Linear algebra and its applications

Vol. 559, p. 194-226

DOI: 10.1016/j.laa.2018.09.009

Date of publication: 2018-12-15

Abstract:

The classification of invariant subspaces is an open problem related to other important ones like the Carlson problem. Here we obtain a reduced form of these invariant subspaces as a new tool to tackle these problems. In particular, it allows us to prove quite easily partial results already known. The key point is assigning to each invariant subspace a marked one (its marked type) in order to partition the set of invariant subspaces in a finite number of subsets (the marked classes), each one containing only one marked subspace. Next, we parametrize (minimally) each marked class by means of the so-called PM reduced families, so that representatives of an invariant subspace (its PM reduced forms) appear in just one of these families.]]>

IEEE Frontiers in Education Conference

p. 1-5

DOI: 10.1109/FIE.2018.8658683

Presentation's date: 2018-10-06

Abstract:

This Innovative Practice Work in Progress Paper presents a pilot test designed to train university students of engineering degrees (mentors) to advise and guide students aged between 15 and 16 (mentees) in technological fields in different teaching centers in Catalonia. The first results suggest the high interest shown by the group in the talks of the mentors; a greater attention and predisposition of the male students are observed in the questions and interventions. The sensitization of the female group on STEM issues is surprising, although there is a low participation and interest shown by the topics discussed. All these aspects must be taken into account in the planning of future interventions.]]>

Congrés Internacional de Docència Universitària i Innovació

p. 1-14

Presentation's date: 2018-07-05

Abstract:

El proyecto piloto t'STEAM coordinado por el l'Institut de Ciències de l'Educació (ICE) de la Universitat Politècnica de Catalunya. UPC·BarcelonaTech (UPC) propone utilizar la mentoría entre mujeres como herramienta para motivar y acompañar las vocaciones científico-técnicas en alumnas de secundaria de 4º de Enseñanza Secundaria Obligatoria (ESO). Se pretenden aprovechar las experiencias de estudiantes de la UPC (mentoras) para establecer una relación con chicas de centros de secundaria (mentoradas) que despierte e incentive su interés en los estudios STEM.]]>

International journal of bifurcation and chaos

Vol. 28, num. 2, p. 1-13

DOI: 10.1142/S0218127418500256

Date of publication: 2018-02-01

Abstract:

We continue the study of the structural stability and the bifurcations of planar continuous bimodal linear dynamical systems (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we complete the study when one of the subsystems is a saddle, leading to a 3D bifurcation diagram where a large catalogue of bifurcations appears: four surfaces of codimension-1 bifurcations; two sequences of surfaces of additional codimension-1 bifurcations; two lines of codimension-2 bifurcations; and one codimension-3 bifurcation.]]>

International journal of bifurcation and chaos

Vol. 27, num. 1, p. 1-13

DOI: 10.1142/S0218127417500055

Date of publication: 2017-01-01

Abstract:

We complete the study of the bifurcations of saddle/spiral bimodal linear systems, depending on the respective traces T and t: one 2-codimensional bifurcation; four kinds of 1-codimensional bifurcations. We stratify the bifurcation set in the (T,t)-plane and we describe the qualitative changes of the dynamical behavior at each bifurcation point.]]>

Mathematical Models for Applied Mechanics

p. 345-350

Presentation's date: 2016-09-29

Abstract:

We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcation.]]>

Meeting of the International Linear Algebra Society

p. 237

Presentation's date: 2016-07-11

Abstract:

We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcation.

We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcation]]>

International journal of mathematical and computational methods

num. 1, p. 345-350

Date of publication: 2016

Abstract:

We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcation]]>

Mathematical Models and Methods in Applied Sciences

p. 90-92

Presentation's date: 2014-09-23

Abstract:

We continue the study of the bifurcations and the structural stability of planar bimodal linear dynamical systems (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here we determine the tangency-saddle singularities in the saddle/spiral case, the only where they can appear.]]>

Meeting of the International Linear Algebra Society

Presentation's date: 2014-08-08

Mathematical problems in engineering

Vol. 2014, num. Article ID 292813, p. 1-9

DOI: 10.1155/2014/292813

Date of publication: 2014-06

Abstract:

