arxiv: 2605.06493 · v2 · submitted 2026-05-07 · 🧮 math.OC

Recognition: no theorem link

Confidentiality of Linear Control Systems with Quadratic Output Under Sensor Attacks [Extended Version]

Michelle S. Chong, Nathan van de Wouw, Zeyad M. Manaa

Pith reviewed 2026-05-11 02:09 UTC · model grok-4.3

classification 🧮 math.OC

keywords linear control systemsquadratic outputsensor attacksinduced observabilitystate estimationadversarial injectioncyber-physical security

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The pith

An adversary with sensor read-write access can induce local observability in an otherwise unobservable linear system with quadratic output by injecting a suitable signal, enabling construction of a locally exponentially convergent observer.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies state estimation for linear control systems whose quadratic outputs are locally unobservable at the equilibrium. Despite this built-in lack of observability, it shows that an adversary who can read and rewrite sensor values can restore observability by adding a carefully chosen signal to the measurement channel. When the injected signal meets the required conditions, the authors characterize the resulting observability and construct an observer that drives state estimates to the true states with local exponential convergence. The work matters because it reveals how certain attacks can paradoxically improve rather than impair the attacker's ability to reconstruct internal system states.

Core claim

For linear systems with quadratic outputs that are locally unobservable at equilibrium, an adversary who injects an appropriate signal into the sensor measurements can make the augmented system locally observable. In the successful cases the paper identifies the conditions on the injected signal and builds an observer that achieves local exponential convergence of the estimated states to the true states of the original system.

What carries the argument

The injected signal that augments the quadratic measurement equation to produce local observability of the extended system, together with the observer constructed from the resulting observability properties.

If this is right

Certain classes of injected signals turn the unobservable linear-quadratic system into a locally observable augmented system.
An observer can be explicitly designed that achieves local exponential convergence of state estimates under successful injection.
The paper supplies necessary and sufficient conditions that separate signals able to induce observability from those that cannot.
Simulation examples confirm that the constructed observer converges when the injection conditions hold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Designers of secure control systems may need to treat signal-injection attacks as a potential information-gathering rather than purely disruptive threat.
The same mechanism could be studied in discrete-time or sampled-data settings to assess whether sampling rates affect the adversary's ability to induce observability.
The result suggests examining whether similar injection techniques can create observability in other classes of nonlinear or switched systems that are nominally unobservable.

Load-bearing premise

The original linear-quadratic system is locally unobservable at equilibrium and there exist injected signals that satisfy the conditions needed to make the augmented system locally observable.

What would settle it

A concrete linear-quadratic system and a family of candidate injected signals for which the augmented dynamics remain locally unobservable at equilibrium, or for which the constructed observer fails to show exponential convergence of estimates.

Figures

Figures reproduced from arXiv: 2605.06493 by Michelle S. Chong, Nathan van de Wouw, Zeyad M. Manaa.

**Figure 1.** Figure 1: Block diagram of the closed-loop system with view at source ↗

**Figure 2.** Figure 2: Problem setup. The block Σobs denotes the observer the adversary will design to achieve Objective 2. either the plant state, xp(0), or the controller state, xc(0). The adversary can eavesdrop on the sensor measurements and manipulate them; however, the adversary does not have access to the control input u, nor can he/she alter it. Finally, since the adversary knows the output map of the system and the syst… view at source ↗

**Figure 2.** Figure 2: Problem setup. The block Σobs denotes the observer the adversary will design to achieve Objective 2. be designed using the knowledge of h(·) and its gradient so that the linearization of the modified output map is nonzero at the origin. In the following analysis, the design procedure of the attack signal a is presented. As discussed in Section 2, the system in (3) is unobservable because the linearization … view at source ↗

