  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.
