  In this paper, a constrained attack-resilient estimation algorithm (CARE) is
developed for stochastic cyber-physical systems. The proposed CARE can
simultaneously estimate the compromised system states and attack signals. It
has improved estimation accuracy and attack detection performance when physical
constraints and operational limitations are available. In particular, CARE is
designed for simultaneous input and state estimation that provides
minimum-variance unbiased estimates, and these estimates are projected onto the
constrained space restricted by inequality constraints subsequently. We prove
that the estimation errors and their covariances from CARE are less than those
from unconstrained algorithms and confirm that this property can further reduce
the false negative rate in attack detection. We show that estimation errors of
CARE are practically exponentially stable in mean square. Finally, an
illustrative example of attacks on a vehicle is given to demonstrate the
improved estimation accuracy and detection performance compared to an existing
unconstrained algorithm.
