Controllable Subspaces in Structured Networks of Hierarchical Directed Acyclic Graphs: Controllability of Individual Nodes

Abstract

This paper introduces the concept of the Fixed Strongly Structurally Controllable Subspace (FSSCS) within the context of structured networks, enabling a comprehensive characterization of controllable subspaces. From a graph-theoretical perspective, we define Fixed Strongly Structurally Controllable (FSSC) nodes, which are nodes (or states) that remain controllable for all network parameters within a structured network. Furthermore, we establish the necessary and sufficient conditions for identifying FSSC nodes in general graphs. This paper also proposes a method to exactly determine the dimension of the Strongly Structurally Controllable Subspace (SSCS) in hierarchical directed acyclic graphs, using a combination of graph-theoretical approaches and controllability matrix analyses. This method facilitates the identification of FSSC nodes and enhances our understanding of the robustness of node controllability against parameter variations in structured networks. © 2025 Elsevier B.V., All rights reserved.