Seminar on Algebraic Systems Theory

organized by Yacine Bouzidi, Adrien Poteaux, Alban Quadrat

University of Lille 1 (Computer Algebra and High Performance Computing) & Inria Lille - Nord Europe (Non-A project)

Seminar
François Boulier University of Lille 1, CFHP	L'équidimensionnalité des idéaux définis par systèmes triangulaires Suite du cours précédent.	16/06/2016 10-12 AM M3 226
François Boulier University of Lille 1, CFHP	L'équidimensionnalité des idéaux définis par systèmes triangulaires Il est prévu que j'assure un mini-cours sur les chaînes différentielles régulières à l'IRMAR vers la fin juin. Comme le sujet est extrêmement vaste, j'ai opté pour la rédaction d'un support de cours accompagné d'exposés sur quelques "morceaux choisis". Dans cet exposé, je me concentrerai sur une propriété essentielle des chaînes différentielles régulières, qui n'apparaît ni chez Ritt, ni chez Kolchin : ce sont des cas particuliers de systèmes polynomiaux triangulaires et les idéaux qu'elles définissent sont équidimensionnels.	02/06/2016 10-12 AM M3, room: salle du conseil (M3ext, 1er étage)
Mihaly Petreczky CNRS, Ecole Centrale de Lille	Realization theory of Nash systems In this talk I will talk about a class of non-linear systems, which I will call Nash systems. In a nutshell, Nash systems are systems whose defining equations are formulated via Nash functions (analytic and semi-algebraic functions). They occur in a wide range of applications, including model of genetic and metabolic networks. I will talk about their structural properties, such as controllabilit, observability, minimality, and about their realization theory (that is, the relationship between the equations used to describe the system and the trajectories). I will propose a number of open problems which are algebraic in nature. Some of these problems are directly related to computer algebra.	24/03/2016 2-3:30 PM M3 226
François Lemaine University of Lille 1, CFHP	Une difficulté pour montrer par l'algèbre que u=int(u) implique u=0 Si une fonction u(x) est égale à sa primitive (s'annulant en x=0), alors l'analyse montre que u(x) est nécessairement la fonction identiquement nulle. Nous verrons que cela est difficile à montrer par des techniques purement algébriques et que cela suggère d'intégrer le théorème de Cauchy-Lipschitz dans le cas algèbrique.	17/03/2016 2-3:30 PM M3 226
Yacine Bouzidi Inria Lille, Non-A	Computer algebra methods for testing the stability of multidimensional systems and differential systems with delay In this presentation, we intend to convince the audience of the usefulness of computer algebra methods for testing efficienty the stability of of two class of systems: n-D discrete linear systems (with n ≥ 2) and differential systems with commensurate delays. We first show how recent computer algebra techniques can be used for testing the structural stability of n-D discrete linear systems (with n ≥ 2). More precisely, starting from the usual stability conditions which resumes to deciding if an hypersurface D(z₁, ..., z_n)=0 has points in the unit complex polydisk, we show that the problem is equivalent to deciding if an algebraic set has real points, and use state-of-the-art algorithms for this purpose. In a second part, we consider the problem of testing the stability of 2-D systems with parameters, and provide an algorithm, which, given a 2-D system that depends on a set of parameters, computes regions of the parameter space in which the considered system is structurally stable. Finally, we present an efficient algorithm for testing the asymptotic stability of differential systems with commensurate delays. This algorithm relies on recent improvements on solving bivariate algebraic systems and computing Puiseux series to speed up the stability test.	03/03/2016 2-3:30 PM Inria, A21
Adrien Poteaux University of Lille 1, CFHP	Troncations pendant l'algorithme de Newton-Puiseux Dans cet exposé, considérons un polynôme bivarié F(X,Y), nous rappellerons la définition des développements de Puiseux (classiques et rationnels), ainsi que l'algorithme de Newton-Puiseux qui permet de calculer ces séries solutions du polynôme F vu comme un polynôme univarié en Y. Nous nous intéresserons ensuite à la troncation possible des puissances de X durant l'exécution de cet algorithme, utile pour accélérer les calculs, mais également pour traiter le cas des quasi-polynômes qui apparait dans certains cas de systèmes à retard.	25/02/2016 2-4 PM M3 226
François Lemaire University of Lille 1, CFHP	Les dernières avancées sur l'intégration de fractions différentielles Je présenterai un résumé de notre article récemment accepté au JSC (papier) en donnant des exemples et quelques définitions clés qui montrent la surprenante difficulté du traitement des fractions rationnelles par rapport aux polynômes différentiels.	04/02/2016 2-4 PM M3 226
Petteri Laakkonen Tampere University of Technology	An algebraic formulation of the internal model principle and the solvability of the robust regulation problem Robustness of a controller means that it achieves its control goal despite some small inaccuracies - e.g. modeling errors or inaccurate parameter estimations - in the system to be controlled. In the robust regulation problem we require the controller to achieve two control goals simultaneously: stabilization of the closed loop and regulation. By regulation we mean that the error between the measured behavior of the system and a given reference signal - the desired behavior - is zeroed out asymptotically. Stability is often a robust property under reasonable assumptions. On the other hand, the famous internal model principle states that including a reduplicated model of the reference signal dynamics into the controller guarantees that the controller achieves regulation for all systems it stabilizes. Thus, a robustly regulating controller is achieved by finding a robustly stabilizing controller that contains an internal model. This talk presents an algebraic approach to the robust regulation. In particular, the aim is to study the internal model principle. The problem is formulated first, and then the internal model principle is presented in algebraic terms. This allows us to give a necessary and sufficient condition for the existence of a robustly regulating controller. Parameterization of all robustly regulating controllers is also discussed.	27/01/2016 10-11 AM Plenary room, INRIA
François Boulier University of Lille 1, CFHP	Sur la généralisation de l'algèbre différentielle à l'algèbre intégro-différentielle L'idée est simple : pour passer de l'algèbre différentielle de Ritt et Kolchin à une algèbre intégro-différentielle, il suffit d'ajouter un opérateur d'intégration qui joue le rôle d'inverse de l'opérateur de dérivation. Elle est intéressante et d'actualité : c'est la continuation logique de résultats très récents obtenus en collaboration avec Georg Regensburger (Linz) et Markus Rosenkranz (Kent) et il y a des applications dans le domaine de l'estimation de paramètres. Ces derniers temps, j'ai donc beaucoup fouillé dans la littérature : d'où viennent les équations intégro-différentielles ? Qui les as inventées ? Pourquoi (quels problèmes amènent de tels modèles) ? Quelles devraient être les règles de calcul symbolique ? Comment résout-on ces équations ? Je suis encore loin d'avoir une vision complète mais j'expliquerai, dans cet exposé, où j'en suis arrivé.	21/01/2016 2-4 PM M3, room: 226
Yacine Bouzidi Inria Lille, Non-A	Solving bivariate algebraic systems and computing topology of plan curves A fundamental problem in computational geometry is the computation of the topology of an algebraic plane curve given by its implicit equation, that is, the computation of a graph lines that approximates the curve while preserving its topology. A critical step in many algorithms computing the topology of a plane curve is the computation of the set of singular and extreme points (wrt x) of this curve, which is equivalent to the computation of the solutions of bivariate systems defined by the curve and some of its partial derivatives. In this presentation, we study form theoretical and practical perspectives the problem of solving systems of bivariate polynomials with integer coefficients. More precisely, we investigate the computation of a Rational Univariate Representation (RUR) of the solutions of a bivariate system, that is, a one-to-one mapping that sends the roots of a univariate polynomial to the solutions of the bivariate system.	14/01/2016 2-4 PM M3, room: 226
Alban Quadrat Inria Lille, Non-A	Artstein's reduction of linear time-delay systems: A constructive algebraic analysis approach Artstein's results (Z. Artstein, "Linear systems with delayed controls: A reduction", IEEE Trans. Autom. Control, 27 (1982), 869--879) show that a first-order linear differential system with delayed inputs is equivalent to a first-order linear differential system without delay under an invertible transformation which includes integral and time-delay operators. Within a constructive algebraic approach, we show how this reduction can be found again, generalized and interpreted as a particular isomorphism between modules over the ring of integro-differential time-delay operators defined by the two above linear systems. Moreover, we prove that Artstein's reduction can be obtained in an automatic way by means of symbolic computation techniques, and thus can be implemented in dedicated computer algebra systems.	07/01/2016 2:30-4:30 PM Amphi Turing (M3, room: 118)

