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Investigating the effect of edge modifications on networked control systems

Gustav Lindmark, Claudio Altafini

TL;DR

This work addresses how edge modifications in networked control systems affect stability and performance. It introduces a delta transfer function $G^\delta$ to quantify output changes from single-edge edits and derives exact ${\mathcal{H}_\infty}$ gains and a ${\mathcal{H}_2}$ lower bound for stable positive networks, along with explicit stability margins. For Laplacian (undirected) dynamics, norms are unbounded, but a displacement-system perspective yields computable ${\mathcal{H}_\infty}$ bounds and an exact ${\mathcal{H}_2}$ result for edge additions under full actuation, plus a coherence interpretation tied to network energy. The results enable efficient, large-scale assessment of topology-based improvements to controllability and robustness, with simulations validating the theory on large random graphs and illustrating practical edge-addition strategies to reduce network coherence.

Abstract

This paper investigates the impact of addition/removal of edges in a complex networked control system, for the purposes of improving its controllability, system performances or robustness to external disturbances. The transfer function formulation we obtain allows to quantify the impact of an edge modification with the $\hinf$ and $\htwo$ norms. For stable networks with positive edge weights, we show that the $\hinf$ norm can be computed exactly for each possible single edge modification, as well as the associated stability margin. For the $\htwo$ norm we instead obtain a lower bound. Since this bound is linked to the trace of the controllability Gramian, it can be used for instance to reduce the energy needed for control. When instead the dynamics is of Laplacian type, then the norms become unbounded. However, the associated displacement systems are stable and for them the effect of edge modifications can be quantified. In particular, in this case we provide an upper bound on the $ \hinf$ norm and compute the exact value of the $ \htwo $ norm for arbitrary edge additions.

Investigating the effect of edge modifications on networked control systems

TL;DR

This work addresses how edge modifications in networked control systems affect stability and performance. It introduces a delta transfer function to quantify output changes from single-edge edits and derives exact gains and a lower bound for stable positive networks, along with explicit stability margins. For Laplacian (undirected) dynamics, norms are unbounded, but a displacement-system perspective yields computable bounds and an exact result for edge additions under full actuation, plus a coherence interpretation tied to network energy. The results enable efficient, large-scale assessment of topology-based improvements to controllability and robustness, with simulations validating the theory on large random graphs and illustrating practical edge-addition strategies to reduce network coherence.

Abstract

This paper investigates the impact of addition/removal of edges in a complex networked control system, for the purposes of improving its controllability, system performances or robustness to external disturbances. The transfer function formulation we obtain allows to quantify the impact of an edge modification with the and norms. For stable networks with positive edge weights, we show that the norm can be computed exactly for each possible single edge modification, as well as the associated stability margin. For the norm we instead obtain a lower bound. Since this bound is linked to the trace of the controllability Gramian, it can be used for instance to reduce the energy needed for control. When instead the dynamics is of Laplacian type, then the norms become unbounded. However, the associated displacement systems are stable and for them the effect of edge modifications can be quantified. In particular, in this case we provide an upper bound on the norm and compute the exact value of the norm for arbitrary edge additions.

Paper Structure

This paper contains 19 sections, 18 theorems, 96 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Network model
  5. System norms and network centrality measures
  6. Edge modifications in networks with stable dynamics
  7. Stability bounds and the ${\mathcal{H}_\infty}$ norm of $G^\delta$
  8. The ${\mathcal{H}_2}$ norm of $G^\delta$
  9. Interpretations
  10. Edge addition in undirected networks with Laplacian dynamics
  11. Transfer function for the displacement system and its ${\mathcal{H}_\infty}$ norm
  12. An upper bound on the ${\mathcal{H}_\infty}$ norm for edge additions
  13. Coherence as ${\mathcal{H}_2}$ norm
  14. The change in coherence from edge additions
  15. Applications
  16. ...and 4 more sections

Key Result

Proposition 1

Let $G(\theta)$ be the frequency function of a stable externally positive LTI system. Then

Figures (3)

  • Figure 1: Block diagram of the delta system $G^\delta$.
  • Figure 2: Numerical computations on a random Erdős–Rényi network with 500 nodes. For all $\mathfrak{s},\mathfrak{t} \in \mathcal{V}$, $\mathfrak{s} \neq \mathfrak{t}$ and weight $w=10$, the plots show (a) the stability margins $1/(\left(I - A\right)^{-1})_{\mathfrak{s}\mathfrak{t} }$, (b) the norm $||G^\delta||_{\mathcal{H}_\infty}$ and the lower bound of $||G^\delta||_{\mathcal{H}_2}$. Edges are ordered along the x-axis in ascending order in both cases. For a sub-selection of 30 edges, also the exact value of $||G^\delta||_{\mathcal{H}_2}$ has been computed.
  • Figure 3: Iterative edge addition in networks with Laplacian dynamics. The original (blue) line network is grown with new edges (red) that efficiently improve the network coherence.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Proposition 1: tanaka2011bounded
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Remark 1
  • ...and 13 more