A Higher-Order Generalized Singular Value Decomposition for Rank-Deficient Matrices

Idris Kempf, Paul J. Goulart and Stephen R. Duncan

@article{KGD:2023,
  author = {Kempf, Idris and Goulart, Paul J. and Duncan, Stephen R.},
  title = {A Higher-Order Generalized Singular Value Decomposition for Rank-Deficient Matrices},
  journal = {SIAM Journal on Matrix Analysis and Applications},
  year = {2023},
  volume = {44},
  number = {3},
  pages = {1047-1072},
  url = {https://doi.org/10.1137/21M1443881},
  doi = {10.1137/21M1443881}
}

Abstract. The higher-order generalized singular value decomposition (HO-GSVD) is a matrix factorization technique that extends the GSVD to \[N \ge 2\] data matrices and can be used to identify common subspaces that are shared across multiple large-scale datasets with different row dimensions. The standard HO-GSVD factors \(N\) matrices \(A\_i\in \mathbb{R}^{m\_i\times n}\) as \(A\_i=U\_i\Sigma\_iV^{\text{T}}\) but requires that each of the matrices \(A\_i\) has full column rank. We propose a modification of the HO-GSVD that extends its applicability to rank-deficient data matrices \(A\_i\) . If the matrix of stacked \(A\_i\) has full rank, we show that the properties of the original HO-GSVD extend to our approach. We extend the notion of common subspaces to isolated subspaces, which identify features that are unique to one \(A\_i\) . We also extend our results to the higher-order cosine-sine decomposition (HO-CSD), which is closely related to the HO-GSVD. Our extension of the standard HO-GSVD allows its application to matrices with \(m\_i\lt n\) or rank \((A\_i)\lt n\) , such as those encountered in bioinformatics, neuroscience, control theory, and classification problems.

Control of Cross-Directional Systems with Approximate Symmetries

Idris Kempf, Paul Goulart and Stephen Duncan

@misc{KGD:2023b,
  author = {Idris Kempf and Paul Goulart and Stephen Duncan},
  title = {Control of Cross-Directional Systems with Approximate Symmetries},
  year = {2023}
}

Structural symmetries of linear dynamical systems can be exploited for decoupling the dynamics and reducing the computational complexity of the controller implementation. However, in practical applications, inexact structural symmetries undermine the ability to decouple the system, resulting in the loss of any potential complexity reduction. To address this, we propose substituting an approximation with exact structural symmetries for the original system model, thereby introducing an approximation error. We focus on internal model controllers for cross-directional systems encountered in large-scale and high-speed control problems of synchrotrons or the process industry and characterise the stability, performance, and robustness properties of the resulting closed loop. While existing approaches replace the original system model with one that minimises the Frobenius norm of the approximation error, we show that this can lead to instability or poor performance. Instead, we propose approximations that are obtained from semidefinite programming problems. We show that our proposed approximations can yield stable systems even when the Frobenius norm approximation does not. The paper concludes with numerical examples and a case study of a synchrotron light source with inexact structural symmetries. Exploiting structural symmetries in large-scale and high-speed systems enables faster sampling times and the use of more advanced control techniques, even when the symmetries are approximate.