Optimal State Estimation Under the Denial-of-Service Attack: an Operator Approach

In this paper, we consider the problem of state-estimation in the presence of Denial-of-Service (DoS) attack. We formulate this problem as an state estimation problem for a plant with switching measured outputs. In the absence of attack, the state-estimator has access to all measured outputs, however, in the presence of attack, only a subset of all measurements are made available to the state-estimator. We seek to find a state-estimator that results in the minimum estimation error for the worst-case attack strategy. First, we parameterize the set of all state-estimators that result in stable estimation error. Then, we will show that any state-estimator in this set can be written as a generalized Luenberger observer with an appropriately defined observer-gain. This observer-gain, in general, can be an operator and possibly unbounded as opposed to the classical static observer-gain. Furthermore, we will show that finding the optimal state-estimator that results in the minimum estimation error can be cast as a convex program over the set of stable factors of the observer operator-gain. This optimization in, in fact, linear programming and tractable.