Computing the Canonical Representation of Constructible Sets

Brunat, Josep M.; Montes, Antônio

doi:10.1007/s11786-016-0248-2

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…in [16], the proposed methods allow one to determine the actual values of the parameters that lead to a loss of the reachability, observability, controllability, constructibility, stabilisability, and detectability properties for parametric (discrete‐time and continuous‐time) systems. The drawback of this more detailed analysis is an increased computational complexity since determining Gröbner covers is more computationally demanding than using the graph‐theoretical techniques given in [16] (see [18–21] for details on the computational complexity of Gröbner covers), which makes the techniques given in this paper not practically applicable to large or networked systems having hundreds of state variables. However, algorithms for algebraic geometry computations are continuously improving, so that it will be possible in the future to solve problems that are today intractable.…”

Section: Discussionmentioning

confidence: 99%

“…In the following, each pair

false(A_{i}, G_{i} false)

i = 1, \dots, ζ

, of a Gröbner cover of a parametric ideal

scriptI

is referred to as a branch of the Gröbner cover, and is constituted by a segment

A_{i}

and by a (not necessarily reduced) Gröbner basis

G_{i}

of the parametric ideal along that segment. See [18–21] for algorithms capable of computing Gröbner covers of parametric polynomial ideals; these algorithms are implemented in the grobcov library of the software Singular [22], an open source computer algebra system, which is used hereafter to carry out all the computations in the examples given in the following, including the computation of the reduced Gröbner basis; the

GRL

order is always used in all these examples, just for reducing the computation times [17].…”

Section: Algebraic Geometry Notionsmentioning

confidence: 99%

See 1 more Smart Citation

Algebraic analysis of the structural properties of parametric linear time‐invariant systems

Menini

Possieri

Tornambè

2020

IET Control Theory & Appl

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By using techniques borrowed from algebraic geometry, some tests are proposed to certify structural properties (such as reachability, observability, controllability, constructibility, stabilisability, and detectability) of discrete‐time and continuous‐time linear time‐invariant systems. These results are then generalised for linear time‐invariant systems depending polynomially on some real parameters, by exploiting the notion of Gröbner cover. The main innovation of the proposed results with respect to existing techniques is that they provide exact certificates for the structural properties of linear time‐invariant systems. Examples of application are given all throughout the study to illustrate and validate the theoretical results.

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Section: Discussionmentioning

confidence: 99%

“…In the following, each pair

false(A_{i}, G_{i} false)

i = 1, \dots, ζ

, of a Gröbner cover of a parametric ideal

scriptI

is referred to as a branch of the Gröbner cover, and is constituted by a segment

A_{i}

and by a (not necessarily reduced) Gröbner basis

G_{i}

GRL

order is always used in all these examples, just for reducing the computation times [17].…”

Section: Algebraic Geometry Notionsmentioning

confidence: 99%

Algebraic analysis of the structural properties of parametric linear time‐invariant systems

Menini

Possieri

Tornambè

2020

IET Control Theory & Appl

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“…We assume access to algorithms that 1. compute unions and intersections of constructible sets, and 2. determine whether a constructible set is empty. See [BM16], for example.…”

Section: Algorithms For Families Of Schemesmentioning

confidence: 99%

When is a Polynomial Ideal Binomial After an Ambient Automorphism?

2018

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Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials x a − λx b with λ ∈ k, or by unital binomials (i.e., with λ = 0 or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates of I over an algebraic group G acting on affine space, these problems are special cases of questions about a family I of ideals over an arbitrary base B. The main results in this general setting are algorithms to find the locus of points in B over which the fiber of I • is contained in the fiber of a second family I ′ of ideals over B;• defines a variety of dimension at least d;• is generated by binomials; or • is generated by unital binomials. A faster containment algorithm is also presented when the fibers of I are prime. The big-fiber algorithm is probabilistic but likely faster than known deterministic ones. Applications include the setting where a second group T acts on affine space, in addition to G, in which case algorithms compute the set of G-translates of I• whose stabilizer subgroups in T have maximal dimension; or • that admit a faithful multigrading by Z r of maximal rank r. Even with no ambient group action given, the final application is an algorithm to• decide whether a normal projective variety is abstractly toric. All of these loci in B and subsets of G are constructible; in some cases they are closed.

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Distance to the loss of structural properties for linear systems under parametric uncertainties

Menini

Possieri

Tornambè

2023

Journal of the Franklin Institute

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Computing the Canonical Representation of Constructible Sets

Abstract: Constructible sets are needed in many algorithms of Computer Algebra, particularly in the Gröbner 1 Cover and other algorithms for parametric polynomial systems. In this paper we review the canonical form of 2 constructible sets and give algorithms for computing it.

Cited by 3 publications

References 13 publications

Algebraic analysis of the structural properties of parametric linear time‐invariant systems

Algebraic analysis of the structural properties of parametric linear time‐invariant systems

When is a Polynomial Ideal Binomial After an Ambient Automorphism?

Distance to the loss of structural properties for linear systems under parametric uncertainties

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