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A low-rank solution method for Riccati equations with indefinite quadratic terms

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Abstract

Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to other established matrix equation solvers, and investigate the applicability and performance in large-scale sparse examples.

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Funding

This work was supported by the German Research Foundation (DFG) Research Training Group 2297 “Mathematical Complexity Reduction (MathCoRe)”, Magdeburg.

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Authors and Affiliations

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

    Peter Benner & Jan Heiland

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

    Peter Benner & Jan Heiland

  3. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    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. Benner is a member of the editorial board of Numerical Algorithms. The authors declare to have no competing interests related to this work.

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Code and data availability

The source code of the implementations used to compute the presented results, the used data and computed results are available at https://doi.org/10.5281/zenodo.6308400 under the BSD-2-Clause license and authored by Steffen W. R. Werner.

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Benner, P., Heiland, J. & Werner, S.W.R. A low-rank solution method for Riccati equations with indefinite quadratic terms. Numer Algor 92, 1083–1103 (2023). https://doi.org/10.1007/s11075-022-01331-w

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