Interplay between discretization and controllability of linear delay systems: An algebraic viewpoint

In this paper, we give an in depth study of linear delay systems controllability preservation/alteration through discretization. We make use of a module theoretic framework acting as a unifying one for most of the existing delay system controllability notions. We propose a formal generic definition of a discretization scheme and illustrate through examples that controllability properties may be lost through discretization. Then, we introduce the notion of preservation (that is, a measure of quantifying the ability of the discretizer to preserve controllability properties) and prove that for a given discretizer, we can always find a delay system for which even the torsion-free controllability (which is the weakest controllability notion) is not preserved. Finally, we reverse the situation, and show that for any given delay system, preserving discretizers exist.