Analysis of second-order modes of linear continuous-time systems under positive-real transformations

Shunsuke Koshita, Yousuke Mizukami, Taketo Konno, Masahide Abe, Masayuki Kawamata

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


This paper discusses the behavior of the second-order modes (Hankel singular values) of linear continuous-time systems under variable transformations with positive-real functions. That is, given a transfer function H(s) and its second-order modes, we analyze the second-order modes of transformed systems H(F(s)), where 1/F(s) is an arbitrary positive-real function. We first discuss the case of lossless positive-real transformations, and show that the second-order modes are invariant under any lossless positive-real transformation. We next consider the case of general positive-real transformations, and reveal that the values of the second-order modes are decreased under any general positive-real transformation. We achieve the derivation of these results by describing the controllability/observability Gramians of transformed systems, with the help of the lossless positive-real lemma, the positive-real lemma, and state-space formulation of transformed systems.

Original languageEnglish
Pages (from-to)575-583
Number of pages9
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number2
Publication statusPublished - 2008 Jan 1


  • Controllability gramian
  • General positive-real transformation
  • Linear continuous-time system
  • Lossless positive-real transformation
  • Observability Gramian
  • Second-order mode (Hankel singular value)

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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