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A Structured Doubling Algorithm for the Numerical Solution of Lur’e Equations

Please always quote using this URN:urn:nbn:de:0296-matheon-7489
  • We introduce a~numerical method for the numerical solution of the Lur'e matrix equations that arise, for instance, in linear-quadratic infinite time horizon optimal control. The method is based on the characterization of the solutions in terms of deflating subspaces of a suitable even matrix pencil. Via a Cayley transformation, the problem is transformed to the discrete-time case. This leaves us with a symplectic problem with several Jordan blocks of eigenvalue 1 and even size, which arise from the remaining eigenvalues at infinity of the original problem. For the solution of this modified problem, we use the {\em structure-preserving doubling algorithm} (SDA), an iterative scheme for the solution of dense continuous- and discrete-time algebraic Riccati equations. Unlike other iterative schemes, this algorithm converges also when the pencil has eigenvalues on the unit circle, as is the case in our problem. Implementation issues such as the choice of the parameter $\gamma$ in the Cayley transform are discussed. The numerical examples presented confirm the effectiveness of this method.

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Metadaten
Author:Timo Reis, Federico Poloni
URN:urn:nbn:de:0296-matheon-7489
Referee:Volker Mehrmann
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2011/01/23
Release Date:2011/01/25
Tag:Lure equation; deflating subspace; iterative solution; matrix pencil
Institute:Technische Universität Berlin
MSC-Classification:15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A22 Matrix pencils [See also 47A56]
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A24 Matrix equations and identities
Preprint Number:758
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