no error bound or only a posteriori error
bounds which are often too expensive to evaluate. In this work we propose a new
balancing procedure that is based on a shift transformation on the state. We first derive
a joint projection reduced-order model in which the part of the system depending only
on the input and the one depending only on the initial value are reduced at once and we
prove an a priori error bound. With this result at hand, we derive a separate projection
procedure in which the two parts are reduced separately. This gives the freedom to
choose different reduction orders for the different subsystems. Moreover, we discuss
how the reduced-order models can be constructed in practice. Since the error bounds
are parameter-dependent we show how they can be optimized efficiently. We conclude
this paper by comparing our results with the ones from the literature by a series of
numerical experiments.

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