Local Decomposition of Kalman Filters and its Application for Secure State Estimation

This article is concerned with the secure state estimation problem of a linear discrete-time Gaussian system in the presence of sparse integrity attacks.
$\mathbf {m}$
sensors are deployed to monitor the state and
$\mathbf {p}$
of them can potentially be compromised by an adversary, whose data can be arbitrarily manipulated by the attacker. We show that the optimal Kalman estimate can be decomposed as a weighted sum of local state estimates. Based on these local estimates, we propose a convex optimization based approach to generate a more secure state estimate. It is proved that our proposed estimator coincides with the Kalman estimator with a certain probability when all sensors are benign. Besides, we establish a sufficient condition under which the proposed estimator is stable against the
$\mathbf {(p,m)}$
-sparse attack. A numerical example is provided to validate the secure state estimation scheme.