Seminar

L'équidimensionnalité des idéaux définis par systèmes triangulaires

Suite du cours précédent.

16/06/2016

10-12 AM

M3 226

L'équidimensionnalité des idéaux définis par systèmes triangulaires

Il est prévu que j'assure un mini-cours sur les chaînes différentielles régulières à l'IRMAR vers la fin juin. Comme le sujet est extrêmement vaste, j'ai opté pour la rédaction d'un support de cours accompagné d'exposés sur quelques "morceaux choisis". Dans cet exposé, je me concentrerai sur une propriété essentielle des chaînes différentielles régulières, qui n'apparaît ni chez Ritt, ni chez Kolchin : ce sont des cas particuliers de systèmes polynomiaux triangulaires et les idéaux qu'elles définissent sont équidimensionnels.

02/06/2016

10-12 AM

M3, room: salle du conseil (M3ext, 1er étage)

Mihaly Petreczky

CNRS, Ecole Centrale de Lille

Realization theory of Nash systems

In this talk I will talk about a class of non-linear systems, which I will call Nash systems. In a nutshell, Nash systems are systems whose defining equations are formulated via Nash functions (analytic and semi-algebraic functions). They occur in a wide range of applications, including model of genetic and metabolic networks. I will talk about their structural properties, such as controllabilit, observability, minimality, and about their realization theory (that is, the relationship between the equations used to describe the system and the trajectories). I will propose a number of open problems which are algebraic in nature. Some of these problems are directly related to computer algebra.

24/03/2016

2-3:30 PM

M3 226

François Lemaine

University of Lille 1, CFHP

Une difficulté pour montrer par l'algèbre que u=int(u) implique u=0

Si une fonction u(x) est égale à sa primitive (s'annulant en x=0), alors l'analyse montre que u(x) est nécessairement la fonction identiquement nulle. Nous verrons que cela est difficile à montrer par des techniques purement algébriques et que cela suggère d'intégrer le théorème de Cauchy-Lipschitz dans le cas algèbrique.

17/03/2016

2-3:30 PM

M3 226

Yacine Bouzidi

Inria Lille, Non-A

Computer algebra methods for testing the stability of multidimensional systems and differential systems with delay

In this presentation, we intend to convince the audience of the usefulness of computer algebra methods for testing efficienty the stability of of two class of systems: n-D discrete linear systems (with n ≥ 2) and differential systems with commensurate delays.

We first show how recent computer algebra techniques can be used for testing the structural stability of n-D discrete linear systems (with n ≥ 2). More precisely, starting from the usual stability conditions which resumes to deciding if an hypersurface D(z₁, ..., z_n)=0 has points in the unit complex polydisk, we show that the problem is equivalent to deciding if an algebraic set has real points, and use state-of-the-art algorithms for this purpose.

In a second part, we consider the problem of testing the stability of 2-D systems with parameters, and provide an algorithm, which, given a 2-D system that depends on a set of parameters, computes regions of the parameter space in which the considered system is structurally stable.

Finally, we present an efficient algorithm for testing the asymptotic stability of differential systems with commensurate delays. This algorithm relies on recent improvements on solving bivariate algebraic systems and computing Puiseux series to speed up the stability test.

03/03/2016

2-3:30 PM

Inria, A21

Adrien Poteaux

University of Lille 1, CFHP

Troncations pendant l'algorithme de Newton-Puiseux

Dans cet exposé, considérons un polynôme bivarié F(X,Y), nous rappellerons la définition des développements de Puiseux (classiques et rationnels), ainsi que l'algorithme de Newton-Puiseux qui permet de calculer ces séries solutions du polynôme F vu comme un polynôme univarié en Y. Nous nous intéresserons ensuite à la troncation possible des puissances de X durant l'exécution de cet algorithme, utile pour accélérer les calculs, mais également pour traiter le cas des quasi-polynômes qui apparait dans certains cas de systèmes à retard.

25/02/2016

2-4 PM

M3 226

François Lemaire

University of Lille 1, CFHP

Les dernières avancées sur l'intégration de fractions différentielles

Je présenterai un résumé de notre article récemment accepté au JSC (papier) en donnant des exemples et quelques définitions clés qui montrent la surprenante difficulté du traitement des fractions rationnelles par rapport aux polynômes différentiels.

04/02/2016

2-4 PM

M3 226

Petteri Laakkonen

Tampere University of Technology

An algebraic formulation of the internal model principle and the solvability of the robust regulation problem

Robustness of a controller means that it achieves its control goal despite some small inaccuracies - e.g. modeling errors or inaccurate parameter estimations - in the system to be controlled. In the robust regulation problem we require the controller to achieve two control goals simultaneously: stabilization of the closed loop and regulation. By regulation we mean that the error between the measured behavior of the system and a given reference signal - the desired behavior - is zeroed out asymptotically. Stability is often a robust property under reasonable assumptions. On the other hand, the famous internal model principle states that including a reduplicated model of the reference signal dynamics into the controller guarantees that the controller achieves regulation for all systems it stabilizes. Thus, a robustly regulating controller is achieved by finding a robustly stabilizing controller that contains an internal model.

This talk presents an algebraic approach to the robust regulation. In particular, the aim is to study the internal model principle. The problem is formulated first, and then the internal model principle is presented in algebraic terms. This allows us to give a necessary and sufficient condition for the existence of a robustly regulating controller. Parameterization of all robustly regulating controllers is also discussed.

27/01/2016

10-11 AM

Plenary room, INRIA

François Boulier

University of Lille 1, CFHP

Sur la généralisation de l'algèbre différentielle à l'algèbre intégro-différentielle

L'idée est simple : pour passer de l'algèbre différentielle de Ritt et Kolchin à une algèbre intégro-différentielle, il suffit d'ajouter un opérateur d'intégration qui joue le rôle d'inverse de l'opérateur de dérivation. Elle est intéressante et d'actualité : c'est la continuation logique de résultats très récents obtenus en collaboration avec Georg Regensburger (Linz) et Markus Rosenkranz (Kent) et il y a des applications dans le domaine de l'estimation de paramètres. Ces derniers temps, j'ai donc beaucoup fouillé dans la littérature : d'où viennent les équations intégro-différentielles ? Qui les as inventées ? Pourquoi (quels problèmes amènent de tels modèles) ? Quelles devraient être les règles de calcul symbolique ? Comment résout-on ces équations ? Je suis encore loin d'avoir une vision complète mais j'expliquerai, dans cet exposé, où j'en suis arrivé.

