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Geometric Control Theory Over Networks: Minimal Node Cardinality Disturbance Decoupling Problems

Luca Claude Gino Lebon, Claudio Altafini

TL;DR

This work reformulates geometric disturbance decoupling for linear networks at the level of node sets, enabling a straightforward graphical interpretation of invariance concepts. It proves that minimal input and (input,output) cardinalities can be computed in polynomial time via min-cut/max-flow on an extended network, and provides explicit feedback laws to achieve decoupling for DDPSF, DDPOF, and DDPDF. The authors introduce maximal controlled invariant and minimal conditioned invariant node-sets, establish duality results, and relate feedback design to edge removals and compensator structures. The resulting framework connects classical geometric control with network topology, enabling scalable design for fault detection, attack resilience, and secure operation in large-scale networks, with concrete algorithms and numerical demonstrations.

Abstract

In this paper we show how to formulate and solve disturbance decoupling problems over networks while choosing a minimal number of input and output nodes. Feedback laws that isolate and eliminate the impact of disturbance nodes on specific target nodes to be protected are provided using state, output, and dynamical feedback. For that, we leverage the fact that when reformulated in terms of sets of nodes rather than subspaces, the controlled and conditional invariance properties admit a simple graphical interpretation. For state and dynamical feedback, the minimal input and output cardinality solutions can be computed exactly in polynomial time, via min-cut/max-flow algorithms.

Geometric Control Theory Over Networks: Minimal Node Cardinality Disturbance Decoupling Problems

TL;DR

This work reformulates geometric disturbance decoupling for linear networks at the level of node sets, enabling a straightforward graphical interpretation of invariance concepts. It proves that minimal input and (input,output) cardinalities can be computed in polynomial time via min-cut/max-flow on an extended network, and provides explicit feedback laws to achieve decoupling for DDPSF, DDPOF, and DDPDF. The authors introduce maximal controlled invariant and minimal conditioned invariant node-sets, establish duality results, and relate feedback design to edge removals and compensator structures. The resulting framework connects classical geometric control with network topology, enabling scalable design for fault detection, attack resilience, and secure operation in large-scale networks, with concrete algorithms and numerical demonstrations.

Abstract

In this paper we show how to formulate and solve disturbance decoupling problems over networks while choosing a minimal number of input and output nodes. Feedback laws that isolate and eliminate the impact of disturbance nodes on specific target nodes to be protected are provided using state, output, and dynamical feedback. For that, we leverage the fact that when reformulated in terms of sets of nodes rather than subspaces, the controlled and conditional invariance properties admit a simple graphical interpretation. For state and dynamical feedback, the minimal input and output cardinality solutions can be computed exactly in polynomial time, via min-cut/max-flow algorithms.
Paper Structure (22 sections, 31 theorems, 15 equations, 3 figures, 3 algorithms)

This paper contains 22 sections, 31 theorems, 15 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Elements of geometric control theory over networks
  3. Notation
  4. Linear systems over networks
  5. Invariance, controlled invariance and conditioned invariance
  6. Maximal controlled invariance, and minimal conditioned invariance
  7. Duality
  8. Disturbance decoupling problems over networks: formulation and solution
  9. Solvability of DDPSF
  10. Solvability of DDPOF
  11. Solvability of DDPDF
  12. Minimal node cardinality DDP
  13. Solving the minimum input cardinality DDPSF
  14. A node mincut/maxflow algorithm
  15. Optimality
  16. ...and 7 more sections

Key Result

Proposition 1

Consider the system eq:lin-syst1. A subset $\mathcal{Z} \subset \mathcal{V}$ is $A$-invariant if and only if any of the following equivalent conditions is met:

Figures (3)

  • Figure 1: Example \ref{['ex:ex1']}. Disturbances are in red, targets in yellow and the control node is in blue.
  • Figure 2: Example \ref{['ex:ex2']}. Disturbances are in red, targets in yellow, control inputs in blue, whereas the output nodes are in purple for the DDPOF and in green for the DDPDF.
  • Figure 3: DDPDF of Example \ref{['ex:ex2']}. Disturbances are in red, targets in yellow, control inputs in blue, output nodes in green whereas the nodes in the observer are in fuchsia. The colored edges reflect the blocks in the closed-loop realization $A_c$.

Theorems & Definitions (65)

  • Remark 1
  • Remark 2
  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Proposition 2
  • proof
  • Definition 4
  • Proposition 3
  • ...and 55 more