Two Theorems in Multi-Weighted Sums, and their Application in the Stability Analysis of Active Linear Systems

摘要

The first theorem in this paper shows that the collection of all normalized, nonnegatively multiweighted sums of a finite set of vectors belonging to an arbitrary finite-dimensional vector space is contained in the convex hull of the vertex vectors of the normalized weights.nThe second theorem states, a sufficient condition under which the collection is complete, i.e. such that there is no point in the convex hull which is not a point of the sum. Two corollaries of this theorem identify particular types of sum for which the collection is complete.nApplication of the theorems in the stability analysis of active linear systems with multi-parameter variance leads to a useful graphical method which might be described as a generalization of Nyquist's criterion for stability.