Do Interpretable Optimal Bounding Ellipsoid Algorithms Converge? Part I - The Long- Awaited Set-Convergence Proof

Abstract

Optimal bounding ellipsoid (OBE) algorithms with interpretable optimization (volume or trace) criteria converge under the condition that the disturbance process visit the error bounds infinitely often. This result is proven in two logical steps. First, the set-membership-stochastic approximation (SM-SA) is presented. SM-SA is a new volume minimization OBE algorithm for which convergence of both the point-estimator and the membership set is proven. Next, it is argued that the salient SM-SA convergence properties apply to all volume-based OBE algorithms. This paper is concerned with proof of convergence for the case of i.i.d. disturbances. In the companion paper (Part II), less constrained white and colored noises are considered, yielding very interesting relationships between OBE algorithms and RLS.