Filtering and Control with set of distributions

A common trait of state estimation techniques is that they assume that the distributions associated with the prior, state transition, and likelihood functions are perfectly known. However, in many practical cases, our information about the system to be modelled may not allow us to characterize these functions with single (precise) distributions. For example, in the Gaussian case, we may only be able to determine an interval that contains the mean of the Gaussian distribution or, in more general cases, we may only be able to state that the distribution of the noise belongs to some set of distributions. This leads to alternative models of representation of uncertainty based on a set of probability distributions and, thus, to robust filtering. The most explored techniques for robust filtering are H-infinity, H2 and set-membership estimation. These techniques deal mainly with two kinds of uncertainties: norm-bounded parametric uncertainty and/or bounded uncertainty in the noise statistics or in the noise intensity. In a recent paper we have proposed a new more general approach to robust filtering that instead focuses attention on the use of closed convex sets of distributions to model the imprecision in the knowledge about the system parameters and probabilistic relationships involved. Norm-bounded parametric uncertainty and/or bounded uncertainty can in fact be seen as special cases of closed convex sets of distributions. For instance, we can solve the filtering problem from the only knowledge of few moments of the noise terms and without introducing additional assumptions on the distributions of the noises (e.g., Gaussianity) or on the final form of the estimator (e.g., linear estimator).

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