The converse of the passivity and small-gain theorems for input-output maps

TL;DR

This paper establishes the necessary conditions for stability in feedback systems involving passive and small-gain properties, extending classical theorems to time-varying and nonlinear contexts.

Contribution

It proves the converse of the passivity and small-gain theorems for time-varying systems, providing necessary conditions for stability in complex feedback interconnections.

Findings

Output strict passivity is necessary for stability with passive environments.

The small-gain condition is necessary for stability in certain time-varying systems.

The results extend classical linear control theorems to nonlinear and time-varying systems.

Abstract

We prove the following converse of the passivity theorem. Consider a causal system given by a sum of a linear time-invariant and a passive linear time-varying input-output map. Then, in order to guarantee stability (in the sense of finite L2-gain) of the feedback interconnection of the system with an arbitrary nonlinear output strictly passive system, the given system must itself be output strictly passive. The proof is based on the S-procedure lossless theorem. We discuss the importance of this result for the control of systems interacting with an output strictly passive, but otherwise completely unknown, environment. Similarly, we prove the necessity of the small-gain condition for closed-loop stability of certain time-varying systems, extending the well-known necessity result in linear robust control.