21/01/2016

2-4 PM

M3, room: 226

Yacine Bouzidi

Inria Lille, Non-A

Solving bivariate algebraic systems and computing topology of plan curves

A fundamental problem in computational geometry is the computation of the topology of an algebraic plane curve given by its implicit equation, that is, the computation of a graph lines that approximates the curve while preserving its topology. A critical step in many algorithms computing the topology of a plane curve is the computation of the set of singular and extreme points (wrt x) of this curve, which is equivalent to the computation of the solutions of bivariate systems defined by the curve and some of its partial derivatives. In this presentation, we study form theoretical and practical perspectives the problem of solving systems of bivariate polynomials with integer coefficients. More precisely, we investigate the computation of a Rational Univariate Representation (RUR) of the solutions of a bivariate system, that is, a one-to-one mapping that sends the roots of a univariate polynomial to the solutions of the bivariate system.

14/01/2016

2-4 PM

M3, room: 226

Alban Quadrat

Inria Lille, Non-A

Artstein's reduction of linear time-delay systems: A constructive algebraic analysis approach

Artstein's results (Z. Artstein, "Linear systems with delayed controls: A reduction", IEEE Trans. Autom. Control, 27 (1982), 869--879) show that a first-order linear differential system with delayed inputs is equivalent to a first-order linear differential system without delay under an invertible transformation which includes integral and time-delay operators. Within a constructive algebraic approach, we show how this reduction can be found again, generalized and interpreted as a particular isomorphism between modules over the ring of integro-differential time-delay operators defined by the two above linear systems. Moreover, we prove that Artstein's reduction can be obtained in an automatic way by means of symbolic computation techniques, and thus can be implemented in dedicated computer algebra systems.

07/01/2016

2:30-4:30 PM

Amphi Turing (M3, room: 118)

Former Seminar on Algebraic Systems Theory

organized by Yacine Bouzidi, Sette Diop, Hugues Mounier, Alban Quadrat

Inria Saclay - Île-de-France and L2S,
Supélec,
3 rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France.

