Minimax stochastic estimation and filtering under unknown covariances

In this paper we consider a minimax approach to the estimation and filtering problems in the stochastic framework, where covariances of the random factors are completely unknown. We introduce a notion of the attenuation level of random factors as a performance measure for both a linear unbiased estimate and a filter. This is the worst- case variance of the estimation error normalized by the sum of variances of all random factors over all nonzero covariance matrices. It is shown that this performance measure is equal to the spectral norm of the "transfer matrix" and therefore the minimax estimate and filter can be computed in terms of linear matrix inequalities (LMIs). Moreover, the explicit formulae for both the minimax estimate and the minimal value of the attenuation level are presented in the estimation problem. In addition, we demonstrate that the LMI technique can be applied to derive the optimal robust estimator and filter, when there is a priori information about convex polyhedral sets which unknown covariance matrices of random factors belong to.