Determining “mixedness” and an application of finite-gain stability results to “mixed” system interconnections

A loss of passivity in the face of certain frequency dynamics (eg: high frequency dynamics) given an otherwise passive system leads to the notion of a “mixed” system. A “mixed” system is one that has a concept of small gain associated with it over those frequency intervals where passivity is lost. In this paper, a test for determining “mixedness” for linear, time-invariant systems is provided and several finite-gain stability results for interconnections of such systems are presented. The “mixedness” test involves examining the spectral characteristics of two Hamiltonian matrices, one associated with the passive aspects of the system and one associated with the notion of small gain. The finite-gain stability results are derived using a dissipative systems framework.