  In this paper, quickest detection of false data injection attack on remote
state estimation is considered. A set of $N$ sensors make noisy linear
observations of a discrete-time linear process with Gaussian noise, and report
the observations to a remote estimator. The challenge is the presence of a few
potentially malicious sensors which can start strategically manipulating their
observations at a random time in order to skew the estimates. The quickest
attack detection problem for a known {\em linear} attack scheme in the Bayesian
setting with a Geometric prior on the attack initiation instant is posed as a
constrained Markov decision process (MDP), in order to minimize the expected
detection delay subject to a false alarm constraint, with the state involving
the probability belief at the estimator that the system is under attack. State
transition probabilities are derived in terms of system parameters, and the
structure of the optimal policy is derived analytically. It turns out that the
optimal policy amounts to checking whether the probability belief exceeds a
threshold. Next, generalized CUSUM based attack detection algorithm is proposed
for the non-Bayesian setting where the attacker chooses the attack initiation
instant in a particularly adversarial manner. It turns out that computing the
statistic for the generalised CUSUM test in this setting relies on the same
techniques developed to compute the state transition probabilities of the MDP.
Numerical results demonstrate significant performance gain under the proposed
algorithms against competing algorithms.
