[Paper Review] Herdability of Linear Systems Based on Sign Patterns and Graph Structures

Sebastian F. Ruf, Magnus Egerstedt+1 more|arXiv (Cornell University)|Apr 17, 2019

Stability and Control of Uncertain Systems45 references4 citations

TL;DR

This paper introduces herdability as a set-based reachability condition for continuous-time linear time-invariant (LTI) systems, determining whether all state components can be driven above a non-negative threshold. It establishes sign-based sufficient conditions for herdability using graph structures and controllability matrix sign patterns, enabling structural analysis without exact edge weights.

ABSTRACT

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.

Motivation & Objective

• To formalize herdability as a set-based reachability condition where all state components must exceed a non-negative threshold, rather than full controllability.
• To investigate how the sign pattern of system matrices A and B, and the underlying graph structure, determine herdability independently of exact edge weights.
• To develop sufficient conditions for complete herdability based on the sign structure of the controllability matrix and graph-based reachability paths.
• To extend structural controllability concepts to herdability by analyzing positive and negative path contributions in directed graphs.
• To identify classes of systems that are sign-herdable based on structural balance and path sign consistency in network topologies.

Proposed method

• Define herdability as the ability to drive all state components above a threshold d ≥ 0 using control inputs, independent of full controllability.
• Use the controllability matrix C and its sign pattern to derive sufficient conditions for herdability, focusing on column signs across walks from inputs to states.
• Introduce the sets 𝒩_d^j and 𝒫_d^j to classify nodes based on the net sign of all walks of length d from input u_j to state x_i.
• Define sign-strict herdability when all walks from an input to a state have consistent sign, ensuring monotonic influence.
• Introduce sign-balance for nodes where conflicting paths are balanced by sign-strictly-herdable nodes, enabling broader herdability analysis.
• Provide algorithms to compute 𝒩_d^j and 𝒫_d^j via graph traversal, with linear-time complexity for DAGs and exponential complexity otherwise.

Experimental results

Research questions

• RQ1 Under what conditions on the sign pattern of the controllability matrix can a linear system be completely herdable?
• RQ2 How does the underlying graph structure of a system influence its herdability, independent of edge weight magnitudes?
• RQ3 Can herdability be determined solely from the sign structure of system matrices A and B, and if so, under what structural assumptions?
• RQ4 What role does structural balance in the graph play in determining sign herdability, and how does it simplify analysis?
• RQ5 What are the computational complexities of verifying herdability based on graph traversal of signed paths?

Key findings

• Herdability of a linear system can be determined by analyzing the sign pattern of the controllability matrix C, with consistent column signs indicating potential herdability.
• A system is completely sign-herdable if every state node is either sign-strictly herdable (all paths from inputs have consistent sign) or sign-balanced (conflicting paths are dominated by sign-strictly-herdable nodes).
• Theorem 9 provides a sufficient condition for complete sign herdability: for each state node, there exists an input and distance d such that the node is reachable via only positive or only negative walks.
• Theorem 10 extends this to sign-balanced nodes, showing that if all nodes are balanced by sign-strictly-herdable nodes, the system is completely sign-herdable.
• For directed acyclic graphs (DAGs), the sets 𝒩_d^j and 𝒫_d^j can be computed in linear time, enabling efficient herdability verification.
• In non-DAGs, the computation may require exponential time due to the need to enumerate all walks, limiting practical scalability to sparse or acyclic topologies.