Open AccessSolvability conditions for the positive real lemma equations in the discrete time
Author(s) -
Ferrante Augusto,
Ntogramatzidis Lorenzo
Publication year - 2017
Publication title -
iet control theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.059
H-Index - 108
eISSN - 1751-8652
pISSN - 1751-8644
DOI - 10.1049/iet-cta.2017.0314
Subject(s) - lemma (botany) , passivity , mathematics , stability theory , discrete time and continuous time , control theory (sociology) , realisation , transfer function , positive systems , function (biology) , pure mathematics , control (management) , mathematical analysis , linear system , computer science , nonlinear system , physics , engineering , ecology , statistics , poaceae , electrical engineering , quantum mechanics , artificial intelligence , evolutionary biology , biology
Passivity theory and the positive real lemma have been recognised as two of the cornerstones of modern systems and control theory. As digital control is pervasive in virtually all control applications, developing a general theory on the discrete‐time positive real lemma appears to be an important issue. While for minimal realisations the relations between passivity, positive‐realness and existence of solutions of the positive real lemma equations is very well understood, it seems fair to say that this is not the case in the discrete‐time case, especially when the realisation is non‐minimal and no conditions are assumed on left‐ and/or right‐invertibility of the transfer function. The purpose of this study is to present a necessary and sufficient condition for existence of solutions of the positive real equations under the only assumption that the state matrix A is asymptotically stable.