# On Structured Lyapunov Functions and Dissipativity in Interconnected LTI Systems

## TL;DR

This paper explores how structured Lyapunov functions relate to dissipativity in interconnected LTI systems, showing that additive quadratic Lyapunov functions imply individual system dissipativity and robustness to link removal.

## Contribution

It establishes a connection between structured Lyapunov functions and dissipativity in acyclic interconnected LTI systems, providing new characterizations of neutral supply functions.

## Key findings

- Additive quadratic Lyapunov functions imply individual system dissipativity.
- Dissipativity is robust to removal of interconnection links.
- Characterizations of neutral supply functions are provided.

## Abstract

In this paper we study connections between structured storage or Lyapunov functions of a class of interconnected systems (dynamical networks) and dissipativity properties of the individual systems. We prove that if a dynamical network, composed as a set of linear time invariant (LTI) systems interconnected over an acyclic graph, admits an additive quadratic Lyapunov function, then the individual systems in the network are dissipative with respect to a (nonempty) set of interconnection neutral supply functions. Each supply function from this set is defined on a single interconnection link in the network. Specific characterizations of neutral supply functions are presented which imply robustness of network stability/dissiptivity to removal of interconnection links.