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

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, lowpasshighpass, 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. key words: continuous-time state-space system, second-order mode (Hankel singular value), frequency transformation, invariance, controllability Gramian, observability Gramian