On strongly unimodal third-order SISO linear systems with applications to pharmacokinetics

Abstract

This paper addresses the problem of characterizing external positivity (equivalently, non-negative impulse response) of third-order single-input, single-output (SISO) linear systems. We show how an exact, geometric solution to this problem follows by first identifying an equivalence between the impulse response of an externally positive system on the one hand, and the probability density function of a non-negative random variable on the other, then drawing on the characterization of matrix exponential distributions, defined as probability distributions for which the Laplace transform is a rational function. The results are then extended to the characterization of strongly unimodal systems, defined as systems in which input signals with a time-derivative that has at most one sign variation (namely, are pulse-like) are mapped to output signals with the same property. The results are applied to a third-order compartmental system arising in pharmacokinetics, in which the properties of non-negativity of the impulse response and the preservation of unimodality from drug adminstration (input) to compartmental drug concentration (output) are of clinical relevance.