Variation diminishing Hankel operators
Christian Grußler, Rodolphe J. Sepulchre · 2020
This paper studies the variation diminishing property of linear time-invariant Hankel k-positive systems, i.e., systems whose Hankel operator maps inputs with k - 1 sign changes to outputs with at most the same variation. Our main result is that these systems have a dominant approximation in the form of a parallel interconnection of k positive lags, that is, first order positive systems. This is shown by expressing the k-positivity of a LTI system as the external positivity (that is, 1-positivity) of k compound LTI systems. Our characterizations are generalizations of the well known properties of positive systems (k = 1) and Hankel totally positive systems (k = ∞).