Representing externally positive systems through minimal eventually positive realizations

Abstract

Abstract-In order to investigate the cases in which an externally positive discrete-time system fails to have a minimal positive realization, in this paper we introduce the notion of minimal eventually positive realization, for which the state update matrix becomes positive after a certain power. This property captures the idea that in the impulse response of an externally positive system the state of a minimal realization may fail to be positive, but only transiently. It is shown in the paper that whenever a minimal eventually positive realization exists, then the sequence of Markov parameters of the impulse response admits decimated subsequences for which minimal positive realizations exist and can be obtained by downsampling the eventually positive realization.

Properties of eventually positive linear input–output systems

“…This implies that there exists a finite i such that K i p ⊂ R n ≥0 , and similar result holds for a dual cone (K i p ) * ⊂ R n ≥0 . These cones can be constructed using, e.g, [9], [17]. However, to our best knowledge it is hard to construct a polyhedral cone, which explicitly depends on the eigenspace of the matrix.…”

“…In this section, we consider the control systems (1) and introduce systems, which eventually behave like internally positive systems. We called them internally eventually positive systems, but we note that these systems were also studied in [9] and called eventually positive realizations.…”

“…Characterizations of externally positive systems through state-space matrices has been derived in [8], and recently received an inflow of new ideas in [9]. Therein the author builds realizations of externally positive systems using the concept of eventual positivity.…”

Minimal eventually positive realizations of externally positive systems

“…With respect to a preliminary version of this manuscript appearing as a conference paper (Altafini, 2015), the constructive procedure proposed here (Algorithm 1) is more general, and recovers the result of Altafini (2015) as a special case (Algorithm 2). Such special case is instrumental to show that a consequence of the existence of a minimal eventually positive realization is that the sequence of Markov parameters that compose the impulse response has decimated subsequences for which a minimal positive realization exists, and can be found downsampling the eventually positive realization.…”

“…In continuous-time, instead, provided the sampling time is chosen sufficiently high, the sampled system obtained from a minimal eventually positive realization is itself positive and minimal. Also in this case (which is not treated in Altafini (2015)), once an eventually positive realization is available, a lower bound on the sampling time leading to a minimal positive sampled realization is known. These results on sampled/downsampled systems can be interpreted as a dual of the usual Nyquist-Shannon theorem: instead of seeking for a sampling frequency sufficiently high so as to preserve all interesting frequencies of the system, if one selects a sampling frequency enough low it is possible to achieve an internal minimal representation of the system which remains positive, because it disregards the high frequency content.…”