⚡ electrical engineering

Observability Blocking in a Linear Synchronization Network with Partial State Measurements

This paper proposes novel control strategies utilizing partial state measurements—specifically output feedback and observer-based designs—to block adversaries from inferring system dynamics in large-scale linear synchronization networks while preserving the system's eigenvalues.

Original authors: Alexis Moreno, Abdullah Al Maruf

Published 2026-08-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexis Moreno, Abdullah Al Maruf

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, interconnected systems that power our modern world—from the electrical grids lighting our cities to the fleets of autonomous vehicles navigating our streets—security is no longer just about keeping intruders out. It is about preventing them from seeing inside. These systems are networks of many individual parts, or nodes, that constantly share information to stay in sync. If an attacker can tap into even a few of these nodes, they can often piece together the behavior of the entire network, predicting its movements and potentially manipulating it. This ability to see the inner workings of a system is called observability. For decades, engineers have focused on making systems easier to observe so they can be controlled effectively. However, a new line of thinking suggests that in a dangerous world, the best defense might be to make the system invisible to those who wish to harm it, without stopping the system from working for those who need it to.

This is the challenge tackled by researchers Alexis Moreno and Abdullah Al Maruf, who have developed a way to blind an attacker to the state of a large network using only limited information. In their study, they imagine a scenario where an adversary has managed to compromise a small group of nodes within a larger network, allowing them to measure the status of those specific points. The goal is to design a control strategy that prevents the attacker from using those measurements to figure out what the rest of the network is doing. The researchers found that while previous methods could achieve this, they required the system operator to have access to the status of every single node in the network—a requirement that is often impossible in large, sprawling systems where sensors are sparse and communication is limited. To solve this, the team proposed two new strategies that work with only a partial view of the system.

The first strategy relies on a direct link between the sensors the operator can see and the controls they can apply. Imagine the operator looking at a few specific nodes and adjusting the inputs at a few other nodes based solely on what they see. The researchers showed that by carefully tuning these adjustments, they could make the network's behavior at the compromised nodes appear completely static or unchanging to the attacker, effectively hiding the true dynamics. However, this method comes with a trade-off: while it successfully blinds the attacker, it inevitably alters the natural rhythm of the network, changing some of its underlying frequencies. This means the system might behave slightly differently than it did before, which could be risky if those natural rhythms are critical for stability. Furthermore, this approach only works if the sensors are placed in very specific locations; if the sensors are too close to the compromised nodes or in the wrong part of the network, the method fails entirely.

To overcome these limitations, the researchers developed a second, more robust approach that acts like a mental reconstruction of the whole system. Instead of just reacting to what is seen, the operator builds a virtual model, or an observer, that uses the limited sensor data to guess the state of every single node in the network. Once this internal picture is clear, the operator uses it to apply controls that block the attacker's view. The beauty of this method is that it preserves the network's original natural rhythms perfectly, leaving the system's fundamental behavior untouched while still hiding its secrets from the intruder. It also offers much greater flexibility, allowing the sensors to be placed in almost any location, even right next to the compromised nodes, without breaking the defense. The researchers demonstrated that this technique works not just for a central controller but can also be distributed, where each control node works with its own local model and talks to its neighbors to maintain the collective blind spot.

To prove their ideas, the team ran detailed computer simulations on a network of eleven nodes. They set up a scenario where two nodes acted as controllers, five as sensors, and four as compromised points for the attacker. In the first test, they used the direct method. The simulation showed that they could successfully hide the network's state from the attacker, but the network's internal frequencies shifted, and the method only worked because the sensors were placed in a specific, favorable arrangement. When they moved the sensors to a less ideal location, the direct method failed completely, unable to generate the necessary controls. However, when they switched to the observer-based method, the result was different. The virtual model quickly learned the true state of the network, and the controls successfully blocked the attacker's view, even with the sensors in the difficult position. Crucially, the network's original natural frequencies remained exactly the same, proving that the system could be secured without compromising its performance.

The researchers also tested a distributed version of this observer-based strategy, where the two control nodes shared their estimates with each other. The simulation showed that even in this decentralized setup, the system could rapidly converge on the true state of the network and maintain the blind spot against the attacker. The results confirmed that by using a virtual reconstruction of the system, operators can secure large networks against inference attacks without needing to monitor every single point or alter the system's fundamental behavior. While these findings are currently limited to computer simulations, they offer a promising path forward for securing the complex, interconnected systems that underpin modern infrastructure, turning the vulnerability of partial visibility into a strength by making the whole system invisible to those who would do it harm.

