Robust estimators for systems with deterministic and stochastic uncertainties

For uncertain systems containing both deterministic and stochastic uncertainties, we consider two problems of optimal estimation. The first is the design of a filter that minimizes an upper bound on the worst-case gain in the mean energy between the noise affecting the system and the estimation error. The second is the design of a filter that minimizes an upper bound on the worst-case asymptotic mean square estimation error when the plant is driven by a white noise process. We present filtering algorithms that solve each of these problems, with the filter parameters determined via convex optimization based on linear matrix inequalities. We demonstrate the performance of these robust algorithms on a numerical example consisting of the design of equalizers for a communication channel.