Filtering and smoothing in an H∞ setting

Consideration is given to the problems of filtering and smoothing for linear systems in an H^∞ setting, i.e. the plant and measurement noises have bounded energies (are in L ₂), but are otherwise arbitrary. Two distinct situations for the initial condition of the system are considered: in one case the initial condition is assumed known; in the other case it is not known, but the initial condition, the plant, and the measurement noise are in some weighted ball of Rⁿ×L₂. Both finite-horizon and infinite-horizon cases are considered. The authors present necessary and sufficient conditions for the existence of estimators (both filters and smoothers) that achieved a prescribed performance bound and develop algorithms that result in performance within the bounds. They also present the optimal smoother. The approach uses basic quadratic optimization theory in a time-domain setting, as a consequence of which time-varying and time-invariant linear systems can be considered with equal ease