Variation diminishing linear time-invariant systems

Abstract

This paper studies the variation diminishing property of k-positive systems, which map inputs with k − 1 sign changes to outputs with at most the same variation. We characterize this property for the Toeplitz and Hankel operators of finite-dimensional linear time invariant systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of k 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 characterization generalizes well known properties of externally positive systems (k = 1) and totally positive systems (k = ∞).

Internally Hankel $k$-positive systems

“…Notice a nonnegative matrix verifies X [1] = X ≥ 0, which is equivalent to X being OVD 0 . This can be generalized through the compound matrix as follows [17,20].…”

“…The OVD k property of LTI systems (1) has been studied in [17], where a distinction is made between LTI systems with OVD k Toeplitz and Hankel operators. The latter are particularly relevant to this work.…”

“…A core result of total positivity is an algebraic characterization: an operator is OVD k if and only its matrix representation is k-positive, that is, all the minors of order up to k in that matrix are nonnegative [17,20]; total positivity refers to the case when this is true for all k. Under this framework, externally positive LTI systems are associated with OVD 0 Hankel and Toeplitz operators, while for internally positive systems x → Ax as well as the controllablity and observability operators are OVD 0 . Despite the link between positive systems and variation-diminishing operators, it was not until very recently that OVD k−1 and k-positivity have been studied as properties of LTI systems when k > 1.…”

“…Despite the link between positive systems and variation-diminishing operators, it was not until very recently that OVD k−1 and k-positivity have been studied as properties of LTI systems when k > 1. New results, with applications in non-linear systems analysis and model order reduction, have so far focused on two distinct cases: the external case [16,17,15], where OVD k is considered as a property of system convolution operators; and related autonomous cases [26,1], which concern k-positivity of the state-space matrix A with b = 0. A main result of [17] characterizes (externally) Hankel k-positive systems, i.e., systems with OVD k−1 Hankel operators, in terms of the external positivity of all so-called j-compound systems, 1 ≤ j ≤ k, whose impulse responses are given by consecutive j-minors of the Hankel operator's matrix representation.…”

“…Non-linear extensions of this problem are of interest both in control [21] and online learning, in the form of (static) regret [29]; we thus envision our work as the basis for possible interdisciplinary applications. Other possible contributions resulting from non-linear extension are discussed in [17].…”