Technical Summary: Observability Blocking in a Linear Synchronization Network with Partial State Measurements

Problem Formulation
This paper addresses the security challenge in large-scale linear synchronization networks where adversaries can infer global system dynamics by monitoring a subset of compromised nodes. While previous work established that state-feedback controllers could block observability at these nodes by modifying specific eigenvectors while preserving the system's eigenstructure, those methods required full state access. In practical large-scale networks, operators often possess only partial state measurements due to distributed sensing and communication constraints. The core problem is to design control strategies that block an adversary's ability to estimate the network state using only partial measurements from designated sensor nodes, while preserving key open-loop dynamical properties.

Methodology
The authors propose two primary control strategies and a distributed extension, all operating under partial state measurements. The network is modeled as a linear synchronization system $\dot{x}(t) = -Lx(t) + Bu(t)$ , where $L$ is the graph Laplacian, $B$ represents actuation nodes, and the adversary observes states via $y_A(t) = C_A x(t)$ .

Output-Feedback Approach:
This method employs a static output-feedback control law $u(t) = -F_1 y_O(t)$ , where $y_O(t)$ are measurements from sensor nodes. The design utilizes an eigenstructure assignment technique to select a subset of open-loop eigenvalues ( $\Lambda_D$ ) to preserve. For a selected eigenvalue $\lambda_{ip}$ , the corresponding eigenvector is modified such that its entries at the compromised nodes become zero. This renders the mode unobservable to the adversary according to the Popov–Belevitch–Hautus (PBH) test. The gain matrix $F_1$ is computed to satisfy these conditions. The authors note that this approach preserves only the selected subset of eigenvalues, potentially altering the rest of the spectrum, and requires the closed-loop system to remain observable at the sensor nodes.
Observer-Based Approach:
To overcome the limitations of output feedback, the authors propose a scheme using a Luenberger state observer to reconstruct the full state $\hat{x}(t)$ from partial measurements, followed by a state-feedback control law $u(t) = -F_2 \hat{x}(t)$ .
- Observer Design: The observer gain $K$ is designed to ensure rapid convergence of the estimation error $e(t) = x(t) - \hat{x}(t)$ to zero, assuming the open-loop pair $(C_O, -L)$ is observable.
- Controller Design: The state-feedback gain $F_2$ is designed to modify a specific eigenvector to zero out entries at compromised nodes, similar to the output-feedback method.
- Advantages: Unlike the output-feedback approach, this method preserves all open-loop eigenvalues. Furthermore, it does not require the closed-loop system to remain observable at the sensor nodes, offering greater flexibility in sensor placement. The separation principle ensures that the observer and controller can be designed independently.
Distributed Framework:
The observer-based approach is extended to a distributed setting where each actuation node utilizes only its own state and its neighbors' states. Actuation nodes communicate their control signals or state estimates to maintain a consistent input across the observer network. This allows for the independent design of local observer gains and a global controller gain, provided the actuation nodes can communicate.

Key Contributions
The paper makes four specific contributions:

Output-Feedback Algorithm: A general algorithm is presented that blocks observability at compromised nodes using only partial measurements, preserving a selected subset of open-loop eigenvalues.
Observer-Based Design: An alternative approach is introduced that uses a state observer and full-state feedback. This method preserves all open-loop eigenvalues and allows for more flexible sensor placement, as it does not rely on the closed-loop system remaining observable at the sensor nodes.
Distributed Extension: The observer-based design is extended to a distributed framework, detailing the construction of local observers and controllers for actuation nodes that rely on local and neighbor information.
Validation: Numerical simulations on an 11-node network demonstrate the design procedures. The results validate that the output-feedback method fails when sensor placement violates specific observability conditions (as predicted by the paper's corollaries), whereas the observer-based method remains effective. The simulations also confirm the rapid convergence of estimation errors and the successful blocking of observability at compromised nodes.

Significance and Claims
The authors claim that their work bridges a critical gap between theoretical observability blocking and practical implementation in large-scale networks where full state access is unavailable. By leveraging graph-theoretic insights and linear-algebraic techniques, the proposed methods enhance privacy and security by preventing adversaries from reconstructing critical states.

The paper highlights that the observer-based approach is superior in terms of flexibility and eigenstructure preservation compared to the output-feedback approach. Specifically, it allows for the blocking of observability even when sensor nodes coincide with compromised nodes, a scenario where output-feedback methods fail. The distributed framework further extends applicability to scenarios where centralized state estimation is impractical. The authors maintain a modest scope, noting that while the designs are general for linear time-invariant dynamics, the specific algorithms rely on the distinctness of eigenvalues and specific controllability/observability conditions. Future work is identified as the development of rigorous algorithms for fully distributed solutions where individual actuation nodes are not individually observable but are collectively observable.

More like this