From d4825ee51a663f01e7f19a43820f0dbd1996e5b8 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 21 Sep 2020 17:04:39 +0200 Subject: [PATCH] Update Content - 2020-09-21 --- content/zettels/singular_value_decomposition.md | 11 ++++++++--- 1 file changed, 8 insertions(+), 3 deletions(-) diff --git a/content/zettels/singular_value_decomposition.md b/content/zettels/singular_value_decomposition.md index 1b7c0cc..a05bb9b 100644 --- a/content/zettels/singular_value_decomposition.md +++ b/content/zettels/singular_value_decomposition.md @@ -10,7 +10,7 @@ Tags ## 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. @@ -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} -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: \\[ 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. This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\). -<./biblio/references.bib> + +## Bibliography {#bibliography} + +Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. . + +Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.