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Set-membership estimation is usually formulated in the context of set-valuedcalculus and no probabilistic calculations are necessary. In this paper, weshow that set-membership estimation can be equivalently formulated in theprobabilistic setting by employing sets of probability measures. Inference inset-membership estimation is thus carried out by computing expectations withrespect to the updated set of probability measures P as in the probabilisticcase. In particular, it is shown that inference can be performed by solving aparticular semi-infinite linear programming problem, which is a special case ofthe truncated moment problem in which only the zero-th order moment is known(i.e., the support). By writing the dual of the above semi-infinite linearprogramming problem, it is shown that, if the nonlinearities in the measurementand process equations are polynomial and if the bounding sets for initialstate, process and measurement noises are described by polynomial inequalities,then an approximation of this semi-infinite linear programming problem canefficiently be obtained by using the theory of sum-of-squares polynomialoptimization. We then derive a smart greedy procedure to compute a polytopicouter-approximation of the true membership-set, by computing the minimum-volumepolytope that outer-bounds the set that includes all the means computed withrespect to P.

A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding