Invariance of Second-Order Modes of Linear Continuous-Time Systems under Typical Frequency Transformations

Masayuki KAWAMATA  Yousuke MIZUKAMI  Shunsuke KOSHITA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E90-A   No.7   pp.1481-1486
Publication Date: 2007/07/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e90-a.7.1481
Print ISSN: 0916-8508
Type of Manuscript: LETTER
Category: Systems and Control
continuous-time state-space system,  second-order mode (Hankel singular value),  frequency transformation,  invariance,  controllability Gramian,  observability Gramian,  

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This paper discusses the behavior of the second-order modes (Hankel singular values) of linear continuous-time systems under typical frequency transformations, such as lowpass-lowpass, lowpass-highpass, lowpass-bandpass, and lowpass-bandstop transformations. Our main result establishes the fact that the second-order modes are invariant under any of these typical frequency transformations. This means that any transformed system that is generated from a prototype system has the same second-order modes as those of the prototype system. We achieve the derivation of this result by describing the state-space equations and the controllability/observability Gramians of transformed systems.