Analysis of Second-Order Modes of Linear Continuous-Time Systems under Positive-Real Transformations

Shunsuke KOSHITA  Yousuke MIZUKAMI  Taketo KONNO  Masahide ABE  Masayuki KAWAMATA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E91-A   No.2   pp.575-583
Publication Date: 2008/02/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e91-a.2.575
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Systems and Control
Keyword: 
linear continuous-time system,  second-order mode (Hankel singular value),  lossless positive-real transformation,  general positive-real transformation,  controllability Gramian,  observability Gramian,  

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Summary: 
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.