Optimal Estimation theory for dynamic systems with set membership uncertainty: an overview

In many problems such as linear and nonlinear regressions, parameter and state estimation of dynamic systems, state-space and time series prediction, interpolation, smoothing and functions approximation, one has to evaluate some unknown variable using available data. The data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables. Typically, the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory. An interesting alternative approach, referred to as set membership or unknown but bounded (UBB) error description, has been investigated since the late 1960s. In this approach, uncertainty is described by an additive noise which is known only to have given integral (typically l2 of l1) or componentwise (l infinity) bounds. In this paper we review the main results of this theory, with special attention to the recent advances obtained in the case of componentwise bounds.