Algebraic and Graph-Theoretic Conditions for the Herdability of Linear Time-Invariant Systems

TL;DR

This paper explores algebraic and graph-theoretic conditions that determine when linear time-invariant systems can be driven into the positive orthant, focusing on systems with specific network structures and leader configurations.

Contribution

It introduces new algebraic and graph-based criteria for herdability, including conditions for systems with a single leader and acyclic networks, and shows that follower-leader weights do not influence herdability.

Findings

Herdability depends on algebraic and graph-theoretic properties.

Follower-leader weights do not affect herdability.

Provided an algorithm for leader selection in acyclic networks.

Abstract

In this paper we investigate a relaxed concept of controllability, known in the literature as herdability, namely the capability of a system to be driven towards the(interior of the) positive orthant. Specifically, we investigate herdability for linear time-invariant systems, both from an algebraic perspective and based on the graph representing the systems interactions. In addition, we focus on linear state-space models corresponding to matrix pairs (A;B) in which the matrix B is a selection matrix that determines the leaders in the network, and we show that the weights that followers give to the leaders do not affect the herdability of the system. We then focus on the herdability problem for systems with a single leader in which interactions are symmetric and the network topology is acyclic, in which case an algorithm for the leader selection is provided. In this context, under some additional conditions on the mutual distances, necessary and sufficient conditions for the herdability of the overall system are given.