On Nonlinear Filters for Mixed H²/H∞ Estimation

We study the problem of mixed least-meansquares /H 1 -optimal (or mixed H 2 =H 1 -optimal) estimation of signals generated by discrete-time, finitedimensional, 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, strictly speaking, the filters do not minimize an H 2 norm subject to an H 1 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 1 ..