Optimal estimation and filtration under unknown covariances of random factors

The general schemes of linear estimation and filtration were considered on assumption of the unknown covariance matrix of random factors such as unknown parameters, measurement errors, and initial and external perturbations. A new criterion was introduced for the quality of estimate or filter. It is the level of damping random perturbations which is defined by the maximal value over all covariance matrices of the root-mean-square error normalized by the sum of variances of all random factors. The level of damping random perturbations was shown to be equal to the square of the spectral norm of the matrix relating the error of estimation and the random factors, and the optimal estimate minimizing this criterion was established. In the problem of filtration, it was shown how the filter parameters that are optimal in the level of damping random perturbations are expressed in terms of the linear matrix inequalities.