Seminar
Hugues Mounier Université Paris Sud, L2S	Mini cours sur la théorie algébrique des systèmes de dimension infinie Une théorie algébrique pour la commande des systèmes à retards et à paramè tres répartis commandés aux bords est présentée. Les systèmes y sont représentés par des modules sur des anneaux d'opérateurs. Plusieurs propriétés de commandabilité sont exposées. Les liens avec la théorie des modules et l'algèbre homologique que ces théories entretiennent seront détaillés. Slides Cours	07/07 2:30-4:30 AM Salle du conseil du L2S
Hugues Mounier Université Paris Sud, L2S	Mini cours sur la théorie algébrique des systèmes de dimension infinie Une théorie algébrique pour la commande des systèmes à retards et à paramè tres répartis commandés aux bords est présentée. Les systèmes y sont représentés par des modules sur des anneaux d'opérateurs. Plusieurs propriétés de commandabilité sont exposées. Les liens avec la théorie des modules et l'algèbre homologique que ces théories entretiennent seront détaillés. Slides Cours	30/06 2:30-5 PM Salle du conseil du L2S
Hugues Mounier Université Paris Sud, L2S	Mini cours sur la théorie algébrique des systèmes de dimension infinie Une théorie algébrique pour la commande des systèmes à retards et à paramè tres répartis commandés aux bords est présentée. Les systèmes y sont représentés par des modules sur des anneaux d'opérateurs. Plusieurs propriétés de commandabilité sont exposées. Les liens avec la théorie des modules et l'algèbre homologique que ces théories entretiennent seront détaillés. Slides Cours	29/06 2:30-5 PM Salle du conseil du L2S
Guillaume Moroz Inria Nancy, VEGAS	Parametric polynomial systems and linkages A linkage can be represented by a graph with distance constraints associated to its edges. The euclidean embeddings of a linkage in the plan can be naturally modeled by a polynomial system where some or all the lengths are considered as parameters. We will see how computer algebra techniques can be used to solve problems involving linkages in mechanical design. Slides	18/02/15 11-12:30 AM C3.11
Islam Boussaada L2S	Tracking the algebraic multiplicity of crossing imaginary roots for time-delay systems: A functional confluent Vandermonde-based framework It is well known that the solutions behavior for linear dynamical systems is strongly related to the associated spectrum. In particular, a standard approach in analyzing the stability of time-delay systems consists in characterizing the associated Crossing Imaginary Roots (CIRs). By characterization it is meant the identification of CIRs as well as their associated multiplicities (algebraic/geometric). Efficient approaches for CIRs identification exist. However, the multiplicity of CIRs was not deeply investigated. In this talk, we provide a functional confluent Vandermonde/Birkhoff-based framework yielding to CIRs multiplicity for retarded differential equations. Thanks to this, we show that the Polya-Szego bound can never be reached for non zero frequencies, providing a sharper bound. We emphasize also that the proposed framework can be applied to neutral systems. This is a joint work with Silviu Niculescu (L2S).	10/02/15 10:30-12:00 AM C3.11
Yacine Bouzidi Inria Saclay, DISCO	Résolution des systèmes algébriques zéro-dimensionnels Suite du précédent exposé.	22/10/14 09:30-12:00 AM D3.01
Yacine Bouzidi Inria Saclay, DISCO	Résolution des systèmes algébriques zéro-dimensionnels Lorsque l'on pense résolution de systèmes algébriques en calcul scientifique, on sous-entend généralement systèmes admettant un nombre fini de solutions complexes (appelés aussi systèmes zéro-dimensionnels). En calcul formel, une approche classique pour résoudre de tel systèmes consiste à calculer une représentation symbolique des solutions (e.g base de Gröbner, représentation triangulaire, représentation univariée rationnelle, etc), puis d'utiliser celle-ci pour obtenir des approximations numériques certifiées de ces solutions. Dans cet exposé, après un court rappel sur quelques notions mathématiques en lien avec le problème de résolutions de systèmes alghébrique, on présentera quelques algorithmes classiques qui permettent de résoudre ces systèmes dans le cas général (n équations, n variables) par le calcul d'une représentation formelle des solutions. On montrera également l'intérêt de tels algorithmes dans la résolution d'un problème fondamental en géométrie algorithmique qui est celui du calcul de topologie de courbes planes. Cette application sera l'occasion d'exposer les avancées récentes concernant la résolution de systèmes algébriques en deux variables.	14/10/14 10:30-12:00 AM D3.01
Yacine Bouzidi Inria Nancy, VEGAS	Solving bivariate systems of equations via Rational Univariate Representations A fundamental problem in computational geometry is the computation of the topology of a real algebraic plane curve given by its implicit equation, that is, the computation of a set of polygonal lines that approximates the curve while preserving its topology. A critical step in many algorithms computing the topology of a plane curve is the computation of the set of singular and extreme points (wrt x) of this curve, which is equivalent to the computation of the solutions of bivariate systems defined by the curve and some of its partial derivatives. In this presentation we address the problem of solving systems of bivariate polynomials with coefficients in Z. Our approach consists in computing a Rational Univariate Representation (RUR) of the solutions, that is, a one-to-one mapping that sends the roots of a univariate polynomial to the solutions of the bivariate system. Such a representation is very useful to compute efficiently with the solutions of a bivariate system. The computation of this representation decomposes in two parts, the computation of a separating linear form, that is, a linear combination of the variables that take different values when evaluated at different solutions of the system, and the polynomials of the RUR associated to this linear form. In this talk, we present new algorithms along with complexity results for both the computation of a separating linear form and the computation of the associated polynomials of the RUR. These new complexity bounds improve the existing ones by several factors.	30/06/14 2:30-4 PM D4.05
Petteri Laakkonen Tampere University of Technology	General factorization approach to frequency domain robust regulation Vidyasagar and others developed frequency domain robust regulation theory for rational transfer matrices around 1980s. This theory is based on coprime factorizations and allows parameterization of all robustly regulating controllers and simple characterization of the famous internal model principle - every robustly regulating controller contains a suitably redublicated model of the dynamics to be controlled. These results of rational matrices have been generalized to algebraic structures suitable for infinite-dimensional systems. The problematic part is that coprime factorizations do not necessarily exist or they may be hard to find in algebraic structures for infinite-dimensional systems. The purpose of this talk is to present some new results on frequency domain robust regulation that use only a left or a right coprime factorization or non-coprime factorizations. We first introduce the basic results for rational functions using coprime factorizations and finally proceed to an abstract algebraic setting where we develop robust regulation theory without using coprime factorizations.	10/06/14 2:30-4 PM C3-12
François Boulier University of Lille I	Differential Algebra: Applications, Software and Theory The differential algebra of Ritt and Kolchin is an algebraic theory for systems of nonlinear differential equations. It has been a very active research topic of the Computer Algebra Group of Lille for 20 years, with a focus on algorithms, software and applications. In this talk, I will present an introduction to the mathematical theory and some of our results: an open source library which can be used through the MAPLE DifferentialAlgebra package, a recent algorithm which is useful for converting differential to integral equations and is thus important for parameter estimation problems (joint work with Inria/Non-A), ... Slides Maple worksheet1 Maple worksheet2	27/05/14 2:30-4 PM C3-12
Mohamed Barakat University of Kaiserlautern	Generalized morphisms, and transforming homological algorithms into formulas Homological algebra is an effective and very powerful organizational tool in mathematics with a wide range of applications. Generalized morphisms streamline parts of this tool even further and turn very complicated looking algorithms into extremely simple formulas.	04/04/14 10-12 AM C3-12
Georg Regensburger RICAM, Linz	Generalized mass action systems and positive solutions of polynomial systems Chemical reaction network theory provides statements about uniqueness, existence, and stability of positive steady states of the related dynamical systems for all rate constants. The relevant conditions depend only on the network structure and the stoichiometric subspace. In terms of polynomial equations, they guarantee existence and uniqueness of positive solutions for all parameters. We will survey some classical results and discuss an extension of several statements to generalized mass action systems where reaction rates are allowed to be power-laws in the concentrations. In this setting, uniqueness and existence additionally depend on sign vectors of the stoichiometric and kinetic-order subspaces. This is joint work with Stefan Müller. Slides	06/03/14 4:00-5:00 PM C3-12
Thomas Cluzeau University of Limoges	An algebraic analysis approach to the simplification of systems of linear functional equations In this talk, I will show how the algebraic analysis approach to linear systems theory can be used to simplify the equations of the system. I will concentrate on three distinct problems: the decomposition problem, i.e., study when a linear system is equivalent to a block-diagonal system, Serre's reduction problem, i.e., study when a linear system can be defined by fewer equations and fewer unknowns, the equivalence problem, i.e., study when two linear systems are equivalent. I will illustrate the different results on examples coming from control theory and mathematical physics. The algorithms presented have been implemented in a Maple package called OreMorphisms and I will demonstrate how it can be used. This work has been done in collaboration with Alban Quadrat (INRIA Saclay - Île-de-France). Slides	11/02/14 2:00-4:30 PM C3-12
Jacques-Arthur Weil University of Limoges	Application des formes réduites à l'intégrabilité de systèmes différentiels non-linéaires Je montrerai comment les techniques de linéarisation permettent de construire des équations variationnelles, équations linéaires "emboitées" de taille croissante qui permettent d'étudier finement les propriétés de systèmes différentiels non-linéaires au voisinage d'une trajectoire. En utilisant la structure de ces équations variationnelles, je montrerai une technique de "réduction", issue de travaux théoriques de Kolchin et Kovacic, qui permet de "lire" sur ces systèmes variationnels des informations sur, par exemple, l'intégrabilité du système non-linéaire. Slides	03/02/14 11:00-12:30 AM E1-18
Jacques-Arthur Weil University of Limoges	Théorie de Galois différentielle constructive et formes réduites de systèmes différentiels linéaires L'exposé présentera des méthodes constructives pour résoudre ou simplifier les systèmes d'équations différentielles linéaires. L'objet classifiant sous-jacent sera le groupe de Galois différentiel ; nous mettrons l'accent particulièrement sur des techniques récentes de réduction de systèmes mettant en jeu des techniques d'algèbres de Lie. Slides	20/01/14 11:00-12:30 AM F3-04
Islam Boussaada L2S	Sur la multiplicité de la valeur spectrale à l'origine pour les systèmes à retard et son lien avec les matrices d'incidence de Birkhoff L'analyse de stabilité des systèmes à retard s'appuie sur l'identification des valeurs spectrales sur l'axe des imaginaires et la compréhension de leurs dynamiques. La multiplicité (algébrique/ géométrique) d'une de ces valeurs joue un rôle essentiel dans la description de son comportement par rapport à une petite variation des paramètres (gains/retards) du système. Le point de départ de cet exposé est un résultat établi en deux étapes [S.-I. Niculescu, W. Michiels, IEEE Trans. on Aut. Cont. 49 (5) (2004)] et [V.Kharitonov, S.-I.Niculescu, J.Moreno,W.Michiels, IEEE Trans. on Aut. Cont. 50 (1) (2005)] affirmant que n retards sont nécessaires et suffisants pour stabiliser une chaîne de n intégrateurs. Ce résultat nous a permis d'établir une approche qui consiste à stabiliser un système en dimension finie ou infinie par des retours d'état retardés grâce au théorème de la variété du centre associée à une valeur spectrale à l'origine. Cette approche met en évidence l'intérêt d'établir une borne pour la multiplicité de cette valeur spectrale. Le lien entre ce problème de multiplicité et les matrices d'incidence de Birkhoff (issues de problèmes d'interpolation) sera présenté. Slides	17/12/13 10:30-12:00 AM E1.16
