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

TL;DR

This paper introduces a probabilistic framework for set-membership filtering, enabling the use of sum-of-squares optimization for polynomial systems to efficiently approximate state sets with polytopes.

Contribution

It reformulates set-membership estimation within a probabilistic context using sets of probability measures and develops an efficient sum-of-squares based approximation method for polynomial systems.

Findings

Probabilistic interpretation of set-membership filtering established.

Sum-of-squares optimization enables efficient approximation for polynomial systems.

Greedy algorithm computes minimal-volume polytopic outer-approximations.

Abstract

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed 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, it is shown that, if the nonlinearities in the measurement and process equations are polynomial 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 sum-of-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 with respect to P.