Abstract
The Determinantal Assignment Problem (DAP) has been introduced as the unifying description of all frequency assignment problems in linear systems and it is studied in a projective space setting. This is a multi-linear nature problem and its solution is equivalent to finding real intersections between a linear space, associated with the polynomials to be assigned, and the Grassmann variety of the projective space. This paper introduces a new relaxed version of the problem where the computation of the approximate solution, referred to as the approximate DAP, is reduced to a distance problem between a point in the projective space from the Grassmann variety Gm(Rn). The cases G2(Rn) and its Hodge-dual Gn−2(Rn) are examined and a closed form solution to the distance problem is given based on the skew-symmetric matrix description of multivectors via the gap metric. A new algorithm for the calculation of the approximate solution is given and stability radius results are used to investigate the acceptability of the resulting perturbed solutions.
Highlights
The Determinantal Assignment Problem (DAP) belongs to the family of algebraic synthesis methods and has emerged as the abstract problem formulation of pole, zero assignment of linear systems [Kar. 1]
The above approach for the study of DAP in a projective, rather than an affine space setting, as in [Bro. 1], [Mart. 1], among others, provides a computational approach that relies on exterior algebra and on the explicit description of the Grassmann variety in terms of the Quadratic Plucker Relations (QPR), which allows its formulation as a distance problem between varieties in the projective space
The approximate determinantal assignment problem has been defined and solved as a distance problem between the Grassmann variety and a linear variety defined by the properties of a desirable polynomial
Summary
The Determinantal Assignment Problem (DAP) belongs to the family of algebraic synthesis methods and has emerged as the abstract problem formulation of pole, zero assignment of linear systems [Kar. 1]. 1], among others, provides a computational approach that relies on exterior algebra and on the explicit description of the Grassmann variety in terms of the QPR, which allows its formulation as a distance problem between varieties in the (real) projective space. This may transform the problem of exact intersection to a problem of “approximate intersection”, i.e., small distance -via a suitable metric- between varieties, transform the exact DAP synthesis method to a DAP design methodology, where approximate solutions to the exact problem are sought. The Grassmann variety of a real projective space P(nq)−1 (R), i.e., the variety of the decomposable vectors of P(nq)−1 (R), is denoted as Gq(Rn)
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