Alban Quadrat Inria Saclay - L2S	Etude des problèmes de stabilisation des systèmes linéaires de dimension infinie Dans les années 80, Vidyasagar, Desoer, Callier, Francis, Zames, ..., ont développé une approche mathématique, appelée "représentation fractionnaire des systèmes", permettant l'étude des problèmes d'analyse et de synthèse pour de larges classes de systèmes linéaires (dimension finie et infinie, continus, discrets). Dans ce cadre, les concepts de stabilisation interne (existence d'un contrôleur stabilisant), stabilisation forte (existence d'un contrôleur stabilisant stable), stabilisation simultantée (existence d'un contrôleur stabilisant une famille finie de systèmes), stabilisation robuste (existence d'un contrôleur stabilisant l'ensemble des systèmes autour d'un système), ..., ont trouvé des caractérisations algébriques. Cette approche est à la base de la commande H^∞ des systèmes de dimension finie. La généralisation de la commande H^∞ pour les systèmes linéaires de dimension infinie (e.g., systèmes différentiels à retard, équations aux dérivées partielles) a longuement été étudiée ces dernières années. Cependant, des problèmes fondamentaux restent encore ouverts. Nous commencerons notre exposé par donner une courte introduction aux problèmes de stabilisation pour les systèmes de dimension infinie définis dans le domaine fréquentiel. Puis, nous montrerons comment l'utilisation de concepts algébriques modernes tels que les idéaux fractionnaires ou les réseaux algébriques sur certaines algèbres de Banach (e.g., H^∞(C+), A, A(D), W₊) permettent de caractériser les problèmes de stabilisation (interne, forte, simultanée, robuste). Cette approche nous permettra de résoudre des problèmes ouverts (e.g., question de Vidyasagar-Francis-Schneider, conjecture de Lin, conjecture de Feintuch). Ces résultats nous permettront de généraliser la célèbre paramétrisation de Youla-Kucera des contrôleurs stabilisants pour la classe de systèmes stabilisants n'admettant pas nécessairement de factorisations doublement copremières. Nous expliquerons pourquoi des problèmes algébriques similaires à ceux étudiés ici pour les problèmes de stabilisation ont été à la base du développement de l'algèbre moderne (certains prenant leur source dans des tentatives de résolution du dernier théorème de Fermat). Finalement, nous montrerons comment les résultats précédents permettent d'étudier les problèmes de stabilisation des systèmes linéaires de dimension infinie par des méthodes venant de la géométrie non commutative d'Alain Connes.	15/10/13 10:30-12:00 AM D1.05
Maris Tõnso Institute of Cybernetics, Tallinn University of Technology	NLControl - package for modelling, analysis and synthesis of nonlinear control systems In this talk we introduce a symbolic computation package, NLControl, meant to assist solving the modelling, analysis and synthesis problems for nonlinear control systems. Most functions in NLControl are based on the algebraic approach of differential one-forms and on the theory on non-commutative skew polynomial rings. To apply the approach of differential one-forms the nonlinear system is globally linearised and described by its infinitesimal representation. Certain subspaces of differential one-forms are computed, providing the base for solution. Skew polynomials are optionally used to simplify the construction of the one-forms. Intrinsic necessary and sufficient conditions for the existence of the solution are formulated in terms of these one-forms. To find the solution, one has to integrate the one-forms, to get back into the level of equations. This entails additional integrability restrictions. Linear algebraic approach provides universal and simple theoretical framework for addressing a wide range of nonlinear control problems, for instance reduction, realization, model matching, tranforming the state equations into observer form, flatness, input-output linearization, disturbance decoupling.	28/05/13 2:00-3:30 PM C3.12
Alin Bostan Inria Saclay	Introduction à la complexité algébrique (seconde partie) Les principaux thèmes de cet exposé sont la conception et l'analyse d'algorithmes algébriques pour la manipulation des polynômes et des matrices. L'accent sera mis principalement sur la complexité asymptotique, et sur l'illustration de paradigmes algorithmiques fondamentaux (diviser-pour-régner, pas de bébé/pas de géant, principe de transposition) et des techniques sous-jacentes (exponentiation binaire, itération de Newton, évaluation-interpolation). Quelques applications seront décrites, concernant la manipulation rapide des fonctions rationnelles, des matrices polynomiales, des nombres algébriques et des suites récurrentes. Slides	25/04/13 2:00-3:30 PM F3.09
Emmanuel Witrant GIPSA-Lab	Control-oriented modeling of inhomogeneous transport One of the major difficulties in the analysis of physics associated with transport phenomena in nonhomogeneous media is the complexity of the process model. This complexity is mainly due to the large-scale aspect, the medium heterogeneity (anisotropic - direction dependent - transport coefficients) and the dynamics of the interconnections (equilibrium, flow exchange, transport and propagation between the nodes). Such models come from hydrodynamics (e.g. transport in porous media), aerodynamics (e.g. Poiseuille flows) or magnetohydrodynamics (e.g. thermonuclear fusion) and are described with partial or functional differential equations. Their use in control strategies imply to take into account the fact that the infinite dimensional aspect of the system (physical continuum) is affected by both space and time variations. This talk is focused on the modeling and identification aspects. Starting from a generic formulation of the conservation laws, the impact and inclusion of the nonhomogeneous property of the surrounding medium is first described. The resulting dynamics are classified depending on the main transport tendency (e.g. diffusive or convective, with or without sink). The importance of dynamical and exogenous inputs is emphasized, in order to formulate the associated identification or control problem. These issues are illustrated on different challenging problems: ventilation control in large plants, profiles control for tokamak plasmas and quantification of the anthropogenic impact on atmospheric composition from firn air measurements. Slides	19/04/13 10:30-12:00 AM C3.12
Alban Quadrat Inria Saclay - L2S	An Introduction to Algebraic Analysis Continuation of the previous talks: We shall give a basic introduction to constructive homological and its applications to mathematical systems theory. Explicit examples will be illustrated with the OreModules package.	08/04/13 2:00-3:30 PM C3.11
Alin Bostan Inria Saclay	Introduction à la complexité algébrique (première partie) Les principaux thèmes de cet exposé sont la conception et l'analyse d'algorithmes algébriques pour la manipulation des polynômes et des matrices. L'accent sera mis principalement sur la complexité asymptotique, et sur l'illustration de paradigmes algorithmiques fondamentaux (diviser-pour-régner, pas de bébé/pas de géant, principe de transposition) et des techniques sous-jacentes (exponentiation binaire, itération de Newton, évaluation-interpolation). Quelques applications seront décrites, concernant la manipulation rapide des fonctions rationnelles, des matrices polynomiales, des nombres algébriques et des suites récurrentes. Slides	04/04/13 3:00-4:30 PM C3.11
Alban Quadrat Inria Saclay - L2S	An Introduction to Algebraic Analysis Continuation of the previous talk: The next talks will be dedicated to algebraic analysis techniques for the study of linear functional systems (ODE, PDE, delays, recurrences, differences). In particular, an introduction to elementary techniques of module theory, homological algebra, constructive algebra, and symbolic computation will be given.	29/03/13 10:00-11:30 AM C2
Alban Quadrat Inria Saclay - L2S	An Introduction to Algebraic Analysis The next talks will be dedicated to algebraic analysis techniques for the study of linear functional systems (ODE, PDE, delays, recurrences, differences). In particular, an introduction to elementary techniques of module theory, homological algebra, constructive algebra, and symbolic computation will be given.	05/03/13 10:30-12:00 AM C3.12
Harish Pillai IIT Bombay	Algebraic Directions for Behaviours I am going to discuss some recent results on nD discrete systems that imitates the idea of initial conditions in 1D systems.	19/12/12 2-3 PM C3.12
Harish Pillai IIT Bombay	Algebraic Directions for Behaviours Continuing from the last talk, I shall introduce the algebraic approach to behaviours. I shall show why algebraic approach to behaviours can help understand some situations in multidimensional systems.	13/12/12 2-3 PM C3.11
Mohamed Barakat University of Kaiserlautern	Derived Categories and System Theory Buchsbaum and Grothendieck invented Abelian categories as the correct categorical generalization of module categories, i.e., as the most abstract way to do linear algebra. Derived categories are certain "completions" of Abelian categories which were later invented by Grothendieck and Verdier to prove the existence of certain operations and dualities (two of Grothendieck's six operations in algebraic geometry and the Verdier duality) which do not exist in the smaller Abelian category (of quasi-coherent sheaves). Nowadays derived categories popup everywhere in mathematics and one of their remarkable features is their ability to connect apparently remote fields of mathematics. The reason for this is that derived categories of very, very different Abelian categories might be equivalent. In the talk I will present several examples of these "tunnel effects". I will end with considering system theory where it would be nice to see the world on the other side of the tunnel. Notes	04/12/12 2-3 PM C3.12
Daniel Robertz RWTH Aachen University	Janet's Algorithm and Systems Theory This talk gives an introduction to Janet bases and some of its applications to systems theory. Originally developed for the algebraic analysis of systems of partial differential equations in the beginning of the 20th century, the algorithm by Maurice Janet is today an efficient alternative for Buchberger's algorithm to compute Gröbner bases for modules over polynomial rings. In this talk I give a modern description of Janet's algorithm and explain nice combinatorial properties of the resulting Janet bases: separation of the variables into multiplicative and non-multiplicative ones for each Janet basis element allows to read off vector space bases for both the submodule and the residue class module. As a consequence, the Hilbert series and polynomial of a (graded) module as well as a free resolution are easily obtained from the Janet basis. A formal algebraic approach to systems of linear functional equations requires the computation of Janet bases in different steps of a structural analysis. Among the fundamental problems that can be solved using Janet bases is the question whether the solution set of a linear system (say of differential time-delay equations or of partial differential equations) can be parametrized, i.e. represented as the image of a linear operator. Slides	28/11/12 2-3 PM C3.12
Harish Pillai IIT Bombay	An Introduction to Behavioural Systems Theory Behavioural Systems theory has been around for about twenty years now, though it may not be very well known. I shall give a basic introduction about behaviours and demonstrate how all linear systems (discrete or continuous) can be viewed in terms of behaviours. I shall show how standard concepts in systems theory like "states, controllability, observability" take on a (what I believe) natural meaning in behaviours. I shall then look at concepts of dissipativity, optimal control and such like. If time permits, I shall also show how module theory gets linked to behavioural theory.	15/11/12 2-3 PM C3.12
Hugues Mounier Universit&eactue; Paris Sud, L2S		24/01/12 Salle des séminaire
Sette Diop CNRS, L2S	Introduction to an algebraic-numerical approach to systems observation problems Continued	17/05/11 Salle des séminaire
Sette Diop CNRS, L2S	Introduction to an algebraic-numerical approach to systems observation problems Continued	12/05/11 Salle des séminaire
Sette Diop CNRS, L2S	Introduction to an algebraic-numerical approach to systems observation problems Continued	06/05/11 Salle des séminaire
Sette Diop CNRS, L2S	Introduction to an algebraic-numerical approach to systems observation problems Observation problems in control systems literature generally refer to problems of estimation of state variables (or identification of model parameters) given two sources of data: dynamic models of systems consisting in first order differential equations relating all system quantities, and online measurements of some of these quantities. For nonlinear systems the classical approach stems from the work of R. E. Kalman on the distinguishability of state space points given the knowledge of time histories of the output and input. Initiated in the late 1980's by J. F. Pommaret and M. Fliess a differential algebra approach was used to revisit observability property, culminating in a quite comprehensive theory in the 1990's capturing observation problems rather in terms trajectory recovery in lieu of the old state space geometric approach. Contributors include S. T. Glad and the author. This differential algebra approach leads in a natural way to have recourse to numerical analysis chapters in attempts to solve the longstanding problems of observer and estimation scheme design. This seminar is an introductory account to this approach.	26/04/11 Salle des séminaire

