Obtaining Structural Network Controllability with Higher-Order Local Dynamics

TL;DR

This work shows that upgrading a subset of nodes in networks of identical first‑order units to higher‑order dynamics can achieve structural output controllability with potentially far fewer modifications than using heterogeneous first‑order subsystems. By linking state controllability in the original network to output controllability in the lifted network, it classifies topologies into X‑networks (amenable to homogeneous higher‑order upgrades) and Y‑networks (requiring heterogeneous upgrades) and provides constructive design procedures. The authors develop graph‑theoretic and PBH‑test tools to analyze and synthesize upgrades, supported by case studies on binary trees and single bifurcations that demonstrate scalable gains in the number of required modifications. The results offer practical guidelines for designing large‑scale networked controllers while preserving homogeneous local dynamics, with potential applications in modular and scalable control systems.

Abstract

We consider a network of identical, first-order linear systems, and investigate how replacing a subset of the systems composing the network with higher-order ones, either taken to be generic or specifically designed, may affect its controllability. After establishing a correspondence between state controllability in networks of first-order systems with output controllability in networks of higher-order systems, we show that adding higher-order dynamics may require significantly fewer subsystem modifications to achieve structural controllability, when compared to first-order heterogeneous subsystems. Furthermore, we characterize the topology of networks (which we call X-networks) in which the introduction of heterogeneous local dynamics is not necessary for structural output controllability, as the latter can be attained by suitable higher-order subsystems with homogeneous internal dynamics.

Figures (5)

• Figure 1: Example of original network (a). Node 3 of (a) is replaced by a subsystem with first-order heterogeneous dynamics (b). Node 3 of (a) is replaced by a subsystem with higher-order heterogeneous dynamics (c). The extended subsystems are marked in red. All the edges have generic weights. The input node is highlighted in violet.
• Figure 2: Examples of X-networks are given in subfigures (a)-(b)-(c), and Y-networks in (d). Each network is covered by a set of paths which exhibit a critical interconnection outlined in Corollary \ref{['cor:ctrb_cases']}. Two non-vertex-disjoint stems originated from different input nodes are depicted in (a), a stem sharing a node with a cycle in (b), two non-vertex-disjoint cycles in (c). (d) presents two stems that share a input and a state node. The input nodes are highlighted in violet. The edges connecting nodes in the considered covering paths are solid, other edges are dotted.
• Figure 3: Examples of higher-order subsystems placement and design described in the proof of Theorem \ref{['thm:hoi']}. The depicted networks are the expanded X-networks corresponding to those illustrated in subfigures \ref{['fig:XY_ex']} (a)-b)-(c). The input nodes are highlighted in violet. Nodes corresponding to the same higher-order subsystem are marked in red. The edges connecting nodes in vertex-disjoint stems and cycles covering all state nodes are solid. The other edges are dotted.
• Figure 4: (a) Binary tree network of height $h$. The red ellipses denote the nodes associated to a higher-order subsystem in ${\rm G}(\hat{\Sigma})$, which include all nodes of the tree except for the leaves (terminal nodes). The state nodes $x_i\in\mathcal{V}$ are denoted simply by $i$ to reduce cluttering.
• Figure 5: (a) Single bifurcation network of height $h$. (b) Extended single bifurcation network. The red ellipses denote the extended nodes, which include the even nodes of the right branch of the original network ($h$ is assumed to be even). The state nodes $x_i\in\mathcal{V}$ are denoted simply by $i$ to reduce cluttering.

Theorems & Definitions (30)

• Remark 1
• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1
• Corollary 1
• proof
• Definition 1