On nonlinear filters for mixed H^2/H^∞ estimation

Abstract

We study the problem of mixed least-mean-squares H^∞ -optimal (or mixed H^2/H^∞-optimal) estimation of signals generated by discrete-time, finite-dimensional, linear state-space models. The major result is that, for finite-horizon problems, and when the stochastic disturbances have Gaussian distributions, the optimal solutions have finite-dimensional (i.e., bounded-order) nonlinear state-space structure of order 2n+1 (where n is the dimension of the underlying state-space model). Being nonlinear, the filters do not minimize an H2 norm subject to an H^∞ constraint, but instead minimize the least-mean-squares estimation error (given a certain a priori probability distribution on the disturbances) subject to a given constraint on the maximum energy gain from disturbances to estimation errors. The mixed filters therefore have the property of yielding the best average (least-mean-squares) performance over all filters that achieve a certain worst-case (H^∞) bound