Seminar

Hugues Mounier

Université Paris Sud, L2S

Mini cours sur la théorie algébrique des systèmes de dimension infinie

Une théorie algébrique pour la commande des systèmes à retards et à paramè tres répartis commandés aux bords est présentée. Les systèmes y sont représentés par des modules sur des anneaux d'opérateurs. Plusieurs propriétés de commandabilité sont exposées. Les liens avec la théorie des modules et l'algèbre homologique que ces théories entretiennent seront détaillés.

Slides

Cours

07/07

2:30-4:30 AM

Salle du conseil du L2S

Hugues Mounier

Université Paris Sud, L2S

Mini cours sur la théorie algébrique des systèmes de dimension infinie

Slides

Cours

30/06

2:30-5 PM

Salle du conseil du L2S

Hugues Mounier

Université Paris Sud, L2S

Mini cours sur la théorie algébrique des systèmes de dimension infinie

Slides

Cours

29/06

2:30-5 PM

Salle du conseil du L2S

Guillaume Moroz

Inria Nancy, VEGAS

Parametric polynomial systems and linkages

A linkage can be represented by a graph with distance constraints associated to its edges. The euclidean embeddings of a linkage in the plan can be naturally modeled by a polynomial system where some or all the lengths are considered as parameters. We will see how computer algebra techniques can be used to solve problems involving linkages in mechanical design.

Slides

18/02/15

11-12:30 AM

C3.11

Islam Boussaada

L2S

Tracking the algebraic multiplicity of crossing imaginary roots for time-delay systems: A functional confluent Vandermonde-based framework

It is well known that the solutions behavior for linear dynamical systems is strongly related to the associated spectrum. In particular, a standard approach in analyzing the stability of time-delay systems consists in characterizing the associated Crossing Imaginary Roots (CIRs). By characterization it is meant the identification of CIRs as well as their associated multiplicities (algebraic/geometric). Efficient approaches for CIRs identification exist. However, the multiplicity of CIRs was not deeply investigated. In this talk, we provide a functional confluent Vandermonde/Birkhoff-based framework yielding to CIRs multiplicity for retarded differential equations. Thanks to this, we show that the Polya-Szego bound can never be reached for non zero frequencies, providing a sharper bound. We emphasize also that the proposed framework can be applied to neutral systems.

This is a joint work with Silviu Niculescu (L2S).

10/02/15

10:30-12:00 AM

C3.11

Yacine Bouzidi

Inria Saclay, DISCO

Résolution des systèmes algébriques zéro-dimensionnels

Suite du précédent exposé.

22/10/14

09:30-12:00 AM

D3.01

Yacine Bouzidi

Inria Saclay, DISCO

Résolution des systèmes algébriques zéro-dimensionnels

Lorsque l'on pense résolution de systèmes algébriques en calcul scientifique, on sous-entend généralement systèmes admettant un nombre fini de solutions complexes (appelés aussi systèmes zéro-dimensionnels). En calcul formel, une approche classique pour résoudre de tel systèmes consiste à calculer une représentation symbolique des solutions (e.g base de Gröbner, représentation triangulaire, représentation univariée rationnelle, etc), puis d'utiliser celle-ci pour obtenir des approximations numériques certifiées de ces solutions.

Dans cet exposé, après un court rappel sur quelques notions mathématiques en lien avec le problème de résolutions de systèmes alghébrique, on présentera quelques algorithmes classiques qui permettent de résoudre ces systèmes dans le cas général (n équations, n variables) par le calcul d'une représentation formelle des solutions. On montrera également l'intérêt de tels algorithmes dans la résolution d'un problème fondamental en géométrie algorithmique qui est celui du calcul de topologie de courbes planes. Cette application sera l'occasion d'exposer les avancées récentes concernant la résolution de systèmes algébriques en deux variables.

