Characterizing nonlinear systems with mixed input-output properties through dissipation inequalities
Sebastiaan van den Eijnden, Thomas Chaffey
TL;DR
Nonlinear systems can exhibit mixed input-output properties, showing finite gain for some inputs and passivity for others. The paper develops a dissipativity-based, storage-function framework inspired by the generalized Kalman–Yakubovich–Popov (KYP) lemma to characterize these mixed properties via specialized supply rates. It provides (i) a constructive storage-function-based characterization (Theorem th:0) and (ii) a feedback interconnection stability result (Theorem th:1) with LMIs, illustrated by examples where classical small-gain or passivity criteria fail. By linking mixed IO behavior to dissipativity and scalable graph concepts, the work enables algorithmic verification and broader applicability to nonlinear stability analysis.
Abstract
Systems that show different characteristics, such as finite-gain and passivity, depending on the nature of the inputs, are said to possess mixed input-output properties. In this paper, we provide a constructive method for characterizing mixed input-output properties of nonlinear systems using a dissipativity framework. Our results take inspiration from the generalized Kalman-Yakubovich-Popov lemma, and show that a system is mixed if it is dissipative with respect to highly specialized supply rates. The mixed input-output characterization is used for assessing stability of feedback interconnections in which the feedback components violate conditions of classical results such as the small-gain and passivity theorem. We showcase applicability of our results through various examples.