• Corpus ID: 17778728

# A Rational Krylov Iteration for Optimal H 2 Model Reduction

@inproceedings{Gugercin2006ARK,
title={A Rational Krylov Iteration for Optimal H 2 Model Reduction},
author={Serkan Gugercin and Athanasios C. Antoulas and Christopher A. Beattie},
year={2006}
}
• Published 2006
In the sequel, we will construct the reduced order models Gr(s) through Krylov projection methods. Toward this end, we construct matrices V ∈ R and Z ∈ R that span certain Krylov subspaces with the property that Z V = Ir. The reduced order model Gr(s) will then be obtained as Ar = Z T AV, Br = Z T B, and Cr = CV. (2) The corresponding oblique projection is given by V Z . Many researchers have worked on the problem (1); see [18], [16], [7], [5], [3], [17], [4] and references therein. Since…

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## References

SHOWING 1-10 OF 23 REFERENCES
An approximate approach to H2 optimal model reduction
• Mathematics, Computer Science
IEEE Trans. Autom. Control.
• 1999
The problem is formulated as that of minimizing the H/sub 2/ model-reduction cost over the Stiefel manifold so that the stability constraint on reduced-order models is automatically satisfied and thus totally avoided in the new problem formulation.
Krylov Projection Methods for Model Reduction
The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation, based on which three algorithms for model reduction are proposed, which are suited for parallel or approximate computations.
The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore
• Mathematics
• 1985
First-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an
Second-order algorithm for optimal model order reduction
• Mathematics
• 1990
A second-order algorithm is given for optimal model order reduction of continuous single-input/single-output systems. Given a transfer function of order n, it finds the transfer function of a model
A new algorithm for L2 optimal model reduction
• Mathematics, Computer Science
Autom.
• 1992
It is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and, on this basis, a two-step iterative algorithm is developed combining a least squares solver with a gradient minimizer.
Lectures on complex approximation
• Mathematics
• 1987
I: Approximation by Series Expansions and by Interpolation.- I. Representation of complex functions by orthogonal series and Faber series.- 1. The Hilbert space L2(G).- A. Definition of L2(G).- B.
Rational L/sub 2/ approximation: a non-gradient algorithm
• Mathematics
[1991] Proceedings of the 30th IEEE Conference on Decision and Control
• 1991
The problem of determining the best rational approximant of a given rational transfer function of higher order according to the L/sub 2/-norm criterion is considered. An efficient algorithm is
Solving large-scale control problems
• P. Benner
• Computer Science
IEEE Control Systems
• 2004
It is concluded that modern tools from numerical linear algebra, along with careful investigation and exploitation of the problem structure, can be used to derive algorithms capable of solving large control problems.
Efficient numerical solution of the LQR‐problem for the heat equation
• Mathematics
• 2004
We discuss how the theory developed by Banks and Kunisch can be applied to a modified version of a controlled heat transfer model introduced by Troltzsch and Unger. In the numerical implementation we
A comparative study of 7 algorithms for model reduction
• Mathematics
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
• 2000
Compares seven model reduction algorithms by applying them to four different dynamical systems. There are four singular value decomposition (SVD) based methods, and three moment matching based