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Mathematical Problems in Engineering
Volume 2016, Article ID 9605464, 11 pages
http://dx.doi.org/10.1155/2016/9605464
Research Article

-Stability of Positive Linear Systems

Department of Automatic Control and Applied Informatics, Technical University “Gheorghe Asachi” of Iasi, 700050 Iasi, Romania

Received 5 August 2015; Revised 11 January 2016; Accepted 14 January 2016

Academic Editor: Asier Ibeas

Copyright © 2016 Octavian Pastravanu and Mihaela-Hanako Matcovschi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this work is to show that the Perron-Frobenius eigenstructure of a positive linear system is involved not only in the characterization of long-term behavior (for which well-known results are available) but also in the characterization of short-term or transient behavior. We address the analysis of the short-term behavior by the help of the “-stability” concept introduced in literature for general classes of dynamics. Our paper exploits this concept relative to Hölder vector -norms, , adequately weighted by scaling operators, focusing on positive linear systems. Given an asymptotically stable positive linear system, for each , we prove the existence of a scaling operator (built from the right and left Perron-Frobenius eigenvectors, with concrete expressions depending on ) that ensures the best possible values for the parameters and , corresponding to an “ideal” short-term (transient) behavior. We provide results that cover both discrete- and continuous-time dynamics. Our analysis also captures the differences between the cases where the system dynamics is defined by matrices irreducible and reducible, respectively. The theoretical developments are applied to the practical study of the short-term behavior for two positive linear systems already discussed in literature by other authors.