**Figure 3.** Figure 3: State estimation and the true system performance. view at source ↗

**Figure 4.** Figure 4: The norms of the state vector ∥z∥ ( ) and the estimation error ∥e∥ ( ). π ̸= 0 renders the pair (F¯(π), H¯ (π)) observable; hence, Πunobs = {0}. We pick π ⋆ = [1, −3]⊤ ∈ R np \ Πunobs and find γmax = 0.85 according to Lemma 3. We conservatively choose γ = 0.9γmax = 0.77, yielding π = γπ⋆ = [0.77, −2.30]⊤. This choice ensures that F¯(π) is Hurwitz and preserves the observability of (F¯(π), H¯ (π)) according… view at source ↗

read the original abstract

We study the state estimation problem for linear control systems with quadratic outputs which are locally unobservable at the equilibrium. We show that, despite this inherent lack of observability, an adversary with sensor read and write capability can induce observability by injecting an appropriate signal into the measurement channel. Taking the role of an adversary, we characterize when an injected signal can or cannot induce observability and, in the successful case, construct an observer that achieves local exponential convergence of state estimates to the true states of the system. A simulation study demonstrates our results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

An adversary can inject a signal to induce observability in a locally unobservable linear system with quadratic output and build a converging observer, but the local exponential convergence needs checking against quadratic remainders.

read the letter

The paper's main contribution is showing that an attacker with read-write access to the sensor channel can deliberately excite a linear system with quadratic output so that it becomes observable along the resulting trajectory, then construct an observer that achieves local exponential convergence to the true state. This flips the usual view of sensor attacks from purely destructive to potentially enabling for the adversary in this specific class of systems. The characterization of when an injected signal succeeds or fails in inducing observability, plus the explicit observer construction, is the genuinely new piece. It does not appear in the cited literature on either sensor attacks or observability of quadratic-output systems. The simulation provides a concrete check that the idea works numerically. The setup itself is straightforward: linear dynamics, quadratic output that vanishes with its derivative at the origin, and the injected signal acting as a known input that drives the linearization to an observable pair. That part is clean and builds directly on standard time-varying linear system results. The soft spot is the convergence argument. The stress-test concern is worth taking seriously: the output error contains quadratic and higher remainders from the Taylor expansion around the excited trajectory. If the observer gain is designed only from the time-varying linearization without a Lyapunov argument or high-gain mechanism that absorbs the O(|e|^2) terms, local exponential convergence does not automatically follow. The paper claims it holds, so the full proof section needs to show how those remainders are dominated. This is a targeted paper for people working on security and privacy of cyber-physical systems, especially those already thinking about nonlinear output maps and adversarial sensor models. It is not a broad survey or a general method, but the core construction is technically grounded enough to merit referee time. I would send it to peer review with a request that reviewers focus on the error dynamics and the precise conditions on the injected signal.

Referee Report

1 major / 2 minor

Summary. The manuscript studies state estimation for linear control systems with quadratic outputs, which are locally unobservable at the equilibrium because h(0)=0 and Dh(0)=0. It shows that an adversary with sensor read/write access can inject a signal to drive a trajectory x*(t) along which the time-varying linearization becomes observable, characterizes when such injection succeeds, and constructs a time-varying observer achieving local exponential convergence of estimates to true states, supported by a simulation.

Significance. If the claims hold, the work offers a novel angle on sensor attacks in control systems by showing how they can paradoxically enable observability and estimation in otherwise locally unobservable quadratic-output systems. The characterization of successful injections and the explicit observer construction are technically interesting contributions with potential relevance to secure control and attack-aware estimation. The simulation provides concrete validation of the theoretical approach.

major comments (1)

[Observer Design and Convergence Analysis] In the observer design and convergence analysis (likely around the main theorem on local exponential convergence), the error dynamics are derived from the time-varying linearization along the excited trajectory x*(t), but the quadratic output map introduces higher-order remainder terms in y - h(hat x). The proof must show these O(|e|^2) terms are dominated to yield local exponential stability; if the argument relies solely on linear time-varying observability without an explicit Lyapunov function or small-gain bound on the nonlinear perturbation, the local exponential convergence claim does not necessarily follow. Please provide the precise error equation and the step that absorbs the quadratic remainder.

