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

TL;DR

This work addresses robust node-level controllability in structured networks by introducing Fixed Strongly Structurally Controllable Subspace (FSSCS) and Fixed Strongly Structurally Controllable (FSSC) nodes, ensuring controllability across all parameter realizations. It merges graph-theoretical constructs with controllability matrices to derive necessary and sufficient conditions for FSSC identification and to compute the exact dimension of SSCS in hierarchical directed acyclic graphs (HDAGs) through a modified controllability matrix and corresponding subgraph. The approach provides a concrete pathway to determine FSSC nodes and offers a rigorous method to quantify the minimal controllable subspace under parameter variations, with implications for task allocation and resilience in hierarchical networks. The results extend the fixed-structure framework to strong structural controllability and establish exact dimension results for HDAGs, while outlining open challenges for general directed acyclic graphs and future work on SSCS dimension determination. All mathematical notation is used to formalize the subspace relationships and the matrix-graph correspondence central to the analysis.

Abstract

Within the context of structured networks, this paper introduces the concept of the Fixed Strongly Structurally Controllable Subspace (FSSCS), enabling a comprehensive characterization of controllable subspaces. From a graph-theoretical viewpoint, the paper defines Fixed Strongly Structurally Controllable (FSSC) nodes based on the FSSCS concept and establishes the necessary and sufficient conditions for their identification. This paper proposes a method for determining the exact dimension of the Strongly Structurally Controllable Subspace (SSCS) in hierarchical directed acyclic graphs, employing a blend of graph-theoretical approaches and controllability matrix analyses. This approach not only facilitates the identification of FSSC nodes but also enhances our understanding of the robustness of node controllability against variations in network parameters within structured networks, marking a significant advancement in the field of strong structural controllability of individual nodes.

Figures (2)

• Figure 1: Graphs satisfying Condition \ref{['unique_stem_condition']}. In the controllability matrix, (a) From Theorem \ref{['thm_integrator']}, either the multi-term of $i_w$ or $i_w$ can be zero. (b) From Lemma \ref{['lem_mul']}, either the multi-term of $i_w$ or the multi-terms of $i_l$ and $j_l$ can be zero.
• Figure 2: A graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ and its corresponding subgraph $\bar{\mathcal{G}}(\mathcal{V},\bar{\mathcal{E}})$ obtained from Algorithm \ref{['alg_sub']}: (a) From Proposition \ref{['thm_hosoe']}, the dimension of SCS for $\mathcal{G}(\mathcal{V},\mathcal{E})$ is 7. (b) From Proposition \ref{['thm_minimum']}, the dimension of SSCS for $\mathcal{G}(\mathcal{V},\mathcal{E})$ is 5.

Theorems & Definitions (6)

• proof
• proof
• proof
• proof
• proof
• proof