Guardian maps for continuous-time systems: A Lie-algebraic approach

Guardian maps are scalar maps that vanish when a matrix or polynomial is on the verge of stability. Several guardian maps have been proposed in the literature for Hurwitz stability based on the Kronecker sum, the second lower Schläflian matrix, and the bialternate sum. It is natural to ask if there is a unifying principle for all these maps. Here, we introduce the Lie-algebraic notion of a guardian representation, and show that all the examples above are instances of this unifying idea. We also show that the bialternate sum coincides with the second additive compound.