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Thomas Dehaeze 2020-09-21 17:04:39 +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" >}}). This is taken from ([Skogestad and Postlethwaite 2007](#org323f388)).
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.
@ -47,7 +47,7 @@ We define \\(u\_1 = \bar{u}\\), \\(v\_1 = \bar{v}\\), \\(u\_k=\ubar{u}\\) and \\
## 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 the book [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}}). This is taken from ([Preumont 2018](#org713967a)).
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 \\]
@ -66,4 +66,9 @@ The conditioning of the Jacobian is measured by the **condition number**:
When \\(c(J)\\) becomes large, the most straightforward way to handle the ill-conditioning is to truncate the smallest singular value out of the sum. When \\(c(J)\\) becomes large, the most straightforward way to handle the ill-conditioning is to truncate the smallest singular value out of the sum.
This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\). This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\).
<./biblio/references.bib>
## Bibliography {#bibliography}
<a id="org713967a"></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="org323f388"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.