  In this paper, we investigate the role of a physical watermarking signal in
quickest detection of a deception attack in a scalar linear control system
where the sensor measurements can be replaced by an arbitrary stationary signal
generated by an attacker. By adding a random watermarking signal to the control
action, the controller designs a sequential test based on a Cumulative Sum
(CUSUM) method that accumulates the log-likelihood ratio of the joint
distribution of the residue and the watermarking signal (under attack) and the
joint distribution of the innovations and the watermarking signal under no
attack. As the average detection delay in such tests is asymptotically (as the
false alarm rate goes to zero) upper bounded by a quantity inversely
proportional to the Kullback-Leibler divergence(KLD) measure between the two
joint distributions mentioned above, we analyze the effect of the watermarking
signal variance on the above KLD. We also analyze the increase in the LQG
control cost due to the watermarking signal, and show that there is a tradeoff
between quick detection of attacks and the penalty in the control cost. It is
shown that by considering a sequential detection test based on the joint
distributions of residue/innovations and the watermarking signal, as opposed to
the distributions of the residue/innovations only, we can achieve a higher KLD,
thus resulting in a reduced average detection delay. Numerical results are
provided to support our claims.
