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

Set-membership estimation is in general referred in literature as the deterministic approach to state estimation, since its solution can be formulated in the context of set-valued calculus and no stochastic calculations are necessary. This turns out not to be entirely true. In this paper, we show that set-membership estimation can be equivalently formulated in the stochastic setting by employing sets of probability measures. Inferences in set-membership estimation are thus carried out by computing expectations w.r.t. the updated set of probability measures P as in the stochastic case. In particular, we show that inferences can be computed by solving a particular semi-infinite linear programming problem, which is a special case of the 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 linear programming problem, we show that, if the nonlinearities in the measurement and process equations are polynomials and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sumof-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed w.r.t. P .