Secure State Estimation against Sensor Attacks in the Presence of Noise

We consider the problem of estimating the state of a noisy linear dynamical system when an unknown subset of sensors is arbitrarily corrupted by an adversary. We propose a secure state estimation algorithm, and derive (optimal) bounds on the achievable state estimation error given an upper bound on the number of attacked sensors. The proposed state estimator involves Kalman filters operating over subsets of sensors to search for a sensor subset which is reliable for state estimation. To further improve the subset search time, we propose Satisfiability Modulo Theory based techniques to exploit the combinatorial nature of searching over sensor subsets. Finally, as a result of independent interest, we give a coding theoretic view of attack detection and state estimation against sensor attacks in a noiseless dynamical system.

💡 Research Summary

This paper addresses the critical challenge of securely estimating the state of a linear dynamical system in the presence of both stochastic noise (process and sensor noise) and adversarial sensor attacks. The adversary can arbitrarily corrupt the outputs of an unknown but fixed subset of up to k sensors. The authors’ key contribution is a secure state estimation algorithm with provable, optimal performance guarantees.

The core problem is formulated around two main tasks: Effective Attack Detection and Optimal Secure State Estimation. The authors introduce the concept of an (ε, s)-effective attack, defined as an attack that, when using sensor set s for estimation via a Kalman filter, causes the sample average of the estimation error covariance trace to exceed the optimal attack-free error trace by more than ε. This definition provides a quantifiable threshold for distinguishing attacks that significantly degrade performance.

The proposed secure state estimator operates by searching for a “reliable” subset of sensors. The algorithm enumerates subsets of sensors of size p-k (since at most k can be faulty). For each candidate subset, it runs a Kalman filter and monitors its innovation sequence or estimation error. It then applies an effective attack detection test to check if the observed error statistics are consistent with the hypothesis of an ε-ineffective attack on that subset (i.e., the attack’s impact is within the ε-bound). The first subset that passes this test is deemed reliable, and its corresponding Kalman filter estimate is used as the secure state estimate.

A fundamental theoretical contribution is the proof of this estimator’s optimality. First, an Impossibility Theorem establishes a lower bound: no estimator, even an oracle that knows which sensors are attacked, can achieve an error covariance trace better than that of the worst-case Kalman filter using only p-k sensors, denoted tr(P*_s_worst,p-k). The authors then prove that their proposed algorithm’s estimation error is guaranteed not to exceed this lower bound, thus achieving optimal performance.

To address the combinatorial complexity of searching over all (p choose p-k) sensor subsets, the paper proposes a method based on Satisfiability Modulo Theory (SMT). By encoding the consistency checks between sensor measurements and Kalman filter predictions as logical constraints, the search for a consistent (reliable) sensor subset can be offloaded to an efficient SMT solver, significantly reducing computation time compared to brute-force enumeration. Numerical experiments demonstrate the efficacy of this approach.

Finally, as an independent insightful contribution, the paper provides a coding-theoretic interpretation of secure estimation in noiseless systems. It shows that the well-known necessary and sufficient conditions for attack detection and state recovery (sparse observability) in noiseless dynamics are equivalent to the Hamming distance requirements for error detection and correction in classical coding theory. This bridges concepts between control theory and information theory.

In summary, this work presents a comprehensive framework for secure state estimation under sensor attacks and noise. It combines a theoretically sound and optimal estimation algorithm with practical computational techniques (SMT) and offers a novel cross-disciplinary perspective linking system security to coding theory.