A mixed passivity/small-gain theorem for Sobolev input-output stability

A stability theorem for the feedback connection of two (possibly infinite-dimensional) time-invariant linear systems is presented. The theorem is formulated in the frequency domain and is in the spirit of combined passivity/small-gain results. It places a mixture of positive realness and small-gain assumptions on the two transfer functions to ensure a certain notion of input-output stability, called Sobolev stability (which includes the classical L2-stability concept as a special case). The result is more general than the classical passivity and small-gain theorems: strong positive realness of either the plant or controller is not required and the small gain condition only needs to hold on a suitable subset of the open right-half plane. We show that the “mixed” stability theorem is applicable in settings where L2-stability of the feedback connection is not possible, such as output regulation and disturbance rejection of certain periodic signals by so-called repetitive control.