Minimax state estimation for linear stochastic systems with noise uncertainty

The problem of minimax linear state estimation for linear stochastic systems driven and observed in noises whose second-order properties are unknown is considered. Two general aspects of this problem are treated: the one-dimensional problem with uncertain noise spectra and the multidimensional problem with uncertain componentwise noise correlation. General minimax results are presented for each of these situations involving characterizations of the minimax filters in terms of least-favorable second-order properties. Explicit solutions are given for the spectral-band uncertainty model in the one-dimensional cases treated and for a matrix-norm neighborhood model in the multidimensional case. Characterization of saddle points in terms of the extremal properties of the noise uncertainty classes is also discussed.