On the Herdability of Linear Time-Invariant Systems with Special Topological Structures

TL;DR

This paper studies the herdability of linear time-invariant systems with specific network topologies, focusing on conditions under which the system can be driven into the positive orthant, especially in multi-agent networks with certain graph structures.

Contribution

It characterizes herdability conditions for LTI systems with adjacency matrices representing multi-agent networks with special topologies, including directed, balanced, and tree structures.

Findings

Herdability depends on the graph's structure and the placement of leaders.

Directed and structurally balanced graphs exhibit specific herdability properties.

Single-leader tree topologies have distinct herdability conditions.

Abstract

In this paper, we investigate the herdability property, namely the capability of a system to be driven towards the (interior of the) positive orthant, for linear time-invariant state-space models. Herdability of certain matrix pairs (A,B), where A is the adjacency matrix of a multi-agent network, and B is a selection matrix that singles out a subset of the agents (the "network leaders"), is explored. The cases when the graph associated with A, G(A), is directed and clustering balanced (in particular, structurally balanced), or it has a tree topology and there is a single leader, are investigated.