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Losing Control of your Network? Try Resilience Theory

Jean-Baptiste Bouvier, Sai Pushpak Nandanoori, Melkior Ornik

TL;DR

The paper addresses resilience of linear cyber-physical networks to partial loss of control authority over actuators, introducing an equivalence condition for resilient stabilizability of the network. It develops quantitative resilience analyses for fully-actuated and underactuated networks, and extends the framework to nonresilient cases to bound their impact on the rest of the system. The authors validate the theory with numerical examples, including islanded microgrids and the linearized IEEE 39-bus system, illustrating finite-time stabilization, boundedness, and the limitations posed by sparse connectivity. The work provides a rigorous foundation for designing robust controllers under actuator compromise with practical implications for power systems and other critical infrastructures.

Abstract

Resilience of cyber-physical networks to unexpected failures is a critical need widely recognized across domains. For instance, power grids, telecommunication networks, transportation infrastructures and water treatment systems have all been subject to disruptive malfunctions and catastrophic cyber-attacks. Following such adverse events, we investigate scenarios where a node of a linear network suffers a loss of control authority over some of its actuators. These actuators are not following the controller's commands and are instead producing undesirable outputs. The repercussions of such a loss of control can propagate and destabilize the whole network despite the malfunction occurring at a single node. To assess system vulnerability, we establish resilience conditions for networks with a subsystem enduring a loss of control authority over some of its actuators. Furthermore, we quantify the destabilizing impact on the overall network when such a malfunction perturbs a nonresilient subsystem. We illustrate our resilience conditions on two academic examples, on an islanded microgrid, and on the linearized IEEE 39-bus system.

Losing Control of your Network? Try Resilience Theory

TL;DR

The paper addresses resilience of linear cyber-physical networks to partial loss of control authority over actuators, introducing an equivalence condition for resilient stabilizability of the network. It develops quantitative resilience analyses for fully-actuated and underactuated networks, and extends the framework to nonresilient cases to bound their impact on the rest of the system. The authors validate the theory with numerical examples, including islanded microgrids and the linearized IEEE 39-bus system, illustrating finite-time stabilization, boundedness, and the limitations posed by sparse connectivity. The work provides a rigorous foundation for designing robust controllers under actuator compromise with practical implications for power systems and other critical infrastructures.

Abstract

Resilience of cyber-physical networks to unexpected failures is a critical need widely recognized across domains. For instance, power grids, telecommunication networks, transportation infrastructures and water treatment systems have all been subject to disruptive malfunctions and catastrophic cyber-attacks. Following such adverse events, we investigate scenarios where a node of a linear network suffers a loss of control authority over some of its actuators. These actuators are not following the controller's commands and are instead producing undesirable outputs. The repercussions of such a loss of control can propagate and destabilize the whole network despite the malfunction occurring at a single node. To assess system vulnerability, we establish resilience conditions for networks with a subsystem enduring a loss of control authority over some of its actuators. Furthermore, we quantify the destabilizing impact on the overall network when such a malfunction perturbs a nonresilient subsystem. We illustrate our resilience conditions on two academic examples, on an islanded microgrid, and on the linearized IEEE 39-bus system.
Paper Structure (16 sections, 22 theorems, 104 equations, 12 figures)

This paper contains 16 sections, 22 theorems, 104 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Networks preliminaries
  3. Stabilizability of resilient networks
  4. Background results
  5. Stabilizability results
  6. Resilient stabilizability results
  7. Stabilizability of nonresilient networks
  8. Fully-actuated networks
  9. Underactuated networks
  10. Numerical examples
  11. Fully actuated 3-component network
  12. Underactuated 3-component network
  13. Microgrid test system
  14. Resilient stabilizability of the IEEE 39-bus system
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 1

If $\bar{\mathcal{U}} \cap \ker(\bar{B}) \neq \emptyset$ and $\mathop{\mathrm{int}}\nolimits(\mathop{\mathrm{co}}\nolimits(\bar{\mathcal{U}})) \neq \emptyset$, then system eq:initial ODE is stabilizable (resp. controllable) if and only if $\mathop{\mathrm{rank}}\nolimits( \mathcal{C}(A,\bar{B}) ) =

Figures (12)

  • Figure 1: Time evolution of states $\chi$ and $x_N$ along with their bounds \ref{['eq: x_N bound incomplete']}, \ref{['eq: X bound']} and \ref{['eq: x_N bound']}.
  • Figure 2: Illustration of bounds \ref{['eq: x_N bound incomplete']}, \ref{['eq: X bound not full rank']} and \ref{['eq: x_N bound underactuated']} on the states $\chi$ and $x_N$.
  • Figure 3: Linear feedback $\hat{u}(t) = -K\chi(t)$.
  • Figure 4: Single-line diagram of the microgrid test system from bidram2013distributed.
  • Figure 5: Topology of the communication digraph.
  • ...and 7 more figures

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1: Brammer's conditions Brammer
  • Theorem 2: Sontag's condition Sontag
  • Proposition 1
  • Theorem 3: Necessary and sufficient condition ECC_extended
  • ...and 37 more