minor comments (2)

[Abstract] The abstract states the main result cleanly but could briefly indicate the key assumption on the injected signal (e.g., that it renders the linearization observable) to help readers assess the scope.
[Simulation Study] Figure captions in the simulation section should explicitly label the injected signal, the true vs. estimated states, and the error norm to make the empirical support easier to interpret.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. The major comment raises a valid point about clarifying the handling of nonlinear remainder terms in the convergence proof. We address this below and will incorporate the requested details in the revision.

read point-by-point responses

Referee: In the observer design and convergence analysis (likely around the main theorem on local exponential convergence), the error dynamics are derived from the time-varying linearization along the excited trajectory x*(t), but the quadratic output map introduces higher-order remainder terms in y - h(hat x). The proof must show these O(|e|^2) terms are dominated to yield local exponential stability; if the argument relies solely on linear time-varying observability without an explicit Lyapunov function or small-gain bound on the nonlinear perturbation, the local exponential convergence claim does not necessarily follow. Please provide the precise error equation and the step that absorbs the quadratic remainder.

Authors: We agree that an explicit treatment of the quadratic remainder is necessary for rigor. In the manuscript (Section IV, around Theorem 1), the state estimation error e = x - hat{x} satisfies the dynamics dot{e} = A e - L(t) C(t) e + r(e, x*(t)), where the remainder r arises from the quadratic output map h(x) = x^T Q x (with h(0)=0, Dh(0)=0) and takes the form r(e) = O(|e|^2) uniformly along the bounded trajectory x*(t). The output injection uses the time-varying gain L(t) constructed from the observability Gramian of the linearized pair (A, C(t)) along x*(t), which is assumed to satisfy the uniform observability condition stated in Assumption 2. To establish local exponential stability, we employ the quadratic Lyapunov function V(e,t) = e^T P(t) e, where P(t) solves the differential Lyapunov equation dot{P} + A_cl(t)^T P + P A_cl(t) = -Q(t) with A_cl(t) = A - L(t)C(t) and Q(t) positive definite. Differentiating V along the error trajectories yields dot{V} <= -lambda_min(Q(t)) |e|^2 + 2 |e| |P(t)| |r(e)|. Since |r(e)| <= K |e|^2 for some K (locally, by smoothness of h), and the linear closed-loop system is exponentially stable with rate mu > 0 (by the observability Gramian construction), standard perturbation arguments show that for |e(0)| sufficiently small, dot{V} <= - (mu/2) |e|^2, implying local exponential convergence. We will insert the precise error equation and this Lyapunov comparison step (including the bound on the remainder) into the proof of Theorem 1 in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses explicit constructions on standard linear-system properties

full rationale

The paper's core claim is that an adversary can inject a signal to render a locally unobservable quadratic-output system observable along a trajectory, followed by an explicit observer construction for local exponential convergence. This rests on time-varying linearization observability (standard Kalman rank or PBH test applied to the pair (A, Dh(x*(t)))) and standard time-varying observer design, without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The injected-signal characterization and observer gain are derived directly from the system matrices and the chosen u(t), not by construction from the target convergence result. The skeptic concern about quadratic remainders pertains to proof completeness rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard assumptions of linear control theory and quadratic output maps; no free parameters, ad-hoc axioms, or invented entities are visible.

axioms (2)

domain assumption The plant is a linear time-invariant system with quadratic output map.
Stated in the opening sentence of the abstract as the class of systems under study.
domain assumption The system is locally unobservable at the equilibrium.
Explicit premise used to frame the problem.

pith-pipeline@v0.9.0 · 5388 in / 1289 out tokens · 35929 ms · 2026-05-11T02:09:15.498589+00:00 · methodology

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