Secure Set-Based State Estimation for Linear Systems under Adversarial Attacks on Sensors

Abstract: Set-based state estimation plays a vital role in the safety verification of dynamical systems, which becomes significantly challenging when the system's sensors are susceptible to cyber-attacks. Existing methods often impose limitations on the attacker's capabilities, restricting the number of attacked sensors to be strictly less than half of the total number of sensors. This paper proposes a Secure Set-Based State Estimation (S3E) algorithm that addresses this limitation. The S3E algorithm guarantees that the true system state is contained within the estimated set, provided the initialization set encompasses the true initial state and the system is redundantly observable from the set of uncompromised sensors. The algorithm gives the estimated set as a collection of constrained zonotopes, which can be employed as robust certificates for verifying whether the system adheres to safety constraints. Furthermore, we demonstrate that the estimated set remains unaffected by attack signals of sufficiently large and also establish sufficient conditions for attack detection, identification, and filtering. This compels the attacker to inject only stealthy signals of small magnitude to evade detection, thus preserving the accuracy of the estimated set. When a few number of sensors (less than half) can be compromised, we prove that the estimated set remains bounded by a contracting set that converges to a ball whose radius is solely determined by the noise magnitude and is independent of the attack signals. To address the computational complexity of the algorithm, we offer several strategies for complexity-performance trade-offs. The efficacy of the proposed algorithm is illustrated through its application to a three-story building model.