34,118 research outputs found
Recommended from our members
Approximate zero polynomials of polynomial matrices and linear systems
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials"
Recommended from our members
Approximate zero polynomials of polynomial matrices and linear systems
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials"
Recommended from our members
Approximate Zero Polynomials of Polynomial Matrices and Linear Systems
The aim of this paper is to extend recent results on the approximate GCD of polynomials [1] and approximate zeros to the case of a polynomial matrices within the framework of exterior algebra [2]. The results provide the means to introduce a new characterization of approximate decoupling zeros and measures for approximate controllability and observability for the case of linear systems
Certifying isolated singular points and their multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construc-tion uses a single linear differential form
defined from the Jacobian matrix of the input, and defines the deflated system
by applying this differential form to the original system. The advantages of
this new deflation is that it does not introduce new variables and the increase
in the number of equations is linear instead of the quadratic increase of
previous methods. The second construction gives the coefficients of the
so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new vari-ables. We show
that the roots of this new system include the original singular root but now
with multiplicity one, and the new variables uniquely determine the
multiplicity structure. Both constructions are "exact", meaning that they
permit one to treat all conjugate roots simultaneously and can be used in
certification procedures for singular roots and their multiplicity structure
with respect to an exact rational polynomial system
- …