Determinantal Assignment Problem Research Articles

Partially Fixed Structure Determinantal Assignment Problems

In this article, 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 multivectors), 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 multivector into a decomposable vector. The approximation of a vector by an optimal decomposable multivector is a nonlinear procedure and has been solved completely using the power method. The method developed in this article, although it produces a suboptimal 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. We apply these results to treat the restricted approximate decomposability problem, which leads to approximate solutions to the pole placement and zero assignment problems.

A Grassmann Matrix Approach for the Computation of Degenerate Solutions for Output Feedback Laws

The paper is concerned with the improvement of the overall sensitivity properties of a method to design feedback laws for multivariable linear systems which can be applied to the whole family of determinantal type frequency assignment problems, expressed by a unified description, the so-called Determinantal Assignment Problem (DAP). By using the exterior algebra/algebraic geometry framework, DAP is reduced to a linear problem (zero assignment of polynomial combinants) and a standard problem of multilinear algebra (decomposability of multivectors) which is characterized by the set of Quadratic Plücker Relations (QPR) that define the Grassmann variety of P. This design method is based on the notion of degenerate compensator, which are the solutions that indicate the boundaries of the control design and they provide the means for linearising asymptotically the nonlinear nature of the problems and hence are used as the starting points to generate linearized feedback laws. A new algorithmic approach is introduced for the computation and the selection of degenerate solutions (decomposable vectors) which allows the computation of static and dynamic feedback laws with reduced sensitivity (and hence more robust solutions). This approach is based on alternative, linear algebra type criterion for decomposability of multivectors to that defined by the QPRs, in terms of the properties of structured matrices, referred to as Grassmann Matrices. The overall problem is transformed to a nonlinear maximization problem where the objective function is expressed via the Grassmann Matrices and the first order conditions for optimality are reduced to a nonlinear eigenvalue-eigenvector problem. Hence, an iterative method similar to the power method for finding the largest modulus eigenvalue and the corresponding eigenvector is proposed as a solution for the above problem.

Zero Assignment Problem in RLC Networks

The paper deals with the problem of zero assignment in RLC networks by selection of appropriate values for the non- dynamical elements, the resistors. For a certain family of network redesign problems by the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem belongs to the family of DAP problems (Determinantal Assignment Problem) and has common features with the pole assignment problem by decentralized output feedback and the zero assignment problem via structured additive perturbations. We demonstrate that the sufficient condition for generic zero assignment by selecting the resistors holds true. This condition is related to the rank of the differential of the related map and holds true generically when the degrees of freedom of the matrix of resistors exceeds the number of frequencies to be assigned (n > p + q). Using this result, we show through a generic example that the sufficient condition for the general zero assignment problems in RLC networks is satisfied and thus, zero assignment can be achieved via resistor determination.

Solution of the determinantal assignment problem using the Grassmann matrices

ABSTRACTThe paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over 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 are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . 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 are 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.

The approximate Determinantal Assignment Problem

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.

Generalised resultants, dynamic polynomial combinants and the minimal design problem

The theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems (DAP), which provides the unifying description of the dynamic, as well as static pole and zero dynamic assignment problems in linear systems. The assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original DAP. This paper demonstrates the origin of dynamic polynomial combinants from linear systems, examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree, and examines their spectral assignment. Central to this study is the link of dynamic combinants to the theory of generalised resultants, which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterisation of combinants and respective generalised resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, which is referred to as the dynamic combinant minimal design (DCMD) problem. An algorithmic approach based on rank tests of Sylvester matrices is given, which produces the minimal order and degree solution in a finite number of steps. Such solutions provide low bounds for the respective dynamic assignment control problems.

Approximate Solutions of the Determinantal Assignment Problem and Distance Problems

The paper introduces the formulation of an exact algebrogeometric problem, the study of the Determinantal Assignment Problem (DAP) in the set up of design, where approximate solutions of the algebraic problem are sought. Integral part of the solution of the Approximate DAP is the computation of distance of a multivector from the Grassmann variety of a projective space. We examine the special case of the calculation of the minimum distance of a multivector in ∧2(ℝ5) from the Grassmann variety G2(ℝ5). This problem is closely related to the problem of decomposing the multivector and finding its best decomposable approximation. We establish the existence of the best decomposition in a closed form and link the problem of distance to the decomposition of multivectors. The uniqueness of this decomposition is then examined and several new alternative decompositions are presented that solve our minimization problem based on the structure of the problem.

Parameterisation of Degenerate Solutions of the Determinantal Assignment Problem

The paper is concerned with defining and parametrising the families of all degenerate compensators (feedback, squaring down etc) emerging in a variety of linear control problems. Such compensators indicate the boundaries of the control design, but they also provide the means for linearising the non-linear nature of the Determinantal Assignment Problems, which provide the unifying description for all frequency assignment problems (pole, zero) under static and dynamic compensation schemes. The conditions provide the means for the selection of appropriate degenerate solutions that allow frequency assignability in the corresponding frequencies.

