From 5a3c9583d3c29a6a70ef63deb7730ff1c754dc94 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 21 Sep 2020 17:11:53 +0200 Subject: [PATCH] Update Content - 2020-09-21 --- .../zettels/singular_value_decomposition.md | 19 ++++++++++--------- 1 file changed, 10 insertions(+), 9 deletions(-) diff --git a/content/zettels/singular_value_decomposition.md b/content/zettels/singular_value_decomposition.md index a05bb9b..4e44a9d 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 ([Skogestad and Postlethwaite 2007](#org323f388)). +This is taken from ([Skogestad and Postlethwaite 2007](#org0179442)). 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. @@ -34,20 +34,21 @@ The input and output directions are related through the singular values: \end{equation} So, if we consider an input in the direction \\(v\_i\\), then the output is in the direction \\(u\_i\\). -Furthermore, since \\(\normtwo{v\_i}=1\\) and \\(\normtwo{u\_i}=1\\), we see that **the singular value \\(\sigma\_i\\) directly gives the gain of the matrix \\(G\\) in this direction**. +Furthermore, since \\(\\|v\_i\\|\_2=1\\) and \\(\\|u\_i\\|\_2=1\\), we see that **the singular value \\(\sigma\_i\\) directly gives the gain of the matrix \\(G\\) in this direction**. The **largest gain** for any input is equal to the **maximum singular value**: -\\[\maxsv(G) \equiv \sigma\_1(G) = \max\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_1}}{\normtwo{v\_1}} \\] +\\[\overline{\sigma}(G) \equiv \sigma\_1(G) = \max\_{d\neq 0}\frac{\\|Gd\\|\_2}{\\|d\\|\_2} = \frac{\\|Gv\_1\\|\_2}{\\|v\_1\\|\_2} \\] The **smallest gain** for any input direction is equal to the **minimum singular value**: -\\[\minsv(G) \equiv \sigma\_k(G) = \min\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_k}}{\normtwo{v\_k}} \\] +\\[ \underline{\sigma}(G) \equiv \sigma\_k(G) = \min\_{d\neq 0}\frac{\\|Gd\\|\_2}{\\|d\\|\_2} = \frac{\\|Gv\_k\\|\_2}{\\|v\_k\\|\_2} \\] -We define \\(u\_1 = \bar{u}\\), \\(v\_1 = \bar{v}\\), \\(u\_k=\ubar{u}\\) and \\(v\_k = \ubar{v}\\). Then is follows that: -\\[ G\bar{v} = \maxsv \bar{u} ; \quad G\ubar{v} = \minsv \ubar{u} \\] +We define \\(u\_1 = \overline{u}\\), \\(v\_1 = \overline{v}\\), \\(u\_k=\underline{u}\\) and \\(v\_k = \underline{v}\\). +Then is follows that: +\\[ G\overline{v} = \overline{\sigma} \cdot \overline{u} ; \quad G\underline{v} = \underline{\sigma} \cdot \underline{u} \\] ## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices} -This is taken from ([Preumont 2018](#org713967a)). +This is taken from ([Preumont 2018](#org37ddb8b)). 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 \\] @@ -69,6 +70,6 @@ This will have usually little impact of the fitting error while reducing conside ## Bibliography {#bibliography} -Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. . +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. +Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.