Generalised resultants, dynamic polynomial combinants and the minimal design problem

Karcanias, N.

doi:10.1080/00207179.2013.791402

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…Remark 1. It is well known that every dynamic DAP can be formulated to an equivalent constant DAP (Karcanias, 2013) and hence the overall study will be focused to the static version of the problem.…”

Section: Problem Statementmentioning

confidence: 99%

Selection of Decentralised Schemes and Parametrisation of the Decentralised Degenerate Compensators

2016

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The design of decentralised control schemes has two major aspects. The selection of the decentralised structure and then the design of the decentralised controller that has a given structure and addresses certain design requirements. This paper deals with the parametrisation and selection of the decentralized structure such that problems such as the decentralised pole assignment may have solutions. We use the approach of global linearisation for the asymptotic linearisation of the pole assignment map around a degenerate compensator. Thus, we examine in depth the case of degenerate compensators and investigate the conditions under which certain degenerate structures exist. This leads to a parametrisation of decentralised structures based on the structural properties of the system.

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Section: Problem Statementmentioning

confidence: 99%

Selection of Decentralised Schemes and Parametrisation of the Decentralised Degenerate Compensators

2016

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“…The multilinear nature of DAP suggests that the natural framework for its study is that of exterior algebra [1], [2]. The study of DAP [4] may be reduced to a linear problem of zero assignment of polynomial combinants [11] and a standard problem of multi-linear algebra, that is the decomposability of multi-vectors [1]. The solution of the linear subproblem, whenever it exists, defines a linear space in a projective space P ρ (R) whereas decomposability is characterised by the set of Quadratic Plücker Relations (QPR), which define the Grassmann variety of P ρ (R) [3].…”

Section: Introductionmentioning

confidence: 99%

Partially Fixed Structure Determinantal Assignment Problems

Leventides

Karcanias

Livada

2021

IEEE Trans. Automat. Contr.

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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.

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“…The multilinear nature of DAP suggests that the natural framework for its study is that of exterior algebra [1]. The study of DAP [4] may be reduced to a linear problem of zero assignment of polynomial combinants [17] and a standard problem of multilinear algebra, that is the decomposability of multivectors [1]. The solution of the linear subproblem, whenever it exists, defines a linear space in a projective space  t P whereas decomposability is characterised by the set of Quadratic Plücker Relations (QPR), which define the Grassmann variety of t P [2].…”

Section: Introductionmentioning

confidence: 99%

Solution of the determinantal assignment problem using the Grassmann matrices

Karcanias

Leventides

2015

International Journal of Control

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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.

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Generalised resultants, dynamic polynomial combinants and the minimal design problem

Cited by 5 publications

References 15 publications

Selection of Decentralised Schemes and Parametrisation of the Decentralised Degenerate Compensators

Selection of Decentralised Schemes and Parametrisation of the Decentralised Degenerate Compensators

Partially Fixed Structure Determinantal Assignment Problems

Solution of the determinantal assignment problem using the Grassmann matrices

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