Herdability of Linear Systems Based on Sign Patterns and Graph Structures

We consider 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. First a number of foundational results on herdability of a continuous time, linear time invariant system are presented. These show that the herdability of a linear system can be determined based on certain matrices, such as the controllability matrix, which arise in the study of controllability of linear systems. Second, the relationship between the sign pattern of the underlying graph structure of a system and the herdability properties of the system is investigated. In doing so the notion of sign herdability is introduced which captures classes of systems whose sign pattern determines their herdability. We identify a set of conditions, first on the sign pattern of the controllability matrix and then on the underlying graph structure, that ensure that the system is sign herdable.