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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 ([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. 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} \end{equation}
So, if we consider an input in the direction \\(v\_i\\), then the output is in the direction \\(u\_i\\). 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**: 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**: 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: We define \\(u\_1 = \overline{u}\\), \\(v\_1 = \overline{v}\\), \\(u\_k=\underline{u}\\) and \\(v\_k = \underline{v}\\).
\\[ G\bar{v} = \maxsv \bar{u} ; \quad G\ubar{v} = \minsv \ubar{u} \\] 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} ## 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: 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 \\]
@ -69,6 +70,6 @@ This will have usually little impact of the fitting error while reducing conside
## Bibliography {#bibliography} ## 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="org37ddb8b"></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. <a id="org0179442"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.