On the Optimality of the Kitanidis Filter

As a natural extension of the Kalman filter to systems subject to arbitrary unknown inputs, the Kitanidis filter has been designed by one-step minimization of the trace of the state estimation error covariance matrix. This optimality does not exclude the possibility that, among the class of unbiased recursive filters, another filter may lead to a lower trace criterion. In this paper, it is shown that the Kitanidis filter is indeed optimal in the sense of the whole gain sequence, thus excluding the aforementioned possibility. Moreover, this gain sequence optimality holds also in a stronger sense, in terms of positive definite matrix inequality, which notably implies that the optimality holds not only in the sense of the trace criterion, but also of the matrix spectral norm criterion.