The Minimal Design Problem on Dynamic Polynomial Combinants

AbstractThe theory of dynamic polynomial combinants is linked to the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the pole and zero dynamic assignment problems in Linear Systems. The fundamentals of the theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Central to this study is the link of dynamic combinants to the theory of “Generalised Resultants“, which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterization of combinants and respective Generalised Resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, referred to as the “Dynamic Combinant Minimal Design“ (DCMD) problem. Such solutions provide low bounds for the respective Dynamic Assignment control problems.

Dynamic Polynomial Combinants and Generalised Resultants: Parameterization according to Order and Degree

The theory of constant polynomial combinants has been well developed and it is linked to the linear part of the constant Determinantal Assignment problem that provides the unifying description of the pole and zero assignment problems in Linear Systems. Considering the case of dynamic pole, zero assignment problems leads to the emergence of dynamic polynomial combinants. This paper aims to develop the fundamentals of the theory of polynomial combinants by examining issues of their parameterization of dynamic polynomial combinants according to the notions of order and degree. Central to this study is the link of dynamic combinants to the theory of “Generalised Resultants”. The paper provides a description of the key spectral assignment problems, derives the conditions for arbitrary assignability of spectrum and introduces a complete parameterization of combinants and respective Generalised Resultants which is crucial for studying the minimal degree and order spectrum assignability.

GRASSMANN MATRICES, DETERMINANTAL ASSIGNMENT PROBLEM AND APPROXIMATE DECOMPOSABILITY

The exterior equation an n—dimensional vector space over F, is an integral part of the study of the Determinantal Assignment Problem (DAP) of linear systems and its solvability (decomposability of ) is characterised by the Quadratic Pliicker Relations (QPR). An alternative new test for decomposability of is given, in terms of the rank properties of the Grassmann matrix, , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if dim = m, where . If is decomposable, then the solution space is simply defined by . The linear algebra formulation of the decomposability problem provides an alternative framework (to that defined by the QPRs) for the study of solvability and computation of solutions of DAP and enables the definition and study of “approximate solutions” of exterior equations as a distance problem. For the case of m = 2, n = 4 a solution to approximate decomposability is given and its properties are linked to the singular values of .

Computation of Solutions of the DAP Using Optimization Methods

Linear Control Theory problems such as those of pole assignment by state, output feedback (centralised or decentralised), and zero assignment by input, or output squaring down may be reduced to a standard common problem known as the Determinantal assignment Problem (DAP) Giannakopoulos (1985). The aim of this paper is to develop and compare optimization algorithms for the computation of solutions of DAP. The developed numerical approaches may be used as a basis of a design technique centered around the frequency assignment problems.

Fas-Fas ligand interactions underlie the pathogenesis of endotoxin-induced liver injury

The exterior equation an n—dimensional vector space over F, is an integral part of the study of the Determinantal Assignment Problem (DAP) of linear systems and its solvability (decomposability of ) is characterised by the Quadratic Pliicker Relations (QPR). An alternative new test for decomposability of is given, in terms of the rank properties of the Grassmann matrix, , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if dim = m, where . If is decomposable, then the solution space is simply defined by . The linear algebra formulation of the decomposability problem provides an alternative framework (to that defined by the QPRs) for the study of solvability and computation of solutions of DAP and enables the definition and study of “approximate solutions” of exterior equations as a distance problem. For the case of m = 2, n = 4 a solution to approximate decomposability is given and its properties are linked to the singular values of .

Decentralized determinantal assignment problem: fixed and almost fixed modes and zeros

The decentralized determinantal assignment problem (DDAP) is defined as the unifying description for the study of pole and zero assignment problems under decentralized output, state feedback (DOF, DSF) and decentralized ‘squaring down’ (DSD), respectively. DDAP is reduced to a linear problem of zero assignment of polynomial combinants and a multilinear problem of restricted decomposability of multivectors. The decentralization characteristic (DC) and the decentralized polynomial Grassmann representative )D — ℝ[s] — GR) of DDAP are defined. The fixed zero polynomial of DDAP is then determined as the zero polynomial of D— ℝ[s]—GR. The canonical D—ℝ[s]—GR, (CD—R[s]—GR) and the decentralized Plucker matrix (DPM) of DDAP are introduced and necessary conditions for arbitrary assignment of the non-fixed zeros are given in terms of the DPM. The family of strongly zero non-assignable (SNA( systems is defined, and for such systems the notion of the fixed zero is extended to that of the ‘almost fixed zero’ (AFZ). An...