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+++ title = "Singular Value Decomposition" author = ["Thomas Dehaeze"] draft = false +++
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SVD of a MIMO system
This is taken from (Skogestad and Postlethwaite 2007).
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.
\begin{equation} 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.
The input and output directions are related through the singular values:
\begin{equation} G v_i = \sigma_i u_i \end{equation}
So, if we consider an input in the direction \(v_i\), then the output is in the direction \(u_i\). 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: \[\overline{\sigma}(G) \equiv \sigma_1(G) = \max_{d\neq 0}\frac{\|Gd\|_2}{\|d\|_2} = \frac{\|Gv_1\|_2}{\|v_1\|_2} \] The smallest gain for any input direction is equal to the minimum singular value: \[ \underline{\sigma}(G) \equiv \sigma_k(G) = \min_{d\neq 0}\frac{\|Gd\|_2}{\|d\|_2} = \frac{\|Gv_k\|_2}{\|v_k\|_2} \]
We define \(u_1 = \overline{u}\), \(v_1 = \overline{v}\), \(u_k=\underline{u}\) 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
This is taken from (Preumont 2018).
The Singular Value Decomposition (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix: \[ J = U \Sigma V^T = \sum_{i=1}^r \sigma_i u_i v_i^T \] With:
- \(U\) 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
- \(\Sigma\) a rectangular diagonal matrix of dimension \(m \times n\) containing the square root of the common non-zero eigenvalues of \(JJ^T\) and \(J^TJ\)
- \(r\) is the number of non-zero singular values of \(J\)
The pseudo-inverse of \(J\) is: \[ J^+ = V\Sigma^+U^T = \sum_{i=1}^r \frac{1}{\sigma_i} v_i u_i^T \]
The conditioning of the Jacobian is measured by the condition number: \[ c(J) = \frac{\sigma_{max}}{\sigma_{min}} \]
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 \(v\).
Bibliography
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.
Skogestad, Sigurd, and Ian Postlethwaite. 2007. Multivariable Feedback Control: Analysis and Design. John Wiley.