Infinite-dimensional Euclidean Distance Degrees in Model Order Reduction

A fundamental problem in LTI system theory is the Model Order Reduction (MOR) problem: Given a higher-order SISO LTI system described by a transfer function of McMillan degree $N$, the goal is to approximate it by a "smaller" morel of degree $n<N$ such that the $\mathcal{H}_2$-norm of the approximation error is minimized. While it has been shown that the MOR problem has finitely many critical points, no general bound for the total number of these points has been available previously.

In this talk we will derive critical equations for this optimization problem, leading to the Walsh polynomial system. We then transform this system to a degeneracy locus of a vector bundle morphism, which allows us to count the number of critical points. The approach works both for discrete- and continuous-time systems and can be seen as an instance of an infinite-dimensional Euclidean Distance Degree. If time permits, we also discuss solutions with higher-order poles.

This is based on joint work with Sibren Lagauw from KU Leuven.</p>