A Stability Result for Interconnections of Nonlinear Systems with "Mixed" Small Gain and Passivity Properties

Abstract

A feedback interconnection consisting of two nonlinear systems is shown to be input-output stable when a "mixed" small gain and passivity assumption is placed on each of the systems. The "mixed" small gain and passivity property captures the well-known notions of passivity and small gain associated with systems: the property can be appropriately reduced to an input and output strictly passive system description; or alternatively can be reduced to a description of a system with small, finite gain. More importantly, the property captures a concept of "blending" of the small gain and passivity ideas. This concept of "blending" can be visualized, for example, by considering linear time-invariant systems that exhibit passivetype properties at, say low frequencies; and lose these passivetype properties but have small gain at high frequencies.