Attack Detection in Dynamic Games with Quadratic Measurements

This paper studies attack detection for discrete-time linear systems with stochastic process noise that produce both a vulnerable (i.e., attackable) linear measurement and a secured (i.e., unattackable) quadratic measurement. The motivating application of this model is a dynamic-game setting where the quadratic measurement is interpreted as a system-level utility or reward, and control inputs into the linear system are interpreted as control policies that, once applied, are known to all game participants and which steer the system towards a game-theoretic equilibrium (e.g., Nash equilibrium). To detect attacks on the linear channel, we develop a novel quadratic-utility-aware observer that leverages the secured quadratic output and enforces measurement consistency via a projection step. We establish three properties for this observer: feasibility of the true state, prox-regularity of the quadratic-constraint set, and a monotone error-reduction guarantee in the noise-free case. To detect adversarial manipulation, we compare linear and quadratic observer trajectories using a wild bootstrap maximum mean discrepancy (MMD) test that provides valid inference under temporal dependence. We validate our framework using numerical experiments of a pursuit-evasion game, where the quadratic observer preserves estimation accuracy under linear-sensor attacks, while the statistical test detects distributional divergence between the observers’ trajectories.

💡 Research Summary

The paper addresses the problem of detecting attacks on a discrete‑time linear system that is equipped with two distinct measurement channels: a conventional linear sensor that can be corrupted by an adversary and a secure quadratic sensor that cannot be tampered with. The quadratic measurement is interpreted as a system‑level utility or reward in a dynamic‑game setting, where control inputs represent equilibrium policies known to all participants.

The authors propose a two‑track observer architecture. The first track is a standard Kalman filter that processes only the vulnerable linear measurements. The second track is a novel quadratic‑utility‑aware observer that processes the secure quadratic measurements. The quadratic observer follows an EKF‑style correction: the nonlinear observation h(x)=xᵀVx is linearized around the prior estimate, yielding a Jacobian Hₖ = (2V x̂ᵠₖ|k−1)ᵀ, and a Kalman‑type gain Kₖ is applied. Because the linearization can be poor when the estimate deviates, the authors introduce a consistency projection step. After the EKF update, the intermediate estimate ˜xᵠₖ|k is projected onto a feasible set ℱₖ defined by linearized constraints derived from the current and past N quadratic measurements. Each constraint has an adaptive bound δₖ,i(x)=ζ+L‖A^{−i}x−x̂ᵠ_{k−i}|k−i−1‖², where L=‖V‖₂. The projection minimizes a covariance‑weighted norm while guaranteeing that the true state always satisfies the constraints.

The paper provides a rigorous theoretical analysis of this observer. Under three mild assumptions—(1) A is invertible, (2) each constraint function is non‑degenerate, and (3) an aggregated constraint qualification holds—the feasible set ℱₖ is shown to be strongly amenable and therefore prox‑regular. Lemma 1 proves that, in the noise‑free case, the true state belongs to ℱₖ for all time, thanks to the adaptive bound design. Lemma 2 and Theorem 1 establish that the projection step never increases the weighted estimation error: the post‑projection error norm is bounded above by the pre‑projection error norm. Consequently, the quadratic observer enjoys a monotone error‑reduction property in the absence of noise.

To detect attacks, the authors compare the trajectories of the linear and quadratic observers using a statistical test based on the Maximum Mean Discrepancy (MMD). Because the estimates are temporally dependent, a wild‑bootstrap version of MMD is employed. The two sets of estimates Xᴸₖ and Xᵠₖ are merged into Zₖ, an RBF kernel is applied, and a centered kernel matrix is formed. Random sign variables vᵢ are used to generate bootstrap replicates of the degenerate V‑statistic, yielding an empirical null distribution that respects temporal dependence. The (1−α) quantile of the bootstrap distribution serves as the critical threshold γ_α; if the observed MMD² exceeds γ_α, the null hypothesis of identical distributions is rejected, indicating an attack on the linear channel.

Numerical experiments are conducted on a pursuit‑evasion game. The system dynamics, control policy, and utility matrix V are chosen to reflect a realistic game scenario. An adversary injects a constant bias into the linear sensor after a chosen attack onset time. Results show that (i) the quadratic observer maintains low root‑mean‑square error despite the attack, confirming its robustness, (ii) the linear observer’s error grows sharply after the attack, and (iii) the wild‑bootstrap MMD test detects a statistically significant divergence between the two observer trajectories immediately after the attack, with p‑values falling below 0.05.

In summary, the paper contributes (1) a novel observer that leverages a secure quadratic measurement, enforces measurement consistency through a projection onto a prox‑regular set, and provides provable error‑reduction guarantees, and (2) a statistically sound, temporally aware detection test that flags attacks on vulnerable linear sensors by exploiting the resilient quadratic observer. The approach opens avenues for secure state estimation in multi‑agent dynamic games, and future work may extend the framework to multiple quadratic sensors, nonlinear dynamics, non‑Gaussian noise, and adaptive bound tuning.