PaperlayerView exact revision on arXiv

Source-linked AI summary

Observability Blocking in a Linear Synchronization Network with Partial State Measurements

Alexis Moreno, Abdullah Al Maruf

arXiv:2608.25259v1eess.SY

TL;DR

The paper addresses observability blocking when operators lack full state access in large synchronization networks. It develops output-feedback, observer-based, and distributed control designs using partial measurements. The approaches preserve selected or all open-loop eigenvalues, with simulations demonstrating their scope and validity.

  • Problem

    Existing state-feedback observability-blocking methods require full system-state access, whereas large networks provide only partial measurements.

  • Method

    The paper develops output-feedback and observer-based controllers using partial measurements, then extends the observer-based design to a distributed framework.

  • Results

    The output-feedback approach preserves a subset of open-loop eigenvalues, while the observer-based approach preserves all open-loop eigenvalues and offers greater sensor-placement flexibility.

  • Takeaways & Limitations

    The proposed designs provide partial-measurement-based alternatives for blocking observability at compromised nodes, including a distributed observer-based framework.

Abstract

from arXiv · show

Large-scale networked systems are increasingly vulnerable to adversaries that can infer system dynamics from a small set of compromised nodes. This paper addresses the problem of blocking such inference using limited state information. While existing state-feedback methods achieve observability blocking with eigenvalue preservation, they require full state access and are impractical for large networks. We propose multiple control strategies that operate under partial state measurements. The first approach employs output feedback to achieve observability blocking while preserving a subset of open-loop eigenvalues. The second approach leverages an observer to reconstruct the system state and enables full-state feedback control, preserving all eigenvalues and providing greater flexibility in sensor placement. We further extend the observer-based design to a distributed framework. Numerical examples demonstrate the scope and validity of the proposed methods.

I. INTRODUCTION

The study addresses the impracticality of full-state state-feedback observability blocking in large networks by developing methods that use partial measurements. It presents output-feedback, observer-based, and distributed strategies that preserve selected or all dynamical properties while blocking adversarial reconstruction.

  • Full-state access makes earlier state-feedback observability-blocking methods impractical for large networks with distributed sensing and communication constraints.
  • The study develops control schemes that use partial measurements to prevent reconstruction of critical states at vulnerable locations.
  • The proposed output-feedback scheme blocks observability at compromised nodes using measurements from designated sensor nodes.
  • The observer-based approach estimates system states, enables state-feedback control, offers greater sensor-placement flexibility, and can preserve all open-loop eigenvalues.
  • A distributed framework extends the observer-based design, and simulation case studies illustrate the procedures and validate the theoretical results.

II. NETWORK MODEL AND PROBLEM FORMULATION

The paper models a weighted directed linear synchronization network with actuation, sensor, and compromised nodes. The operator controls the system using sensor measurements to make the closed-loop dynamics unobservable to adversarial measurements.

  • The network is a continuous-time linear synchronization system of n interconnected nodes represented by a weighted directed graph.
  • Actuation nodes receive control inputs, sensor nodes provide operator measurements, and compromised nodes provide measurements available to an adversary.
  • The dynamics use the graph Laplacian as the state matrix and an input matrix selecting the actuation nodes.
  • The model assumes the pair (−L, B) is controllable.
  • The operator seeks feedback controls based only on sensor measurements that make the closed-loop model unobservable through compromised-node measurements.

III. MAIN RESULTS

The main results develop output-feedback, observer-based, and distributed approaches to block observability. These methods construct controllers using partial measurements while targeting preservation of desired network dynamics.

  • The paper first presents an output-feedback approach for blocking observability at compromised nodes.
  • The observer-based approach constructs a state observer from sensor measurements and uses estimated states in a state-feedback controller.
  • The observer-based design is further leveraged to develop a distributed control framework for observability blocking.
  • The methods rely on Laplacian eigenstructure and eigenstructure assignment to impose unobservability on selected modes through feedback.

