Linear differential-algebraic systems are generically controllable

TL;DR

This paper explores the topological properties of controllable differential-algebraic systems, establishing generic conditions for various controllability concepts using algebraic and polynomial matrix analysis.

Contribution

It provides necessary and sufficient conditions for the generic controllability of differential-algebraic systems across five different concepts, using polynomial matrix rank properties.

Findings

The set of controllable systems is generic under certain algebraic conditions.

Necessary and sufficient conditions for controllability are derived for each concept.

Polynomial matrix rank criteria are key to characterizing controllability.

Abstract

In the present work we investigate topological properties of the set of controllable differential-algebraic systems of the form $\tfrac{\text{d}}{\text{d}t}Ex = Ax+Bu$ with real matrices $E,A\in\mathbb{R}^{\ell\times n}$ and $B\in\mathbb{R}^{\ell\times m}$. We consider the five controllability concepts free initializability (controllability at infinity), impulse controllability, controllability in the behavioural sense, complete controllability and strong controllability. To be able to make use of the already known algebraic characterizations of these concepts, we first consider block matrices whose entries are real polynomials in one indeterminant. We find necessary and sufficient conditions under which the set of such block matrices, whose rank is "full" in the field of rational functions or even on the whole complex plane, is generic. Using these results, we can then for each of the five controllability concepts mentioned above find necessary and sufficient conditions at $\ell,n$ and $m$, respectively, under which the set of controllable systems is generic.