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Thomas Dehaeze
2020-09-21 19:10:23 +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](#org0179442)).
This is taken from ([Skogestad and Postlethwaite 2007](#org4953f60)).
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.
@@ -18,14 +18,9 @@ We are interested by the physical interpretation of the SVD when applied to the
G = U \Sigma V^H
\end{equation}
\\(\Sigma\\)
: is an \\(l \times m\\) matrix with \\(k = \min\\{l, m\\}\\) non-negative **singular values** \\(\sigma\_i\\), arranged in descending order along its main diagonal, the other entries are zero.
\\(U\\)
: is an \\(l \times l\\) unitary matrix. The columns of \\(U\\), denoted \\(u\_i\\), represent the **output directions** of the plant. They are orthonormal.
\\(V\\)
: is an \\(m \times m\\) unitary matrix. The columns of \\(V\\), denoted \\(v\_i\\), represent the **input directions** of the plant. They are orthonormal.
- \\(\Sigma\\): is an \\(l \times m\\) matrix with \\(k = \min\\{l, m\\}\\) non-negative **singular values** \\(\sigma\_i\\), arranged in descending order along its main diagonal, the other entries are zero.
- \\(U\\): is an \\(l \times l\\) unitary matrix. The columns of \\(U\\), denoted \\(u\_i\\), represent the **output directions** of the plant. They are orthonormal.
- \\(V\\): is an \\(m \times m\\) unitary matrix. The columns of \\(V\\), denoted \\(v\_i\\), represent the **input directions** of the plant. They are orthonormal.
The input and output directions are related through the singular values:
@@ -48,7 +43,7 @@ Then is follows that:
## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices}
This is taken from ([Preumont 2018](#org37ddb8b)).
This is taken from ([Preumont 2018](#org6558f35)).
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 \\]
@@ -70,6 +65,6 @@ This will have usually little impact of the fitting error while reducing conside
## Bibliography {#bibliography}
<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="org6558f35"></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="org0179442"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
<a id="org4953f60"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.