MOR Wiki - User contributions [en]

File:IRKA Example.zip

2016-03-17T15:43:18Z

Fehr: Fehr uploaded a new version of "File:IRKA Example.zip"

A very rudimentary and simple implementation of the iteratively corrected rational Krylov algorithm for $H_2$ model reduction
proposed by Gugercin, Antoulas and Beattie [GugercinAntoulasBeattie08].
Part of this work is inspired from the work of Heiko K.F. Panzer [Panzer14].

This implementation was made for the Paper
BEST PRACTICES FOR NUMERICAL REPLICABILITY, REPRODUCIBILITY AND REUSABILITY EXEMPLIFIED BY
MODEL REDUCTION SOFTWARE

This implementation either runs with Matlab or with Octave

# Dependencies
For MATLAB as well as octave the control toolbox needs to be available

# RunMe

1. change into installation directory
2. run TestIRKA.m

Octave version >=3.8 with control-package needs to be installed

# Installation

1. no installation is required

Literature:

[Panzer14] H. Panzer; Model Order Reduction by Krylov Subspace Methods with Global Error Bounds and
Automatic Choice of Parameters. Doctoral Thesis Technische Universität München, 2014.

[GugercinAntoulasBeattie08] S. Gugercin; A.C. Antoulas; C. Beattie: "H2 Model Reduction for Large-
Scale Linear Dynamical Systems", SIAM. J. Matrix Anal. & Appl., vol.30, no.2,
pp.609-638, 2008.

Iterative Rational Krylov Algorithm

2016-03-17T15:38:34Z

Fehr: /* Description */

[[Category:method]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:linear algebra]]

==Description==

'''IRKA''' (iterative rational Krylov algorithm) is an interpolation-based model reduction method for [[List_of_abbreviations#SISO|SISO]] linear time invariant systems

:<equation id="gensys" shownumber>
<math>
\dot{x}(t)=A x(t)+b u(t), \quad
y(t)=c^Tx(t),\quad A\in\mathbb{R}^{n\times n},~b\in\mathbb{R}^{n},~c\in\mathbb{R}^{n}.\qquad (1)
</math>
</equation>

For a given system <math>G </math> and a prescribed reduced system order <math>r</math>, the goal of the algorithm is to find a local minimizer <math>\hat{G} </math> for the <math> H_2 </math> model reduction problem

:<math>
||G-\hat{G} ||_{H_2} = \min_{\text{dim}(\tilde{G})=r} ||G-\tilde{G}||_{H_2}.
</math>

Initially investigated in <ref name="MeiL67"></ref>, first order necessary conditions for a local minimizer <math>\hat{G}</math> imply that its rational transfer function <math>\hat{G}(s)=\hat{c}^T (sI-\hat{A})^{-1}b</math> is a Hermite interpolant of the original transfer function at its reflected system poles, i.e.,

:<math>
G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,
</math>
where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation at the reflected reduced system poles is ensured. In pseudocode, the classical '''IRKA''' algorithm from <ref name="GugAB08"></ref> looks like

1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.
2. Choose <math>V_r </math> and <math> W_r</math> so that <math>\text{Ran}(V_r) =\{(\sigma_1 I -A)^{-1}b,\dots,(\sigma_rI-A)^{-1}b \} </math>, <math>\text{Ran}(W_r) =\{(\sigma_1 I -A^T)^{-1}c,\dots, (\sigma_rI-A^T)^{-1}c \} </math> and <math> W_r^TV_r=I</math>.
3. while (relative change in <math>\{\sigma_i\} > tol</math>)
(a) <math>\hat{A} = W_r^TAV_r</math>
(b) Assign <math>\sigma_i \leftarrow -\lambda_i(\hat{A}),</math> for <math> i=1,\dots,r</math>
(c) Update <math>V_r</math> and <math>W_r</math> so that <math>\text{Ran}(V_r) =\{(\sigma_1 I -A)^{-1}b,\dots,(\sigma_rI-A)^{-1}b \} </math>, <math>\text{Ran}(W_r) =\{(\sigma_1 I -A^T)^{-1}c,\dots, (\sigma_rI-A^T)^{-1}c \} </math> and <math> W_r^TV_r=I</math>.
4. <math>\hat{A} = W_r^TAV_r, \hat{b}= W_r^Tb, \hat{c}^T = c^TV_r.</math>

Although a rigorous convergence proof so far has only be given for symmetric state space systems <ref name="FlaBG12"></ref>, numerous experiments have shown that the algorithm often converges rapidly. Moreover, the algorithm has been extended to, e.g., multiple-input-multiple output, discrete time and differential algebraic systems.

== A minimal example ==

For the lecture Model [http://www.itm.uni-stuttgart.de/courses/model_reduction/model_reduction_en.php MOR] of Mechanical System from the Institute of Engineering and Computational Mechanics University of Stuttgart a very simple example of the IRKA Algorithm were written.

The implementation with two basic exampless can be found here

[[Media:IRKA_Example.zip]]

This code is published under the BSD3-Clause License
All rights reserved. (c) 2015, Joerg.Fehr, University of Stuttgart

==References==
<references>

<ref name="MeiL67"> L. Meier, D.G. Luenberger, "[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1098680&tag=1 Approximation of linear constant systems]", IEEE Transactions on Automatic Control, vol.12, no.5, pp.585-588, 1967.</ref>

<ref name="GugAB08"> S. Gugercin, A.C. Antoulas, C. Beattie "[http://epubs.siam.org/doi/abs/10.1137/060666123 H2 Model Reduction for Large-Scale Linear Dynamical Systems]", SIAM. J. Matrix Anal. & Appl., vol.30, no.2, pp.609-638, 2008.</ref>

<ref name="FlaBG12"> G. Flagg, C. Beattie, S. Gugercin "[http://www.sciencedirect.com/science/article/pii/S0167691112000576 Convergence of the Iterative Rational Krylov Algorithm]", Systems & Control Letters, vol.61, no.6, pp.688-691, 2012.</ref>

</ references>

File:IRKA Example.zip

2016-03-17T15:24:18Z

Fehr:

File:IRKA Example.zip

2016-03-17T13:47:23Z

Fehr: Example files of an IRKA implementation includes a Test File, a p-coded version of Frequency Response matrix, and an IRKA implementation for SISO LTI systems

Example files of an IRKA implementation
includes a Test File, a p-coded version of Frequency Response matrix, and an IRKA implementation for SISO LTI systems