14/10/14

10:30-12:00 AM

D3.01

Yacine Bouzidi

Inria Nancy, VEGAS

Solving bivariate systems of equations via Rational Univariate Representations

A fundamental problem in computational geometry is the computation of the topology of a real algebraic plane curve given by its implicit equation, that is, the computation of a set of polygonal lines that approximates the curve while preserving its topology. A critical step in many algorithms computing the topology of a plane curve is the computation of the set of singular and extreme points (wrt x) of this curve, which is equivalent to the computation of the solutions of bivariate systems defined by the curve and some of its partial derivatives.

In this presentation we address the problem of solving systems of bivariate polynomials with coefficients in Z. Our approach consists in computing a Rational Univariate Representation (RUR) of the solutions, that is, a one-to-one mapping that sends the roots of a univariate polynomial to the solutions of the bivariate system. Such a representation is very useful to compute efficiently with the solutions of a bivariate system. The computation of this representation decomposes in two parts, the computation of a separating linear form, that is, a linear combination of the variables that take different values when evaluated at different solutions of the system, and the polynomials of the RUR associated to this linear form. In this talk, we present new algorithms along with complexity results for both the computation of a separating linear form and the computation of the associated polynomials of the RUR. These new complexity bounds improve the existing ones by several factors.

30/06/14

2:30-4 PM

D4.05

Petteri Laakkonen

Tampere University of Technology

General factorization approach to frequency domain robust regulation

Vidyasagar and others developed frequency domain robust regulation theory for rational transfer matrices around 1980s. This theory is based on coprime factorizations and allows parameterization of all robustly regulating controllers and simple characterization of the famous internal model principle - every robustly regulating controller contains a suitably redublicated model of the dynamics to be controlled. These results of rational matrices have been generalized to algebraic structures suitable for infinite-dimensional systems. The problematic part is that coprime factorizations do not necessarily exist or they may be hard to find in algebraic structures for infinite-dimensional systems.

The purpose of this talk is to present some new results on frequency domain robust regulation that use only a left or a right coprime factorization or non-coprime factorizations. We first introduce the basic results for rational functions using coprime factorizations and finally proceed to an abstract algebraic setting where we develop robust regulation theory without using coprime factorizations.

10/06/14

2:30-4 PM

C3-12

François Boulier

University of Lille I

Differential Algebra: Applications, Software and Theory

The differential algebra of Ritt and Kolchin is an algebraic theory for systems of nonlinear differential equations. It has been a very active research topic of the Computer Algebra Group of Lille for 20 years, with a focus on algorithms, software and applications.

In this talk, I will present an introduction to the mathematical theory and some of our results: an open source library which can be used through the MAPLE DifferentialAlgebra package, a recent algorithm which is useful for converting differential to integral equations and is thus important for parameter estimation problems (joint work with Inria/Non-A), ...

Slides

Maple worksheet1

University of Limoges

An algebraic analysis approach to the simplification of systems of linear functional equations

In this talk, I will show how the algebraic analysis approach to linear systems theory can be used to simplify the equations of the system. I will concentrate on three distinct problems:

the decomposition problem, i.e., study when a linear system is equivalent to a block-diagonal system,

Serre's reduction problem, i.e., study when a linear system can be defined by fewer equations and fewer unknowns,

the equivalence problem, i.e., study when two linear systems are equivalent.

I will illustrate the different results on examples coming from control theory and mathematical physics. The algorithms presented have been implemented in a Maple package called OreMorphisms and I will demonstrate how it can be used.

L'analyse de stabilité des systèmes à retard s'appuie sur l'identification des valeurs spectrales sur l'axe des imaginaires et la compréhension de leurs dynamiques. La multiplicité (algébrique/ géométrique) d'une de ces valeurs joue un rôle essentiel dans la description de son comportement par rapport à une petite variation des paramètres (gains/retards) du système.

Le point de départ de cet exposé est un résultat établi en deux étapes [S.-I. Niculescu, W. Michiels, IEEE Trans. on Aut. Cont. 49 (5) (2004)] et [V.Kharitonov, S.-I.Niculescu, J.Moreno,W.Michiels, IEEE Trans. on Aut. Cont. 50 (1) (2005)] affirmant que n retards sont nécessaires et suffisants pour stabiliser une chaîne de n intégrateurs.

Ce résultat nous a permis d'établir une approche qui consiste à stabiliser un système en dimension finie ou infinie par des retours d'état retardés grâce au théorème de la variété du centre associée à une valeur spectrale à l'origine.

Cette approche met en évidence l'intérêt d'établir une borne pour la multiplicité de cette valeur spectrale. Le lien entre ce problème de multiplicité et les matrices d'incidence de Birkhoff (issues de problèmes d'interpolation) sera présenté.

Slides

17/12/13

10:30-12:00 AM

E1.16

Alban Quadrat

Inria Saclay - L2S

Etude des problèmes de stabilisation des systèmes linéaires de dimension infinie

Dans les années 80, Vidyasagar, Desoer, Callier, Francis, Zames, ..., ont développé une approche mathématique, appelée "représentation fractionnaire des systèmes", permettant l'étude des problèmes d'analyse et de synthèse pour de larges classes de systèmes linéaires (dimension finie et infinie, continus, discrets). Dans ce cadre, les concepts de stabilisation interne (existence d'un contrôleur stabilisant), stabilisation forte (existence d'un contrôleur stabilisant stable), stabilisation simultantée (existence d'un contrôleur stabilisant une famille finie de systèmes), stabilisation robuste (existence d'un contrôleur stabilisant l'ensemble des systèmes autour d'un système), ..., ont trouvé des caractérisations algébriques. Cette approche est à la base de la commande H^∞ des systèmes de dimension finie.

La généralisation de la commande H^∞ pour les systèmes linéaires de dimension infinie (e.g., systèmes différentiels à retard, équations aux dérivées partielles) a longuement été étudiée ces dernières années. Cependant, des problèmes fondamentaux restent encore ouverts.

Nous commencerons notre exposé par donner une courte introduction aux problèmes de stabilisation pour les systèmes de dimension infinie définis dans le domaine fréquentiel. Puis, nous montrerons comment l'utilisation de concepts algébriques modernes tels que les idéaux fractionnaires ou les réseaux algébriques sur certaines algèbres de Banach (e.g., H^∞(C+), A, A(D), W₊) permettent de caractériser les problèmes de stabilisation (interne, forte, simultanée, robuste). Cette approche nous permettra de résoudre des problèmes ouverts (e.g., question de Vidyasagar-Francis-Schneider, conjecture de Lin, conjecture de Feintuch). Ces résultats nous permettront de généraliser la célèbre paramétrisation de Youla-Kucera des contrôleurs stabilisants pour la classe de systèmes stabilisants n'admettant pas nécessairement de factorisations doublement copremières.

Nous expliquerons pourquoi des problèmes algébriques similaires à ceux étudiés ici pour les problèmes de stabilisation ont été à la base du développement de l'algèbre moderne (certains prenant leur source dans des tentatives de résolution du dernier théorème de Fermat). Finalement, nous montrerons comment les résultats précédents permettent d'étudier les problèmes de stabilisation des systèmes linéaires de dimension infinie par des méthodes venant de la géométrie non commutative d'Alain Connes.

