Duality and Network Theory in Passivity-based Cooperative Control

TL;DR

This paper establishes a novel connection between passivity-based cooperative control and convex network optimization, introducing maximal equilibrium independent passivity and demonstrating how system dynamics relate to network duality, with applications to traffic models.

Contribution

It introduces the concept of maximal equilibrium independent passivity and links passivity-based control to convex network optimization, providing a unified analysis framework.

Findings

Networks of systems with this passivity property approach dual optimization solutions

Duality relations from convex optimization translate to control variables

Application to nonlinear traffic dynamics shows asymptotic clustering

Abstract

This paper presents a class of passivity-based cooperative control problems that have an explicit connection to convex network optimization problems. The new notion of maximal equilibrium independent passivity is introduced and it is shown that networks of systems possessing this property asymptotically approach the solutions of a dual pair of network optimization problems, namely an optimal potential and an optimal flow problem. This connection leads to an interpretation of the dynamic variables, such as system inputs and outputs, to variables in a network optimization framework, such as divergences and potentials, and reveals that several duality relations known in convex network optimization theory translate directly to passivity-based cooperative control problems. The presented results establish a strong and explicit connection between passivity-based cooperative control theory on the one side and network optimization theory on the other, and they provide a unifying framework for network analysis and optimal design. The results are illustrated on a nonlinear traffic dynamics model that is shown to be asymptotically clustering.