A dissipativity theorem for p-dominant systems

TL;DR

This paper generalizes the classical dissipativity theorem to analyze p-dominance, enabling stability analysis for systems converging to low-dimensional attractors, thus broadening the scope of nonlinear system analysis.

Contribution

It introduces a generalized dissipativity framework for p-dominance, extending classical stability theory to systems with low-dimensional attractors.

Findings

Provides a dissipativity-based criterion for p-dominance

Enables analysis of systems converging to p-dimensional attractors

Connects p-dominance with a generalized contraction property

Abstract

We revisit the classical dissipativity theorem of linear-quadratic theory in a generalized framework where the quadratic storage is negative definite in a p-dimensional subspace and positive definite in a complementary subspace. The classical theory assumes p = 0 and provides an inter- connection theory for stability analysis, i.e. convergence to a zero dimensional attractor. The generalized theory is shown to provide an interconnection theory for p-dominance analysis, i.e. convergence to a p-dimensional dominant subspace. In turn, this property is the differential characterization of a generalized contraction property for nonlinear systems. The proposed generalization opens a novel avenue for the analysis of interconnected systems with low-dimensional attractors.