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

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 boundedreal 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. key words: discrete-time state-space system, second-order mode (Hankel singular value), bounded-real transformation, controllability Gramian, observability Gramian, discrete-time bounded-real lemma