A “Mixed” Small Gain and Passivity Theorem for an Interconnection of Linear Time-Invariant Systems

We show that the negative feedback interconnection of two causal, stable, linear time-invariant systems with a “mixed” small gain and passivity frequency domain property is guaranteed to be finite-gain stable. This “mixed” small gain and passivity property refers to the characteristic that the frequency range −∞ < ω < ∞ can be divided into intervals for which the two systems in the interconnection are both: a) “input and output strictly passive” (and one or both of the systems may or may not have “gain less than one”); or b) “input and output strictly passive and with gain less than one”; or c) “with gain less than one” (and one or both of the systems may or may not be “input and output strictly passive”). The “mixed” small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite-gain stability of the interconnection is proven via a stability result for dissipative interconnected systems.