Minimal eventually positive realizations of externally positive systems. (June 2016)

• Record Type: Journal Article
• Authors: Altafini, Claudio
• Is Part Of: Automatica. Volume 68(2016)
• Page Start: 140
• Page End: 147
• Publication Date: 2016-06
• Subjects: Positive linear systems -- Minimal realization -- Eventually positive matrices -- Perron–Frobenius theorem; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2016.01.072
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: It is a well-known fact that externally positive linear systems may fail to have a minimal positive realization. In order to investigate these cases, we introduce the notion of minimal eventually positive realization, for which the state update matrix becomes positive after a certain power. Eventually positive realizations capture 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. As a consequence, we show that in discrete-time it is possible to use downsampling to obtain minimal positive realizations matching decimated sequences of Markov coefficients of the impulse response. In continuous-time, instead, if the sampling time is chosen sufficiently long, a minimal eventually positive realization leads always to a sampled realization which is minimal and positive.