The positive real lemma and construction of all realizations of generalized positive rational functions

Daniel Alpay, Izchak Lewkowicz

Research output: Contribution to journalArticlepeer-review

Abstract

We here extend the well known positive real lemma (also known as the KalmanYakubovichPopov lemma) to a complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, this approach enables us to characterize systems which can be brought, through a static output feedback, to be generalized positive.

Original languageAmerican English
Pages (from-to)985-993
Number of pages9
JournalSystems and Control Letters
Volume60
Issue number12
DOIs
StatePublished - 1 Dec 2011

Keywords

  • Convex invertible cones
  • Generalized positive real functions
  • Linear matrix inequalities
  • Lyapunov inclusion
  • Positive real functions
  • Positive real lemma
  • State space realization
  • Static output-feedback

All Science Journal Classification (ASJC) codes

  • Mechanical Engineering
  • General Computer Science
  • Electrical and Electronic Engineering
  • Control and Systems Engineering

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