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G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r, |
G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r, |
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</math> |
</math> |
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| − | 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 |
+ | 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 is ensured. In pseudocode, the classical algorithm (IRKA) 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>. |
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>. |
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Revision as of 08:10, 30 May 2013
Description
The iterative rational Krylov algorithm (IRKA) is an interpolation-based model reduction method for single-input-single-output linear time invariant systems
- \( \dot{x}(t)=A x(t)+b u(t), \quad y(t)=c^Tx(t),\quad E,A\in\mathbb{R}^{n\times n},~b\in\mathbb{R}^{n},~c\in\mathbb{R}^{n}.\qquad (1) \) (1)
For a given system \(G \) and a prescribed reduced system order \(r\), the goal of the algorithm is to find a local minimizer \(\hat{G} \) for the \( H_2 \) model reduction problem
\[ ||G-\hat{G} ||_{H_2} = \min_{\text{dim}(\tilde{G})=r} ||G-\hat{G}||_{H_2}. \]
Initially investigated in [1], first order necessary conditions for a local minimizer \(\hat{G}\) imply that its rational transfer function \(\hat{G}(s)=\hat{c}^T (sI-\hat{A})^{-1}b\) is a Hermite interpolant of the original transfer function at its reflected system poles, i.e.,
\[ G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r, \] where \(\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} \) are assumed to be the simple poles of \( \hat{G} \). Based on the idea of rational interpolation by rational Krylov subspaces, in [2] the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation is ensured. In pseudocode, the classical algorithm (IRKA) from [2] looks like
1. Make an initial selection of \(\sigma_i \) for \(i=1,\dots,r \) that is closed under conjugation and fix a convergence tolerance \(tol\).
2. Choose \(V_r \) and \( W_r\) so that \(\text{Ran}(V_r) =\{(\sigma_1 I -A)^{-1}b,\dots,(\sigma_rI-A)^{-1}b \} \), \(\text{Ran}(W_r) =\{(\sigma_1 I -A^T)^{-1}c,\dots, (\sigma_rI-A^T)^{-1}c \} \) and \( W_r^TV_r=I\).
3. while (relative change in \(\{\sigma_i\} > tol\))
(a) \(\hat{A} = W_r^TAV_r\)
(b) Assign \(\sigma_i \leftarrow -\lambda_i(\hat{A}),\) for \( i=1,\dots,r\)
(c) Update \(V_r\) and \(W_r\) so that \(\text{Ran}(V_r) =\{(\sigma_1 I -A)^{-1}b,\dots,(\sigma_rI-A)^{-1}b \} \), \(\text{Ran}(W_r) =\{(\sigma_1 I -A^T)^{-1}c,\dots, (\sigma_rI-A^T)^{-1}c \} \) and \( W_r^TV_r=I\).
4. \(\hat{A} = W_r^TAV_r, \hat{b}= W_r^Tb, \hat{c}^T = c^TV_r.\)
Although a rigorous convergence proof so far has only be given for symmetric state space systems [3], numerous experiments have shown that the algorithm often converges rapidly.
References
<references>
</ references>
- ↑ 1.0 1.1 L. Meier, D.G. Luenberger, "Approximation of linear constant systems", IEEE Transactions on Automatic Control, vol.12, no.5, pp.585-588 1967
- ↑ 2.0 2.1 2.2 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
- ↑ 3.0 3.1 G. Flagg, C. Beattie, S. Gugercin "Convergence of the Iterative Rational Krylov Algorithm", Systems & Control Letters, vol.61, no.6, pp.688-691 2012