Solution of the determinantal assignment problem using the Grassmann matrices

Abstract

This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: http://openaccess.city.ac.uk/12469/ Link to published version: http://dx. Abstract The paper provides a direct solution to the Determinantal Assignment Problem (DAP) which unifies all frequency assignment problems of Linear Control Theory. The current approach is based on the solvability of the exterior equation 12 m …z v v v      where m i z v     , is an n  dimensional vector space over F which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of z , and it is in turn characterised by the set of Quadratic Plücker Relations (QPR) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector z are given in terms of the rank properties of the Grassmann matrix, () m n z  of the vector z, which is constructed by the coordinates of m z . It is shown that the exterior equation is solvable (z is decomposable), if and only if () m n dim z m  where () = { ()} mm n r n zz  ; the solution space for a decomposable z , is the space () { } m nz i z sp i m v    . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations is examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP, (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist.

Structure assignment problems in linear systems: Algebraic and geometric methods

“…A direct solution to the computation of exact, as well as approximate solutions of DAP, has been proposed recently in Leventides et al (2014c), Karcanias and Leventides (2015). The exact DAP is to find a decomposable l-vector k t that satisfies (20) and is an intersection problem between a linear variety and the Grassmann variety.…”

“…The Hodge-Grassmann matrix The Hodge-Grassmann matrix is the Grassmann matrix of the Hodge dual of the multivector z and its properties are dual to those of the Grassmann matrix. In fact decomposability turns out to be an image problem for the transpose of the Hodge-Grassmann matrix (Karcanias and Leventides, 2015).…”

“…The affine geometry approach deals with generic cases only and it does not provide computations for exact, as well as, approximate problems. The DAP approach has the advantage of introducing new system invariants of rational vector spaces in terms Grassmann vectors and Plücker matrices (Karcanias and Giannakopoulos, 1984) providing a matrix characterisation of decomposability in terms of the Grassmann matrices , (Karcanias and Leventides, 2015) and developing a novel computational framework based on the technique of Global Linearisation (GL) (Leventides and Karcanias, 1995). GL is based on the notion of degenerate feedback (Brockett and Byrnes, 1981) and apart from establishing solvability conditions (Leventides and Karcanias, 1995), also provides a linearisation of the inherently nonlinear equations and leads to the computation of solutions (when such solutions exist).…”

“…A significant advantage of the DAP framework is that it introduces a new approach for the computation of exact and approximate solutions of DAP. This is based on an alternative, linear algebra type, criterion for decomposability of multivectors to that defined by the QPRs, in terms of the rank properties of structured matrices, referred to as Grassmann matrices , (Karcanias and Leventides, 2015). The development of the new computational framework requires the study of the properties of Grassmann matrices, which are further developed by using the Hodge duality (Hodge and Pedoe, 1952) leading to the definition of the Hodge-Grassmann matrix (Karcanias and Leventides, 2015).…”

“…This is based on an alternative, linear algebra type, criterion for decomposability of multivectors to that defined by the QPRs, in terms of the rank properties of structured matrices, referred to as Grassmann matrices , (Karcanias and Leventides, 2015). The development of the new computational framework requires the study of the properties of Grassmann matrices, which are further developed by using the Hodge duality (Hodge and Pedoe, 1952) leading to the definition of the Hodge-Grassmann matrix (Karcanias and Leventides, 2015). Computing solutions (exact, or approximate) to DAP requires the investigation of distance problems, such as: (i) distance of a point from the Grassmann variety; (ii) distance of a linear variety from the Grassmann variety; (iii) parametrisation of families of linear varieties with a given distance from the Grassmann variety; (iv) relating the latter distance problems with properties of the stability domain.…”