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

Shunsuke KOSHITA; Yousuke MIZUKAMI; Taketo KONNO; Masahide ABE; Masayuki KAWAMATA

doi:10.1093/ietfec/e91-a.2.575

Regular Section

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

Shunsuke KOSHITA, Yousuke MIZUKAMI, Taketo KONNO, Masahide ABE, Masayuki KAWAMATA

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Keywords: 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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2008 Volume E91.A Issue 2 Pages 575-583

DOI https://doi.org/10.1093/ietfec/e91-a.2.575

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Abstract

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.

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