Filtering and smoothing in an H∞ setting

The problems of filtering and smoothing are considered for linear systems in an H^∞ setting, i.e. the plant and measurement noises have bounded energies (are in L₂), but are otherwise arbitrary. Two distinct situations for the initial condition of the system are considered; the initial condition is assumed known in one case, while in the other the initial condition is not known but the initial condition, the plant, and measurement noise are in some weighted ball of RⁿXL₂. Finite-horizon and infinite-horizon cases are considered. Necessary and sufficient conditions are presented for the existence of estimators (both filters and smoothers) that achieve a prescribed performance bound, and algorithms that result in performance within the bounds are developed. In case of smoothers, the optimal smoother is also presented. The approach uses basic quadratic optimization theory in a time-domain setting, as a consequence of which both linear time-varying and time-invariant systems can be considered with equal ease. (In the smoothing problem, for linear time-varying systems, one considers only the finite-horizon case)