We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along it. For a differentiable family of planar bimodal linear control systems, we obtain its stratification diagram and, if controllability holds for each value of the parameters, we construct a differentiable family of feedbacks which stabilizes both subsystems for each value of the parameters.]]>

Mathematical problems in engineering

Vol. 2014, p. 1-8

DOI: 10.1155/2014/892948

Date of publication: 2014-01-01

Abstract:

Structural stability ensures that the qualitative behavior of a system is preserved under small perturbations. We study it for planar bimodal linear dynamical systems, that is, systems consisting of two linear dynamics acting on each side of a given hyperplane and assuming continuity along the separating hyperplane. We describe which one of these systems is structurally stable when (real) spiral does not appear and when it does we give necessary and sufficient conditions concerning finite periodic orbits and saddle connections. In particular, we study the finite periodic orbits and the homoclinic orbits in the saddle/spiral case.]]>

Mathematical methods in the applied sciences

Vol. 37, num. 1, p. 20-31

DOI: 10.1002/mma.2780

Date of publication: 2014

Abstract:

Given the set of vertical pairs of matrices ${\cal M}\subset M_{m,n}(\mathbb C)\times M_n(\mathbb C)$ keeping the subspace $\mathbb C^d\times\{0\}\subset\mathbb C^n$ invariant,we compute miniversal deformations of a given pair when it is observable, and the subspace $\mathbb C^d\times\{0\}$ is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs, and study the effect of each deformation parameter. Copyright © 2013 JohnWiley & Sons, Ltd.]]>

International Conference on Numerical Analysis and Applied Mathematics

p. 2205-2208

DOI: 10.1063/1.4825976

Presentation's date: 2013-09

Abstract:

We consider bimodal linear dynamical systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. Focusing in the planar case, we describe which of these systems are structurally stable

We consider bimodal linear dynamical systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. Focusing in the planar case, we describe which of these systems are structurally stable.]]>

International Conference on Numerical Analysis and Applied Mathematics

p. 2068-2071

DOI: 10.1063/1.4825942

Presentation's date: 2013-09

Abstract:

We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. For a differentiable family of controllable planar ones, we construct a differentiable family of feedbacks which point wise stabilizes both subsystems

We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. For a differentiable family of controllable planar ones, we construct a differentiable family of feedbacks which pointwise stabilizes both subsystems.]]>

Mathematical problems in engineering

Vol. 2013, p. 1-14

DOI: 10.1155/2013/342548

Date of publication: 2013-05-01

Abstract:

We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions

We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions.]]>

DOI: DOI: 10.5772/2327

Date of publication: 2012-07

SIAM Conference on Applied Linear Algebra

p. 146

Presentation's date: 2012-06-21

Mathematical methods in the applied sciences

Vol. 35, num. 3, p. 249-371

DOI: 10.1002/mma.1553

Date of publication: 2012-02

Abstract:

Given the set of matrix pairsM Mm,n.C/ Mn.C/ keeping a subspace S Cn invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset ofM. It generalizes the known results when S is a supplementary subspace of the unobservable one.]]>

Date: 2012

Abstract:

Given the set of vertical pairs of matrices M¿ Mm,n(C)×Mn(C) keeping the subspace Cd×{0} ¿ Cn invariant, we compute miniversal deformations of a given pair when it is observable and the subspace Cd × {0} is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs and study the effect of each deformation parameter.]]>

Electronic journal of linear algebra

Vol. 22, p. 1112-1128

Date of publication: 2011-11

Abstract:

Given a pair of matrices representing a controllable linear system, its equivalence classes by the single or combined action of feedbacks, change of state and input variables, as well as their intersection are studied. In particular, it is proved that they are differentiable manifolds and their dimensions are computed. Some remarks concerning the effect of different kinds of feedbacks are derived.]]>

Meeting of the International Linear Algebra Society

p. 57

Presentation's date: 2011-08-23

Mathematical methods in the applied sciences

Vol. 34, num. 3, p. 309-314

DOI: 10.1002/mma.1357

Date of publication: 2011

Abstract:

