Conditioning of the stable, discrete-time Lyapunov operator (Q2706241)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Conditioning of the stable, discrete-time Lyapunov operator |
scientific article |
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19 March 2001
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Lyapunov matrix equation
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condition estimates
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large-scale systems
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radius of stability
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Schatten \(p\)-norm
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singular values
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ill-conditioning
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Conditioning of the stable, discrete-time Lyapunov operator (English)
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The Schatten \(p\)-norm condition of the Lyapunov operator \({\mathcal L}_A\) is studied. \(\|{\mathcal L}_A\|_p\) is bounded in terms of the singular values of the matrix \(A\). A lower bound for \(\|{\mathcal L}^{-1}_A\|_p\) dependent on \({\mathcal L}_1(A)\) is given, generalizing the results of \textit{P. M. Gahinet}, \textit{A. J. Laub}, \textit{C. S. Kenney} and \textit{G. A. Hewer} [IEEE Trans. Autom. Control 35, No. 11, 1209-1217 (1990; Zbl 0721.93064)]. Lower bounds for \(\|{\mathcal L}^{-1}_A\|_1\) and \(\|{\mathcal L}^{-1}_A\|_\infty\) are presented in terms of the singular values of \(A\). An upper bound for \(\|{\mathcal L}^{-1}_A\|_p\) depends on the radius of stability. Three examples illustrate how ill-conditioning of \({\mathcal L}_A\) leads to a low-rank approximation.
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