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A unifying framework for tangential interpolation of structured bilinear control systems

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Abstract

In this paper, we consider the structure-preserving model order reduction problem for multi-input/multi-output bilinear control systems by tangential interpolation. We propose a new type of tangential interpolation problem for structured bilinear systems, for which we develop a new structure-preserving interpolation framework. This new framework extends and generalizes different formulations of tangential interpolation for bilinear systems from the literature and also provides a unifying framework. We then derive explicit conditions on the projection spaces to enforce tangential interpolation in different settings, including conditions for tangential Hermite interpolation. The analysis is illustrated by means of three numerical examples.

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Code and data availability

The source code, data and results of the numerical experiments are open source/open access and available at [41].

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Acknowledgements

We would like to thank Jens Saak for providing the data for the bilinear semi-discretized steel profile and inspiring discussions about the interpretation of tangential interpolation for linear systems.

Funding

Benner and Werner were supported by the German Research Foundation (DFG) Research Training Group 2297 “Mathematical Complexity Reduction (MathCoRe)”, Magdeburg. Gugercin was supported in parts by National Science Foundation under Grant No. DMS-1819110. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 and by the Simons Foundation Grant No. 50736 while all authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Model and dimension reduction in uncertain and dynamic systems” program.

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Authors and Affiliations

  1. Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106, Magdeburg, Germany

    Peter Benner

  2. Faculty of Mathematics, Otto von Guericke University, Universitätsplatz 2, 39106, Magdeburg, Germany

    Peter Benner

  3. Department of Mathematics and Division of Computational Modeling and Data Analytics, Academy of Data Science, Virginia Tech, Blacksburg, VA, 24061, USA

    Serkan Gugercin & Steffen W. R. Werner

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  1. Peter Benner
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Correspondence to Steffen W. R. Werner.

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Parts of this work were carried out while Werner was at the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany and at the Courant Institute of Mathematical Sciences, New York University, USA. The authors declare to have no competing interests related to this work.

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Benner, P., Gugercin, S. & Werner, S.W.R. A unifying framework for tangential interpolation of structured bilinear control systems. Numer. Math. 155, 445–483 (2023). https://doi.org/10.1007/s00211-023-01380-w

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