15/10/13

10:30-12:00 AM

D1.05

Maris Tõnso

Institute of Cybernetics, Tallinn University of Technology

NLControl - package for modelling, analysis and synthesis of nonlinear control systems

In this talk we introduce a symbolic computation package, NLControl, meant to assist solving the modelling, analysis and synthesis problems for nonlinear control systems. Most functions in NLControl are based on the algebraic approach of differential one-forms and on the theory on non-commutative skew polynomial rings. To apply the approach of differential one-forms the nonlinear system is globally linearised and described by its infinitesimal representation. Certain subspaces of differential one-forms are computed, providing the base for solution. Skew polynomials are optionally used to simplify the construction of the one-forms. Intrinsic necessary and sufficient conditions for the existence of the solution are formulated in terms of these one-forms. To find the solution, one has to integrate the one-forms, to get back into the level of equations. This entails additional integrability restrictions. Linear algebraic approach provides universal and simple theoretical framework for addressing a wide range of nonlinear control problems, for instance reduction, realization, model matching, tranforming the state equations into observer form, flatness, input-output linearization, disturbance decoupling.

28/05/13

2:00-3:30 PM

C3.12

Alin Bostan

Inria Saclay

Introduction à la complexité algébrique (seconde partie)

Les principaux thèmes de cet exposé sont la conception et l'analyse d'algorithmes algébriques pour la manipulation des polynômes et des matrices. L'accent sera mis principalement sur la complexité asymptotique, et sur l'illustration de paradigmes algorithmiques fondamentaux (diviser-pour-régner, pas de bébé/pas de géant, principe de transposition) et des techniques sous-jacentes (exponentiation binaire, itération de Newton, évaluation-interpolation). Quelques applications seront décrites, concernant la manipulation rapide des fonctions rationnelles, des matrices polynomiales, des nombres algébriques et des suites récurrentes.

Slides

25/04/13

2:00-3:30 PM

F3.09

Emmanuel Witrant

GIPSA-Lab

Control-oriented modeling of inhomogeneous transport

One of the major difficulties in the analysis of physics associated with transport phenomena in nonhomogeneous media is the complexity of the process model. This complexity is mainly due to the large-scale aspect, the medium heterogeneity (anisotropic - direction dependent - transport coefficients) and the dynamics of the interconnections (equilibrium, flow exchange, transport and propagation between the nodes). Such models come from hydrodynamics (e.g. transport in porous media), aerodynamics (e.g. Poiseuille flows) or magnetohydrodynamics (e.g. thermonuclear fusion) and are described with partial or functional differential equations. Their use in control strategies imply to take into account the fact that the infinite dimensional aspect of the system (physical continuum) is affected by both space and time variations.

This talk is focused on the modeling and identification aspects. Starting from a generic formulation of the conservation laws, the impact and inclusion of the nonhomogeneous property of the surrounding medium is first described. The resulting dynamics are classified depending on the main transport tendency (e.g. diffusive or convective, with or without sink). The importance of dynamical and exogenous inputs is emphasized, in order to formulate the associated identification or control problem. These issues are illustrated on different challenging problems: ventilation control in large plants, profiles control for tokamak plasmas and quantification of the anthropogenic impact on atmospheric composition from firn air measurements.

Slides

19/04/13

10:30-12:00 AM

C3.12

Alban Quadrat

Inria Saclay - L2S

An Introduction to Algebraic Analysis

Continuation of the previous talks: We shall give a basic introduction to constructive homological and its applications to mathematical systems theory. Explicit examples will be illustrated with the OreModules package.

08/04/13

2:00-3:30 PM

C3.11

Alin Bostan

Inria Saclay

Introduction à la complexité algébrique (première partie)

Slides

04/04/13

3:00-4:30 PM

C3.11

Alban Quadrat

Inria Saclay - L2S

An Introduction to Algebraic Analysis

Continuation of the previous talk: The next talks will be dedicated to algebraic analysis techniques for the study of linear functional systems (ODE, PDE, delays, recurrences, differences). In particular, an introduction to elementary techniques of module theory, homological algebra, constructive algebra, and symbolic computation will be given.

29/03/13

10:00-11:30 AM

Alban Quadrat

Inria Saclay - L2S

An Introduction to Algebraic Analysis

The next talks will be dedicated to algebraic analysis techniques for the study of linear functional systems (ODE, PDE, delays, recurrences, differences). In particular, an introduction to elementary techniques of module theory, homological algebra, constructive algebra, and symbolic computation will be given.

05/03/13

10:30-12:00 AM

C3.12

Harish Pillai

IIT Bombay

Algebraic Directions for Behaviours

I am going to discuss some recent results on nD discrete systems that imitates the idea of initial conditions in 1D systems.

19/12/12

2-3 PM

C3.12

Harish Pillai

IIT Bombay

Algebraic Directions for Behaviours

Continuing from the last talk, I shall introduce the algebraic approach to behaviours. I shall show why algebraic approach to behaviours can help understand some situations in multidimensional systems.

13/12/12

2-3 PM

C3.11

Mohamed Barakat

University of Kaiserlautern

Derived Categories and System Theory

Buchsbaum and Grothendieck invented Abelian categories as the correct categorical generalization of module categories, i.e., as the most abstract way to do linear algebra. Derived categories are certain "completions" of Abelian categories which were later invented by Grothendieck and Verdier to prove the existence of certain operations and dualities (two of Grothendieck's six operations in algebraic geometry and the Verdier duality) which do not exist in the smaller Abelian category (of quasi-coherent sheaves). Nowadays derived categories popup everywhere in mathematics and one of their remarkable features is their ability to connect apparently remote fields of mathematics. The reason for this is that derived categories of very, very different Abelian categories might be equivalent. In the talk I will present several examples of these "tunnel effects". I will end with considering system theory where it would be nice to see the world on the other side of the tunnel.

Notes

04/12/12

2-3 PM

C3.12

Daniel Robertz

RWTH Aachen University

Janet's Algorithm and Systems Theory

This talk gives an introduction to Janet bases and some of its applications to systems theory.

Originally developed for the algebraic analysis of systems of partial differential equations in the beginning of the 20th century, the algorithm by Maurice Janet is today an efficient alternative for Buchberger's algorithm to compute Gröbner bases for modules over polynomial rings.

In this talk I give a modern description of Janet's algorithm and explain nice combinatorial properties of the resulting Janet bases: separation of the variables into multiplicative and non-multiplicative ones for each Janet basis element allows to read off vector space bases for both the submodule and the residue class module. As a consequence, the Hilbert series and polynomial of a (graded) module as well as a free resolution are easily obtained from the Janet basis.

A formal algebraic approach to systems of linear functional equations requires the computation of Janet bases in different steps of a structural analysis. Among the fundamental problems that can be solved using Janet bases is the question whether the solution set of a linear system (say of differential time-delay equations or of partial differential equations) can be parametrized, i.e. represented as the image of a linear operator.

Slides

28/11/12

2-3 PM

C3.12

Harish Pillai

IIT Bombay

An Introduction to Behavioural Systems Theory

Behavioural Systems theory has been around for about twenty years now, though it may not be very well known. I shall give a basic introduction about behaviours and demonstrate how all linear systems (discrete or continuous) can be viewed in terms of behaviours. I shall show how standard concepts in systems theory like "states, controllability, observability" take on a (what I believe) natural meaning in behaviours. I shall then look at concepts of dissipativity, optimal control and such like. If time permits, I shall also show how module theory gets linked to behavioural theory.

15/11/12

2-3 PM