Ellipsoid Set Refinement by Simultaneous Use of Multiple Hyperplane Cuts

Abstract

Let Hn+, Hn-, n = 1,2,… denote a sequence of pairs of parallel hyperplanes in Rm, and, for each n, let Hn denote the set of points on the open "hyperstrip" between the planes Hn+ andHn-. It is assumed that the polytopic set Pk = ∩n=1kHn is not empty for any k. Further let εn, n = 1,2,… denote a corresponding sequence of hyperellipsoids, and let εn ∈ Rm denote the open hyperellipsoidal set bounded by εn. This paper derives a mathematical formulation for fitting a new, smaller hypervolume, hyperellipsoid, εn, to the intersection ε1 ∩ H1 ∩ H2 ∩ … ∩ Hk = ε1 ∩ Pk. The prevailing method for solving this problem involves sequential refinement of ellipsoids as hyperstrips are considered one at a time. The simultaneous use of multiple hyperstrips results in a better fit to the polytope Pk than that achieved by sequential refinement. The problem arises in numerous control and signal processing problems - in particular the broad class of ellipsoid bounding algorithms used for identification and classification.