Convex Incremental Dissipativity Analysis of Nonlinear Systems - Extended version

TL;DR

This paper introduces a convex incremental dissipativity analysis framework for nonlinear systems that extends stability and performance analysis beyond equilibrium points using matrix inequalities and quadratic functions.

Contribution

It develops a novel convex analysis method linking incremental, differential, and general dissipativity for nonlinear systems, extending classical concepts like $ ext{L}_2$-gain and passivity.

Findings

Provides a matrix inequalities-based convex analysis method

Links incremental, differential, and general dissipativity concepts

Extends classical stability and performance measures to incremental frameworks

Abstract

Efficiently computable stability and performance analysis of nonlinear systems becomes increasingly more important in practical applications. Dissipativity can express stability and performance jointly, but existing results are limited to the regions around the equilibrium points of these nonlinear systems. The incremental framework, based on the convergence of the system trajectories, removes this limitation. We investigate how stability and performance characterizations of nonlinear systems in the incremental framework are linked to dissipativity, and how general performance characterization beyond the $\mathcal{L}_2$-gain concept can be understood in this framework. This paper presents a matrix inequalities-based convex incremental dissipativity analysis for nonlinear systems via quadratic storage and supply functions. The proposed dissipativity analysis links the notions of incremental, differential, and general dissipativity. We show that through differential dissipativity, incremental and general dissipativity of the nonlinear system can be guaranteed. These results also lead to the incremental extensions of the $\mathcal{L}_2$-gain, the generalized $\mathcal{H}_2$-norm, the $\mathcal{L}_\infty$-gain, and passivity of nonlinear systems.