Interconnections of nonlinear systems with “mixed” small gain and passivity properties and associated input–output stability results

Abstract

A negative feedback interconnection consisting of two causal, 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 gain. More importantly, the property captures the concept of a “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 passivity-type properties at, say, low frequencies; and lose these passivity-type properties but have small gain at high frequencies.