A fractional representation approach to the robust regulation problem for SISO systems

The purpose of this article is to develop a new approach to the robust regulation problem for plants which do not necessarily admit coprime factorizations. The approach is purely algebraic and allows us dealing with a very general class of systems in a unique simple framework. We formulate the famous internal model principle in a form suitable for plants defined by fractional representations which are not necessarily coprime factorizations. By using the internal model principle, we are able to give necessary and sufficient solvability conditions for the robust regulation problem and to parameterize all robustly regulating controllers.Comment: 13 pages, 1 figure, to appear in Systems & Control Letter

About Dynamical Systems Appearing in the Microscopic Traffic Modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.Comment: PDF 38 page

Serre's reduction of linear partial differential systems with holonomic adjoints

Given a linear functional system (e.g., ordinary/partial differential systems, differential time-delay systems, difference systems), Serre's reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre's reduction of underdetermined linear systems of partial differential equations with either polynomial, formal power series or analytic coefficients and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial differential systems can be defined by means of a single linear partial differential equation. In the case of polynomial coefficients, we give an algorithm to compute the corresponding equation.Etant donné un système fonctionnel linéaire (e.g., système d'équations différentielles ordinaires, système d'équations aux dérivées partielles, système d'équations différentielles à retard, système d'équations aux différences), la réduction de Serre a pour but de trouver un système fonctionnel linéaire équivalent contenant moins d'équations et d'inconnues. L'objectif de ce papier est l'étude de la réduction de Serre des systèmes linéaires sous-déterminés d'équations aux dérivées partielles à coefficients polynomiaux, séries formelles ou séries localement convergentes, dont les adjoints sont holonomes au sens de l'analyse algébrique. Nous prouvons que de tels systèmes peuvent être définis par une seule équation aux dérivées partielles. Dans le cas des coefficients polynomiaux, nous donnons un algorithme permettant de calculer l'équation correspondante

Serre's reduction of linear systems of partial differential equations with holonomic adjoints

Given a linear functional system (e.g., ordinary/partial di erential system, di erential time-delay system, di erence system), Serre's reduction aims at nding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre's reduction of underdetermined linear systems of partial di erential equations with either polynomial, formal power series or analytic coe cients and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial di erential systems can be de ned by means of a single linear partial di erential equation. In the case of polynomial coe cients, we give an algorithm to compute the corresponding equation

Symbolic preconditioning techniques for linear systems of partial differential equations

Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. This is very important since numerical simulations heavily rely on solving systems of PDEs. For the large-scale problems we deal with in today's standard applications, it is necessary to rely on iterative Krylov methods that are scalable (i.e., weakly dependent on the number of degrees on freedom and number of subdomains) and have limited memory requirements. They are preconditioned by domain decomposition methods, incomplete factorizations and multigrid preconditioners. These techniques are well understood and efficient for scalar symmetric equations (e.g., Laplacian, biLaplacian) and to some extent for non-symmetric equations (e.g., convection-diffusion). But they have poor performances and lack robustness when used for symmetric systems of PDEs, and even more so for non-symmetric complex systems (fluid mechanics, porous media ...). As a general rule, the study of iterative solvers for systems of PDEs as opposed to scalar PDEs is an underdeveloped subject. We aim at building new robust and efficient solvers, such as domain decomposition methods and preconditioners for some linear and well-known systems of PDEs

The Dynamic Equations of the Tree Morphogenesis GreenLab Model

We explicit the dynamic equations followed by a tree during its growth according to the Greenlab model. In a first part we explicit the Organogenesis equations. In the second part, we recall the equations which, using a macroscopic photosynthesis point of view, describe with a great precision the evolution of the organ sizes. In a third part we establish the morphogenesis equations describing the geometry of a tree. The three parts are illustrated with an example

Symbolic methods for developing new domain decomposition algorithms

The purpose of this work is to show how algebraic and symbolic techniques such as Smith normal forms and Gröbner basis techniques can be used to develop new Schwarz-like algorithms and preconditioners for linear systems of partial differential equationsL'objet de ce travail est de monter comment les techniques algébriques et symboliques telles que les formes normales de Smith et les techniques de bases de Gröbner peuvent être utilisées pour développer de nouveaux algorithmes de type Schwarz et des préconditionneurs pour les systèmes linéaires d'équations aux dérivées partielles

A constructive study of the module structure of rings of partial differential operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. © 2014 Springer Science+Business Media

Simple modelling and control of plasma current profile

International audienceThe purpose of this paper is to present a simplified model and control law of the current and temperature profile in a tokamak plasma. Based on a description of the plasma as a magnetised uid, the model is expressed in the form of coupled one dimensional transport-diffusion equations. A simple feedback is used to obtain a given stationary profile. The numerical simulations are done in the Scilab/Scicos environment

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc