71 lines
3.7 KiB
Markdown
71 lines
3.7 KiB
Markdown
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title = "Singular Value Decomposition"
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author = ["Thomas Dehaeze"]
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draft = false
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## SVD of a MIMO system {#svd-of-a-mimo-system}
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This is taken from ([Skogestad and Postlethwaite 2007](#org4953f60)).
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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.
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\begin{equation}
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G = U \Sigma V^H
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\end{equation}
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- \\(\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.
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- \\(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.
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- \\(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.
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The input and output directions are related through the singular values:
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\begin{equation}
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G v\_i = \sigma\_i u\_i
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\end{equation}
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So, if we consider an input in the direction \\(v\_i\\), then the output is in the direction \\(u\_i\\).
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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**.
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The **largest gain** for any input is equal to the **maximum singular value**:
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\\[\overline{\sigma}(G) \equiv \sigma\_1(G) = \max\_{d\neq 0}\frac{\\|Gd\\|\_2}{\\|d\\|\_2} = \frac{\\|Gv\_1\\|\_2}{\\|v\_1\\|\_2} \\]
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The **smallest gain** for any input direction is equal to the **minimum singular value**:
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\\[ \underline{\sigma}(G) \equiv \sigma\_k(G) = \min\_{d\neq 0}\frac{\\|Gd\\|\_2}{\\|d\\|\_2} = \frac{\\|Gv\_k\\|\_2}{\\|v\_k\\|\_2} \\]
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We define \\(u\_1 = \overline{u}\\), \\(v\_1 = \overline{v}\\), \\(u\_k=\underline{u}\\) and \\(v\_k = \underline{v}\\).
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Then is follows that:
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\\[ G\overline{v} = \overline{\sigma} \cdot \overline{u} ; \quad G\underline{v} = \underline{\sigma} \cdot \underline{u} \\]
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## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices}
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This is taken from ([Preumont 2018](#org6558f35)).
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The **Singular Value Decomposition** (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix:
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\\[ J = U \Sigma V^T = \sum\_{i=1}^r \sigma\_i u\_i v\_i^T \\]
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With:
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- \\(U\\) and \\(V\\) orthogonal matrices. The columns \\(u\_i\\) and \\(v\_i\\) of \\(U\\) and \\(V\\) are the eigenvectors of the square matrices \\(JJ^T\\) and \\(J^TJ\\) respectively
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- \\(\Sigma\\) a rectangular diagonal matrix of dimension \\(m \times n\\) containing the square root of the common non-zero eigenvalues of \\(JJ^T\\) and \\(J^TJ\\)
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- \\(r\\) is the number of non-zero singular values of \\(J\\)
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The pseudo-inverse of \\(J\\) is:
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\\[ J^+ = V\Sigma^+U^T = \sum\_{i=1}^r \frac{1}{\sigma\_i} v\_i u\_i^T \\]
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The conditioning of the Jacobian is measured by the **condition number**:
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\\[ c(J) = \frac{\sigma\_{max}}{\sigma\_{min}} \\]
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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.
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This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\).
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## Bibliography {#bibliography}
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<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>.
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<a id="org4953f60"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
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