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bibliography: => #+BIBLIOGRAPHY: here

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Thomas Dehaeze
2021-05-02 22:18:30 +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}
This is taken from ([Skogestad and Postlethwaite 2007](#orga80f5ed)).
This is taken from ([Skogestad and Postlethwaite 2007](#org8e4f47e)).
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.
@@ -43,7 +43,7 @@ Then is follows that:
## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices}
This is taken from ([Preumont 2018](#orgb521567)).
This is taken from ([Preumont 2018](#org6d4589f)).
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 \\]
@@ -63,8 +63,9 @@ When \\(c(J)\\) becomes large, the most straightforward way to handle the ill-co
This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\).
## Bibliography {#bibliography}
<a id="orgb521567"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<a id="org6d4589f"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<a id="orga80f5ed"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
<a id="org8e4f47e"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.