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Thomas Dehaeze 2020-09-21 17:03:45 +02:00
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content/zettels/singular_value_decomposition.md
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## SVD of a MIMO system {#svd-of-a-mimo-system} ## SVD of a MIMO system {#svd-of-a-mimo-system}
This is taken from the book [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}}).
We are interested by the physical interpretation of the SVD when applied to the frequency response of a MIMO system \\(G(s)\\) with \\(m\\) inputs and \\(l\\) outputs. We are interested by the physical interpretation of the SVD when applied to the frequency response of a MIMO system \\(G(s)\\) with \\(m\\) inputs and \\(l\\) outputs.
\begin{equation} \begin{equation}
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## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices} ## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices}
This is taken from [Singular Value Decomposition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}}). This is taken from the book [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}}).
The **Singular Value Decomposition** (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix: The **Singular Value Decomposition** (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix:
\\[ J = U \Sigma V^T = \sum\_{i=1}^r \sigma\_i u\_i v\_i^T \\] \\[ J = U \Sigma V^T = \sum\_{i=1}^r \sigma\_i u\_i v\_i^T \\]