Probability-Constrained robust estimation for a random parameter with stochastic model uncertainties

This paper investigates the problem of estimating a random parameter in a linear observation model when the transformation matrix suffers stochastic uncertainties. In practice, the uncertainties can be either the bounded uncertainties or the Gaussian uncertainties. The existing minimax robust estimation method is only applicable to the bounded uncertainties, and minimizes the worst-case mean squared error (MSE) over the region of uncertainties, irrespective of the probability of such worst-case scenario, so it tends to be overly conservative. A probability-constrained (PC) robust linear estimation method is proposed in this paper to minimize the MSE with a guaranteed probability, which can handle bounded uncertainties as well as Gaussian uncertainties. However, the formulated problem is not analytically tractable due to the unspecified distribution of the uncertainties, so two approximate PC robust estimators are derived by tackling two upper bounds of the original optimization problem and converting them to semidefinite programming problems with the safe tractable approximation techniques. The comparative analysis reveals that there is a tradeoff between the MSE performance and computation complexity in the proposed two PC estimators. The Monte Carlo simulations are used to corroborate the theoretical results, which demonstrate that the proposed PC estimators are robust to the uncertainties compared to the nominal linear minimum mean squared error (LMMSE) estimator and the nominal best linear unbiased estimator (BLUE), and are less conservative compared to the traditional minimax robust estimator. (c) 2021 Elsevier B.V. All rights reserved.