Computing standard canonical forms of regular linear time varying DAEs via a preliminary stage

TL;DR

This paper introduces a method to compute standard canonical forms for linear time-varying DAEs by identifying a suitable block structure and applying an iterative process, enabling systematic transformations and new representations.

Contribution

It presents a novel approach to compute canonical forms of linear time-varying DAEs through a preliminary block structure stage and an iterative algorithm.

Findings

The iterative process terminates finitely due to nilpotency.

Transformation matrices can be systematically derived.

Applicable to structured DAEs in applications.

Abstract

For regular linear time-invariant DAEs the corresponding matrix pencil is regular and the computation of a standard canonical form is well-understood. Although the investigation of linear DAEs with time-varying coefficients is more complex, it is analogously related to the examination of pairs of time-dependent matrix functions. We show how the computation of a standard canonical form (SCF) for linear time-varying DAEs becomes possible if a suitable block structure of the pair of matrix functions is found in a preliminary stage. Starting from this preliminary stage, an iterative process delivers an SCF. This iteration terminates in finitely many steps due to the nilpotency of an involved matrix. The corresponding transformation matrix functions can be provided systematically, which leads also to new representations of the canonical subspaces and projectors related to the original DAE. We demonstrate how the structured canonical forms resulting in the tractability and strangeness frameworks and some DAEs in Hessenberg form from applications can be transformed into this preliminary stage and discuss some examples.