Differential dissipativity theory for dominance analysis

TL;DR

This paper introduces a differential dissipativity framework for analyzing dominance in high-dimensional nonlinear systems, enabling simplified analysis of multistability and oscillations through linear dissipation inequalities.

Contribution

It extends classical dissipativity theory to a nonlinear setting for dominance analysis, providing a new approach to study multistability and oscillatory behaviors.

Findings

Dominance can be characterized via linear dissipation inequalities.

The framework generalizes stability analysis to multistability and oscillations.

Provides a tractable method for analyzing complex nonlinear systems.

Abstract

High-dimensional systems that have a low-dimensional dominant behavior allow for model reduction and simplified analysis. We use differential analysis to formalize this important concept in a nonlinear setting. We show that dominance can be studied through linear dissipation inequalities and an interconnection theory that closely mimics the classical analysis of stability by means of dissipativity theory. In this approach, stability is seen as the limiting situation where the dominant behavior is 0-dimensional. The generalization opens novel tractable avenues to study multistability through 1-dominance and limit cycle oscillations through 2-dominance.