Partially Fixed Structure Determinantal Assignment Problems

Abstract

We deal with the study of the Determinantal Assignment Problem (DAP) when the parameters of the compensator are not entirely free, but some of them are fixed. The problem is reduced to a restricted form of an exterior algebra problem (decomposability of multi-vectors) which is referred to as Partial Decomposability problem. We study this problem and in case that this problem has no solution, we examine the problem of approximate partial decomposability. We treat the problem of exact or partial decomposability into a vector and a multivector of lower dimension. If this procedure is repeated then this results in an approximation of the initial multi-vector into a decomposable vector. The approximation of a vector by an optimal decomposable multi-vector is a non linear procedure and has been solved completely using the Power Method. The method developed in this paper although it produces a sub-optimal solution, can be used alternatively for the solution of DAP or the Approximate DAP, as a shorter and easier approach, because it is based on known tools as the Singular Value Decomposition (SVD). We apply these results to treat the Restricted -Approximate Decomposability problem, which leads to approximate solutions to the pole placement and zero assignment problems.