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, referred to as set membership or unknown but bounded (UBB) error description, has been investigated since the late 60s. In this approach, uncertainty is described by an additive noise which is known only to have given integral (typically l 1 or l 2) or componentwise (l ∞) bounds. In this chapter the main results of this theory are reviewed, with special attention to the most recent advances obtained in the case of componentwise bounds.