A mathematical preprint describes three ways to make a selected mode of a network unobservable at measurements taken by compromised nodes. The methods use static output feedback, an observer that reconstructs the system state, or distributed local controllers. The guarantees are not identical: the output-feedback result preserves a designated set of open-loop eigenvalues and their associated eigenvectors, while the observer-based and distributed results report preservation of all open-loop eigenvalues under their respective conditions.
The work is an arXiv preprint about a continuous-time linear synchronization model. Each node has a scalar state, and the model distinguishes actuation nodes, sensor nodes and compromised nodes, with those groups allowed to overlap.
A narrower guarantee from measured outputs
The first route is static output feedback. It is driven by the operator's measured output, with the controller using that output directly. Its theorem is conditional on distinct Laplacian eigenvalues, at least m + 2 actuation nodes, an invertible CO Vh matrix and controllability. If those conditions hold, the theorem states that a selected open-loop mode is unobservable to compromised nodes while every eigenvalue in the designated set, along with its associated eigenvector, is preserved.
The paper reports a smaller output-feedback actuation requirement under cutset conditions: at least |Vcut| + 2 actuation nodes. When all eigenvalues of L are real, the reported count is m + 1 actuation nodes in general and |Vcut| + 1 in the cutset case.
Sensor placement changes the options
Sensor placement is a direct feasibility constraint for that first route. Output feedback requires CO Vh to be invertible. Under the stated sensor and cutset conditions, Algorithm 1 fails when the matrix is not invertible, and no output-feedback controller can impose the selected unobservability.
The observer-based route uses a different arrangement. It reconstructs the full state from operator sensor measurements and then applies state feedback to the estimated state through a Luenberger observer, a state estimator. Unlike output feedback, it does not require the closed-loop model to remain observable at the sensor nodes, so sensor placement can be more flexible.
Under its stated assumptions, including distinct eigenvalues, at least m + 2 actuation nodes, open-loop sensor observability and controllability, the observer theorem reports rapid estimation-error convergence and independently designed feedback. It also reports a selected mode that is unobservable at compromised nodes while all open-loop eigenvalues are preserved.
A test on one modeled network
The numerical illustration uses one 11-node undirected network with edge weight 1. Its actuation nodes are Nodes 1 and 2, while its compromised nodes are Nodes 7, 9, 10 and 11.
In the sensor placement for which CO Vh was not invertible, the reported simulation says Algorithm 1 failed and no output-feedback controller existed under the stated corollary. In the same numerical comparison, the observer-based simulation reported rapid convergence of the estimation error.
Local control, with a boundary
The study also extends the observer design to distributed control. Each actuation node is given only its own state and the states of its neighbors when constructing its local control input.
With distinct eigenvalues, at least m + 2 actuation nodes, observable local sensor pairs, controllability and communication between actuators, the distributed theorem says local observers and feedback gains can be designed independently. Under those conditions, the closed loop has a selected mode unobservable at compromised nodes and preserves all open-loop eigenvalues.
The fully distributed case based on collective observability has a stated boundary. The paper says a joint observer design may be needed and leaves rigorous, provable algorithms for that case to future work.
What the evidence covers
Taken together, the results are conditional control constructions backed by analysis and a single numerical case study. The 11-node network is a modeled illustration rather than a statistical sample, and the observer-based and distributed guarantees depend on their stated observability, controllability, actuation and communication conditions.
The output-feedback result is narrower in another way: its preservation guarantee covers only the designated eigenvalues and associated eigenvectors, and its feasibility also depends on the invertibility of CO Vh. The observer-based and distributed theorems report preservation of all open-loop eigenvalues, but only under their own stated conditions.
Paper data and sources
Original title: Observability Blocking in a Linear Synchronization Network with Partial State Measurements
Authors: Alexis Moreno, Abdullah Al Maruf
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text