A. Output-Feedback Based Approach

The output-feedback approach assigns a closed-loop eigenstructure using only operator measurements, making a selected mode unobservable at compromised nodes while preserving a desired eigenvalue set.

  • A. Output-Feedback Based Approach: Static output feedback uses sensor measurements to design a controller that makes the closed-loop dynamics unobservable to compromised nodes.
  • A. Output-Feedback Based Approach: The method selects a self-conjugate set ΛD of h open-loop eigenvalues for preservation and chooses one associated mode for modification.
  • A. Output-Feedback Based Approach: The selected eigenvector is modified so its compromised-node entries become zero, making the corresponding mode unobservable by the PBH test.
  • A. Output-Feedback Based Approach: The algorithm forms desired eigenvectors, extracts actuation-related subvectors, partitions L, and computes the output-feedback gain F1.
  • A. Output-Feedback Based Approach: The controller preserves all eigenvalues in ΛD and their associated eigenvectors except the selected modified mode, including its conjugate when complex.
  • A. Output-Feedback Based Approach: Under distinct eigenvalues, q ≥ m + 2, invertible COVh, and controllable (−L, B), the controller makes the compromised-node pair unobservable.

L11 L12 L21 L22

The output-feedback design blocks a selected mode from compromised-node measurements using partial state measurements while preserving a chosen subset of the open-loop eigenstructure. Its feasibility depends on actuation, observability, controllability, and graph-cut conditions, with important limits on sensor placement and stability.

  • Graph-structured reduction: A graph cutset can reduce the required actuation nodes to q ≥ |V_cut| + 2 when the selected eigenvalue is absent from L_V2V2.The resulting controller makes the selected mode unobservable for measurements at the cutset nodes.
  • Feasibility limits: Output feedback fails when all sensors lie in V_cut ∪ V_2 and the cutset nodes are compromised under the stated observability assumptions.In this configuration, blocking compromised-node observability would also make the mode unobservable at sensor nodes, conflicting with output-observability invariance.
  • Trade-offs: Only h open-loop eigenvalues are preserved, so the remaining n − h eigenvalues may change and can potentially destabilize the closed-loop system.More sensor nodes permit more eigenstructure to be retained; for real eigenvalues, the actuation requirement becomes q = m + 1.
  • Scope: The design applies beyond Laplacian synchronization models to linear time-invariant dynamics satisfying the theorem’s conditions.Its generality does not depend on a specific state matrix or graph topology.

B. Observer Based Approach

The observer-based approach reconstructs the network state from partial measurements and applies independently designed state feedback to block compromised-node observability. Unlike output feedback, it preserves all open-loop eigenvalues and remains applicable under broader sensor-placement conditions.

  • Observer-based design: A Luenberger observer estimates the full network state from sensor measurements, and state feedback uses the estimate for actuation.The observer gain K reconstructs the state, while the controller gain F2 acts on the estimated state.
  • Observer construction: When (C_O, −L) is observable, observer pole placement makes the estimation error converge rapidly and independently of the state-feedback design.The convergence rate is governed by the eigenvalue of L + KC_O with the smallest real part.
  • Main result: Theorem 2 blocks an arbitrary selected mode at compromised nodes while preserving all open-loop eigenvalues.The observer and state-feedback gains can be designed independently under distinct-eigenvalue, actuation, observability, and controllability assumptions.
  • Actuation requirements: A vertex cutset can reduce the required actuation nodes to q ≥ |V_cut| + 2 under the stated eigenvalue condition.For real Laplacian eigenvalues, the requirement reduces to q = m + 1, and the cutset bound reduces correspondingly.
  • Security property: The observer-based system remains unobservable to the adversary at all times, because estimation error acts as an unknown external input before convergence.Thus, blocking does not depend on waiting for the estimation error to become zero.
  • Sensor-placement flexibility: Unlike output feedback, the observer-based design does not require the closed-loop sensor model to remain observable and can apply when sensor and compromised nodes coincide.It remains applicable in that extreme case when the open-loop sensor model is observable and Theorem 2 conditions hold.