Given S an (A,B)-invariant subspace, we prove that the set of friend feedbacks is a linear variety, which can be considered as the direct sum of the feedbacks of the restriction to S and the co-restriction to S ⊥. In particular, when the natural controllability hypothesis hold, both pole assignments are simultaneously possible bymeans of a convenient friend feedback. Copyright]]>

Linear algebra and its applications

Vol. 434, num. 5, p. 1325-1335

DOI: 10.1016/j.laa.2010.11.009

Date of publication: 2011

Abstract:

It is well known that, when a full rank observable pair (C,A) is slightly perturbed, the new observability indices k′ are majorized by the initial ones k, k≻k′. Conversely, any indices k′ majorized by k can be obtained by perturbing (C,A). The aim of this paper is the explicit construction of perturbations of (C,A) which have the desired indices k′ by means of a sequence of uniparametrical versal perturbations. Even more, using versal deformations we refine this construction in such a way that the perturbation has the maximum possible number of zeros and no parameters in the square part.]]>

IEEE Frontiers in Education Conference

p. 1-6

DOI: 10.1109/FIE.2010.5673349

Presentation's date: 2010-10-27

Abstract:

The Bologna process is a good opportunity to bring together first-year mathematics courses of engineering degrees and technology courses offered in subsequent years. In fact, the Faculty Council has decided that 20% of the credits from basic courses must be related to technological applications. To this end, during the past academic year a mathematical engineering seminar was held with each session dealing with one technological discipline. The main goal of the seminar, which relied on the presence of speakers from both mathematics and engineering departments, was to identify the most commonly used mathematical tools. Furthermore, a set of exercises and some guidelines addressed to faculty lacking an engineering background were created. Here, we present some of this material: first, a summary of the collection of exercises illustrating the use of Linear Algebra in different engineering areas such as Mechanical Engineering, Control and Automation, and second, some exercises]]>

Frontiers in education conference

num. S1J, p. 1-6

DOI: 10.1109/FIE.2010.5673349

Date of publication: 2010-10

Abstract:

The Bologna process is a good opportunity to bring together first-year mathematics courses of engineering degrees and technology courses offered in subsequent years. In fact, the Faculty Council has decided that 20% of the credits from basic courses must be related to technological applications. To this end, during the past academic year a mathematical engineering seminar was held with each session dealing with one technological discipline. The main goal of the seminar, which relied on the presence of speakers from both mathematics and engineering departments, was to identify the most commonly used mathematical tools. Furthermore, a set of exercises and some guidelines addressed to faculty lacking an engineering background were created. Here, we present some of this material: first, a summary of the collection of exercises illustrating the use of Linear Algebra in different engineering areas such as Mechanical Engineering, Control and Automation, and second, some exercises and the guideline for Electrical Engineering.]]>

International Conference on Circuits, Systems, Signals

p. 204-210

Presentation's date: 2010-09

Abstract:

Given the set of matrix pairs M ⊂ Mm,n(C) × Mn(C) keeping a subspace S ⊂ Cn invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of M. It generalizes the known results when S is a supplementary subspace of the unobservable one.]]>

International Conference on Circuits, Systems, Signals

p. 211-218

Presentation's date: 2010-09

Abstract:

Given a pair of matrices representing a controllable linear system, we study its equivalence classes by the single or combined action of feedbacks and change of state and input variables, as well as their intersections. In particular, we prove that they are differentiable manifolds and we compute their dimensions. Some remarks concerning the effect of different kinds of feedbacks are derived.]]>

Joint ALAMA-GAMM/ANLA Meeting

p. 1-6

Presentation's date: 2010-06

Abstract:

Keywords: Bimodal piecewise linear system, miniversal deformations, reduced forms.

