Computation of the nearest structured matrix triplet with common null space

TL;DR

This paper develops computational methods to determine the minimal perturbations that cause singularity, high index, or instability in dissipative Hamiltonian differential-algebraic systems, leveraging explicit characterizations for these structured matrices.

Contribution

It introduces new algorithms based on differential equations that efficiently compute distances to singularity, high index, and instability for structured DAEs with dissipative Hamiltonian form.

Findings

Algorithms successfully compute minimal perturbations for structured DAEs.

Methods converge to smallest perturbations that alter system properties.

Explicit characterizations enable efficient computation for structured systems.

Abstract

We study computational methods for computing the distance to singularity, the distance to the nearest high index problem, and the distance to instability for linear differential-algebraic systems (DAEs) with dissipative Hamiltonian structure. While for general unstructured DAEs the characterization of these distances is very difficult, and partially open, it has been recently shown that for dissipative Hamiltonian systems and related matrix pencils there exist explicit characterizations. We will use these characterizations for the development of computational methods to compute these distances via methods that follow the flow of a differential equation converging to the smallest perturbation that destroys the property of regularity, index one or stability.