C. A Distributed Framework

The distributed extension assigns each actuation node a local observer using its own and neighboring states, while coordinated communication supports observer-controller separation. Under the theorem’s assumptions, it achieves the same gain as the centralized observer-based design.

  • Local information and control: Each actuation node observes its own state and neighboring states, constructs a local observer, and uses its estimate to compute its control input.The local output matrix C_Oi includes node i and its neighbors, and K_i is the corresponding observer gain.
  • Coordination assumption: The distributed design assumes actuation nodes communicate applied inputs or state estimates so every observer uses the same aggregate input.This communication is required for the separation principle to hold in the distributed setting.
  • Conditions: Theorem 3 requires distinct Laplacian eigenvalues, q ≥ m + 2, individual local observability, controllability of (−L, B), and communication among actuation nodes.Under these assumptions, local observers and state-feedback gains can be designed independently.
  • Equivalence: The distributed and centralized observer-based approaches produce the same controller gain matrix under the same network, eigenvalue, and eigenvector selections.The distributed gain F is identical to the centralized gain F2.
  • Actuation reduction: A cutset can reduce the required actuation nodes to q ≥ |V_cut| + 1, with one further reduction when all Laplacian eigenvalues are real.The formal reduction is stated for the distributed theorem under the corresponding graph-separation condition.
  • Collective observability: Individual local observers need not each be observable if the combined observer model is collectively observable.In that case, the observers update their equations using the collectively available information.

O1 CT

The distributed observer framework uses local-neighbor information and consensus communication to drive state estimates toward the true state, subject to a Hurwitz stability condition.

  • Each actuation node accesses its own state and neighboring states to design its local control input.
  • Actuation nodes must share their control signals or state estimates so every observer uses the same input vector.This communication assumption is required for the separation principle in the distributed setting.
  • A consensus term coupling neighboring estimates drives the distributed estimates toward the true state when the combined observer matrix is Hurwitz.

IV. SIMULATION RESULTS

Simulations on an 11-node network demonstrate output-feedback, centralized observer-based, and distributed observability-blocking designs. The observer-based design succeeds where output feedback fails and preserves the open-loop eigenstructure except for the blocked mode.

  • The simulation uses an 11-node undirected network with unit edge weights, actuation at Nodes 1 and 2, and compromised Nodes 7, 9, 10, and 11.Node 6 is a single vertex-cutset separating actuation and compromised nodes.
  • With output feedback, selecting λip = 0.1983 modifies its eigenvector to zero at compromised nodes, making that mode unobservable there.The same mode is also unobservable from any node in {6, 7, 8, 9, 10, 11}.
  • Output feedback preserves the selected eigenvalues in ΛD, while the remaining eigenvalues change.
  • For sensor Nodes 6, 7, and 8, output feedback fails because COVh is not invertible, but the observer-based method achieves rapidly converging estimation errors.The observer poles are placed at {-1, -2, · · ·, -11}; faster placement increases gain values and may increase transient control overshoot.
  • The centralized observer-based controller preserves all open-loop eigenvalues and all eigenvectors except the one associated with λp = 0.1983.
  • In the distributed implementation, individually observable local observers are built at Nodes 1 and 2, and the resulting controller gains match the centralized-observer case.

V. CONCLUSION

The paper develops output-feedback and observer-based strategies for blocking observability with partial state information. The observer-based design preserves all open-loop eigenvalues and offers greater flexibility, while output feedback is simpler and preserves partial eigenstructure; the observer design is also extended distributively.

  • Output-feedback and observer-based controls block observability in networked systems using partial state information.
  • The observer-based approach preserves all open-loop eigenvalues and offers greater flexibility, whereas output feedback is simpler and preserves partial eigenstructure.
  • The observer-based design is extended to a distributed control framework.
Loading 2608.25259v1…