Spectral characterization and enforcement of negative imaginariness for descriptor systems / Peter Benner, Matthias Voigt

Abstract

Abstract: Systems with counterclockwise input-output dynamics (or negative imaginary transfer functions) arise in various applications such as the modeling of flexible mechanical structures or electrical circuits where certain kinds of measurements are taken. In this paper we introduce descriptor systems with such an additional structure. We state various of their properties and prove algebraic characterizations of negative imaginariness in terms of spectral conditions of certain structured matrix pencils. For this purpose we also analyze particular boundary cases which are characterized by properties of a structured Kronecker canonical form. Finally, we describe a method which can be used to restore the negative imaginary property in case that this is lost. This happens, e.g., when a system with theoretically negative imaginary transfer function is obtained by, e.g., model order reduction methods, linearization, or other approximations. The method is illustrated by numerical examples.