Herdable Systems Over Signed, Directed Graphs

TL;DR

This paper introduces the concept of herdability in linear systems, analyzing how the structure of signed, directed graphs influences the ability to control system states to exceed thresholds, with theoretical characterizations and classifications.

Contribution

It provides a necessary and sufficient condition for herdability and characterizes herdable nodes based on graph structure, especially for systems with a directed out-branching.

Findings

Identifies conditions for herdability in signed, directed graphs.

Classifies nodes based on walk length and sign for herdability.

Characterizes herdable nodes in systems with a directed out-branching.

Abstract

This paper considers the notion of herdability, a set-based reachability condition, which asks whether the state of a system can be controlled to be element-wise larger than a non-negative threshold. The basic theory of herdable systems is presented, including a necessary and sufficient condition for herdability. This paper then considers the impact of the underlying graph structure of a linear system on the herdability of the system, for the case where the graph is represented as signed and directed. By classifying nodes based on the length and sign of walks from an input, we find a class of completely herdable systems as well as provide a complete characterization of nodes that can be herded in systems with an underlying graph that is a directed out-branching rooted at a single input.