  This paper studies the problem of secure state estimation of a linear
time-invariant (LTI) system with bounded noise in the presence of sparse
attacks on an unknown, time-varying set of sensors. In other words, at each
time, the attacker has the freedom to choose an arbitrary set of no more that
$p$ sensors and manipulate their measurements without restraint. To this end,
we propose a secure state estimation scheme and guarantee a bounded estimation
error subject to $2p$-sparse observability and a mild, technical assumption
that the system matrix has no degenerate eigenvalues. The proposed scheme
comprises a design of decentralized observer for each sensor based on the local
observable subspace decomposition. At each time step, the local estimates of
sensors are fused by solving an optimization problem to obtain a secure
estimation, which is then followed by a local detection-and-resetting process
of the decentralized observers. The estimation error is shown to be
upper-bounded by a constant which is determined only by the system parameters
and noise magnitudes. Moreover, we optimize the detector threshold to ensure
that the benign sensors do not trigger the detector. The efficacy of the
proposed algorithm is demonstrated by its application on a benchmark example of
IEEE 14-bus system.
