Analysis of Second-Order Modes of Linear Discrete-Time Systems under Bounded-Real Transformations

Shunsuke KOSHITA  Masahide ABE  Masayuki KAWAMATA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E90-A   No.11   pp.2510-2515
Publication Date: 2007/11/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e90-a.11.2510
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Systems and Control
discrete-time state-space system,  second-order mode (Hankel singular value),  bounded-real transformation,  controllability Gramian,  observability Gramian,  discrete-time bounded-real lemma,  

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This paper discusses the behavior of the second-order modes (Hankel singular values) of linear discrete-time systems under bounded-real transformations, where the transformations are given by arbitrary transfer functions with magnitude bounded by unity. Our main result reveals that the values of the second-order modes are decreased under any of the above-mentioned transformations. This result is the generalization of the theory of Mullis and Roberts, who proved that the second-order modes are invariant under any allpass transformation, i.e. any lossless bounded-real transformation. We derive our main result by describing the controllability/observability Gramians of transformed systems with the help of the discrete-time bounded-real lemma.