Bimodal linear systems are those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane. In this work, we focus on the derivation of (orthogonal) miniversal deformations, by using reduced forms.]]>

Linear and multilinear algebra

Vol. 58, num. 1, p. 45-59

DOI: 10.1080/03081080802086296

Date of publication: 2010-01

American Mathematical Society. National Meeting

Meeting of the International Linear Algebra Society

p. 16

International journal of bifurcation and chaos

Vol. 20, num. 9, p. 2795-2808

DOI: 10.1142/S0218127410027362

Date of publication: 2010

Boletín de la Sociedad Española de Matemática Aplicada

Vol. 51, p. 55-63

Date of publication: 2010

Linear and multilinear algebra

Vol. 58, num. 5, p. 563-569

DOI: 10.1080/03081080902745304

Date of publication: 2010

Abstract:

Given a general local differentiable family of pairs of matrices, we obtain a local differentiable family of feedbacks solving the pole assignment problem, that is to say, shifting the spectrum into a prefixed one. We point out that no additional hypothesis is needed. In fact, simple approaches work in particular cases (controllable pairs, constancy of the dimension of the controllable subspace, and so on). Here the general case is proved by means of Arnold’s techniques: the key point is to reduce the construction to a versal deformation of the central pair; in fact to a quite singular miniversal one for which the family of feedbacks can be explicitly constructed. As a direct application, a differentiable family of stabilizing feedbacks is obtained, provided that the central pair is stabilizable.]]>

Mathematical problems in engineering

Vol. 2010, num. ID 698548, p. 1-9

DOI: 10.1155/2010/698548

Date of publication: 2010

Abstract:

Given a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold.]]>

Mediterranean journal of mathematics

Vol. 6, num. 4, p. 471-481

DOI: 10.1007/s00009-009-0019-2

Date of publication: 2009-12

Abstract:

Given a pair of matrices (A,B) we study the Lipschitz stability of its controlled invariant subspaces. A sufficient condition is derived from the geometry of the set formed by the quadruples (A,B, F, S) where S is an (A,B)-invariant subspace and F a corresponding feedback.]]>

Mathematical methods in the applied sciences

Vol. 32, num. 14, p. 1753-1767

DOI: 10.1002/mma.1108

Date of publication: 2009-09

Int. conference of non-autonomous and stochastic dynamical systems, and multidisciplinary applications (NSDS'09)

p. 30

Presentation's date: 2009-06

Linear algebra and its applications

Vol. 430, num. 5-6, p. 1574-1589

DOI: 10.1016/j.laa.2008.04.033

Date of publication: 2009-03

Abstract:

Specific algorithms, such as those involving the supremal of the invariant subspaces contained in a suitable subspace, are known to be able to test whether a disturbance decoupling problem (DDP) is solvable. Here, by reducing the system to its Molinari form, we obtain an alternative description of this supremal object and compute its dimension. Hence we have a general result for solving the decoupling provided that a Molinari basis is known. In particular, a necessary numerical condition for it is derived. The same technique is applied to the DDPS, that is, when stability of the decoupled closed loop system is required.]]>

AIP Conference proceedings

Vol. 1168, p. 49-52

DOI: 10.1063/1.3241507

Date of publication: 2009

Abstract:

Given a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold.]]>

AIP Conference proceedings

Vol. 1168, num. 3, p. 564-567

DOI: 10.1063/1.3241524

Date of publication: 2009

Abstract:

Given S an (A;B)-invariant subspace, we prove that the set of friend feedbacks is a (nm¡md+dq)-dimensional linear variety, which can be considered as the direct sum of the feedbacks of the restriction to S and the co-restriction to S?. In particular, if (A;B) is controllable and S is a controllability subspace, both pole assignments are simultaneously possible by means of a convenient friend feedback.]]>

Date of publication: 2008-07

Abstract:

Consideramos en el espacio de parejas de tensores tension y deformacion la relacion de equivalencia que se corresponde con cambios de base ortonormales. Identificandolas con parejas de matrices cuadradas, podemos utilizar la tecnica de las deformaciones miniversales para averiguar, dada una pareja de tensores cualquiera, cuales son las parejas de tensores que se pueden obtener al perturbar ligeramente la dada, puesto que las clases de equivalencia se pueden identificar con las orbitas que resultan al actuar un grupo de Lie sobre la variedad diferenciable de las parejas de matrices simetricas y son, por lo tanto, variedades diferenciables. Presentamos también las dimensiones que son posibles para dichas orbitas.]]>

Date of publication: 2008-07

Date of publication: 2008-07

Date: 2008-04

International Workshop on dynamical systems and multidisciplinary applications